Subgroups — #11756400

Order 960.

424 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	C₃×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#91835	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#1220	C₂	1	1	yes	maximal, characteristic
5	#91815	C₂	1	1	yes	maximal, characteristic, complemented
6	#91787	C₂	1	2	yes	maximal, complemented
7	#91781	C₂	1	1	yes	maximal, characteristic, complemented
8	#91777	C₂	1	1	yes	maximal, characteristic, complemented
9	#91776	C₂	1	2	yes	maximal, complemented
10	#91774	C₂	1	2	yes	maximal
11	#91768	C₂	1	2	yes	maximal, complemented
12	#65975	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂×#1220	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₀	C₂²	1	2	yes	—
18	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
21	#6719	C₂²	1	1	yes	characteristic
22	#6719	C₂²	1	2	yes	—
23	#6714	C₂²	1	2	yes	—
24	#6714	C₂²	1	2	yes	—
25	#6714	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₁₅	C₂²	1	2	yes	—
27	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
28	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
29	#6712	C₂²	1	2	yes	—
30	#6712	C₂²	1	2	yes	—
31	#6712	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₅	C₂²	1	2	yes	—
33	C₄×Dic₁₅	C₂²	1	2	yes	—
34	C₄×Dic₁₅	C₂²	1	2	yes	—
35	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
36	#5641	—	5	1	no	maximal, Hall
37	C₅×C₂²⋊Q₈	D₃	1	1	yes	characteristic, complemented
38	#4092	—	3	1	no	—
39	Q₈×Dic₅	—	3	1	no	—
40	C₄×C₂²⋊D₅	—	3	1	no	—
41	#4072	—	3	1	no	—
42	#4044	—	3	2	no	—
43	#4038	—	3	1	no	—
44	#4034	—	3	1	no	—
45	#4033	—	3	2	no	—
46	#4031	—	3	2	no	—
47	#4025	—	3	2	no	—
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₂²×D₁₅	C₂³	1	1	yes	characteristic
50	C₃×C₅×Q₈	—	2	2	no	—
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	#1220	—	2	2	no	—
57	C₂×Dic₁₅	C₂³	1	2	yes	—
58	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
59	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
60	C₂×Dic₁₅	C₂³	1	2	yes	—
61	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
62	C₄×D₁₅	—	2	2	no	—
63	Dic₃₀	—	2	4	no	—
64	C₃×C₂²⋊Q₈	D₅	1	1	yes	characteristic, complemented
65	#946	—	5	1	no	—
66	Q₈×Dic₃	—	5	1	no	—
67	C₄×#67	—	5	1	no	—
68	#926	—	5	1	no	—
69	#898	—	5	2	no	—
70	#892	—	5	1	no	—
71	#888	—	5	1	no	—
72	#887	—	5	2	no	—
73	#885	—	5	2	no	—
74	#879	—	5	2	no	—
75	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
77	C₂×C₂²⋊D₅	—	3	1	no	—
78	C₂×C₄×D₅	—	3	1	no	—
79	C₂×Dic₁₀	—	3	2	no	—
80	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
81	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
82	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
83	C₂²⋊Dic₅	—	3	1	no	—
84	C₂²⋊Dic₅	—	3	2	no	—
85	#685	—	3	2	no	—
86	#685	—	3	2	no	—
87	#685	—	3	1	no	—
88	C₄⋊Dic₅	—	3	2	no	—
89	C₄⋊Dic₅	—	3	1	no	—
90	C₄⋊Dic₅	—	3	1	no	—
91	#683	—	3	2	no	—
92	#683	—	3	2	no	—
93	#683	—	3	1	no	—
94	C₄×Dic₅	—	3	2	no	—
95	C₄×Dic₅	—	3	2	no	—
96	C₄×Dic₅	—	3	2	no	—
