Subgroups — C₅×#68755 #14300529

Order 1920.

752 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×D₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5524	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#825	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#805	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#339	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4532	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1871	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₃×C₅×#117	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
15	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₅×#208	C₂²	1	1	yes	characteristic
17	C₅×D₃×M₁₆	—	2	2	no	—
18	C₅×D₃×M₁₆	—	2	2	no	—
19	C₂×C₅×ΓL(1,𝔽₂₅)	—	2	1	no	—
20	C₂×C₅×D₃×C₈	—	2	1	no	—
21	C₅×#897	C₄	1	1	yes	characteristic, complemented
22	C₅×#897	C₄	1	1	yes	characteristic, complemented
23	C₅×#895	C₄	1	1	yes	characteristic, complemented
24	C₅×#895	C₄	1	1	yes	characteristic, complemented
25	C₅×#886	—	2	1	no	—
26	C₅×#882	—	2	1	no	—
27	C₄×C₅×D₁₂	—	2	1	no	—
28	C₄×C₅×D₁₂	—	2	1	no	—
29	C₅×C₃⋊#95	—	2	1	no	—
30	C₅×#825	—	2	2	no	—
31	C₅×#805	—	2	2	no	—
32	#68755	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
33	C₂²×C₅×M₁₆	—	3	1	no	—
34	C₅×#525	—	3	1	no	—
35	C₂×C₅×#104	—	3	1	no	—
36	C₂×C₅×#104	—	3	1	no	—
37	C₄×C₅×C₂²⋊C₄	—	3	1	no	—
38	C₅×#339	—	3	1	no	—
39	C₅×#339	D₃	1	1	yes	characteristic, complemented
40	C₂³×C₅×D₃	D₄	1	1	yes	characteristic
41	C₂²×C₃×C₄×C₅	—	2	1	no	—
42	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₅×#67	—	2	1	no	—
44	C₂²×C₅×Dic₃	—	2	1	no	—
45	C₂²×C₅×Dic₃	D₄	1	1	yes	characteristic
46	C₅×#235	—	2	1	no	—
47	C₅×#235	—	2	1	no	—
48	C₅×#235	—	2	1	no	—
49	C₅×#235	—	2	1	no	—
50	C₂×C₅×D₁₂	—	2	1	no	—
51	C₂×C₅×D₁₂	—	2	1	no	—
52	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×C₅×D₃	—	2	1	no	—
54	C₂×C₄×C₅×D₃	—	2	1	no	—
55	C₂×C₄×C₅×D₃	—	2	1	no	—
56	C₂×C₄×C₅×D₃	—	2	1	no	—
57	C₂×C₄×C₅×D₃	—	2	1	no	—
58	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
61	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
62	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
63	C₃×C₅×M₁₆	—	2	1	no	—
64	C₃×C₅×M₁₆	D₄	1	2	yes	complemented
65	C₂×C₃×C₅×C₈	—	2	1	no	—
66	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
67	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
68	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
69	C₃×C₄²×C₅	—	2	1	no	—
70	C₃×C₄²×C₅	—	2	1	no	—
71	C₃×C₄²×C₅	—	2	1	no	—
72	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
73	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
