Subgroups — C₅×#78659 #14415457

Order 1920.

1004 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5828	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×D₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#408	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5119	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×(C₂×D₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#5101	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4730	C₂	1	2	yes	maximal, complemented
9	C₅×#4729	C₂	1	2	yes	maximal, complemented
10	C₅×#4728	C₂	1	2	yes	maximal, complemented
11	C₅×#67⋊C₈	C₂	1	2	yes	maximal, complemented
12	C₅×#2933	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#948	—	2	4	no	—
18	C₅×#946	C₄	1	1	yes	characteristic
19	C₅×#946	C₄	1	1	yes	characteristic
20	C₅×#938	C₄	1	1	yes	characteristic
21	C₅×#938	C₄	1	1	yes	characteristic
22	C₅×#937	C₄	1	2	yes	—
23	C₅×#935	C₄	1	2	yes	—
24	C₄×C₅×#67	C₂²	1	2	yes	—
25	C₄×C₅×#67	C₂²	1	2	yes	—
26	C₅×#926	C₂²	1	1	yes	characteristic
27	C₂×C₅×#208	C₂²	1	1	yes	characteristic
28	C₂×C₅×#208	C₂²	1	2	yes	complemented
29	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
30	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	complemented
31	C₅×D₃×M₁₆	—	2	4	no	—
32	C₂×C₅×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
33	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
34	C₃×C₅×C₂²⋊C₈	C₂²	1	2	yes	complemented
35	C₃×C₅×C₂²⋊C₈	C₂²	1	2	yes	complemented
36	C₅×(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
37	C₅×(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
38	C₅×(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
39	C₅×(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
40	C₅×#815	C₂²	1	2	yes	complemented
41	C₅×Dic₃⋊C₈	C₂²	1	2	yes	complemented
42	C₅×Dic₃⋊C₈	C₂²	1	2	yes	complemented
43	C₅×C₈×Dic₃	C₂²	1	2	yes	complemented
44	#78659	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
45	C₂×C₅×#131	—	3	1	no	—
46	C₂²×C₅×M₁₆	—	3	1	no	—
47	C₅×#525	—	3	1	no	—
48	C₅×#436	—	3	2	no	—
49	C₅×#435	—	3	2	no	—
50	C₅×#435	—	3	2	no	—
51	C₅×C₈×D₄	—	3	2	no	—
52	C₅×#424	—	3	1	no	—
53	C₅×#408	D₃	1	1	yes	characteristic, complemented
54	C₂×C₅×C₂²⋊C₈	—	3	1	no	—
55	C₅×#405	—	3	1	no	—
56	C₂³×C₅×D₃	C₂×C₄	1	1	yes	characteristic
57	C₃×C₅×#41	—	2	2	no	—
58	C₃×C₅×#41	—	2	2	no	—
59	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
61	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
63	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
64	C₂×C₅×#67	C₂×C₄	1	2	yes	—
65	C₂×C₅×#67	C₂×C₄	1	2	yes	—
66	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×C₅×D₃	—	2	2	no	—
68	C₂×C₄×C₅×D₃	—	2	2	no	—
69	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
