Subgroups — #11757048

Order 960.

1620 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#92098	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#1007	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#713	C₂	1	1	yes	maximal, characteristic, complemented
4	#92035	C₂	1	4	yes	maximal, complemented
5	#92034	C₂	1	4	yes	maximal, complemented
6	D₃×#713	C₂	1	4	yes	maximal, complemented
7	#92028	C₂	1	4	yes	maximal, complemented
8	#92017	C₂	1	6	yes	maximal, complemented
9	D₅×#235	C₂	1	6	yes	maximal, complemented
10	#66287	—	3	1	no	maximal, Hall
11	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
12	#6821	C₂²	1	4	yes	complemented
13	Q₈×D₁₅	C₂²	1	4	yes	complemented
14	#6817	C₂²	1	6	yes	complemented
15	C₅×#239	C₂²	1	4	yes	complemented
16	C₅×D₃×Q₈	C₂²	1	4	yes	complemented
17	C₅×#235	C₂²	1	6	yes	complemented
18	C₃×#713	C₂²	1	4	yes	complemented
19	C₃×#713	C₂²	1	4	yes	complemented
20	C₂×C₃×D₂₀	C₂²	1	3	yes	complemented
21	C₂×C₃×C₄×D₅	C₂²	1	3	yes	complemented
22	D₅×#67	C₂²	1	12	yes	complemented
23	#6782	C₂²	1	4	yes	complemented
24	#6777	C₂²	1	12	yes	complemented
25	D₃×D₂₀	C₂²	1	12	yes	complemented
26	D₅×D₁₂	C₂²	1	6	yes	complemented
27	C₄×D₃×D₅	C₂²	1	12	yes	complemented
28	#6773	C₂²	1	6	yes	complemented
29	#6772	C₂²	1	6	yes	complemented
30	#6771	C₂²	1	12	yes	complemented
31	#6766	C₂²	1	12	yes	complemented
32	#6765	C₂²	1	12	yes	complemented
33	D₅×Dic₆	C₂²	1	6	yes	complemented
34	#5964	—	5	1	no	maximal, Hall
35	C₅×2₋¹⁺⁴	—	3	1	no	—
36	#4149	—	3	4	no	—
37	#4149	—	3	6	no	—
38	D₅×#41	—	3	4	no	—
39	D₅×#41	—	3	6	no	—
40	#4147	—	3	1	no	—
41	#4147	—	3	4	no	—
42	C₂×#713	—	3	4	no	—
43	C₂×#713	D₃	1	1	yes	characteristic, complemented
44	C₂²×C₃×D₅	C₂³	1	3	yes	—
45	C₂×D₃×D₅	C₂³	1	12	yes	—
46	C₃×C₅×Q₈	C₂³	1	4	yes	—
47	C₂×C₃×C₄×C₅	C₂³	1	3	yes	—
48	#1220	C₂³	1	4	yes	—
49	D₆₀	C₂³	1	6	yes	—
50	C₄×D₁₅	C₂³	1	12	yes	—
51	Dic₃₀	C₂³	1	6	yes	—
52	C₅×#67	C₂³	1	4	yes	—
53	C₅×D₁₂	C₂³	1	6	yes	—
54	C₄×C₅×D₃	C₂³	1	12	yes	—
55	C₅×Dic₆	C₂³	1	6	yes	—
56	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
57	C₃×D₂₀	C₂³	1	12	yes	—
58	C₃×C₄×D₅	C₂³	1	6	yes	—
59	C₃×C₄×D₅	C₂³	1	6	yes	—
60	#1204	C₂³	1	4	yes	—
61	#1203	C₂³	1	4	yes	—
62	#1202	C₂³	1	12	yes	—
63	#1201	C₂³	1	12	yes	—
64	#1200	C₂³	1	4	yes	—
65	D₃×Dic₅	C₂³	1	4	yes	—
66	D₅×Dic₃	C₂³	1	12	yes	—
67	C₂×C₃×#41	—	5	1	no	—
68	#1010	—	5	4	no	—
69	#1009	—	5	4	no	—
70	D₃×#41	—	5	4	no	—
71	D₃×#41	—	5	4	no	—