74	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
75	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
76	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
77	C₅×C₄⋊Dic₃	—	2	1	no	—
78	C₅×C₄⋊Dic₃	—	2	1	no	—
79	C₄×C₅×Dic₃	—	2	1	no	—
80	C₄×C₅×Dic₃	—	2	1	no	—
81	C₅×#208	—	2	1	no	—
82	C₅×#208	D₄	1	2	yes	complemented
83	C₂×C₅×C₃⋊C₈	—	2	1	no	—
84	C₅×ΓL(1,𝔽₂₅)	—	2	4	no	—
85	C₅×D₃×C₈	—	2	4	no	—
86	C₂×D₃×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
87	#5524	C₂×C₅	1	1	yes	characteristic
88	C₂×#825	C₂×C₅	1	1	yes	characteristic, complemented
89	C₄×(C₂×D₃)⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
90	C₂×#805	C₂×C₅	1	1	yes	characteristic, complemented
91	C₃×#339	C₂×C₅	1	1	yes	characteristic, complemented
92	#4532	C₂×C₅	1	1	yes	characteristic, complemented
93	C₂×C₅×#41	—	3	1	no	—
94	C₂³×C₄×C₅	—	3	1	no	—
95	C₂×C₅×M₁₆	—	3	1	no	—
96	C₂×C₅×M₁₆	—	6	1	no	—
97	C₂×C₅×M₁₆	—	6	2	no	—
98	C₂×C₅×M₁₆	—	6	2	no	—
99	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
100	C₂²×C₅×C₈	—	6	1	no	—
101	C₅×#123	—	6	1	no	—
102	C₅×C₂²⋊Q₈	—	3	1	no	—
103	C₅×C₂²⋊Q₈	—	3	1	no	—
104	C₅×#121	—	3	1	no	—
105	C₅×#121	—	3	1	no	—
106	C₅×#120	—	6	1	no	—
107	C₄×C₅×D₄	—	6	1	no	—
108	C₄×C₅×D₄	—	6	1	no	—
109	C₅×#117	C₂×D₃	1	1	yes	characteristic
110	C₂×C₅×C₂²⋊C₄	—	6	1	no	—
111	C₂×C₄²×C₅	—	3	1	no	—
112	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
113	C₅×#104	—	6	2	no	—
114	C₅×#104	—	6	2	no	—
115	C₅×#95	—	6	1	no	—
116	#1871	—	3	1	no	Sylow
117	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₅×D₃	—	2	1	no	—
119	C₂²×C₅×D₃	—	2	1	no	—
120	C₂²×C₅×D₃	—	2	2	no	—
121	C₂²×C₅×D₃	—	2	1	no	—
122	C₂²×C₅×D₃	—	4	1	no	—
123	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₃×C₄×C₅	—	2	2	no	—
126	C₂×C₃×C₄×C₅	—	2	1	no	—
127	C₂×C₃×C₄×C₅	—	2	1	no	—
128	C₂×C₃×C₄×C₅	—	4	1	no	—
129	C₂×C₃×C₄×C₅	—	4	1	no	—
130	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
132	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
134	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
135	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
136	C₅×#67	—	4	1	no	—
137	C₅×#67	—	4	1	no	—
138	C₂×C₅×Dic₃	—	2	1	no	—
139	C₂×C₅×Dic₃	—	2	1	no	—
140	C₂×C₅×Dic₃	—	2	2	no	—
141	C₂×C₅×Dic₃	—	2	1	no	—
142	C₂×C₅×Dic₃	—	4	1	no	—
143	C₂×C₅×Dic₃	—	4	1	no	—
144	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
146	C₅×D₁₂	—	2	1	no	—
147	C₅×D₁₂	—	2	1	no	—
148	C₅×D₁₂	—	4	2	no	—
149	C₅×D₁₂	—	4	2	no	—
150	C₄×C₅×D₃	—	2	1	no	—
151	C₄×C₅×D₃	—	2	1	no	—
152	C₄×C₅×D₃	—	2	1	no	—
153	C₄×C₅×D₃	—	2	1	no	—
154	C₄×C₅×D₃	—	2	1	no	—