72	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
73	C₃×C₅×M₁₆	—	2	2	no	—
74	C₂×C₃×C₅×C₈	—	2	2	no	—
75	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
76	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
77	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
78	C₅×#217	C₂×C₄	1	2	yes	—
79	C₅×#217	C₂×C₄	1	2	yes	—
80	C₅×#217	C₂×C₄	1	2	yes	—
81	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
82	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
83	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
84	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
85	C₅×Dic₃⋊C₄	C₂×C₄	1	2	yes	—
86	C₅×Dic₃⋊C₄	C₂×C₄	1	2	yes	—
87	C₄×C₅×Dic₃	C₂³	1	2	yes	—
88	C₅×#208	—	2	4	no	—
89	C₅×#208	D₄	1	4	yes	complemented
90	C₂×C₅×C₃⋊C₈	—	2	2	no	—
91	C₂×C₅×C₃⋊C₈	—	2	4	no	—
92	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
93	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
94	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
95	C₅×ΓL(1,𝔽₂₅)	—	2	4	no	—
96	C₅×D₃×C₈	—	2	4	no	—
97	#5828	C₂×C₅	1	1	yes	characteristic
98	C₂×#948	C₂×C₅	1	1	yes	characteristic, complemented
99	C₂×D₃×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
100	C₃×#408	C₂×C₅	1	1	yes	characteristic, complemented
101	#5119	C₂×C₅	1	1	yes	characteristic, complemented
102	C₂×(C₂×D₃)⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
103	#5101	C₂×C₅	1	1	yes	characteristic, complemented
104	#4730	C₂×C₅	1	2	yes	complemented
105	#4729	C₂×C₅	1	2	yes	complemented
106	#4728	C₂×C₅	1	2	yes	complemented
107	#67⋊C₈	C₂×C₅	1	2	yes	complemented
108	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
109	C₂³×C₄×C₅	—	3	1	no	—
110	C₅×#131	—	6	4	no	—
111	C₂×C₅×M₁₆	—	3	1	no	—
112	C₂×C₅×M₁₆	—	3	2	no	—
113	C₂×C₅×M₁₆	—	3	2	no	—
114	C₂×C₅×M₁₆	—	6	4	no	—
115	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
116	C₂²×C₅×C₈	—	3	1	no	—
117	C₂²×C₅×C₈	—	3	2	no	—
118	C₂²×C₅×C₈	—	3	2	no	—
119	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
120	C₅×#123	—	3	2	no	—
121	C₅×C₂²⋊Q₈	—	3	1	no	—
122	C₅×C₂²⋊Q₈	—	3	1	no	—
123	C₅×#121	—	3	1	no	—
124	C₅×#121	—	3	1	no	—
125	C₅×#120	—	3	2	no	—
126	C₄×C₅×D₄	—	3	2	no	—
127	C₄×C₅×D₄	—	3	2	no	—
128	C₅×#117	—	3	1	no	—
129	C₅×C₄⋊C₈	—	3	2	no	—
130	C₅×C₄⋊C₈	—	3	2	no	—
131	C₅×C₂²⋊C₈	—	3	1	no	—
132	C₅×C₂²⋊C₈	—	3	1	no	—
133	C₅×C₂²⋊C₈	—	3	1	no	—
134	C₅×C₂²⋊C₈	—	3	1	no	—
135	C₅×C₂²⋊C₈	C₂×D₃	1	2	yes	complemented
136	C₅×C₂²⋊C₈	C₂×D₃	1	2	yes	complemented
137	C₅×#97	—	3	2	no	—
138	C₄×C₅×C₈	—	3	2	no	—
139	#2933	—	3	1	no	Sylow
140	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
141	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₅×D₃	—	2	2	no	—
143	C₂²×C₅×D₃	—	2	2	no	—
144	C₂²×C₅×D₃	—	4	1	no	—
145	C₂²×C₅×D₃	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₅×D₃	C₂²×C₄	1	1	yes	characteristic
