Subgroups — C₅×#83912 #14424639

Order 1920.

1528 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#1009	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5833	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#949	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5232	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#5187	C₂	1	2	yes	maximal, complemented
7	C₅×#5172	C₂	1	2	yes	maximal, complemented
8	C₅×#5169	C₂	1	2	yes	maximal, complemented
9	C₅×#5156	C₂	1	2	yes	maximal, complemented
10	C₂×C₅×#821	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#5110	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#3012	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic, complemented
15	C₅×#1009	—	2	4	no	—
16	C₅×#1009	—	2	4	no	—
17	C₂×C₅×#239	C₂²	1	2	yes	—
18	C₂×C₅×D₃×D₄	—	2	2	no	—
19	C₂×C₅×D₃×D₄	C₂²	1	2	yes	—
20	C₂×C₅×#235	—	2	1	no	—
21	C₂×C₅×#235	C₂²	1	1	yes	characteristic
22	C₂²×C₅×D₁₂	—	2	1	no	—
23	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
24	C₃×C₅×#135	—	2	4	no	—
25	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₃×C₅×SD₁₆	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₃×C₅×SD₁₆	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic, complemented
29	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
30	C₅×#949	—	2	4	no	—
31	C₅×#946	C₂²	1	2	yes	complemented
32	C₂×C₅×#215	C₂²	1	2	yes	complemented
33	C₅×#939	—	2	2	no	—
34	C₅×#938	C₂²	1	2	yes	complemented
35	C₂×C₅×#213	C₂²	1	2	yes	complemented
36	C₅×#930	—	2	1	no	—
37	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
38	C₅×#924	—	2	1	no	—
39	C₂×C₅×#208	C₂²	1	1	yes	characteristic, complemented
40	C₅×#835	C₂²	1	2	yes	complemented
41	C₅×#832	C₂²	1	2	yes	complemented
42	C₅×#821	C₂²	1	2	yes	complemented
43	C₅×#821	C₂²	1	2	yes	complemented
44	C₅×(C₂×D₃)⋊C₈	C₂²	1	2	yes	complemented
45	C₅×(C₂×D₃)⋊C₈	C₂²	1	2	yes	complemented
46	#83912	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
47	C₂×C₅×H(5,𝔽₂)	—	3	1	no	—
48	C₂×C₅×AGL(1,ℤ/8)	—	3	1	no	—
49	C₂×C₅×#135	D₃	1	1	yes	characteristic, complemented
50	C₅×#537	—	3	1	no	—
51	C₅×#452	—	3	2	no	—
52	C₅×#450	—	3	2	no	—
53	C₅×#449	—	3	2	no	—
54	C₅×#447	—	3	2	no	—
55	C₅×#417	—	3	1	no	—
56	C₂×C₅×#102	—	3	1	no	—
57	C₅×#408	—	3	1	no	—
58	C₂³×C₅×D₃	—	2	1	no	—
59	C₂³×C₅×D₃	—	2	2	no	—
60	C₃×C₅×#41	—	2	2	no	—
61	C₃×C₅×#41	—	2	2	no	—
62	C₃×C₅×#41	—	2	2	no	—
63	C₃×C₅×#41	D₄	1	4	yes	complemented
64	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
65	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
66	C₂×C₃×C₅×D₄	—	2	1	no	—
67	C₂×C₃×C₅×D₄	—	2	1	no	—
68	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
69	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
70	C₂²×C₃×C₄×C₅	—	2	1	no	—
71	C₂²×C₃×C₄×C₅	—	2	1	no	—
72	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
73	C₂×C₅×#67	—	2	2	no	—
74	C₂×C₅×#67	—	2	2	no	—
75	C₂×C₅×#67	D₄	1	2	yes	—
76	C₂²×C₅×Dic₃	—	2	1	no	—
77	C₅×#239	—	2	4	no	—
78	C₅×#239	—	2	4	no	—
79	C₅×D₃×D₄	—	2	16	no	—
80	C₅×D₃×D₄	—	2	4	no	—
81	C₅×D₃×D₄	—	2	4	no	—
82	C₅×#235	—	2	8	no	—
