Subgroups — #11756492

Order 960.

432 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#91836	C₂	1	2	yes	maximal, complemented
3	#91835	C₂	1	2	yes	maximal, complemented
4	#91787	C₂	1	2	yes	maximal, complemented
5	#91786	C₂	1	1	yes	maximal, characteristic, complemented
6	#91784	C₂	1	1	yes	maximal, characteristic
7	#91783	C₂	1	2	yes	maximal, complemented
8	#91782	C₂	1	1	yes	maximal, characteristic, complemented
9	#91776	C₂	1	1	yes	maximal, characteristic, complemented
10	#91764	C₂	1	1	yes	maximal, characteristic, complemented
11	#91761	C₂	1	1	yes	maximal, characteristic, complemented
12	#66067	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
14	C₂×D₆₀	C₂²	1	1	yes	characteristic
15	C₂×D₆₀	C₂²	1	2	yes	—
16	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₁₅	C₂²	1	2	yes	—
18	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
23	#6714	C₂²	1	1	yes	characteristic
24	#6714	C₂²	1	1	yes	characteristic
25	#6714	C₂²	1	2	yes	—
26	#6714	C₂²	1	2	yes	—
27	#6714	C₂²	1	2	yes	—
28	#6714	C₂²	1	2	yes	—
29	#6714	C₂²	1	2	yes	—
30	C₄⋊Dic₁₅	C₂²	1	2	yes	—
31	#6712	C₂²	1	1	yes	characteristic
32	#6712	C₂²	1	1	yes	characteristic
33	#6712	C₂²	1	2	yes	—
34	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₅	C₂²	1	2	yes	—
36	#5733	—	5	1	no	maximal, Hall
37	C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
38	#4093	—	3	2	no	—
39	#4092	—	3	2	no	—
40	#4044	—	3	2	no	—
41	#4043	—	3	1	no	—
42	#4041	—	3	1	no	—
43	#4040	—	3	2	no	—
44	#4039	—	3	1	no	—
45	#4033	—	3	1	no	—
46	#4021	—	3	1	no	—
47	#4018	—	3	1	no	—
48	C₂²×D₁₅	C₂³	1	1	yes	characteristic
49	C₂²×D₁₅	C₂³	1	2	yes	—
50	C₃×C₅×Q₈	—	4	2	no	—
51	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
57	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
58	C₂×Dic₁₅	C₂³	1	2	yes	—
59	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
60	D₆₀	—	2	2	no	—
61	D₆₀	—	4	2	no	—
62	C₄×D₁₅	—	2	2	no	—
63	C₄×D₁₅	—	4	2	no	—
64	Dic₃₀	—	2	2	no	—
65	C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
66	#947	—	5	2	no	—
67	#946	—	5	2	no	—
68	#898	—	5	2	no	—
69	#897	—	5	1	no	—
70	#895	—	5	1	no	—
71	#894	—	5	2	no	—
72	#893	—	5	1	no	—
73	#887	—	5	1	no	—
74	#875	—	5	1	no	—
75	#872	—	5	1	no	—
76	C₂×C₅×Q₈	C₂×D₃	1	2	yes	—
77	C₂×D₂₀	—	3	1	no	—
78	C₂×D₂₀	—	3	2	no	—
79	C₂×C₄×D₅	—	3	1	no	—
80	C₂×C₄×D₅	—	3	2	no	—
81	C₂×Dic₁₀	—	3	1	no	—
82	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
83	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
84	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
85	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
86	#685	—	3	1	no	—
87	#685	—	3	1	no	—
88	#685	—	3	2	no	—
89	#685	—	3	2	no	—
90	#685	—	3	2	no	—
91	#685	—	3	2	no	—
92	#685	—	3	2	no	—
93	C₄⋊Dic₅	—	3	2	no	—
94	#683	—	3	1	no	—
95	#683	—	3	1	no	—
96	#683	—	3	2	no	—
97	C₄×Dic₅	—	3	1	no	—
98	C₄×Dic₅	—	3	2	no	—
99	#542	—	15	1	no	Sylow
100	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