147	C₃×C₅×Q₈	—	2	2	no	—
148	C₃×C₅×D₄	—	2	4	no	—
149	C₃×C₅×D₄	—	2	2	no	—
150	C₂×C₃×C₄×C₅	—	2	2	no	—
151	C₂×C₃×C₄×C₅	—	2	2	no	—
152	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
153	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
154	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
155	C₂×C₃×C₄×C₅	#41	1	2	yes	—
156	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
157	C₅×#67	—	2	4	no	—
158	C₅×#67	—	4	2	no	—
159	C₂×C₅×Dic₃	—	2	2	no	—
160	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
161	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
162	C₂×C₅×Dic₃	C₂²×C₄	1	1	yes	characteristic
163	C₂×C₅×Dic₃	C₂²×C₄	1	1	yes	characteristic
164	C₄×C₅×D₃	—	2	2	no	—
165	C₄×C₅×D₃	—	2	2	no	—
166	C₄×C₅×D₃	—	2	1	no	—
167	C₄×C₅×D₃	—	2	1	no	—
168	C₄×C₅×D₃	—	2	1	no	—
169	C₄×C₅×D₃	—	2	1	no	—
170	C₃×C₅×C₈	—	2	2	no	—
171	C₃×C₅×C₈	—	2	1	no	—
172	C₃×C₅×C₈	—	2	1	no	—
173	C₅×C₃⋊C₈	—	2	1	no	—
174	C₅×C₃⋊C₈	—	2	1	no	—
175	C₅×C₃⋊C₈	C₂×D₄	1	4	yes	—
176	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
177	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
178	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
179	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
180	#948	—	2	4	no	—
181	#946	C₄×C₅	1	1	yes	characteristic
182	#946	C₄×C₅	1	1	yes	characteristic
183	#938	C₄×C₅	1	1	yes	characteristic
184	#938	C₄×C₅	1	1	yes	characteristic
185	#937	C₄×C₅	1	2	yes	—
186	#935	C₄×C₅	1	2	yes	—
187	C₄×#67	C₂²×C₅	1	2	yes	—
188	C₄×#67	C₂²×C₅	1	2	yes	—
189	#926	C₂²×C₅	1	1	yes	characteristic
190	C₂×#208	C₂²×C₅	1	1	yes	characteristic
191	C₂×#208	C₂²×C₅	1	2	yes	complemented
192	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
193	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	complemented
194	D₃×M₁₆	—	2	4	no	—
195	C₂×ΓL(1,𝔽₂₅)	C₂²×C₅	1	2	yes	—
196	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
197	C₃×C₂²⋊C₈	C₂²×C₅	1	2	yes	complemented
198	C₃×C₂²⋊C₈	C₂²×C₅	1	2	yes	complemented
199	(C₂×D₃)⋊C₈	C₂²×C₅	1	1	yes	characteristic
200	(C₂×D₃)⋊C₈	C₂²×C₅	1	1	yes	characteristic
201	(C₂×D₃)⋊C₈	C₂²×C₅	1	1	yes	characteristic
202	(C₂×D₃)⋊C₈	C₂²×C₅	1	1	yes	characteristic
203	#815	C₂²×C₅	1	2	yes	complemented
204	Dic₃⋊C₈	C₂²×C₅	1	2	yes	complemented
205	Dic₃⋊C₈	C₂²×C₅	1	2	yes	complemented
206	C₈×Dic₃	C₂²×C₅	1	2	yes	complemented
207	C₂⁴×C₅	—	3	1	no	—
208	C₅×#41	—	2	2	no	—
209	C₅×#41	—	2	2	no	—
210	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
211	C₂×C₅×D₄	—	3	2	no	—
212	C₂×C₅×D₄	—	3	2	no	—
213	C₂×C₅×D₄	C₄×D₃	1	2	yes	—
214	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
215	C₂²×C₄×C₅	—	3	1	no	—
216	C₂²×C₄×C₅	—	3	1	no	—
217	C₂²×C₄×C₅	—	3	1	no	—
218	C₂²×C₄×C₅	—	3	1	no	—