83	C₅×#235	—	2	2	no	—
84	C₅×#235	—	2	2	no	—
85	C₂×C₅×D₁₂	—	2	8	no	—
86	C₂×C₅×D₁₂	—	2	2	no	—
87	C₂×C₅×D₁₂	—	2	1	no	—
88	C₂×C₅×D₁₂	—	2	2	no	—
89	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
90	C₂×C₅×D₁₂	D₄	1	4	yes	complemented
91	C₂×C₅×D₁₂	C₂³	1	2	yes	—
92	C₂×C₄×C₅×D₃	—	2	2	no	—
93	C₂×C₄×C₅×D₃	C₂³	1	2	yes	—
94	C₂×C₅×Dic₆	D₄	1	1	yes	characteristic
95	C₃×C₅×Q₁₆	—	2	2	no	—
96	C₃×C₅×SD₁₆	—	2	2	no	—
97	C₃×C₅×SD₁₆	—	2	2	no	—
98	C₃×C₅×D₈	—	2	2	no	—
99	C₂×C₃×C₅×C₈	—	2	2	no	—
100	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
101	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
102	C₅×#217	—	2	2	no	—
103	C₅×#215	—	2	4	no	—
104	C₅×#213	—	2	4	no	—
105	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
106	C₅×C₄⋊Dic₃	—	2	1	no	—
107	C₅×C₄⋊Dic₃	C₂³	1	2	yes	—
108	C₅×Dic₃⋊C₄	—	2	2	no	—
109	C₅×#208	—	2	2	no	—
110	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
111	C₂×#1009	C₂×C₅	1	1	yes	characteristic, complemented
112	C₂×C₃×#135	C₂×C₅	1	1	yes	characteristic, complemented
113	#5833	C₂×C₅	1	1	yes	characteristic, complemented
114	C₂×#949	C₂×C₅	1	1	yes	characteristic, complemented
115	#5232	C₂×C₅	1	1	yes	characteristic, complemented
116	#5187	C₂×C₅	1	2	yes	complemented
117	#5172	C₂×C₅	1	2	yes	complemented
118	#5169	C₂×C₅	1	2	yes	complemented
119	#5156	C₂×C₅	1	2	yes	complemented
120	C₂×#821	C₂×C₅	1	1	yes	characteristic, complemented
121	#5110	C₂×C₅	1	1	yes	characteristic, complemented
122	C₅×H(5,𝔽₂)	—	6	4	no	—
123	C₅×H(5,𝔽₂)	—	6	4	no	—
124	C₂×C₅×#41	—	3	1	no	—
125	C₂×C₅×#41	—	3	2	no	—
126	C₂×C₅×#41	—	6	1	no	—
127	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
128	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic, complemented
129	C₂²×C₅×D₄	—	3	2	no	—
130	C₂²×C₅×D₄	—	3	1	no	—
131	C₂²×C₅×D₄	—	6	1	no	—
132	C₂²×C₅×D₄	—	6	2	no	—
133	C₅×AGL(1,ℤ/8)	—	6	4	no	—
134	C₅×#135	—	2	4	no	—
135	C₂×C₅×Q₁₆	C₂×D₃	1	1	yes	characteristic, complemented
136	C₂×C₅×SD₁₆	—	3	2	no	—
137	C₂×C₅×SD₁₆	C₂×D₃	1	1	yes	characteristic, complemented
138	C₂×C₅×SD₁₆	C₂×D₃	1	1	yes	characteristic, complemented
139	C₂×C₅×D₈	—	3	2	no	—
140	C₂×C₅×D₈	C₂×D₃	1	1	yes	characteristic, complemented
141	C₂×C₅×M₁₆	—	3	1	no	—
142	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic, complemented
143	C₅×C₂²⋊Q₈	—	3	2	no	—
144	C₅×C₂²⋊Q₈	—	6	1	no	—
145	C₅×#121	—	3	2	no	—
146	C₅×#121	—	6	1	no	—
147	C₅×#121	—	6	2	no	—
148	C₂×C₅×C₄⋊C₄	—	3	1	no	—
149	C₅×#103	—	3	2	no	—
150	C₅×#102	—	3	2	no	—
151	C₅×#102	—	3	2	no	—
152	C₅×#102	—	3	2	no	—
153	C₅×C₂²⋊C₈	—	3	2	no	—
154	C₅×C₂²⋊C₈	—	3	2	no	—
155	#3012	—	3	1	no	Sylow
156	C₂³×C₃×C₅	—	2	1	no	—
157	C₂³×C₃×C₅	—	2	1	no	—
158	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₅×D₃	—	2	2	no	—
160	C₂²×C₅×D₃	—	2	8	no	—
161	C₂²×C₅×D₃	—	2	4	no	—
162	C₂²×C₅×D₃	—	8	2	no	—
163	C₂²×C₅×D₃	—	8	2	no	—
164	C₂²×C₅×D₃	—	8	1	no	—
165	C₂²×C₅×D₃	—	8	1	no	—
166	C₂²×C₅×D₃	C₂×D₄	1	2	yes	—
167	C₃×C₅×Q₈	—	2	1	no	—
168	C₃×C₅×Q₈	—	2	1	no	—
169	C₃×C₅×Q₈	—	2	1	no	—
170	C₃×C₅×Q₈	C₂×D₄	1	2	yes	—
171	C₃×C₅×D₄	—	2	4	no	—
172	C₃×C₅×D₄	—	2	1	no	—
173	C₃×C₅×D₄	—	2	1	no	—
174	C₃×C₅×D₄	—	2	1	no	—