101	C₂×D₁₅	—	2	1	no	—
102	C₂×D₁₅	—	2	2	no	—
103	C₂×D₁₅	—	2	2	no	—
104	C₂×D₁₅	—	2	1	no	—
105	C₂×D₁₅	—	2	1	no	—
106	C₂×D₁₅	—	2	2	no	—
107	C₃×C₄×C₅	—	2	2	no	—
108	C₃×C₄×C₅	—	2	2	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	2	1	no	—
111	C₃×C₄×C₅	#41	1	2	yes	—
112	Dic₁₅	—	2	2	no	—
113	Dic₁₅	—	2	1	no	—
114	Dic₁₅	—	2	1	no	—
115	Dic₁₅	—	2	1	no	—
116	C₂×C₃×Q₈	C₂×D₅	1	2	yes	—
117	C₂×D₁₂	—	5	1	no	—
118	C₂×D₁₂	—	5	2	no	—
119	C₂×C₄×D₃	—	5	1	no	—
120	C₂×C₄×D₃	—	5	2	no	—
121	C₂×Dic₆	—	5	1	no	—
122	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
123	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
124	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
125	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
126	(C₂×D₃)⋊C₄	—	5	1	no	—
127	(C₂×D₃)⋊C₄	—	5	1	no	—
128	(C₂×D₃)⋊C₄	—	5	2	no	—
129	(C₂×D₃)⋊C₄	—	5	2	no	—
130	(C₂×D₃)⋊C₄	—	5	2	no	—
131	(C₂×D₃)⋊C₄	—	5	2	no	—
132	(C₂×D₃)⋊C₄	—	5	2	no	—
133	C₄⋊Dic₃	—	5	2	no	—
134	Dic₃⋊C₄	—	5	1	no	—
135	Dic₃⋊C₄	—	5	1	no	—
136	Dic₃⋊C₄	—	5	2	no	—
137	C₄×Dic₃	—	5	1	no	—
138	C₄×Dic₃	—	5	2	no	—
139	C₂²×D₅	—	3	2	no	—
140	C₂²×D₅	—	3	1	no	—
141	C₅×Q₈	—	4	2	no	—
142	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
143	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
146	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
147	C₂×Dic₅	—	3	1	no	—
148	C₂×Dic₅	—	3	1	no	—
149	C₂×Dic₅	—	3	2	no	—
150	C₂×Dic₅	—	3	1	no	—
151	D₂₀	—	6	2	no	—
152	D₂₀	—	12	2	no	—
153	C₄×D₅	—	6	2	no	—
154	C₄×D₅	—	12	2	no	—
155	Dic₁₀	—	6	2	no	—
156	C₄⋊Q₈	D₁₅	1	1	yes	characteristic, complemented
157	#126	—	15	2	no	—
158	#124	—	15	2	no	—
159	#124	—	15	1	no	—
160	#123	—	15	2	no	—
161	C₂²⋊Q₈	—	15	2	no	—
162	C₂²⋊Q₈	—	15	1	no	—
163	#121	—	15	1	no	—
164	C₄×Q₈	—	15	1	no	—
165	C₄×D₄	—	15	1	no	—
166	#117	—	15	1	no	—
167	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
168	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
170	D₁₅	—	4	1	no	—
171	D₁₅	—	4	2	no	—
172	C₂²×D₃	—	5	2	no	—
173	C₂²×D₃	—	5	1	no	—
174	C₃×Q₈	—	4	2	no	—
175	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
176	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
179	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
180	C₂×Dic₃	—	5	1	no	—
181	C₂×Dic₃	—	5	1	no	—
182	C₂×Dic₃	—	5	2	no	—
183	C₂×Dic₃	—	5	1	no	—
184	D₁₂	—	10	2	no	—
185	D₁₂	—	20	2	no	—
186	C₄×D₃	—	10	2	no	—
187	C₄×D₃	—	20	2	no	—
188	Dic₆	—	10	2	no	—
189	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
190	C₂×D₅	—	6	2	no	—
191	C₂×D₅	—	6	1	no	—
192	C₂×D₅	—	6	1	no	—
193	C₂×D₅	—	6	2	no	—
194	C₂×D₅	—	6	2	no	—
195	C₂×D₅	—	6	1	no	—
196	C₄×C₅	—	2	2	no	—
197	C₄×C₅	—	2	2	no	—
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	#239	1	2	yes	—
201	Dic₅	—	6	1	no	—
202	Dic₅	—	6	2	no	—
203	Dic₅	—	6	1	no	—
204	Dic₅	—	6	1	no	—
205	C₂×Q₈	—	15	1	no	—
206	C₂×Q₈	C₂×D₁₅	1	2	yes	—
207	C₂×D₄	—	15	1	no	—
208	C₂×D₄	—	15	2	no	—
209	C₂²×C₄	—	15	1	no	—
210	C₂²×C₄	—	15	2	no	—
211	C₄⋊C₄	—	15	1	no	—