219	C₂²×C₄×C₅	—	3	1	no	—
220	C₂²×C₄×C₅	—	6	2	no	—
221	C₂²×C₄×C₅	—	6	2	no	—
222	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
223	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
224	C₅×M₁₆	—	2	2	no	—
225	C₅×M₁₆	—	3	4	no	—
226	C₅×M₁₆	—	6	4	no	—
227	C₅×M₁₆	—	6	4	no	—
228	C₂×C₅×C₈	—	2	2	no	—
229	C₂×C₅×C₈	—	3	2	no	—
230	C₂×C₅×C₈	—	3	1	no	—
231	C₂×C₅×C₈	—	3	1	no	—
232	C₂×C₅×C₈	—	6	4	no	—
233	C₂×C₅×C₈	—	6	4	no	—
234	C₂×C₅×C₈	—	6	2	no	—
235	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
236	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
237	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
238	C₅×C₄⋊C₄	—	3	1	no	—
239	C₅×C₄⋊C₄	—	3	1	no	—
240	C₅×C₄⋊C₄	—	3	2	no	—
241	C₅×C₄⋊C₄	—	3	2	no	—
242	C₅×C₂²⋊C₄	—	3	2	no	—
243	C₅×C₂²⋊C₄	—	3	2	no	—
244	C₅×C₂²⋊C₄	—	3	2	no	—
245	C₅×C₂²⋊C₄	—	3	2	no	—
246	C₅×C₂²⋊C₄	—	3	2	no	—
247	C₄²×C₅	—	3	2	no	—
248	C₂×#131	—	3	1	no	—
249	C₂²×M₁₆	—	3	1	no	—
250	#525	—	3	1	no	—
251	#436	—	3	2	no	—
252	#435	—	3	2	no	—
253	#435	—	3	2	no	—
254	C₈×D₄	—	3	2	no	—
255	#424	—	3	1	no	—
256	#408	C₅×D₃	1	1	yes	characteristic, complemented
257	C₂×C₂²⋊C₈	—	3	1	no	—
258	#405	—	3	1	no	—
259	C₂²×C₃×C₅	—	2	1	no	—
260	C₂²×C₃×C₅	—	2	1	no	—
261	C₂²×C₃×C₅	—	2	2	no	—
262	C₂²×C₃×C₅	—	2	2	no	—
263	C₂²×C₃×C₅	—	2	2	no	—
264	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
265	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
266	C₂×C₅×D₃	—	2	1	no	—
267	C₂×C₅×D₃	—	2	1	no	—
268	C₂×C₅×D₃	—	2	1	no	—
269	C₂×C₅×D₃	—	2	1	no	—
270	C₂×C₅×D₃	—	2	1	no	—
271	C₂×C₅×D₃	—	4	4	no	—
272	C₂×C₅×D₃	#131	1	2	yes	complemented
273	C₃×C₄×C₅	—	2	2	no	—
274	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
275	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
276	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
277	C₅×Dic₃	—	2	1	no	—
278	C₅×Dic₃	—	2	1	no	—
279	C₅×Dic₃	—	2	2	no	—
280	C₅×Dic₃	—	2	2	no	—
281	C₂³×D₃	C₂×C₄×C₅	1	1	yes	characteristic
282	C₃×#41	—	2	2	no	—
283	C₃×#41	—	2	2	no	—
284	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
285	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
286	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
287	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
288	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
289	C₂×#67	C₂×C₄×C₅	1	2	yes	—
290	C₂×#67	C₂×C₄×C₅	1	2	yes	—
291	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
292	C₂×C₄×D₃	—	2	2	no	—
293	C₂×C₄×D₃	—	2	2	no	—
294	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
295	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
296	C₂×C₄×D₃	C₂³×C₅	1	1	yes	characteristic
297	C₂×C₄×D₃	C₂³×C₅	1	1	yes	characteristic
298	C₃×M₁₆	—	2	2	no	—
299	C₂×C₃×C₈	—	2	2	no	—
300	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