175	C₃×C₅×D₄	—	4	2	no	—
176	C₃×C₅×D₄	C₂×D₄	1	2	yes	—
177	C₂×C₃×C₄×C₅	—	2	1	no	—
178	C₂×C₃×C₄×C₅	—	2	4	no	—
179	C₂×C₃×C₄×C₅	—	2	1	no	—
180	C₂×C₃×C₄×C₅	—	4	2	no	—
181	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
182	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
183	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
184	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
185	C₅×#67	—	2	8	no	—
186	C₅×#67	—	2	4	no	—
187	C₅×#67	—	8	2	no	—
188	C₂×C₅×Dic₃	—	2	2	no	—
189	C₂×C₅×Dic₃	—	8	1	no	—
190	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
191	C₅×D₁₂	—	2	8	no	—
192	C₅×D₁₂	—	2	2	no	—
193	C₅×D₁₂	—	2	4	no	—
194	C₅×D₁₂	—	2	2	no	—
195	C₅×D₁₂	C₂×D₄	1	4	yes	—
196	C₄×C₅×D₃	—	2	8	no	—
197	C₄×C₅×D₃	—	2	2	no	—
198	C₄×C₅×D₃	—	2	2	no	—
199	C₅×Dic₆	—	2	2	no	—
200	C₃×C₅×C₈	—	2	1	no	—
201	C₃×C₅×C₈	—	2	1	no	—
202	C₅×C₃⋊C₈	—	2	2	no	—
203	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
204	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic, complemented
205	#1009	—	2	4	no	—
206	#1009	—	2	4	no	—
207	C₂×#239	C₂²×C₅	1	2	yes	—
208	C₂×D₃×D₄	—	2	2	no	—
209	C₂×D₃×D₄	C₂²×C₅	1	2	yes	—
210	C₂×#235	—	2	1	no	—
211	C₂×#235	C₂²×C₅	1	1	yes	characteristic
212	C₂²×D₁₂	—	2	1	no	—
213	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
214	C₃×#135	—	2	4	no	—
215	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
216	C₂×C₃×SD₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
217	C₂×C₃×SD₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
218	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic, complemented
219	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic, complemented
220	#949	—	2	4	no	—
221	#946	C₂²×C₅	1	2	yes	complemented
222	C₂×#215	C₂²×C₅	1	2	yes	complemented
223	#939	—	2	2	no	—
224	#938	C₂²×C₅	1	2	yes	complemented
225	C₂×#213	C₂²×C₅	1	2	yes	complemented
226	#930	—	2	1	no	—
227	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
228	#924	—	2	1	no	—
229	C₂×#208	C₂²×C₅	1	1	yes	characteristic, complemented
230	#835	C₂²×C₅	1	2	yes	complemented
231	#832	C₂²×C₅	1	2	yes	complemented
232	#821	C₂²×C₅	1	2	yes	complemented
233	#821	C₂²×C₅	1	2	yes	complemented
234	(C₂×D₃)⋊C₈	C₂²×C₅	1	2	yes	complemented
235	(C₂×D₃)⋊C₈	C₂²×C₅	1	2	yes	complemented
236	C₂⁴×C₅	—	6	2	no	—
237	C₂⁴×C₅	—	6	1	no	—
238	C₅×#41	—	2	2	no	—
239	C₅×#41	—	2	2	no	—
240	C₅×#41	—	2	2	no	—
241	C₅×#41	—	6	8	no	—
242	C₅×#41	—	6	4	no	—
243	C₅×#41	—	6	4	no	—
244	C₅×#41	—	6	2	no	—
245	C₅×#41	—	6	2	no	—
246	C₅×#41	#67	1	4	yes	complemented
247	C₂×C₅×Q₈	—	3	1	no	—
248	C₂×C₅×Q₈	C₂²×D₃	1	1	yes	characteristic
249	C₂×C₅×Q₈	C₂²×D₃	1	1	yes	characteristic
250	C₂×C₅×D₄	—	2	1	no	—
251	C₂×C₅×D₄	—	2	1	no	—
252	C₂×C₅×D₄	—	3	2	no	—
253	C₂×C₅×D₄	—	3	4	no	—
254	C₂×C₅×D₄	—	3	1	no	—
255	C₂×C₅×D₄	—	3	2	no	—
256	C₂×C₅×D₄	—	6	2	no	—
257	C₂×C₅×D₄	—	6	8	no	—
258	C₂×C₅×D₄	—	6	2	no	—
259	C₂×C₅×D₄	—	6	4	no	—
260	C₂×C₅×D₄	—	6	16	no	—
261	C₂×C₅×D₄	—	6	4	no	—
262	C₂×C₅×D₄	—	6	2	no	—
263	C₂×C₅×D₄	—	6	1	no	—
264	C₂×C₅×D₄	—	6	2	no	—
265	C₂×C₅×D₄	C₂²×D₃	1	1	yes	characteristic
