Subgroups — #11755229

Order 960.

560 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×D₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#91833	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4092	C₂	1	1	yes	maximal, characteristic, complemented
5	#91441	C₂	1	2	yes	maximal
6	C₄×#1201	C₂	1	1	yes	maximal, characteristic
7	#91421	C₂	1	2	yes	maximal, complemented
8	#91365	C₂	1	1	yes	maximal, characteristic, complemented
9	#91352	C₂	1	2	yes	maximal, complemented
10	#91344	C₂	1	1	yes	maximal, characteristic, complemented
11	#91329	C₂	1	2	yes	maximal, complemented
12	Q₈×C₂²⋊D₅	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
15	C₅×D₃×Q₈	—	2	4	no	—
16	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
18	C₂×C₅×Dic₆	C₂²	1	2	yes	—
19	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
21	C₂×#1204	C₂²	1	2	yes	—
22	C₂×#1201	C₂²	1	1	yes	characteristic
23	C₂×D₅×Dic₃	C₂²	1	2	yes	—
24	D₅×Dic₆	—	2	4	no	—
25	#6712	C₂²	1	2	yes	—
26	#6712	C₂²	1	2	yes	—
27	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
28	C₃×#685	C₂²	1	2	yes	—
29	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
30	C₃×#683	C₂²	1	2	yes	—
31	Dic₃⋊Dic₅	C₂²	1	2	yes	—
32	#6669	C₂²	1	1	yes	characteristic
33	#6668	C₂²	1	2	yes	—
34	(C₂×D₃)⋊Dic₅	C₂²	1	1	yes	characteristic
35	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
36	#6665	C₂²	1	1	yes	characteristic
37	Dic₃×Dic₅	C₂²	1	2	yes	—
38	#5632	—	5	1	no	maximal, Hall
39	C₂²×C₅×Q₈	—	3	1	no	—
40	C₂×Q₈×D₅	—	3	1	no	—
41	#4092	—	3	2	no	—
42	#4092	D₃	1	1	yes	characteristic, complemented
43	Q₈×Dic₅	—	3	1	no	—
44	#4090	—	3	1	no	—
45	#4090	—	3	2	no	—
46	C₄×C₂²⋊D₅	—	3	1	no	—
47	C₄×C₂²⋊D₅	—	3	2	no	—
48	#4070	—	3	1	no	—
49	#4070	—	3	2	no	—
50	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
51	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
52	C₃×C₅×Q₈	—	2	2	no	—
53	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
56	C₂×Dic₁₅	C₂³	1	2	yes	—
57	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
58	Dic₃₀	—	2	2	no	—
59	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₂×C₅×Dic₃	C₂³	1	2	yes	—
61	C₄×C₅×D₃	—	2	2	no	—
62	C₄×C₅×D₃	—	2	4	no	—
63	C₅×Dic₆	—	2	4	no	—
64	C₅×Dic₆	D₄	1	4	yes	complemented
65	C₂×C₃×Dic₅	C₂³	1	2	yes	—
66	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
67	C₃×C₄×D₅	—	2	2	no	—
68	#1204	—	2	4	no	—
69	#1201	Q₈	1	4	yes	complemented
70	D₅×Dic₃	—	2	4	no	—
71	C₂×D₃×Q₈	D₅	1	1	yes	characteristic, complemented
72	C₂²×Dic₆	—	5	1	no	—
73	C₃×C₂²⋊Q₈	—	5	1	no	—
74	#944	—	5	1	no	—
75	C₄×#67	—	5	1	no	—
76	#897	—	5	1	no	—