155	C₄×C₅×D₃	—	2	1	no	—
156	C₄×C₅×D₃	—	2	1	no	—
157	C₄×C₅×D₃	—	2	1	no	—
158	C₄×C₅×D₃	—	4	1	no	—
159	C₄×C₅×D₃	—	4	1	no	—
160	C₅×Dic₆	—	2	1	no	—
161	C₅×Dic₆	—	2	1	no	—
162	C₃×C₅×C₈	—	2	2	no	—
163	C₅×C₃⋊C₈	—	2	2	no	—
164	C₂×#235	C₂²×C₅	1	1	yes	characteristic
165	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
166	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
167	C₃×#117	C₂²×C₅	1	1	yes	characteristic
168	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
169	C₂×(C₂×D₃)⋊C₄	—	2	1	no	—
170	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
171	C₂×#208	C₂²×C₅	1	1	yes	characteristic
172	D₃×M₁₆	—	2	2	no	—
173	D₃×M₁₆	—	2	2	no	—
174	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
175	C₂×D₃×C₈	—	2	1	no	—
176	#897	C₄×C₅	1	1	yes	characteristic, complemented
177	#897	C₄×C₅	1	1	yes	characteristic, complemented
178	#895	C₄×C₅	1	1	yes	characteristic, complemented
179	#895	C₄×C₅	1	1	yes	characteristic, complemented
180	#886	—	2	1	no	—
181	#882	—	2	1	no	—
182	C₄×D₁₂	—	2	1	no	—
183	C₄×D₁₂	—	2	1	no	—
184	C₃⋊#95	—	2	1	no	—
185	#825	—	2	2	no	—
186	#805	—	2	2	no	—
187	C₂⁴×C₅	—	3	1	no	—
188	C₅×#41	—	6	1	no	—
189	C₅×#41	—	6	1	no	—
190	C₅×#41	—	6	1	no	—
191	C₅×#41	—	6	1	no	—
192	C₂×C₅×Q₈	—	3	1	no	—
193	C₂×C₅×D₄	—	3	1	no	—
194	C₂×C₅×D₄	—	6	1	no	—
195	C₂×C₅×D₄	—	6	1	no	—
196	C₂×C₅×D₄	—	6	1	no	—
197	C₂²×C₄×C₅	—	2	1	no	—
198	C₂²×C₄×C₅	—	3	1	no	—
199	C₂²×C₄×C₅	—	3	1	no	—
200	C₂²×C₄×C₅	—	3	1	no	—
201	C₂²×C₄×C₅	—	3	1	no	—
202	C₂²×C₄×C₅	—	3	1	no	—
203	C₂²×C₄×C₅	—	6	1	no	—
204	C₂²×C₄×C₅	—	6	1	no	—
205	C₂²×C₄×C₅	—	6	1	no	—
206	C₂²×C₄×C₅	—	6	1	no	—
207	C₂²×C₄×C₅	—	6	1	no	—
208	C₂²×C₄×C₅	—	6	1	no	—
209	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
210	C₅×M₁₆	—	2	1	no	—
211	C₅×M₁₆	—	3	2	no	—
212	C₅×M₁₆	—	6	4	no	—
213	C₅×M₁₆	—	6	1	no	—
214	C₅×M₁₆	D₁₂	1	2	yes	complemented
215	C₂×C₅×C₈	—	2	1	no	—
216	C₂×C₅×C₈	—	6	4	no	—
217	C₂×C₅×C₈	—	6	1	no	—
218	C₅×C₄⋊C₄	—	6	1	no	—
219	C₅×C₄⋊C₄	—	6	1	no	—
220	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
221	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
222	C₅×C₂²⋊C₄	—	2	1	no	—
223	C₅×C₂²⋊C₄	—	6	1	no	—
224	C₅×C₂²⋊C₄	—	6	1	no	—
225	C₅×C₂²⋊C₄	—	6	1	no	—
226	C₅×C₂²⋊C₄	—	6	1	no	—
227	C₅×C₂²⋊C₄	—	6	4	no	—
228	C₄²×C₅	—	2	1	no	—
229	C₄²×C₅	—	2	1	no	—
230	C₄²×C₅	—	2	1	no	—
231	C₄²×C₅	—	6	1	no	—
232	C₄²×C₅	—	6	1	no	—
233	C₂²×M₁₆	—	3	1	no	—
234	#525	—	3	1	no	—
235	C₂×#104	—	3	1	no	—
236	C₂×#104	—	3	1	no	—
237	C₄×C₂²⋊C₄	—	3	1	no	—
238	#339	—	3	1	no	—
239	#339	C₅×D₃	1	1	yes	characteristic, complemented
240	C₂²×C₃×C₅	—	2	1	no	—