301	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
302	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
303	#217	C₂×C₄×C₅	1	2	yes	—
304	#217	C₂×C₄×C₅	1	2	yes	—
305	#217	C₂×C₄×C₅	1	2	yes	—
306	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	2	yes	—
307	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	2	yes	—
308	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
309	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
310	Dic₃⋊C₄	C₂×C₄×C₅	1	2	yes	—
311	Dic₃⋊C₄	C₂×C₄×C₅	1	2	yes	—
312	C₄×Dic₃	C₂³×C₅	1	2	yes	—
313	#208	—	2	4	no	—
314	#208	C₅×D₄	1	4	yes	complemented
315	C₂×C₃⋊C₈	—	2	2	no	—
316	C₂×C₃⋊C₈	—	2	4	no	—
317	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
318	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
319	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
320	ΓL(1,𝔽₂₅)	—	2	4	no	—
321	D₃×C₈	—	2	4	no	—
322	C₂³×C₅	—	3	1	no	—
323	C₂³×C₅	—	3	1	no	—
324	C₂³×C₅	—	6	2	no	—
325	C₂³×C₅	—	6	2	no	—
326	C₂³×C₅	—	12	1	no	—
327	C₂³×C₅	C₂×C₄×D₃	1	2	yes	—
328	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
329	C₅×Q₈	—	2	2	no	—
330	C₅×D₄	—	2	4	no	—
331	C₅×D₄	—	2	2	no	—
332	C₅×D₄	—	6	4	no	—
333	C₅×D₄	—	12	2	no	—
334	C₂×C₄×C₅	—	2	2	no	—
335	C₂×C₄×C₅	—	2	2	no	—
336	C₂×C₄×C₅	—	3	2	no	—
337	C₂×C₄×C₅	—	3	2	no	—
338	C₂×C₄×C₅	—	3	1	no	—
339	C₂×C₄×C₅	—	3	1	no	—
340	C₂×C₄×C₅	—	6	2	no	—
341	C₂×C₄×C₅	—	6	2	no	—
342	C₂×C₄×C₅	—	6	2	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	1	no	—
347	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
348	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
349	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
350	C₂×C₄×C₅	#237	1	2	yes	—
351	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
352	C₅×C₈	—	2	2	no	—
353	C₅×C₈	—	2	1	no	—
354	C₅×C₈	—	2	1	no	—
355	C₅×C₈	—	3	4	no	—
356	C₅×C₈	—	6	1	no	—
357	C₅×C₈	—	6	1	no	—
358	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
359	C₂³×C₄	—	3	1	no	—
360	#131	—	6	4	no	—
361	C₂×M₁₆	—	3	1	no	—
362	C₂×M₁₆	—	3	2	no	—
363	C₂×M₁₆	—	3	2	no	—
364	C₂×M₁₆	—	6	4	no	—
365	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
366	C₂²×C₈	—	3	1	no	—
367	C₂²×C₈	—	3	2	no	—
368	C₂²×C₈	—	3	2	no	—
369	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
370	#123	—	3	2	no	—
371	C₂²⋊Q₈	—	3	1	no	—
372	C₂²⋊Q₈	—	3	1	no	—
373	#121	—	3	1	no	—
374	#121	—	3	1	no	—
375	#120	—	3	2	no	—
376	C₄×D₄	—	3	2	no	—
377	C₄×D₄	—	3	2	no	—
378	#117	—	3	1	no	—
379	C₄⋊C₈	—	3	2	no	—
380	C₄⋊C₈	—	3	2	no	—
381	C₂²⋊C₈	—	3	1	no	—
382	C₂²⋊C₈	—	3	1	no	—
383	C₂²⋊C₈	—	3	1	no	—
384	C₂²⋊C₈	—	3	1	no	—
385	C₂²⋊C₈	C₂×C₅×D₃	1	2	yes	complemented
386	C₂²⋊C₈	C₂×C₅×D₃	1	2	yes	complemented
387	#97	—	3	2	no	—
388	C₄×C₈	—	3	2	no	—
389	C₂×C₃×C₅	—	2	2	no	—