77	#896	—	5	2	no	—
78	#888	—	5	2	no	—
79	#887	—	5	1	no	—
80	#67⋊C₄	—	5	2	no	—
81	#878	—	5	2	no	—
82	C₂×C₅×Q₈	—	3	1	no	—
83	C₂×C₅×Q₈	—	3	2	no	—
84	C₂×C₅×Q₈	—	6	4	no	—
85	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
86	C₂²×C₄×C₅	—	3	2	no	—
87	C₂²×C₄×C₅	—	3	1	no	—
88	C₂×C₂²⋊D₅	—	3	1	no	—
89	Q₈×D₅	—	6	4	no	—
90	C₂×C₄×D₅	—	3	2	no	—
91	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
92	C₂×Dic₁₀	—	3	1	no	—
93	C₂×Dic₁₀	—	3	2	no	—
94	C₂²⋊Dic₅	—	3	1	no	—
95	C₂²⋊Dic₅	—	3	2	no	—
96	#685	—	3	1	no	—
97	#685	C₂×D₃	1	2	yes	—
98	C₄⋊Dic₅	—	3	2	no	—
99	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
100	#683	—	3	2	no	—
101	#683	—	3	2	no	—
102	#683	—	3	2	no	—
103	#683	—	3	1	no	—
104	#683	C₂×D₃	1	2	yes	—
105	C₄×Dic₅	—	3	1	no	—
106	C₄×Dic₅	—	3	2	no	—
107	D₄×Q₈	—	15	1	no	Sylow
108	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
109	C₂×C₅×D₃	—	2	1	no	—
110	C₂×C₅×D₃	—	2	1	no	—
111	C₂×C₅×D₃	C₂×Q₈	1	2	yes	complemented
112	C₂×C₃×D₅	—	2	1	no	—
113	C₂×C₃×D₅	—	2	1	no	—
114	C₂×C₃×D₅	C₂×Q₈	1	2	yes	complemented
115	C₃×C₄×C₅	—	2	2	no	—
116	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
117	Dic₁₅	—	2	2	no	—
118	Dic₁₅	—	2	1	no	—
119	Dic₁₅	C₂×Q₈	1	2	yes	complemented
120	C₃×Dic₅	—	2	1	no	—
121	C₃×Dic₅	—	2	2	no	—
122	C₅×Dic₃	—	2	1	no	—
123	C₅×Dic₃	C₂×D₄	1	4	yes	—
124	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
125	C₂²×C₃×C₄	—	5	1	no	—
126	C₂×#67	—	5	1	no	—
127	C₂²×Dic₃	—	5	2	no	—
128	D₃×Q₈	—	2	4	no	—
129	C₂×C₄×D₃	C₂×D₅	1	2	yes	—
130	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
131	C₂×Dic₆	—	5	1	no	—
132	C₂×Dic₆	—	5	2	no	—
133	C₂×Dic₆	—	10	4	no	—
134	C₂×Dic₆	C₂×D₅	1	2	yes	—
135	C₂×Dic₆	C₂×D₅	1	1	yes	characteristic
136	C₃×C₄⋊C₄	—	5	2	no	—
137	C₃×C₄⋊C₄	—	5	1	no	—
138	C₃×C₂²⋊C₄	—	5	2	no	—
139	#217	—	5	1	no	—
140	(C₂×D₃)⋊C₄	—	5	2	no	—
141	(C₂×D₃)⋊C₄	—	5	1	no	—
142	C₄⋊Dic₃	—	5	2	no	—
143	Dic₃⋊C₄	—	5	2	no	—
144	Dic₃⋊C₄	—	5	2	no	—
145	Dic₃⋊C₄	—	5	1	no	—
146	Dic₃⋊C₄	—	5	2	no	—
147	C₄×Dic₃	—	5	1	no	—
148	C₄×Dic₃	—	5	2	no	—
149	C₂³×C₅	—	3	1	no	—
150	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
151	C₅×Q₈	—	2	2	no	—
152	C₅×Q₈	—	3	4	no	—
153	C₅×Q₈	—	6	4	no	—
154	C₂×C₄×C₅	—	3	2	no	—
155	C₂×C₄×C₅	—	3	1	no	—
156	C₂×C₄×C₅	—	6	4	no	—
157	C₂×C₄×C₅	—	6	2	no	—
158	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
159	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
160	C₂²⋊D₅	—	3	4	no	—
161	C₂×Dic₅	—	3	1	no	—
162	C₂×Dic₅	—	3	1	no	—
163	C₂×Dic₅	—	3	2	no	—
164	C₂×Dic₅	C₂²×D₃	1	2	yes	—
165	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