72	#1007	—	5	1	no	—
73	#1007	D₅	1	1	yes	characteristic, complemented
74	#1004	—	5	6	no	—
75	C₂×#235	—	5	6	no	—
76	C₅×#41	—	3	4	no	—
77	C₅×#41	—	3	6	no	—
78	C₂×C₅×Q₈	—	3	4	no	—
79	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
80	#713	—	3	4	no	—
81	#713	—	3	12	no	—
82	#713	—	3	6	no	—
83	#713	C₂×D₃	1	4	yes	complemented
84	#713	C₂×D₃	1	4	yes	complemented
85	Q₈×D₅	—	3	4	no	—
86	Q₈×D₅	—	3	6	no	—
87	#711	—	3	4	no	—
88	#711	—	3	6	no	—
89	D₄×D₅	—	3	12	no	—
90	D₄×D₅	—	3	12	no	—
91	D₄×D₅	—	3	6	no	—
92	#709	—	3	6	no	—
93	#709	—	3	12	no	—
94	#709	—	3	12	no	—
95	C₂×D₂₀	—	3	12	no	—
96	C₂×D₂₀	C₂×D₃	1	3	yes	complemented
97	C₂×C₄×D₅	—	3	12	no	—
98	C₂×C₄×D₅	C₂×D₃	1	3	yes	complemented
99	#585	—	15	1	no	Sylow
100	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
101	C₂×D₁₅	C₂⁴	1	4	yes	—
102	C₂×C₅×D₃	C₂⁴	1	4	yes	—
103	C₂×C₃×D₅	—	4	3	no	—
104	C₂×C₃×D₅	C₂⁴	1	6	yes	—
105	D₃×D₅	—	4	12	no	—
106	C₃×C₄×C₅	C₂⁴	1	6	yes	—
107	Dic₁₅	C₂⁴	1	4	yes	—
108	C₃×Dic₅	C₂⁴	1	1	yes	characteristic
109	C₃×Dic₅	C₂⁴	1	1	yes	characteristic
110	C₅×Dic₃	C₂⁴	1	4	yes	—
111	C₃×#41	—	5	4	no	—
112	C₃×#41	—	5	4	no	—
113	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
114	C₂×C₃×D₄	—	5	3	no	—
115	C₂²×C₃×C₄	—	5	3	no	—
116	C₂×#67	—	5	12	no	—
117	#239	—	5	4	no	—
118	#239	C₂×D₅	1	4	yes	complemented
119	D₃×Q₈	—	5	4	no	—
120	D₃×Q₈	C₂×D₅	1	4	yes	complemented
121	#237	—	5	12	no	—
122	#237	—	5	12	no	—
123	D₃×D₄	—	5	12	no	—
124	D₃×D₄	—	5	12	no	—
125	#235	—	5	6	no	—
126	#235	—	5	4	no	—
127	#235	—	5	6	no	—
128	#235	—	5	12	no	—
129	#235	—	5	6	no	—
130	#235	C₂×D₅	1	6	yes	complemented
131	C₂×D₁₂	—	5	6	no	—
132	C₂×C₄×D₃	—	5	12	no	—
133	C₂×Dic₆	—	5	6	no	—
134	C₂²×D₅	—	3	12	no	—
135	C₂²×D₅	C₂²×D₃	1	3	yes	—
136	C₅×Q₈	—	3	6	no	—
137	C₅×Q₈	C₂²×D₃	1	4	yes	—
138	C₅×D₄	—	3	4	no	—
139	C₅×D₄	—	3	6	no	—
140	C₂×C₄×C₅	—	3	12	no	—
141	C₂×C₄×C₅	C₂²×D₃	1	3	yes	—
142	C₂²⋊D₅	—	3	4	no	—
143	C₂²⋊D₅	—	3	12	no	—
144	C₂²⋊D₅	—	3	4	no	—
145	C₂×Dic₅	—	3	4	no	—
146	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
147	D₂₀	—	3	12	no	—
148	D₂₀	—	3	6	no	—
149	D₂₀	C₂²×D₃	1	12	yes	—
150	C₄×D₅	—	3	12	no	—
151	C₄×D₅	—	3	12	no	—
152	C₄×D₅	—	3	4	no	—
153	C₄×D₅	C₂²×D₃	1	6	yes	—
154	C₄×D₅	C₂²×D₃	1	6	yes	—
155	Dic₁₀	—	3	6	no	—
156	Dic₁₀	—	3	4	no	—
157	2₋¹⁺⁴	—	3	1	no	—
158	2₋¹⁺⁴	—	15	4	no	—
159	2₋¹⁺⁴	—	15	1	no	—
160	H(5,𝔽₂)	—	15	6	no	—
161	H(5,𝔽₂)	—	15	4	no	—
162	C₂×#41	—	5	1	no	—
163	C₂×#41	—	15	6	no	—