390	C₂×C₃×C₅	—	4	2	no	—
391	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
392	C₂×C₃×C₅	C₂×#131	1	1	yes	characteristic
393	C₂×C₃×C₅	#568	1	1	yes	characteristic
394	C₅×D₃	—	2	2	no	—
395	C₅×D₃	—	4	1	no	—
396	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
397	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
398	C₂²×D₃	—	2	2	no	—
399	C₂²×D₃	—	2	2	no	—
400	C₂²×D₃	—	4	1	no	—
401	C₂²×D₃	C₂²×C₄×C₅	1	1	yes	characteristic
402	C₂²×D₃	C₂²×C₄×C₅	1	1	yes	characteristic
403	C₃×Q₈	—	2	2	no	—
404	C₃×D₄	—	2	4	no	—
405	C₃×D₄	—	2	2	no	—
406	C₂×C₃×C₄	—	2	2	no	—
407	C₂×C₃×C₄	—	2	2	no	—
408	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
409	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
410	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
411	C₂×C₃×C₄	C₅×#41	1	2	yes	—
412	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
413	#67	—	2	4	no	—
414	#67	—	4	2	no	—
415	C₂×Dic₃	—	2	2	no	—
416	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
417	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
418	C₂×Dic₃	C₂²×C₄×C₅	1	1	yes	characteristic
419	C₂×Dic₃	C₂²×C₄×C₅	1	1	yes	characteristic
420	C₄×D₃	—	2	2	no	—
421	C₄×D₃	—	2	2	no	—
422	C₄×D₃	—	2	1	no	—
423	C₄×D₃	—	2	1	no	—
424	C₄×D₃	—	2	1	no	—
425	C₄×D₃	—	2	1	no	—
426	C₃×C₈	—	2	2	no	—
427	C₃×C₈	—	2	1	no	—
428	C₃×C₈	—	2	1	no	—
429	C₃⋊C₈	—	2	1	no	—
430	C₃⋊C₈	—	2	1	no	—
431	C₃⋊C₈	C₂×C₅×D₄	1	4	yes	—
432	C₂²×C₅	—	2	1	no	—
433	C₂²×C₅	—	2	1	no	—
434	C₂²×C₅	—	2	2	no	—
435	C₂²×C₅	—	2	2	no	—
436	C₂²×C₅	—	2	2	no	—
437	C₂²×C₅	—	3	2	no	—
438	C₂²×C₅	—	6	1	no	—
439	C₂²×C₅	—	6	1	no	—
440	C₂²×C₅	—	6	1	no	—
441	C₂²×C₅	—	6	1	no	—
442	C₂²×C₅	—	6	1	no	—
443	C₂²×C₅	—	12	4	no	—
444	C₂²×C₅	#67⋊C₄	1	2	yes	—
445	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
446	C₄×C₅	—	2	2	no	—
447	C₄×C₅	—	6	1	no	—
448	C₄×C₅	—	6	1	no	—
449	C₄×C₅	—	6	2	no	—
450	C₄×C₅	—	6	2	no	—
451	C₄×C₅	#67⋊C₄	1	2	yes	—
452	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
453	C₄×C₅	C₂×#237	1	1	yes	characteristic
454	C₂⁴	—	3	1	no	—
455	#41	—	2	2	no	—
456	#41	—	2	2	no	—
457	C₂×Q₈	C₄×C₅×D₃	1	1	yes	characteristic
458	C₂×D₄	—	3	2	no	—
459	C₂×D₄	—	3	2	no	—
460	C₂×D₄	C₄×C₅×D₃	1	2	yes	—
461	C₂×D₄	C₄×C₅×D₃	1	1	yes	characteristic
462	C₂²×C₄	—	3	1	no	—
463	C₂²×C₄	—	3	1	no	—
464	C₂²×C₄	—	3	1	no	—
465	C₂²×C₄	—	3	1	no	—
466	C₂²×C₄	—	3	1	no	—
467	C₂²×C₄	—	6	2	no	—
468	C₂²×C₄	—	6	2	no	—
469	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
470	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
471	M₁₆	—	2	2	no	—
472	M₁₆	—	3	4	no	—
473	M₁₆	—	6	4	no	—
474	M₁₆	—	6	4	no	—
475	C₂×C₈	—	2	2	no	—
476	C₂×C₈	—	3	2	no	—
477	C₂×C₈	—	3	1	no	—
478	C₂×C₈	—	3	1	no	—
479	C₂×C₈	—	6	4	no	—
480	C₂×C₈	—	6	4	no	—
481	C₂×C₈	—	6	2	no	—