166	C₄×D₅	—	2	2	no	—
167	C₄×D₅	—	6	4	no	—
168	Dic₁₀	—	6	4	no	—
169	Dic₁₀	—	6	2	no	—
170	C₂²×Q₈	—	3	1	no	—
171	C₂²×Q₈	—	15	1	no	—
172	C₄⋊Q₈	—	15	2	no	—
173	C₄⋊Q₈	—	15	1	no	—
174	C₂²⋊Q₈	—	5	1	no	—
175	C₂²⋊Q₈	—	15	2	no	—
176	C₂²⋊Q₈	—	15	2	no	—
177	C₂²⋊Q₈	—	15	1	no	—
178	C₄×Q₈	—	15	1	no	—
179	C₄×D₄	—	15	2	no	—
180	C₄×D₄	—	15	1	no	—
181	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
183	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
184	C₃×D₅	—	2	2	no	—
185	C₅×D₃	—	2	2	no	—
186	C₂³×C₃	—	5	1	no	—
187	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
188	C₃×Q₈	—	2	2	no	—
189	C₂×C₃×C₄	—	5	2	no	—
190	C₂×C₃×C₄	—	5	1	no	—
191	C₂×C₃×C₄	—	10	2	no	—
192	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
193	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
194	#67	—	5	4	no	—
195	C₂×Dic₃	—	5	1	no	—
196	C₂×Dic₃	—	5	1	no	—
197	C₂×Dic₃	—	5	2	no	—
198	C₂×Dic₃	—	10	4	no	—
199	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
200	C₂×Dic₃	C₂²×D₅	1	2	yes	—
201	C₄×D₃	—	2	2	no	—
202	C₄×D₃	—	2	4	no	—
203	Dic₆	—	2	4	no	—
204	Dic₆	—	10	2	no	—
205	Dic₆	—	10	4	no	—
206	Dic₆	C₂²⋊D₅	1	4	yes	complemented
207	C₂²×C₅	—	3	2	no	—
208	C₂²×C₅	—	6	1	no	—
209	C₂²×C₅	—	6	1	no	—
210	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
211	C₂×D₅	—	2	1	no	—
212	C₂×D₅	—	2	1	no	—
213	C₂×D₅	D₃×Q₈	1	2	yes	complemented
214	C₄×C₅	—	2	2	no	—
215	C₄×C₅	—	3	4	no	—
216	C₄×C₅	—	6	1	no	—
217	C₄×C₅	D₃×D₄	1	2	yes	—
218	Dic₅	—	2	1	no	—
219	Dic₅	—	2	2	no	—
220	Dic₅	—	3	2	no	—
221	Dic₅	—	6	2	no	—
222	Dic₅	—	6	1	no	—
223	C₂×Q₈	—	3	1	no	—
224	C₂×Q₈	—	3	2	no	—
225	C₂×Q₈	—	6	4	no	—
226	C₂×Q₈	—	15	2	no	—
227	C₂×Q₈	—	15	1	no	—
228	C₂×Q₈	—	30	4	no	—
229	C₂×Q₈	D₃×D₅	1	1	yes	characteristic
230	C₂×D₄	—	15	1	no	—
231	C₂²×C₄	—	3	2	no	—
232	C₂²×C₄	—	3	1	no	—
233	C₂²×C₄	—	5	1	no	—
234	C₂²×C₄	—	15	2	no	—
235	C₄⋊C₄	—	5	1	no	—
236	C₄⋊C₄	—	5	2	no	—
237	C₄⋊C₄	—	15	1	no	—
238	C₄⋊C₄	—	15	2	no	—
239	C₄⋊C₄	—	15	2	no	—
240	C₄⋊C₄	—	15	2	no	—
241	C₄⋊C₄	—	15	2	no	—
242	C₂²⋊C₄	—	5	2	no	—
243	C₂²⋊C₄	—	15	1	no	—
244	C₂²⋊C₄	—	15	2	no	—
245	C₂²⋊C₄	—	15	1	no	—
246	C₄²	—	15	2	no	—
247	C₄²	—	15	1	no	—
248	C₃×C₅	D₄×Q₈	1	1	yes	characteristic, complemented, Hall
249	C₂²×C₃	—	5	2	no	—
250	C₂²×C₃	—	10	1	no	—
251	C₂²×C₃	—	10	1	no	—
252	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
253	C₂×D₃	—	2	1	no	—
254	C₂×D₃	—	2	1	no	—
255	C₂×D₃	Q₈×D₅	1	2	yes	complemented
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	10	2	no	—
258	C₃×C₄	—	10	1	no	—
259	C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
260	Dic₃	—	2	1	no	—