266	C₂×C₅×D₄	C₂²×D₃	1	1	yes	characteristic
267	C₂²×C₄×C₅	—	2	1	no	—
268	C₂²×C₄×C₅	—	2	1	no	—
269	C₂²×C₄×C₅	—	3	2	no	—
270	C₂²×C₄×C₅	—	6	2	no	—
271	C₂²×C₄×C₅	—	6	1	no	—
272	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
273	C₅×Q₁₆	—	2	2	no	—
274	C₅×SD₁₆	—	2	2	no	—
275	C₅×SD₁₆	—	2	2	no	—
276	C₅×SD₁₆	—	6	4	no	—
277	C₅×D₈	—	2	2	no	—
278	C₅×D₈	—	6	4	no	—
279	C₅×M₁₆	—	6	2	no	—
280	C₂×C₅×C₈	—	2	2	no	—
281	C₂×C₅×C₈	—	3	2	no	—
282	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
283	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
284	C₅×C₄⋊C₄	—	3	2	no	—
285	C₅×C₄⋊C₄	—	6	2	no	—
286	C₅×C₄⋊C₄	—	6	1	no	—
287	C₅×C₂²⋊C₄	—	6	2	no	—
288	C₅×C₂²⋊C₄	—	6	2	no	—
289	C₂×H(5,𝔽₂)	—	3	1	no	—
290	C₂×AGL(1,ℤ/8)	—	3	1	no	—
291	C₂×#135	C₅×D₃	1	1	yes	characteristic, complemented
292	#537	—	3	1	no	—
293	#452	—	3	2	no	—
294	#450	—	3	2	no	—
295	#449	—	3	2	no	—
296	#447	—	3	2	no	—
297	#417	—	3	1	no	—
298	C₂×#102	—	3	1	no	—
299	#408	—	3	1	no	—
300	C₂²×C₃×C₅	—	2	1	no	—
301	C₂²×C₃×C₅	—	2	1	no	—
302	C₂²×C₃×C₅	—	2	2	no	—
303	C₂²×C₃×C₅	—	4	1	no	—
304	C₂²×C₃×C₅	—	4	1	no	—
305	C₂²×C₃×C₅	—	4	1	no	—
306	C₂²×C₃×C₅	—	4	1	no	—
307	C₂²×C₃×C₅	—	4	1	no	—
308	C₂²×C₃×C₅	#120	1	2	yes	—
309	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
310	C₂×C₅×D₃	—	2	2	no	—
311	C₂×C₅×D₃	—	2	4	no	—
312	C₂×C₅×D₃	—	2	2	no	—
313	C₂×C₅×D₃	—	2	2	no	—
314	C₂×C₅×D₃	—	4	2	no	—
315	C₂×C₅×D₃	—	4	2	no	—
316	C₂×C₅×D₃	—	8	4	no	—
317	C₂×C₅×D₃	—	8	8	no	—
318	C₃×C₄×C₅	—	2	2	no	—
319	C₃×C₄×C₅	—	4	1	no	—
320	C₃×C₄×C₅	#120	1	2	yes	—
321	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
322	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
323	C₅×Dic₃	—	2	2	no	—
324	C₅×Dic₃	—	4	2	no	—
325	C₂³×D₃	—	2	1	no	—
326	C₂³×D₃	—	2	2	no	—
327	C₃×#41	—	2	2	no	—
328	C₃×#41	—	2	2	no	—
329	C₃×#41	—	2	2	no	—
330	C₃×#41	C₅×D₄	1	4	yes	complemented
331	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
332	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
333	C₂×C₃×D₄	—	2	1	no	—
334	C₂×C₃×D₄	—	2	1	no	—
335	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
336	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
337	C₂²×C₃×C₄	—	2	1	no	—
338	C₂²×C₃×C₄	—	2	1	no	—
339	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
340	C₂×#67	—	2	2	no	—
341	C₂×#67	—	2	2	no	—
342	C₂×#67	C₅×D₄	1	2	yes	—
343	C₂²×Dic₃	—	2	1	no	—
344	#239	—	2	4	no	—
345	#239	—	2	4	no	—
346	D₃×D₄	—	2	16	no	—
347	D₃×D₄	—	2	4	no	—
348	D₃×D₄	—	2	4	no	—
349	#235	—	2	8	no	—
350	#235	—	2	2	no	—
351	#235	—	2	2	no	—
352	C₂×D₁₂	—	2	8	no	—
353	C₂×D₁₂	—	2	2	no	—
354	C₂×D₁₂	—	2	1	no	—
355	C₂×D₁₂	—	2	2	no	—
356	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
357	C₂×D₁₂	C₅×D₄	1	4	yes	complemented
358	C₂×D₁₂	C₂³×C₅	1	2	yes	—
359	C₂×C₄×D₃	—	2	2	no	—
360	C₂×C₄×D₃	C₂³×C₅	1	2	yes	—
361	C₂×Dic₆	C₅×D₄	1	1	yes	characteristic
362	C₃×Q₁₆	—	2	2	no	—
363	C₃×SD₁₆	—	2	2	no	—
364	C₃×SD₁₆	—	2	2	no	—
365	C₃×D₈	—	2	2	no	—
366	C₂×C₃×C₈	—	2	2	no	—
367	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
368	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
369	#217	—	2	2	no	—