241	C₂²×C₃×C₅	—	2	1	no	—
242	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
243	C₂²×C₃×C₅	#121	1	1	yes	characteristic
244	C₂²×C₃×C₅	#121	1	1	yes	characteristic
245	C₂×C₅×D₃	—	2	1	no	—
246	C₂×C₅×D₃	—	2	1	no	—
247	C₂×C₅×D₃	—	2	1	no	—
248	C₂×C₅×D₃	—	2	1	no	—
249	C₂×C₅×D₃	—	2	1	no	—
250	C₂×C₅×D₃	—	4	2	no	—
251	C₂×C₅×D₃	—	4	2	no	—
252	C₂×C₅×D₃	—	4	1	no	—
253	C₂×C₅×D₃	—	4	1	no	—
254	C₂×C₅×D₃	—	4	1	no	—
255	C₂×C₅×D₃	#104	1	2	yes	complemented
256	C₃×C₄×C₅	—	4	2	no	—
257	C₃×C₄×C₅	—	4	1	no	—
258	C₃×C₄×C₅	—	4	1	no	—
259	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
260	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
261	C₃×C₄×C₅	#120	1	1	yes	characteristic
262	C₃×C₄×C₅	#123	1	1	yes	characteristic
263	C₅×Dic₃	—	2	1	no	—
264	C₅×Dic₃	—	2	1	no	—
265	C₅×Dic₃	—	4	2	no	—
266	C₅×Dic₃	—	4	1	no	—
267	C₂³×D₃	C₅×D₄	1	1	yes	characteristic
268	C₂²×C₃×C₄	—	2	1	no	—
269	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
270	C₂×#67	—	2	1	no	—
271	C₂²×Dic₃	—	2	1	no	—
272	C₂²×Dic₃	C₅×D₄	1	1	yes	characteristic
273	#235	—	2	1	no	—
274	#235	—	2	1	no	—
275	#235	—	2	1	no	—
276	#235	—	2	1	no	—
277	C₂×D₁₂	—	2	1	no	—
278	C₂×D₁₂	—	2	1	no	—
279	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic
280	C₂×C₄×D₃	—	2	1	no	—
281	C₂×C₄×D₃	—	2	1	no	—
282	C₂×C₄×D₃	—	2	1	no	—
283	C₂×C₄×D₃	—	2	1	no	—
284	C₂×C₄×D₃	—	2	1	no	—
285	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
286	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
287	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
288	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
289	C₂×Dic₆	C₂×C₄×C₅	1	1	yes	characteristic
290	C₃×M₁₆	—	2	1	no	—
291	C₃×M₁₆	C₅×D₄	1	2	yes	complemented
292	C₂×C₃×C₈	—	2	1	no	—
293	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
294	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
295	C₃×C₂²⋊C₄	—	2	1	no	—
296	C₃×C₄²	—	2	1	no	—
297	C₃×C₄²	—	2	1	no	—
298	C₃×C₄²	—	2	1	no	—
299	(C₂×D₃)⋊C₄	—	2	4	no	—
300	(C₂×D₃)⋊C₄	—	2	1	no	—
301	(C₂×D₃)⋊C₄	—	2	1	no	—
302	(C₂×D₃)⋊C₄	—	2	1	no	—
303	(C₂×D₃)⋊C₄	—	2	1	no	—
304	C₄⋊Dic₃	—	2	1	no	—
305	C₄⋊Dic₃	—	2	1	no	—
306	C₄×Dic₃	—	2	1	no	—
307	C₄×Dic₃	—	2	1	no	—
308	#208	—	2	1	no	—
309	#208	C₅×D₄	1	2	yes	complemented
310	C₂×C₃⋊C₈	—	2	1	no	—
311	ΓL(1,𝔽₂₅)	—	2	4	no	—
312	D₃×C₈	—	2	4	no	—
313	C₂³×C₅	—	3	1	no	—
314	C₂³×C₅	—	3	1	no	—
315	C₂³×C₅	—	6	2	no	—
316	C₂³×C₅	—	6	1	no	—
317	C₂³×C₅	—	6	1	no	—
318	C₂³×C₅	—	6	1	no	—
319	C₂³×C₅	—	12	1	no	—
320	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
321	C₅×Q₈	—	6	1	no	—
322	C₅×Q₈	—	6	1	no	—
323	C₅×D₄	—	6	1	no	—