261	Dic₃	—	5	2	no	—
262	Dic₃	—	10	1	no	—
263	Dic₃	—	10	2	no	—
264	Dic₃	C₂×C₂²⋊D₅	1	4	yes	—
265	C₂×C₅	—	6	2	no	—
266	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
267	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
268	C₂×C₅	#1010	1	1	yes	characteristic
269	D₅	—	2	2	no	—
270	C₂³	—	3	1	no	—
271	C₂³	—	5	1	no	—
272	Q₈	—	2	2	no	—
273	Q₈	—	3	4	no	—
274	Q₈	—	6	4	no	—
275	Q₈	—	30	4	no	—
276	Q₈	—	30	2	no	—
277	D₄	—	15	4	no	—
278	C₂×C₄	—	3	2	no	—
279	C₂×C₄	—	3	1	no	—
280	C₂×C₄	—	5	2	no	—
281	C₂×C₄	—	5	1	no	—
282	C₂×C₄	—	6	4	no	—
283	C₂×C₄	—	6	2	no	—
284	C₂×C₄	—	10	2	no	—
285	C₂×C₄	—	15	1	no	—
286	C₂×C₄	—	15	1	no	—
287	C₂×C₄	—	15	2	no	—
288	C₂×C₄	—	30	4	no	—
289	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
290	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
291	C₂×C₃	—	10	2	no	—
292	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
293	C₂×C₃	#4147	1	1	yes	characteristic
294	C₂×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
295	D₃	—	2	2	no	—
296	C₅	#5632	1	1	yes	characteristic, complemented, Sylow
297	C₂²	—	3	2	no	—
298	C₂²	—	5	2	no	—
299	C₂²	—	6	1	no	—
300	C₂²	—	6	1	no	—
301	C₂²	—	10	1	no	—
302	C₂²	—	10	1	no	—
303	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	3	4	no	—
306	C₄	—	6	1	no	—
307	C₄	—	10	2	no	—
308	C₄	—	10	1	no	—
309	C₄	—	15	2	no	—
310	C₄	—	30	1	no	—
311	C₄	—	30	2	no	—
312	C₄	D₃×C₂²⋊D₅	1	2	yes	—
313	C₃	Q₈×C₂²⋊D₅	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	6	2	no	—
315	C₂	—	10	2	no	—
316	C₂	#92028	1	1	yes	characteristic
317	C₂	D₃×Q₈×D₅	1	1	yes	characteristic
318	C₂	C₂×D₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₂×C₃×C₅×Q₈	C₂²
prime radical	C₂×#1201	C₂²
hypercenter	C₂×Q₈	D₃×D₅
hypocenter	C₃×C₅	D₄×Q₈
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×D₃×Q₈	C₂	semidirect
2. C₂×D₅×Dic₆	C₂	semidirect
3. #91833	C₂	semidirect
4. C₃×#4092	C₂	semidirect
7. #91421	C₂	semidirect
8. #91365	C₂	semidirect
9. #91352	C₂	semidirect
10. #91344	C₂	semidirect
11. #91329	C₂	semidirect
42. #4092	D₃	semidirect
64. C₅×Dic₆	D₄	semidirect
69. #1201	Q₈	semidirect
71. C₂×D₃×Q₈	D₅	semidirect
111. C₂×C₅×D₃	C₂×Q₈	semidirect
114. C₂×C₃×D₅	C₂×Q₈	semidirect
119. Dic₁₅	C₂×Q₈	semidirect
206. Dic₆	C₂²⋊D₅	semidirect
213. C₂×D₅	D₃×Q₈	semidirect
248. C₃×C₅	D₄×Q₈	semidirect
255. C₂×D₃	Q₈×D₅	semidirect
296. C₅	#5632	semidirect
313. C₃	Q₈×C₂²⋊D₅	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×Q₈	2
C₂×D₅×Dic₆	2
#91833	2
C₃×#4092	2
#91441	2
C₄×#1201	2
#91421	2
#91365	2
#91352	2
#91344	2
#91329	2
Q₈×C₂²⋊D₅	3
#5632	5