370	#215	—	2	4	no	—
371	#213	—	2	4	no	—
372	(C₂×D₃)⋊C₄	—	2	2	no	—
373	C₄⋊Dic₃	—	2	1	no	—
374	C₄⋊Dic₃	C₂³×C₅	1	2	yes	—
375	Dic₃⋊C₄	—	2	2	no	—
376	#208	—	2	2	no	—
377	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
378	C₂³×C₅	—	2	1	no	—
379	C₂³×C₅	—	2	1	no	—
380	C₂³×C₅	—	3	2	no	—
381	C₂³×C₅	—	6	8	no	—
382	C₂³×C₅	—	6	2	no	—
383	C₂³×C₅	—	6	4	no	—
384	C₂³×C₅	—	24	1	no	—
385	C₂³×C₅	—	24	1	no	—
386	C₂³×C₅	—	24	2	no	—
387	C₂³×C₅	—	24	2	no	—
388	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
389	C₅×Q₈	—	2	1	no	—
390	C₅×Q₈	—	2	1	no	—
391	C₅×Q₈	—	2	1	no	—
392	C₅×Q₈	—	6	2	no	—
393	C₅×Q₈	C₂×#67	1	2	yes	—
394	C₅×D₄	—	2	4	no	—
395	C₅×D₄	—	2	1	no	—
396	C₅×D₄	—	2	1	no	—
397	C₅×D₄	—	2	1	no	—
398	C₅×D₄	—	3	4	no	—
399	C₅×D₄	—	4	2	no	—
400	C₅×D₄	—	6	8	no	—
401	C₅×D₄	—	6	8	no	—
402	C₅×D₄	—	6	4	no	—
403	C₅×D₄	—	6	4	no	—
404	C₅×D₄	—	6	2	no	—
405	C₅×D₄	—	6	2	no	—
406	C₅×D₄	—	24	2	no	—
407	C₅×D₄	C₂×#67	1	2	yes	—
408	C₂×C₄×C₅	—	2	1	no	—
409	C₂×C₄×C₅	—	2	4	no	—
410	C₂×C₄×C₅	—	2	1	no	—
411	C₂×C₄×C₅	—	3	2	no	—
412	C₂×C₄×C₅	—	4	2	no	—
413	C₂×C₄×C₅	—	6	8	no	—
414	C₂×C₄×C₅	—	6	2	no	—
415	C₂×C₄×C₅	—	6	2	no	—
416	C₂×C₄×C₅	—	6	2	no	—
417	C₂×C₄×C₅	—	24	1	no	—
418	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
419	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
420	C₂×C₄×C₅	C₂×#67	1	2	yes	—
421	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
422	C₅×C₈	—	2	1	no	—
423	C₅×C₈	—	2	1	no	—
424	C₅×C₈	—	6	2	no	—
425	H(5,𝔽₂)	—	6	4	no	—
426	H(5,𝔽₂)	—	6	4	no	—
427	C₂×#41	—	3	1	no	—
428	C₂×#41	—	3	2	no	—
429	C₂×#41	—	6	1	no	—
430	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
431	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic, complemented
432	C₂²×D₄	—	3	2	no	—
433	C₂²×D₄	—	3	1	no	—
434	C₂²×D₄	—	6	1	no	—
435	C₂²×D₄	—	6	2	no	—
436	AGL(1,ℤ/8)	—	6	4	no	—
437	#135	—	2	4	no	—
438	C₂×Q₁₆	C₂×C₅×D₃	1	1	yes	characteristic, complemented
439	C₂×SD₁₆	—	3	2	no	—
440	C₂×SD₁₆	C₂×C₅×D₃	1	1	yes	characteristic, complemented
441	C₂×SD₁₆	C₂×C₅×D₃	1	1	yes	characteristic, complemented
442	C₂×D₈	—	3	2	no	—
443	C₂×D₈	C₂×C₅×D₃	1	1	yes	characteristic, complemented
444	C₂×M₁₆	—	3	1	no	—
445	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic, complemented
446	C₂²⋊Q₈	—	3	2	no	—
447	C₂²⋊Q₈	—	6	1	no	—
448	#121	—	3	2	no	—
449	#121	—	6	1	no	—
450	#121	—	6	2	no	—
451	C₂×C₄⋊C₄	—	3	1	no	—
452	#103	—	3	2	no	—
453	#102	—	3	2	no	—
454	#102	—	3	2	no	—
455	#102	—	3	2	no	—
456	C₂²⋊C₈	—	3	2	no	—
457	C₂²⋊C₈	—	3	2	no	—
458	C₂×C₃×C₅	—	2	2	no	—
459	C₂×C₃×C₅	—	4	2	no	—
460	C₂×C₃×C₅	—	8	1	no	—
461	C₂×C₃×C₅	C₂×#120	1	1	yes	characteristic
462	C₂×C₃×C₅	#576	1	1	yes	characteristic
463	C₂×C₃×C₅	#577	1	1	yes	characteristic
464	C₅×D₃	—	4	2	no	—
465	C₅×D₃	—	4	4	no	—
466	C₂³×C₃	—	2	1	no	—
467	C₂³×C₃	—	2	1	no	—
468	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
469	C₂²×D₃	—	2	2	no	—
470	C₂²×D₃	—	2	8	no	—
471	C₂²×D₃	—	2	4	no	—
472	C₂²×D₃	—	8	2	no	—
473	C₂²×D₃	—	8	2	no	—
474	C₂²×D₃	—	8	1	no	—
475	C₂²×D₃	—	8	1	no	—
476	C₂²×D₃	C₂×C₅×D₄	1	2	yes	—
477	C₃×Q₈	—	2	1	no	—
478	C₃×Q₈	—	2	1	no	—
479	C₃×Q₈	—	2	1	no	—