324	C₅×D₄	—	6	1	no	—
325	C₅×D₄	—	12	2	no	—
326	C₅×D₄	—	12	2	no	—
327	C₅×D₄	—	12	1	no	—
328	C₅×D₄	—	12	1	no	—
329	C₂×C₄×C₅	—	2	2	no	—
330	C₂×C₄×C₅	—	2	1	no	—
331	C₂×C₄×C₅	—	2	1	no	—
332	C₂×C₄×C₅	—	3	1	no	—
333	C₂×C₄×C₅	—	3	1	no	—
334	C₂×C₄×C₅	—	4	1	no	—
335	C₂×C₄×C₅	—	4	1	no	—
336	C₂×C₄×C₅	—	6	1	no	—
337	C₂×C₄×C₅	—	6	1	no	—
338	C₂×C₄×C₅	—	6	1	no	—
339	C₂×C₄×C₅	—	6	1	no	—
340	C₂×C₄×C₅	—	6	1	no	—
341	C₂×C₄×C₅	—	6	1	no	—
342	C₂×C₄×C₅	—	6	1	no	—
343	C₂×C₄×C₅	—	6	1	no	—
344	C₂×C₄×C₅	—	6	1	no	—
345	C₂×C₄×C₅	—	6	1	no	—
346	C₂×C₄×C₅	—	6	2	no	—
347	C₂×C₄×C₅	—	6	1	no	—
348	C₂×C₄×C₅	—	12	1	no	—
349	C₂×C₄×C₅	—	12	1	no	—
350	C₂×C₄×C₅	—	12	1	no	—
351	C₂×C₄×C₅	—	12	1	no	—
352	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
353	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
354	C₂×C₄×C₅	#235	1	1	yes	characteristic
355	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
356	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
357	C₂×C₄×C₅	#239	1	1	yes	characteristic
358	C₅×C₈	—	2	2	no	—
359	C₅×C₈	—	6	2	no	—
360	C₂×#41	—	3	1	no	—
361	C₂³×C₄	—	3	1	no	—
362	C₂×M₁₆	—	3	1	no	—
363	C₂×M₁₆	—	6	1	no	—
364	C₂×M₁₆	—	6	2	no	—
365	C₂×M₁₆	—	6	2	no	—
366	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
367	C₂²×C₈	—	6	1	no	—
368	#123	—	6	1	no	—
369	C₂²⋊Q₈	—	3	1	no	—
370	C₂²⋊Q₈	—	3	1	no	—
371	#121	—	3	1	no	—
372	#121	—	3	1	no	—
373	#120	—	6	1	no	—
374	C₄×D₄	—	6	1	no	—
375	C₄×D₄	—	6	1	no	—
376	#117	C₂×C₅×D₃	1	1	yes	characteristic
377	C₂×C₂²⋊C₄	—	6	1	no	—
378	C₂×C₄²	—	3	1	no	—
379	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
380	#104	—	6	2	no	—
381	#104	—	6	2	no	—
382	#95	—	6	1	no	—
383	C₂×C₃×C₅	—	2	1	no	—
384	C₂×C₃×C₅	—	2	1	no	—
385	C₂×C₃×C₅	#386	1	1	yes	characteristic
386	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
387	C₂×C₃×C₅	#421	1	1	yes	characteristic
388	C₅×D₃	—	2	2	no	—
389	C₅×D₃	—	4	1	no	—
390	C₅×D₃	—	8	1	no	—
391	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
392	C₂²×D₃	—	2	1	no	—
393	C₂²×D₃	—	2	1	no	—
394	C₂²×D₃	—	2	2	no	—
395	C₂²×D₃	—	2	1	no	—
396	C₂²×D₃	—	4	1	no	—
397	C₂²×D₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
398	C₂²×D₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
399	C₂×C₃×C₄	—	2	2	no	—
400	C₂×C₃×C₄	—	2	1	no	—
401	C₂×C₃×C₄	—	2	1	no	—
402	C₂×C₃×C₄	—	4	1	no	—
403	C₂×C₃×C₄	—	4	1	no	—
404	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
405	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
406	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
407	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