97	C₄×Dic₅	—	3	1	no	—
98	#556	—	15	1	no	Sylow
99	C₂²×C₃×C₅	—	2	1	no	—
100	C₂²×C₃×C₅	—	2	1	no	—
101	C₂²×C₃×C₅	—	2	1	no	—
102	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
103	C₂×D₁₅	—	2	1	no	—
104	C₂×D₁₅	—	2	1	no	—
105	C₂×D₁₅	—	2	1	no	—
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	—	2	1	no	—
108	C₃×C₄×C₅	—	2	2	no	—
109	C₃×C₄×C₅	#41	1	2	yes	—
110	Dic₁₅	—	2	2	no	—
111	Dic₁₅	—	2	1	no	—
112	Dic₁₅	—	2	2	no	—
113	Dic₁₅	—	2	1	no	—
114	Dic₁₅	#41	1	2	yes	—
115	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
117	C₂×#67	—	5	1	no	—
118	C₂×C₄×D₃	—	5	1	no	—
119	C₂×Dic₆	—	5	2	no	—
120	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
121	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
122	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
123	#217	—	5	1	no	—
124	#217	—	5	2	no	—
125	(C₂×D₃)⋊C₄	—	5	2	no	—
126	(C₂×D₃)⋊C₄	—	5	2	no	—
127	(C₂×D₃)⋊C₄	—	5	1	no	—
128	C₄⋊Dic₃	—	5	2	no	—
129	C₄⋊Dic₃	—	5	1	no	—
130	C₄⋊Dic₃	—	5	1	no	—
131	Dic₃⋊C₄	—	5	2	no	—
132	Dic₃⋊C₄	—	5	2	no	—
133	Dic₃⋊C₄	—	5	1	no	—
134	C₄×Dic₃	—	5	2	no	—
135	C₄×Dic₃	—	5	2	no	—
136	C₄×Dic₃	—	5	2	no	—
137	C₄×Dic₃	—	5	1	no	—
138	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
139	C₂²×D₅	—	3	1	no	—
140	C₅×Q₈	—	2	2	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
144	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
145	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
146	C₂²⋊D₅	—	6	2	no	—
147	C₂×Dic₅	—	3	1	no	—
148	C₂×Dic₅	—	3	2	no	—
149	C₂×Dic₅	—	3	1	no	—
150	C₂×Dic₅	—	3	1	no	—
151	C₂×Dic₅	—	3	2	no	—
152	C₄×D₅	—	6	2	no	—
153	Dic₁₀	—	6	4	no	—
154	C₄⋊Q₈	—	15	1	no	—
155	#126	—	15	2	no	—
156	#126	—	15	2	no	—
157	#124	—	15	2	no	—
158	C₂²⋊Q₈	—	15	1	no	—
159	C₂²⋊Q₈	D₁₅	1	1	yes	characteristic, complemented
160	C₄×Q₈	—	15	1	no	—
161	C₄×Q₈	—	15	2	no	—
162	C₄×D₄	—	15	1	no	—
163	#117	—	15	1	no	—
164	#117	—	15	1	no	—
165	C₂×C₃×C₅	—	4	1	no	—
166	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
167	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
168	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
169	D₁₅	—	4	1	no	—
170	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
171	C₂²×D₃	—	5	1	no	—
172	C₃×Q₈	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
176	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
177	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
178	#67	—	10	2	no	—
179	C₂×Dic₃	—	5	1	no	—
180	C₂×Dic₃	—	5	2	no	—
181	C₂×Dic₃	—	5	1	no	—
182	C₂×Dic₃	—	5	1	no	—
183	C₂×Dic₃	—	5	2	no	—
184	C₄×D₃	—	10	2	no	—
185	Dic₆	—	10	4	no	—
186	C₂²×C₅	—	2	1	no	—
187	C₂²×C₅	—	2	1	no	—
188	C₂²×C₅	—	2	1	no	—
189	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
190	C₂×D₅	—	6	1	no	—
191	C₂×D₅	—	6	1	no	—
192	C₂×D₅	—	6	1	no	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	#237	1	2	yes	—
197	Dic₅	—	3	2	no	—
198	Dic₅	—	6	1	no	—
199	Dic₅	—	6	2	no	—
200	Dic₅	—	6	1	no	—
201	Dic₅	—	6	2	no	—
202	C₂×Q₈	—	15	2	no	—
203	C₂×Q₈	C₂×D₁₅	1	1	yes	characteristic