480	C₃×Q₈	C₂×C₅×D₄	1	2	yes	—
481	C₃×D₄	—	2	4	no	—
482	C₃×D₄	—	2	1	no	—
483	C₃×D₄	—	2	1	no	—
484	C₃×D₄	—	2	1	no	—
485	C₃×D₄	—	4	2	no	—
486	C₃×D₄	C₂×C₅×D₄	1	2	yes	—
487	C₂×C₃×C₄	—	2	1	no	—
488	C₂×C₃×C₄	—	2	4	no	—
489	C₂×C₃×C₄	—	2	1	no	—
490	C₂×C₃×C₄	—	4	2	no	—
491	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
492	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
493	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
494	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
495	#67	—	2	8	no	—
496	#67	—	2	4	no	—
497	#67	—	8	2	no	—
498	C₂×Dic₃	—	2	2	no	—
499	C₂×Dic₃	—	8	1	no	—
500	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
501	D₁₂	—	2	8	no	—
502	D₁₂	—	2	2	no	—
503	D₁₂	—	2	4	no	—
504	D₁₂	—	2	2	no	—
505	D₁₂	C₂×C₅×D₄	1	4	yes	—
506	C₄×D₃	—	2	8	no	—
507	C₄×D₃	—	2	2	no	—
508	C₄×D₃	—	2	2	no	—
509	Dic₆	—	2	2	no	—
510	C₃×C₈	—	2	1	no	—
511	C₃×C₈	—	2	1	no	—
512	C₃⋊C₈	—	2	2	no	—
513	C₂²×C₅	—	2	1	no	—
514	C₂²×C₅	—	2	1	no	—
515	C₂²×C₅	—	2	2	no	—
516	C₂²×C₅	—	4	1	no	—
517	C₂²×C₅	—	4	1	no	—
518	C₂²×C₅	—	4	1	no	—
519	C₂²×C₅	—	4	1	no	—
520	C₂²×C₅	—	4	1	no	—
521	C₂²×C₅	—	6	2	no	—
522	C₂²×C₅	—	6	2	no	—
523	C₂²×C₅	—	6	4	no	—
524	C₂²×C₅	—	6	2	no	—
525	C₂²×C₅	—	12	2	no	—
526	C₂²×C₅	—	12	2	no	—
527	C₂²×C₅	—	24	4	no	—
528	C₂²×C₅	—	24	8	no	—
529	C₂²×C₅	#937	1	2	yes	—
530	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
531	C₄×C₅	—	2	2	no	—
532	C₄×C₅	—	4	1	no	—
533	C₄×C₅	—	6	2	no	—
534	C₄×C₅	—	12	2	no	—
535	C₄×C₅	#937	1	2	yes	—
536	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
537	C₄×C₅	C₂²×#67	1	1	yes	characteristic
538	C₂⁴	—	6	2	no	—
539	C₂⁴	—	6	1	no	—
540	#41	—	2	2	no	—
541	#41	—	2	2	no	—
542	#41	—	2	2	no	—
543	#41	—	6	8	no	—
544	#41	—	6	4	no	—
545	#41	—	6	4	no	—
546	#41	—	6	2	no	—
547	#41	—	6	2	no	—
548	#41	C₅×#67	1	4	yes	complemented
549	C₂×Q₈	—	3	1	no	—
550	C₂×Q₈	C₂²×C₅×D₃	1	1	yes	characteristic
551	C₂×Q₈	C₂²×C₅×D₃	1	1	yes	characteristic
552	C₂×D₄	—	2	1	no	—
553	C₂×D₄	—	2	1	no	—
554	C₂×D₄	—	3	2	no	—
555	C₂×D₄	—	3	4	no	—
556	C₂×D₄	—	3	1	no	—
557	C₂×D₄	—	3	2	no	—
558	C₂×D₄	—	6	2	no	—
559	C₂×D₄	—	6	8	no	—
560	C₂×D₄	—	6	2	no	—
561	C₂×D₄	—	6	4	no	—
562	C₂×D₄	—	6	16	no	—
563	C₂×D₄	—	6	4	no	—
564	C₂×D₄	—	6	2	no	—
565	C₂×D₄	—	6	1	no	—
566	C₂×D₄	—	6	2	no	—
567	C₂×D₄	C₂²×C₅×D₃	1	1	yes	characteristic
568	C₂×D₄	C₂²×C₅×D₃	1	1	yes	characteristic
569	C₂²×C₄	—	2	1	no	—
570	C₂²×C₄	—	2	1	no	—
571	C₂²×C₄	—	3	2	no	—
572	C₂²×C₄	—	6	2	no	—
573	C₂²×C₄	—	6	1	no	—
574	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
575	Q₁₆	—	2	2	no	—
576	SD₁₆	—	2	2	no	—
577	SD₁₆	—	2	2	no	—
578	SD₁₆	—	6	4	no	—
579	D₈	—	2	2	no	—
580	D₈	—	6	4	no	—
581	M₁₆	—	6	2	no	—
582	C₂×C₈	—	2	2	no	—
583	C₂×C₈	—	3	2	no	—
584	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
585	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
586	C₄⋊C₄	—	3	2	no	—
587	C₄⋊C₄	—	6	2	no	—
588	C₄⋊C₄	—	6	1	no	—
589	C₂²⋊C₄	—	6	2	no	—
590	C₂²⋊C₄	—	6	2	no	—
591	C₃×C₅	#3012	1	1	yes	characteristic, complemented, Hall
592	C₂²×C₃	—	2	1	no	—
593	C₂²×C₃	—	2	1	no	—
594	C₂²×C₃	—	2	2	no	—
595	C₂²×C₃	—	4	1	no	—