408	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
409	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
410	#67	—	4	1	no	—
411	#67	—	4	1	no	—
412	C₂×Dic₃	—	2	1	no	—
413	C₂×Dic₃	—	2	1	no	—
414	C₂×Dic₃	—	2	2	no	—
415	C₂×Dic₃	—	2	1	no	—
416	C₂×Dic₃	—	4	1	no	—
417	C₂×Dic₃	—	4	1	no	—
418	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
419	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
420	D₁₂	—	2	1	no	—
421	D₁₂	—	2	1	no	—
422	D₁₂	—	4	2	no	—
423	D₁₂	—	4	2	no	—
424	C₄×D₃	—	2	1	no	—
425	C₄×D₃	—	2	1	no	—
426	C₄×D₃	—	2	1	no	—
427	C₄×D₃	—	2	1	no	—
428	C₄×D₃	—	2	1	no	—
429	C₄×D₃	—	2	1	no	—
430	C₄×D₃	—	2	1	no	—
431	C₄×D₃	—	2	1	no	—
432	C₄×D₃	—	4	1	no	—
433	C₄×D₃	—	4	1	no	—
434	Dic₆	—	2	1	no	—
435	Dic₆	—	2	1	no	—
436	C₃×C₈	—	2	2	no	—
437	C₃⋊C₈	—	2	2	no	—
438	C₂²×C₅	—	2	1	no	—
439	C₂²×C₅	—	2	1	no	—
440	C₂²×C₅	—	3	2	no	—
441	C₂²×C₅	—	6	1	no	—
442	C₂²×C₅	—	6	1	no	—
443	C₂²×C₅	—	6	1	no	—
444	C₂²×C₅	—	6	1	no	—
445	C₂²×C₅	—	6	1	no	—
446	C₂²×C₅	—	12	2	no	—
447	C₂²×C₅	—	12	2	no	—
448	C₂²×C₅	—	12	1	no	—
449	C₂²×C₅	—	12	1	no	—
450	C₂²×C₅	—	12	1	no	—
451	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
452	C₂²×C₅	#884	1	1	yes	characteristic
453	C₂²×C₅	#895	1	1	yes	characteristic
454	C₄×C₅	—	4	2	no	—
455	C₄×C₅	—	4	1	no	—
456	C₄×C₅	—	4	1	no	—
457	C₄×C₅	—	6	1	no	—
458	C₄×C₅	—	6	1	no	—
459	C₄×C₅	—	12	2	no	—
460	C₄×C₅	—	12	1	no	—
461	C₄×C₅	C₄×D₁₂	1	1	yes	characteristic
462	C₄×C₅	#882	1	1	yes	characteristic
463	C₄×C₅	#893	1	1	yes	characteristic
464	C₄×C₅	#894	1	1	yes	characteristic
465	C₂⁴	—	3	1	no	—
466	#41	—	6	1	no	—
467	#41	—	6	1	no	—
468	#41	—	6	1	no	—
469	#41	—	6	1	no	—
470	C₂×Q₈	—	3	1	no	—
471	C₂×D₄	—	3	1	no	—
472	C₂×D₄	—	6	1	no	—
473	C₂×D₄	—	6	1	no	—
474	C₂×D₄	—	6	1	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	3	1	no	—
477	C₂²×C₄	—	3	1	no	—
478	C₂²×C₄	—	3	1	no	—
479	C₂²×C₄	—	3	1	no	—
480	C₂²×C₄	—	3	1	no	—
481	C₂²×C₄	—	6	1	no	—
482	C₂²×C₄	—	6	1	no	—
483	C₂²×C₄	—	6	1	no	—
484	C₂²×C₄	—	6	1	no	—
485	C₂²×C₄	—	6	1	no	—
486	C₂²×C₄	—	6	1	no	—
487	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
488	M₁₆	—	2	1	no	—
489	M₁₆	—	3	2	no	—
490	M₁₆	—	6	4	no	—
491	M₁₆	—	6	1	no	—
492	M₁₆	C₅×D₁₂	1	2	yes	complemented
493	C₂×C₈	—	2	1	no	—
494	C₂×C₈	—	6	4	no	—
495	C₂×C₈	—	6	1	no	—
496	C₄⋊C₄	—	6	1	no	—
497	C₄⋊C₄	—	6	1	no	—
498	C₄⋊C₄	C₄×C₅×D₃	1	1	yes	characteristic
499	C₄⋊C₄	C₄×C₅×D₃	1	1	yes	characteristic
500	C₂²⋊C₄	—	2	1	no	—
501	C₂²⋊C₄	—	6	1	no	—
502	C₂²⋊C₄	—	6	1	no	—
503	C₂²⋊C₄	—	6	1	no	—
504	C₂²⋊C₄	—	6	1	no	—
505	C₂²⋊C₄	—	6	4	no	—
506	C₄²	—	2	1	no	—
507	C₄²	—	2	1	no	—