482	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
483	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
484	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
485	C₄⋊C₄	—	3	1	no	—
486	C₄⋊C₄	—	3	1	no	—
487	C₄⋊C₄	—	3	2	no	—
488	C₄⋊C₄	—	3	2	no	—
489	C₂²⋊C₄	—	3	2	no	—
490	C₂²⋊C₄	—	3	2	no	—
491	C₂²⋊C₄	—	3	2	no	—
492	C₂²⋊C₄	—	3	2	no	—
493	C₂²⋊C₄	—	3	2	no	—
494	C₄²	—	3	2	no	—
495	C₃×C₅	#2933	1	1	yes	characteristic, complemented, Hall
496	C₂²×C₃	—	2	1	no	—
497	C₂²×C₃	—	2	1	no	—
498	C₂²×C₃	—	2	2	no	—
499	C₂²×C₃	—	2	2	no	—
500	C₂²×C₃	—	2	2	no	—
501	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
502	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
503	C₂×D₃	—	2	1	no	—
504	C₂×D₃	—	2	1	no	—
505	C₂×D₃	—	2	1	no	—
506	C₂×D₃	—	2	1	no	—
507	C₂×D₃	—	2	1	no	—
508	C₂×D₃	—	4	4	no	—
509	C₂×D₃	C₅×#131	1	2	yes	complemented
510	C₃×C₄	—	2	2	no	—
511	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
512	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
513	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
514	Dic₃	—	2	1	no	—
515	Dic₃	—	2	1	no	—
516	Dic₃	—	2	2	no	—
517	Dic₃	—	2	2	no	—
518	C₂×C₅	—	2	2	no	—
519	C₂×C₅	—	4	2	no	—
520	C₂×C₅	—	6	2	no	—
521	C₂×C₅	—	12	1	no	—
522	C₂×C₅	C₂×#67⋊C₄	1	1	yes	characteristic
523	C₂×C₅	D₃×#131	1	1	yes	characteristic
524	C₂×C₅	#5750	1	1	yes	characteristic
525	C₂³	—	3	1	no	—
526	C₂³	—	3	1	no	—
527	C₂³	—	6	2	no	—
528	C₂³	—	6	2	no	—
529	C₂³	—	12	1	no	—
530	C₂³	C₂×C₄×C₅×D₃	1	2	yes	—
531	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
532	Q₈	—	2	2	no	—
533	D₄	—	2	4	no	—
534	D₄	—	2	2	no	—
535	D₄	—	6	4	no	—
536	D₄	—	12	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	3	2	no	—
540	C₂×C₄	—	3	2	no	—
541	C₂×C₄	—	3	1	no	—
542	C₂×C₄	—	3	1	no	—
543	C₂×C₄	—	6	2	no	—
544	C₂×C₄	—	6	2	no	—
545	C₂×C₄	—	6	2	no	—
546	C₂×C₄	—	6	1	no	—
547	C₂×C₄	—	6	1	no	—
548	C₂×C₄	—	6	1	no	—
549	C₂×C₄	—	6	1	no	—
550	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
551	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
552	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
553	C₂×C₄	C₅×#237	1	2	yes	—
554	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
555	C₈	—	2	2	no	—
556	C₈	—	2	1	no	—
557	C₈	—	2	1	no	—
558	C₈	—	3	4	no	—
559	C₈	—	6	1	no	—
560	C₈	—	6	1	no	—
561	C₂×C₃	—	2	2	no	—
562	C₂×C₃	—	4	2	no	—
563	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
564	C₂×C₃	C₂×C₅×#131	1	1	yes	characteristic
565	C₂×C₃	C₅×#568	1	1	yes	characteristic
566	D₃	—	2	2	no	—
567	D₃	—	4	1	no	—
568	C₅	#78659	1	1	yes	characteristic, direct product factor, Sylow
569	C₂²	—	2	1	no	—
570	C₂²	—	2	1	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	3	2	no	—
575	C₂²	—	6	1	no	—
576	C₂²	—	6	1	no	—
577	C₂²	—	6	1	no	—