212	C₄⋊C₄	—	15	1	no	—
213	C₄⋊C₄	—	15	2	no	—
214	C₄⋊C₄	—	15	2	no	—
215	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
216	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
217	C₄⋊C₄	C₂×D₁₅	1	2	yes	—
218	C₂²⋊C₄	—	15	1	no	—
219	C₂²⋊C₄	—	15	1	no	—
220	C₂²⋊C₄	—	15	2	no	—
221	C₂²⋊C₄	—	15	2	no	—
222	C₂²⋊C₄	—	15	2	no	—
223	C₂²⋊C₄	—	15	2	no	—
224	C₂²⋊C₄	—	15	2	no	—
225	C₄²	—	15	1	no	—
226	C₄²	—	15	2	no	—
227	C₄²	C₂×D₁₅	1	1	yes	characteristic
228	C₃×C₅	#542	1	1	yes	characteristic, complemented, Hall
229	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
230	C₂×D₃	—	10	2	no	—
231	C₂×D₃	—	10	1	no	—
232	C₂×D₃	—	10	1	no	—
233	C₂×D₃	—	10	2	no	—
234	C₂×D₃	—	10	2	no	—
235	C₂×D₃	—	10	1	no	—
236	C₃×C₄	—	2	2	no	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	#713	1	2	yes	—
241	Dic₃	—	10	1	no	—
242	Dic₃	—	10	2	no	—
243	Dic₃	—	10	1	no	—
244	Dic₃	—	10	1	no	—
245	C₂×C₅	#1004	1	1	yes	characteristic
246	C₂×C₅	C₂×#239	1	1	yes	characteristic
247	C₂×C₅	#1007	1	1	yes	characteristic
248	D₅	—	12	2	no	—
249	D₅	—	12	1	no	—
250	C₂³	—	15	2	no	—
251	C₂³	—	15	1	no	—
252	Q₈	—	4	2	no	—
253	Q₈	—	30	2	no	—
254	D₄	—	30	2	no	—
255	D₄	—	60	2	no	—
256	C₂×C₄	—	15	1	no	—
257	C₂×C₄	—	15	2	no	—
258	C₂×C₄	—	15	1	no	—
259	C₂×C₄	—	15	1	no	—
260	C₂×C₄	—	30	2	no	—
261	C₂×C₄	—	60	2	no	—
262	C₂×C₄	C₂²×D₁₅	1	2	yes	—
263	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
265	C₂×C₄	C₂²×D₁₅	1	2	yes	—
266	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
267	C₂×C₃	#4144	1	1	yes	characteristic
268	C₂×C₃	C₂×#713	1	1	yes	characteristic
269	C₂×C₃	#4147	1	1	yes	characteristic
270	D₃	—	20	2	no	—
271	D₃	—	20	1	no	—
272	C₅	#5733	1	1	yes	characteristic, complemented, Sylow
273	C₂²	—	30	2	no	—
274	C₂²	—	30	1	no	—
275	C₂²	—	30	1	no	—
276	C₂²	—	30	2	no	—
277	C₂²	—	30	2	no	—
278	C₂²	—	30	1	no	—
279	C₂²	C₂³×D₁₅	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	30	1	no	—
285	C₄	—	30	1	no	—
286	C₄	—	30	2	no	—
287	C₄	—	30	1	no	—
288	C₄	#6821	1	2	yes	—
289	C₃	#66067	1	1	yes	characteristic, complemented, Sylow
290	C₂	—	60	2	no	—
291	C₂	—	60	1	no	—
292	C₂	#92095	1	1	yes	characteristic
293	C₂	C₂×#6821	1	1	yes	characteristic
294	C₂	#92098	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₅×C₄⋊Q₈	C₂
prime radical	#91784	C₂
hypercenter	C₄⋊Q₈	D₁₅
hypocenter	C₃×C₅	#542
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₄⋊Q₈	C₂	semidirect
2. #91836	C₂	semidirect
3. #91835	C₂	semidirect
4. #91787	C₂	semidirect
5. #91786	C₂	semidirect
7. #91783	C₂	semidirect
8. #91782	C₂	semidirect
9. #91776	C₂	semidirect
10. #91764	C₂	semidirect
11. #91761	C₂	semidirect
37. C₅×C₄⋊Q₈	D₃	semidirect
65. C₃×C₄⋊Q₈	D₅	semidirect
156. C₄⋊Q₈	D₁₅	semidirect
228. C₃×C₅	#542	semidirect
272. C₅	#5733	semidirect
289. C₃	#66067	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₄⋊Q₈	2
#91836	2
#91835	2
#91787	2
#91786	2
#91784	2
#91783	2
#91782	2
#91776	2
#91764	2
#91761	2
#66067	3
#5733	5