164	C₂×#41	—	15	4	no	—
165	C₂×#41	—	15	4	no	—
166	C₂×C₃×C₅	—	2	1	no	—
167	C₂×C₃×C₅	C₂⁵	1	1	yes	characteristic
168	D₁₅	—	2	4	no	—
169	C₃×D₅	—	2	6	no	—
170	C₅×D₃	—	2	4	no	—
171	C₂³×C₃	—	5	3	no	—
172	C₂²×D₃	—	5	12	no	—
173	C₃×Q₈	C₂²×D₅	1	4	yes	—
174	C₃×D₄	—	5	12	no	—
175	C₂×C₃×C₄	—	5	1	no	—
176	C₂×C₃×C₄	—	5	6	no	—
177	C₂×C₃×C₄	—	5	6	no	—
178	C₂×C₃×C₄	C₂²×D₅	1	3	yes	—
179	#67	—	5	4	no	—
180	#67	—	5	12	no	—
181	#67	—	5	12	no	—
182	#67	C₂²×D₅	1	4	yes	—
183	C₂×Dic₃	—	5	12	no	—
184	D₁₂	—	5	4	no	—
185	D₁₂	—	5	6	no	—
186	D₁₂	C₂²×D₅	1	6	yes	—
187	C₄×D₃	—	5	4	no	—
188	C₄×D₃	—	5	12	no	—
189	C₄×D₃	—	5	4	no	—
190	C₄×D₃	C₂²×D₅	1	12	yes	—
191	Dic₆	—	5	6	no	—
192	Dic₆	—	5	4	no	—
193	Dic₆	C₂²×D₅	1	6	yes	—
194	C₂²×C₅	—	3	4	no	—
195	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
196	C₂×D₅	—	3	4	no	—
197	C₂×D₅	—	4	3	no	—
198	C₂×D₅	—	12	12	no	—
199	C₂×D₅	C₂³×D₃	1	6	yes	—
200	C₄×C₅	—	3	4	no	—
201	C₄×C₅	C₂³×D₃	1	6	yes	—
202	Dic₅	—	3	4	no	—
203	Dic₅	C₂³×D₃	1	1	yes	characteristic
204	Dic₅	C₂³×D₃	1	1	yes	characteristic
205	#41	—	3	4	no	—
206	#41	—	3	6	no	—
207	#41	—	5	4	no	—
208	#41	—	5	4	no	—
209	#41	—	15	12	no	—
210	#41	—	15	12	no	—
211	#41	—	15	4	no	—
212	#41	—	15	6	no	—
213	#41	—	15	6	no	—
214	#41	—	15	12	no	—
215	#41	—	15	4	no	—
216	#41	—	15	6	no	—
217	C₂×Q₈	—	3	4	no	—
218	C₂×Q₈	—	15	6	no	—
219	C₂×Q₈	—	15	4	no	—
220	C₂×Q₈	D₃×D₅	1	1	yes	characteristic, complemented
221	C₂×D₄	—	5	3	no	—
222	C₂×D₄	—	15	12	no	—
223	C₂×D₄	—	15	6	no	—
224	C₂×D₄	—	15	12	no	—
225	C₂×D₄	—	15	12	no	—
226	C₂²×C₄	—	5	3	no	—
227	C₂²×C₄	—	15	12	no	—
228	C₃×C₅	#585	1	1	yes	characteristic, complemented, Hall
229	C₂²×C₃	—	5	6	no	—
230	C₂²×C₃	—	20	3	no	—
231	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
232	C₂×D₃	—	5	4	no	—
233	C₂×D₃	—	20	12	no	—
234	C₂×D₃	C₂³×D₅	1	4	yes	—
235	C₃×C₄	—	5	1	no	—
236	C₃×C₄	—	5	1	no	—
237	C₃×C₄	C₂³×D₅	1	6	yes	—
238	Dic₃	—	5	4	no	—
239	Dic₃	C₂³×D₅	1	4	yes	—
240	C₂×C₅	—	2	1	no	—
241	C₂×C₅	—	6	4	no	—
242	C₂×C₅	C₂⁴×D₃	1	1	yes	characteristic
243	D₅	—	2	6	no	—
244	D₅	—	6	4	no	—
245	C₂³	—	5	3	no	—
246	C₂³	—	15	12	no	—
247	Q₈	—	3	6	no	—
248	Q₈	—	15	4	no	—
249	Q₈	—	15	6	no	—
250	Q₈	C₂×D₃×D₅	1	4	yes	—
251	D₄	—	3	4	no	—
252	D₄	—	3	6	no	—
253	D₄	—	5	12	no	—
254	D₄	—	15	12	no	—
255	D₄	—	15	4	no	—
256	D₄	—	15	12	no	—
257	D₄	—	15	4	no	—
258	D₄	—	15	6	no	—
259	C₂×C₄	—	3	12	no	—
260	C₂×C₄	—	5	1	no	—
261	C₂×C₄	—	5	6	no	—
262	C₂×C₄	—	5	6	no	—
263	C₂×C₄	—	15	4	no	—