508	C₄²	—	2	1	no	—
509	C₄²	—	6	1	no	—
510	C₄²	—	6	1	no	—
511	C₃×C₅	#1871	1	1	yes	characteristic, complemented, Hall
512	C₂²×C₃	—	2	1	no	—
513	C₂²×C₃	—	2	1	no	—
514	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
515	C₂²×C₃	C₅×#121	1	1	yes	characteristic
516	C₂²×C₃	C₅×#121	1	1	yes	characteristic
517	C₂×D₃	—	2	1	no	—
518	C₂×D₃	—	2	1	no	—
519	C₂×D₃	—	2	1	no	—
520	C₂×D₃	—	2	1	no	—
521	C₂×D₃	—	2	1	no	—
522	C₂×D₃	—	4	2	no	—
523	C₂×D₃	—	4	2	no	—
524	C₂×D₃	—	4	1	no	—
525	C₂×D₃	—	4	1	no	—
526	C₂×D₃	—	4	1	no	—
527	C₂×D₃	C₅×#104	1	2	yes	complemented
528	C₃×C₄	—	4	2	no	—
529	C₃×C₄	—	4	1	no	—
530	C₃×C₄	—	4	1	no	—
531	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
532	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
533	C₃×C₄	C₅×#120	1	1	yes	characteristic
534	C₃×C₄	C₅×#123	1	1	yes	characteristic
535	Dic₃	—	2	1	no	—
536	Dic₃	—	2	1	no	—
537	Dic₃	—	4	2	no	—
538	Dic₃	—	4	1	no	—
539	C₂×C₅	—	2	1	no	—
540	C₂×C₅	—	2	1	no	—
541	C₂×C₅	—	6	2	no	—
542	C₂×C₅	—	12	1	no	—
543	C₂×C₅	—	24	1	no	—
544	C₂×C₅	#4665	1	1	yes	characteristic
545	C₂×C₅	D₃×#104	1	1	yes	characteristic
546	C₂×C₅	#4821	1	1	yes	characteristic
547	C₂³	—	3	1	no	—
548	C₂³	—	3	1	no	—
549	C₂³	—	6	2	no	—
550	C₂³	—	6	1	no	—
551	C₂³	—	6	1	no	—
552	C₂³	—	6	1	no	—
553	C₂³	—	12	1	no	—
554	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
555	Q₈	—	6	1	no	—
556	Q₈	—	6	1	no	—
557	D₄	—	6	1	no	—
558	D₄	—	6	1	no	—
559	D₄	—	12	2	no	—
560	D₄	—	12	2	no	—
561	D₄	—	12	1	no	—
562	D₄	—	12	1	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	3	1	no	—
567	C₂×C₄	—	3	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	6	1	no	—
571	C₂×C₄	—	6	1	no	—
572	C₂×C₄	—	6	1	no	—
573	C₂×C₄	—	6	1	no	—
574	C₂×C₄	—	6	1	no	—
575	C₂×C₄	—	6	1	no	—
576	C₂×C₄	—	6	1	no	—
577	C₂×C₄	—	6	1	no	—
578	C₂×C₄	—	6	1	no	—
579	C₂×C₄	—	6	1	no	—
580	C₂×C₄	—	6	2	no	—
581	C₂×C₄	—	6	1	no	—
582	C₂×C₄	—	12	1	no	—
583	C₂×C₄	—	12	1	no	—
584	C₂×C₄	—	12	1	no	—
585	C₂×C₄	—	12	1	no	—
586	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
587	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
588	C₂×C₄	C₅×#235	1	1	yes	characteristic
589	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
590	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
591	C₂×C₄	C₅×#239	1	1	yes	characteristic
592	C₈	—	2	2	no	—
593	C₈	—	6	2	no	—
594	C₂×C₃	—	2	1	no	—
595	C₂×C₃	—	2	1	no	—
596	C₂×C₃	C₅×#386	1	1	yes	characteristic
597	C₂×C₃	C₂×C₅×#104	1	1	yes	characteristic
598	C₂×C₃	C₅×#421	1	1	yes	characteristic
599	D₃	—	2	2	no	—
600	D₃	—	4	1	no	—
601	D₃	—	8	1	no	—