596	C₂²×C₃	—	4	1	no	—
597	C₂²×C₃	—	4	1	no	—
598	C₂²×C₃	—	4	1	no	—
599	C₂²×C₃	—	4	1	no	—
600	C₂²×C₃	C₅×#120	1	2	yes	—
601	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
602	C₂×D₃	—	2	2	no	—
603	C₂×D₃	—	2	4	no	—
604	C₂×D₃	—	2	2	no	—
605	C₂×D₃	—	2	2	no	—
606	C₂×D₃	—	4	2	no	—
607	C₂×D₃	—	4	2	no	—
608	C₂×D₃	—	8	4	no	—
609	C₂×D₃	—	8	8	no	—
610	C₃×C₄	—	2	2	no	—
611	C₃×C₄	—	4	1	no	—
612	C₃×C₄	C₅×#120	1	2	yes	—
613	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
614	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
615	Dic₃	—	2	2	no	—
616	Dic₃	—	4	2	no	—
617	C₂×C₅	—	2	2	no	—
618	C₂×C₅	—	4	2	no	—
619	C₂×C₅	—	8	1	no	—
620	C₂×C₅	—	12	2	no	—
621	C₂×C₅	—	12	4	no	—
622	C₂×C₅	#5769	1	1	yes	characteristic
623	C₂×C₅	#5770	1	1	yes	characteristic
624	C₂×C₅	C₂×#937	1	1	yes	characteristic
625	C₂³	—	2	1	no	—
626	C₂³	—	2	1	no	—
627	C₂³	—	3	2	no	—
628	C₂³	—	6	8	no	—
629	C₂³	—	6	2	no	—
630	C₂³	—	6	4	no	—
631	C₂³	—	24	1	no	—
632	C₂³	—	24	1	no	—
633	C₂³	—	24	2	no	—
634	C₂³	—	24	2	no	—
635	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
636	Q₈	—	2	1	no	—
637	Q₈	—	2	1	no	—
638	Q₈	—	2	1	no	—
639	Q₈	—	6	2	no	—
640	Q₈	C₂×C₅×#67	1	2	yes	—
641	D₄	—	2	4	no	—
642	D₄	—	2	1	no	—
643	D₄	—	2	1	no	—
644	D₄	—	2	1	no	—
645	D₄	—	3	4	no	—
646	D₄	—	4	2	no	—
647	D₄	—	6	8	no	—
648	D₄	—	6	8	no	—
649	D₄	—	6	4	no	—
650	D₄	—	6	4	no	—
651	D₄	—	6	2	no	—
652	D₄	—	6	2	no	—
653	D₄	—	24	2	no	—
654	D₄	C₂×C₅×#67	1	2	yes	—
655	C₂×C₄	—	2	1	no	—
656	C₂×C₄	—	2	4	no	—
657	C₂×C₄	—	2	1	no	—
658	C₂×C₄	—	3	2	no	—
659	C₂×C₄	—	4	2	no	—
660	C₂×C₄	—	6	8	no	—
661	C₂×C₄	—	6	2	no	—
662	C₂×C₄	—	6	2	no	—
663	C₂×C₄	—	6	2	no	—
664	C₂×C₄	—	24	1	no	—
665	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
666	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
667	C₂×C₄	C₂×C₅×#67	1	2	yes	—
668	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
669	C₈	—	2	1	no	—
670	C₈	—	2	1	no	—
671	C₈	—	6	2	no	—
672	C₂×C₃	—	2	2	no	—
673	C₂×C₃	—	4	2	no	—
674	C₂×C₃	—	8	1	no	—
675	C₂×C₃	C₂×C₅×#120	1	1	yes	characteristic
676	C₂×C₃	C₅×#576	1	1	yes	characteristic
677	C₂×C₃	C₅×#577	1	1	yes	characteristic
678	D₃	—	4	2	no	—
679	D₃	—	4	4	no	—
680	C₅	#83912	1	1	yes	characteristic, direct product factor, Sylow
681	C₂²	—	2	1	no	—
682	C₂²	—	2	1	no	—
683	C₂²	—	2	2	no	—
684	C₂²	—	4	1	no	—
685	C₂²	—	4	1	no	—
686	C₂²	—	4	1	no	—
687	C₂²	—	4	1	no	—
688	C₂²	—	4	1	no	—
689	C₂²	—	6	2	no	—
690	C₂²	—	6	2	no	—
691	C₂²	—	6	4	no	—
692	C₂²	—	6	2	no	—
693	C₂²	—	12	2	no	—
694	C₂²	—	12	2	no	—
695	C₂²	—	24	4	no	—
696	C₂²	—	24	8	no	—
697	C₂²	C₅×#937	1	2	yes	—
698	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
699	C₄	—	2	2	no	—
700	C₄	—	4	1	no	—
701	C₄	—	6	2	no	—
702	C₄	—	12	2	no	—
703	C₄	C₅×#937	1	2	yes	—
704	C₄	C₂×C₅×D₃×D₄	1	1	yes	characteristic
705	C₄	C₂²×C₅×#67	1	1	yes	characteristic
706	C₃	C₅×#3012	1	1	yes	characteristic, complemented, Sylow
707	C₂	—	2	2	no	—
708	C₂	—	4	2	no	—
709	C₂	—	8	1	no	—
710	C₂	—	12	2	no	—
711	C₂	—	12	4	no	—
712	C₂	C₅×#5769	1	1	yes	characteristic
713	C₂	C₅×#5770	1	1	yes	characteristic