264	C₂×C₄	—	15	12	no	—
265	C₂×C₄	—	15	4	no	—
266	C₂×C₄	—	15	12	no	—
267	C₂×C₄	C₂×D₃×D₅	1	3	yes	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	10	6	no	—
270	C₂×C₃	C₂⁴×D₅	1	1	yes	characteristic
271	D₃	—	2	4	no	—
272	D₃	—	10	4	no	—
273	C₅	#5964	1	1	yes	characteristic, complemented, Sylow
274	C₂²	—	3	4	no	—
275	C₂²	—	5	6	no	—
276	C₂²	—	15	4	no	—
277	C₂²	—	20	3	no	—
278	C₂²	—	60	12	no	—
279	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
280	C₄	—	3	4	no	—
281	C₄	—	5	1	no	—
282	C₄	—	5	1	no	—
283	C₄	—	15	4	no	—
284	C₄	C₂²×D₃×D₅	1	6	yes	—
285	C₃	#66287	1	1	yes	characteristic, complemented, Sylow
286	C₂	—	2	1	no	—
287	C₂	—	6	4	no	—
288	C₂	—	10	6	no	—
289	C₂	—	30	4	no	—
290	C₂	C₂³×D₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₃×D₅
commutator, socle, cosocle	C₂×C₃×C₅	C₂⁵
Fitting	C₂×C₃×C₅×Q₈	C₂²
hypercenter	C₂×Q₈	D₃×D₅
hypocenter	C₃×C₅	#585
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #92098	C₂	semidirect
2. C₅×#1007	C₂	semidirect
3. C₂×C₃×#713	C₂	semidirect
4. #92035	C₂	semidirect
5. #92034	C₂	semidirect
6. D₃×#713	C₂	semidirect
7. #92028	C₂	semidirect
8. #92017	C₂	semidirect
9. D₅×#235	C₂	semidirect
11. C₂×C₃×C₅×Q₈	C₂²	semidirect
12. #6821	C₂²	semidirect
13. Q₈×D₁₅	C₂²	semidirect
14. #6817	C₂²	semidirect
15. C₅×#239	C₂²	semidirect
16. C₅×D₃×Q₈	C₂²	semidirect
17. C₅×#235	C₂²	semidirect
18. C₃×#713	C₂²	semidirect
19. C₃×#713	C₂²	semidirect
20. C₂×C₃×D₂₀	C₂²	semidirect
21. C₂×C₃×C₄×D₅	C₂²	semidirect
22. D₅×#67	C₂²	semidirect
23. #6782	C₂²	semidirect
24. #6777	C₂²	semidirect
25. D₃×D₂₀	C₂²	semidirect
26. D₅×D₁₂	C₂²	semidirect
27. C₄×D₃×D₅	C₂²	semidirect
28. #6773	C₂²	semidirect
29. #6772	C₂²	semidirect
30. #6771	C₂²	semidirect
31. #6766	C₂²	semidirect
32. #6765	C₂²	semidirect
33. D₅×Dic₆	C₂²	semidirect
43. C₂×#713	D₃	semidirect
73. #1007	D₅	semidirect
79. C₂×C₅×Q₈	C₂×D₃	semidirect
83. #713	C₂×D₃	semidirect
84. #713	C₂×D₃	semidirect
96. C₂×D₂₀	C₂×D₃	semidirect
98. C₂×C₄×D₅	C₂×D₃	semidirect
113. C₂×C₃×Q₈	C₂×D₅	semidirect
118. #239	C₂×D₅	semidirect
120. D₃×Q₈	C₂×D₅	semidirect
130. #235	C₂×D₅	semidirect
220. C₂×Q₈	D₃×D₅	semidirect
228. C₃×C₅	#585	semidirect
273. C₅	#5964	semidirect
285. C₃	#66287	semidirect

Maximal subgroups

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

subgroup	index
#92098	2
C₅×#1007	2
C₂×C₃×#713	2
#92035	2
#92034	2
D₃×#713	2
#92028	2
#92017	2
D₅×#235	2
#66287	3
#5964	5