602	C₅	#68755	1	1	yes	characteristic, direct product factor, Sylow
603	C₂²	—	2	1	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	3	2	no	—
606	C₂²	—	6	1	no	—
607	C₂²	—	6	1	no	—
608	C₂²	—	6	1	no	—
609	C₂²	—	6	1	no	—
610	C₂²	—	6	1	no	—
611	C₂²	—	12	2	no	—
612	C₂²	—	12	2	no	—
613	C₂²	—	12	1	no	—
614	C₂²	—	12	1	no	—
615	C₂²	—	12	1	no	—
616	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
617	C₂²	C₅×#884	1	1	yes	characteristic
618	C₂²	C₅×#895	1	1	yes	characteristic
619	C₄	—	4	2	no	—
620	C₄	—	4	1	no	—
621	C₄	—	4	1	no	—
622	C₄	—	6	1	no	—
623	C₄	—	6	1	no	—
624	C₄	—	12	2	no	—
625	C₄	—	12	1	no	—
626	C₄	C₄×C₅×D₁₂	1	1	yes	characteristic
627	C₄	C₅×#882	1	1	yes	characteristic
628	C₄	C₅×#893	1	1	yes	characteristic
629	C₄	C₅×#894	1	1	yes	characteristic
630	C₃	C₅×#1871	1	1	yes	characteristic, complemented, Sylow
631	C₂	—	2	1	no	—
632	C₂	—	2	1	no	—
633	C₂	—	6	2	no	—
634	C₂	—	12	1	no	—
635	C₂	—	24	1	no	—
636	C₂	C₅×#4665	1	1	yes	characteristic
637	C₂	C₅×D₃×#104	1	1	yes	characteristic
638	C₂	C₅×#4821	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#339	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₅×#5524	C₂
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#339	D₃
hypocenter	C₃	C₅×#1871
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#339

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×D₃×M₁₆	C₂	semidirect
3. C₂×C₅×#825	C₂	semidirect
4. C₄×C₅×(C₂×D₃)⋊C₄	C₂	semidirect
5. C₂×C₅×#805	C₂	semidirect
6. C₃×C₅×#339	C₂	semidirect
7. C₅×#4532	C₂	semidirect
21. C₅×#897	C₄	semidirect
22. C₅×#897	C₄	semidirect
23. C₅×#895	C₄	semidirect
24. C₅×#895	C₄	semidirect
32. #68755	C₅	direct (N × H)
39. C₅×#339	D₃	semidirect
64. C₃×C₅×M₁₆	D₄	semidirect
82. C₅×#208	D₄	semidirect
86. C₂×D₃×M₁₆	C₂×C₅	semidirect
88. C₂×#825	C₂×C₅	semidirect
89. C₄×(C₂×D₃)⋊C₄	C₂×C₅	semidirect
90. C₂×#805	C₂×C₅	semidirect
91. C₃×#339	C₂×C₅	semidirect
92. #4532	C₂×C₅	semidirect
176. #897	C₄×C₅	semidirect
177. #897	C₄×C₅	semidirect
178. #895	C₄×C₅	semidirect
179. #895	C₄×C₅	semidirect
214. C₅×M₁₆	D₁₂	semidirect
239. #339	C₅×D₃	semidirect
255. C₂×C₅×D₃	#104	semidirect
291. C₃×M₁₆	C₅×D₄	semidirect
309. #208	C₅×D₄	semidirect
492. M₁₆	C₅×D₁₂	semidirect
511. C₃×C₅	#1871	semidirect
527. C₂×D₃	C₅×#104	semidirect
602. C₅	#68755	direct (N × H)
630. C₃	C₅×#1871	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×D₃×M₁₆	2
C₅×#5524	2
C₂×C₅×#825	2
C₄×C₅×(C₂×D₃)⋊C₄	2
C₂×C₅×#805	2
C₃×C₅×#339	2
C₅×#4532	2
C₅×#1871	3
#68755	5

Subgroups — C5×#68755 #14300529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68755 #14300529