204	C₂×D₄	—	15	1	no	—
205	C₂²×C₄	—	15	1	no	—
206	C₂²×C₄	C₂×D₁₅	1	1	yes	characteristic
207	C₄⋊C₄	—	15	1	no	—
208	C₄⋊C₄	—	15	1	no	—
209	C₄⋊C₄	—	15	1	no	—
210	C₄⋊C₄	—	15	2	no	—
211	C₄⋊C₄	—	15	2	no	—
212	C₄⋊C₄	—	15	2	no	—
213	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
214	C₄⋊C₄	C₂×D₁₅	1	2	yes	—
215	C₂²⋊C₄	—	15	2	no	—
216	C₂²⋊C₄	—	15	1	no	—
217	C₂²⋊C₄	—	15	2	no	—
218	C₂²⋊C₄	—	15	1	no	—
219	C₂²⋊C₄	—	15	2	no	—
220	C₂²⋊C₄	C₂×D₁₅	1	2	yes	—
221	C₄²	—	15	2	no	—
222	C₄²	—	15	2	no	—
223	C₄²	—	15	1	no	—
224	C₄²	—	15	2	no	—
225	C₃×C₅	#556	1	1	yes	characteristic, complemented, Hall
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
230	C₂×D₃	—	10	1	no	—
231	C₂×D₃	—	10	1	no	—
232	C₂×D₃	—	10	1	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	#711	1	2	yes	—
237	Dic₃	—	5	2	no	—
238	Dic₃	—	10	1	no	—
239	Dic₃	—	10	2	no	—
240	Dic₃	—	10	1	no	—
241	Dic₃	—	10	2	no	—
242	C₂×C₅	—	4	1	no	—
243	C₂×C₅	C₂×#237	1	1	yes	characteristic
244	C₂×C₅	#1007	1	1	yes	characteristic
245	C₂×C₅	D₃×#41	1	1	yes	characteristic
246	D₅	—	12	1	no	—
247	C₂³	—	15	1	no	—
248	C₂³	C₂²×D₁₅	1	1	yes	characteristic
249	Q₈	—	2	2	no	—
250	Q₈	—	30	4	no	—
251	D₄	—	30	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	15	1	no	—
254	C₂×C₄	—	15	2	no	—
255	C₂×C₄	—	15	1	no	—
256	C₂×C₄	—	15	1	no	—
257	C₂×C₄	—	15	2	no	—
258	C₂×C₄	—	30	2	no	—
259	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₁₅	1	2	yes	—
261	C₂×C₄	C₂²×D₁₅	1	2	yes	—
262	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
263	C₂×C₃	—	4	1	no	—
264	C₂×C₃	C₂×#711	1	1	yes	characteristic
265	C₂×C₃	#4147	1	1	yes	characteristic
266	C₂×C₃	D₅×#41	1	1	yes	characteristic
267	D₃	—	20	1	no	—
268	C₅	#5641	1	1	yes	characteristic, complemented, Sylow
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	30	1	no	—
273	C₂²	—	30	1	no	—
274	C₂²	—	30	1	no	—
275	C₂²	C₂³×D₁₅	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	15	2	no	—
280	C₄	—	30	1	no	—
281	C₄	—	30	2	no	—
282	C₄	—	30	1	no	—
283	C₄	—	30	2	no	—
284	C₄	#6819	1	2	yes	—
285	C₃	#65975	1	1	yes	characteristic, complemented, Sylow
286	C₂	—	4	1	no	—
287	C₂	—	60	1	no	—
288	C₂	C₂×#6819	1	1	yes	characteristic
289	C₂	#92098	1	1	yes	characteristic
290	C₂	#41×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₅×C₂²⋊Q₈	C₂
prime radical	C₂×#1220	C₂²
hypercenter	C₂²⋊Q₈	D₁₅
hypocenter	C₃×C₅	#556
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₂²⋊Q₈	C₂	semidirect
2. #91835	C₂	semidirect
3. Q₈×Dic₁₅	C₂	semidirect
5. #91815	C₂	semidirect
6. #91787	C₂	semidirect
7. #91781	C₂	semidirect
8. #91777	C₂	semidirect
9. #91776	C₂	semidirect
11. #91768	C₂	semidirect
37. C₅×C₂²⋊Q₈	D₃	semidirect
64. C₃×C₂²⋊Q₈	D₅	semidirect
159. C₂²⋊Q₈	D₁₅	semidirect
225. C₃×C₅	#556	semidirect
268. C₅	#5641	semidirect
285. C₃	#65975	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₂²⋊Q₈	2
#91835	2
Q₈×Dic₁₅	2
C₄×#1220	2
#91815	2
#91787	2
#91781	2
#91777	2
#91776	2
#91774	2
#91768	2
#65975	3
#5641	5