578	C₂²	—	6	1	no	—
579	C₂²	—	6	1	no	—
580	C₂²	—	12	4	no	—
581	C₂²	C₅×#67⋊C₄	1	2	yes	—
582	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
583	C₄	—	2	2	no	—
584	C₄	—	6	1	no	—
585	C₄	—	6	1	no	—
586	C₄	—	6	2	no	—
587	C₄	—	6	2	no	—
588	C₄	C₅×#67⋊C₄	1	2	yes	—
589	C₄	C₂×C₅×D₃×D₄	1	1	yes	characteristic
590	C₄	C₂×C₅×#237	1	1	yes	characteristic
591	C₃	C₅×#2933	1	1	yes	characteristic, complemented, Sylow
592	C₂	—	2	2	no	—
593	C₂	—	4	2	no	—
594	C₂	—	6	2	no	—
595	C₂	—	12	1	no	—
596	C₂	C₂×C₅×#67⋊C₄	1	1	yes	characteristic
597	C₂	C₅×D₃×#131	1	1	yes	characteristic
598	C₂	C₅×#5750	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#408	C₂
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₅×#5828	C₂
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₅×#408	D₃
hypocenter	C₃	C₅×#2933
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×#408

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#948	C₂	semidirect
3. C₂×C₅×D₃×M₁₆	C₂	semidirect
4. C₃×C₅×#408	C₂	semidirect
5. C₅×#5119	C₂	semidirect
6. C₂×C₅×(C₂×D₃)⋊C₈	C₂	semidirect
7. C₅×#5101	C₂	semidirect
8. C₅×#4730	C₂	semidirect
9. C₅×#4729	C₂	semidirect
10. C₅×#4728	C₂	semidirect
11. C₅×#67⋊C₈	C₂	semidirect
28. C₂×C₅×#208	C₂²	semidirect
30. C₂²×C₅×C₃⋊C₈	C₂²	semidirect
34. C₃×C₅×C₂²⋊C₈	C₂²	semidirect
35. C₃×C₅×C₂²⋊C₈	C₂²	semidirect
40. C₅×#815	C₂²	semidirect
41. C₅×Dic₃⋊C₈	C₂²	semidirect
42. C₅×Dic₃⋊C₈	C₂²	semidirect
43. C₅×C₈×Dic₃	C₂²	semidirect
44. #78659	C₅	direct (N × H)
53. C₅×#408	D₃	semidirect
89. C₅×#208	D₄	semidirect
98. C₂×#948	C₂×C₅	semidirect
99. C₂×D₃×M₁₆	C₂×C₅	semidirect
100. C₃×#408	C₂×C₅	semidirect
101. #5119	C₂×C₅	semidirect
102. C₂×(C₂×D₃)⋊C₈	C₂×C₅	semidirect
103. #5101	C₂×C₅	semidirect
104. #4730	C₂×C₅	semidirect
105. #4729	C₂×C₅	semidirect
106. #4728	C₂×C₅	semidirect
107. #67⋊C₈	C₂×C₅	semidirect
135. C₅×C₂²⋊C₈	C₂×D₃	semidirect
136. C₅×C₂²⋊C₈	C₂×D₃	semidirect
191. C₂×#208	C₂²×C₅	semidirect
193. C₂²×C₃⋊C₈	C₂²×C₅	semidirect
197. C₃×C₂²⋊C₈	C₂²×C₅	semidirect
198. C₃×C₂²⋊C₈	C₂²×C₅	semidirect
203. #815	C₂²×C₅	semidirect
204. Dic₃⋊C₈	C₂²×C₅	semidirect
205. Dic₃⋊C₈	C₂²×C₅	semidirect
206. C₈×Dic₃	C₂²×C₅	semidirect
256. #408	C₅×D₃	semidirect
272. C₂×C₅×D₃	#131	semidirect
314. #208	C₅×D₄	semidirect
385. C₂²⋊C₈	C₂×C₅×D₃	semidirect
386. C₂²⋊C₈	C₂×C₅×D₃	semidirect
495. C₃×C₅	#2933	semidirect
509. C₂×D₃	C₅×#131	semidirect
568. C₅	#78659	direct (N × H)
591. C₃	C₅×#2933	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5828	2
C₂×C₅×#948	2
C₂×C₅×D₃×M₁₆	2
C₃×C₅×#408	2
C₅×#5119	2
C₂×C₅×(C₂×D₃)⋊C₈	2
C₅×#5101	2
C₅×#4730	2
C₅×#4729	2
C₅×#4728	2
C₅×#67⋊C₈	2
C₅×#2933	3
#78659	5

Subgroups — C5×#78659 #14415457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#78659 #14415457