714	C₂	C₂×C₅×#937	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×#135	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₂×C₅×#135	D₃
hypocenter	C₃	C₅×#3012
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#135

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₅×#1009	C₂	semidirect
2. C₂×C₃×C₅×#135	C₂	semidirect
3. C₅×#5833	C₂	semidirect
4. C₂×C₅×#949	C₂	semidirect
5. C₅×#5232	C₂	semidirect
6. C₅×#5187	C₂	semidirect
7. C₅×#5172	C₂	semidirect
8. C₅×#5169	C₂	semidirect
9. C₅×#5156	C₂	semidirect
10. C₂×C₅×#821	C₂	semidirect
11. C₅×#5110	C₂	semidirect
14. C₂×C₃×C₅×#41	C₂²	semidirect
25. C₂×C₃×C₅×Q₁₆	C₂²	semidirect
26. C₂×C₃×C₅×SD₁₆	C₂²	semidirect
27. C₂×C₃×C₅×SD₁₆	C₂²	semidirect
28. C₂×C₃×C₅×D₈	C₂²	semidirect
29. C₂²×C₃×C₅×C₈	C₂²	semidirect
31. C₅×#946	C₂²	semidirect
32. C₂×C₅×#215	C₂²	semidirect
34. C₅×#938	C₂²	semidirect
35. C₂×C₅×#213	C₂²	semidirect
37. C₂×C₅×C₄⋊Dic₃	C₂²	semidirect
39. C₂×C₅×#208	C₂²	semidirect
40. C₅×#835	C₂²	semidirect
41. C₅×#832	C₂²	semidirect
42. C₅×#821	C₂²	semidirect
43. C₅×#821	C₂²	semidirect
44. C₅×(C₂×D₃)⋊C₈	C₂²	semidirect
45. C₅×(C₂×D₃)⋊C₈	C₂²	semidirect
46. #83912	C₅	direct (N × H)
49. C₂×C₅×#135	D₃	semidirect
63. C₃×C₅×#41	D₄	semidirect
90. C₂×C₅×D₁₂	D₄	semidirect
111. C₂×#1009	C₂×C₅	semidirect
112. C₂×C₃×#135	C₂×C₅	semidirect
113. #5833	C₂×C₅	semidirect
114. C₂×#949	C₂×C₅	semidirect
115. #5232	C₂×C₅	semidirect
116. #5187	C₂×C₅	semidirect
117. #5172	C₂×C₅	semidirect
118. #5169	C₂×C₅	semidirect
119. #5156	C₂×C₅	semidirect
120. C₂×#821	C₂×C₅	semidirect
121. #5110	C₂×C₅	semidirect
128. C₂×C₅×#41	C₂×D₃	semidirect
135. C₂×C₅×Q₁₆	C₂×D₃	semidirect
137. C₂×C₅×SD₁₆	C₂×D₃	semidirect
138. C₂×C₅×SD₁₆	C₂×D₃	semidirect
140. C₂×C₅×D₈	C₂×D₃	semidirect
142. C₂²×C₅×C₈	C₂×D₃	semidirect
204. C₂×C₃×#41	C₂²×C₅	semidirect
215. C₂×C₃×Q₁₆	C₂²×C₅	semidirect
216. C₂×C₃×SD₁₆	C₂²×C₅	semidirect
217. C₂×C₃×SD₁₆	C₂²×C₅	semidirect
218. C₂×C₃×D₈	C₂²×C₅	semidirect
219. C₂²×C₃×C₈	C₂²×C₅	semidirect
221. #946	C₂²×C₅	semidirect
222. C₂×#215	C₂²×C₅	semidirect
224. #938	C₂²×C₅	semidirect
225. C₂×#213	C₂²×C₅	semidirect
227. C₂×C₄⋊Dic₃	C₂²×C₅	semidirect
229. C₂×#208	C₂²×C₅	semidirect
230. #835	C₂²×C₅	semidirect
231. #832	C₂²×C₅	semidirect
232. #821	C₂²×C₅	semidirect
233. #821	C₂²×C₅	semidirect
234. (C₂×D₃)⋊C₈	C₂²×C₅	semidirect
235. (C₂×D₃)⋊C₈	C₂²×C₅	semidirect
246. C₅×#41	#67	semidirect
291. C₂×#135	C₅×D₃	semidirect
330. C₃×#41	C₅×D₄	semidirect
357. C₂×D₁₂	C₅×D₄	semidirect
431. C₂×#41	C₂×C₅×D₃	semidirect
438. C₂×Q₁₆	C₂×C₅×D₃	semidirect
440. C₂×SD₁₆	C₂×C₅×D₃	semidirect
441. C₂×SD₁₆	C₂×C₅×D₃	semidirect
443. C₂×D₈	C₂×C₅×D₃	semidirect
445. C₂²×C₈	C₂×C₅×D₃	semidirect
548. #41	C₅×#67	semidirect
591. C₃×C₅	#3012	semidirect
680. C₅	#83912	direct (N × H)
706. C₃	C₅×#3012	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#1009	2
C₂×C₃×C₅×#135	2
C₅×#5833	2
C₂×C₅×#949	2
C₅×#5232	2
C₅×#5187	2
C₅×#5172	2
C₅×#5169	2
C₅×#5156	2
C₂×C₅×#821	2
C₅×#5110	2
C₅×#3012	3
#83912	5

Subgroups — C5×#83912 #14424639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#83912 #14424639