Subgroups — #11754512

Order 960.

476 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×#67	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4076	C₂	1	1	yes	maximal, characteristic, complemented
4	#91571	C₂	1	2	yes	maximal, complemented
5	#91559	C₂	1	2	yes	maximal, complemented
6	#91456	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
8	#91414	C₂	1	2	yes	maximal, complemented
9	#91381	C₂	1	1	yes	maximal, characteristic, complemented
10	#91362	C₂	1	2	yes	maximal, complemented
11	#91338	C₂	1	1	yes	maximal, characteristic, complemented
12	#65899	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₅	C₂²	1	2	yes	—
15	C₂×C₅×#67	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
18	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
19	C₂×#1201	C₂²	1	2	yes	—
20	C₂×D₅×Dic₃	C₂²	1	2	yes	—
21	C₄⋊Dic₁₅	C₂²	1	2	yes	—
22	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
23	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
24	C₅×#217	C₂²	1	1	yes	characteristic
25	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
26	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
27	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
28	C₃×#685	C₂²	1	2	yes	—
29	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
30	#6669	C₂²	1	2	yes	—
31	#6668	C₂²	1	2	yes	—
32	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
33	#6665	C₂²	1	2	yes	—
34	#6665	C₂²	1	2	yes	—
35	Dic₃×Dic₅	C₂²	1	2	yes	—
36	#5618	—	5	1	no	maximal, Hall
37	C₄×C₅×D₄	—	3	1	no	—
38	#4085	—	3	2	no	—
39	#4076	—	3	1	no	—
40	#4076	D₃	1	1	yes	characteristic, complemented
41	#4072	—	3	1	no	—
42	C₂×C₄⋊Dic₅	—	3	1	no	—
43	#4044	—	3	2	no	—
44	#4029	—	3	2	no	—
45	#4027	—	3	2	no	—
46	C₄×D₂₀	—	3	1	no	—
47	#4016	—	3	1	no	—
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
50	C₂²×C₃×D₅	C₂³	1	2	yes	—
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	1	yes	characteristic
54	C₂×Dic₁₅	—	4	2	no	—
55	C₂×Dic₁₅	C₂³	1	2	yes	—
56	C₂×Dic₁₅	C₂³	1	2	yes	—
57	C₅×#67	—	2	2	no	—
58	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
61	C₄×C₅×D₃	—	2	2	no	—
62	C₃×C₂²⋊D₅	—	2	4	no	—
63	C₂×C₃×Dic₅	C₂³	1	2	yes	—
64	C₃×D₂₀	—	2	2	no	—
65	#1201	—	4	2	no	—
66	D₅×Dic₃	—	4	2	no	—
67	C₃×#121	—	5	1	no	—
68	#939	—	5	2	no	—
69	#938	—	5	1	no	—
70	D₄×Dic₃	—	5	2	no	—
71	D₄×Dic₃	—	5	1	no	—
72	C₄×#67	D₅	1	1	yes	characteristic, complemented
73	C₂×C₄⋊Dic₃	—	5	1	no	—
74	#892	—	5	1	no	—
75	#890	—	5	1	no	—
76	#886	—	5	2	no	—
77	#879	—	5	2	no	—
78	C₂×C₅×D₄	—	3	1	no	—
79	C₂²×C₄×C₅	—	3	1	no	—
80	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
81	C₂×C₂²⋊D₅	—	3	2	no	—
82	C₂×C₂²⋊D₅	C₂×D₃	1	2	yes	—
83	C₂²×Dic₅	—	3	2	no	—
84	C₂×D₂₀	C₂×D₃	1	1	yes	characteristic
85	C₂×C₄×D₅	—	3	2	no	—
86	C₅×C₄⋊C₄	—	3	1	no	—
87	C₅×C₂²⋊C₄	—	3	1	no	—
88	C₅×C₂²⋊C₄	—	3	1	no	—
89	C₄²×C₅	—	3	1	no	—
90	C₂²⋊Dic₅	—	3	2	no	—
91	#685	—	3	2	no	—
92	#685	—	3	2	no	—
93	#685	C₂×D₃	1	2	yes	—
94	C₄⋊Dic₅	—	3	1	no	—
95	C₄⋊Dic₅	—	3	1	no	—
96	C₄⋊Dic₅	—	3	2	no	—
97	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
98	#683	—	3	2	no	—
99	#683	—	3	2	no	—
100	C₄×Dic₅	—	3	2	no	—
101	#553	—	15	1	no	Sylow
102	C₂²×C₃×C₅	—	2	1	no	—
103	C₂²×C₃×C₅	—	2	1	no	—
104	C₂²×C₃×C₅	#41	1	2	yes	—
105	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
106	C₂×C₅×D₃	—	2	1	no	—
107	C₂×C₅×D₃	—	2	1	no	—
108	C₂×C₅×D₃	—	2	1	no	—
109	C₂×C₃×D₅	—	2	2	no	—
110	C₂×C₃×D₅	—	2	2	no	—
111	C₂×C₃×D₅	—	2	2	no	—
112	C₃×C₄×C₅	—	2	1	no	—
113	C₃×C₄×C₅	#41	1	2	yes	—
114	Dic₁₅	—	2	2	no	—
115	Dic₁₅	—	2	2	no	—
116	C₃×Dic₅	—	2	2	no	—
117	C₅×Dic₃	—	2	1	no	—
118	C₅×Dic₃	—	2	1	no	—
119	C₅×Dic₃	—	2	1	no	—
120	C₂×C₃×D₄	—	5	2	no	—
121	C₂×C₃×D₄	—	5	1	no	—
122	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
123	C₂×#67	—	5	2	no	—
124	C₂×#67	C₂×D₅	1	1	yes	characteristic
125	C₂²×Dic₃	—	5	2	no	—
126	C₂²×Dic₃	—	5	2	no	—
127	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
128	C₃×C₄⋊C₄	—	5	1	no	—
129	C₃×C₂²⋊C₄	—	5	2	no	—
130	#217	—	5	2	no	—
131	#217	—	5	2	no	—
132	#217	C₂×D₅	1	1	yes	characteristic
133	(C₂×D₃)⋊C₄	—	5	2	no	—
134	(C₂×D₃)⋊C₄	C₂×D₅	1	1	yes	characteristic
135	C₄⋊Dic₃	—	5	2	no	—
136	C₄⋊Dic₃	—	5	1	no	—
137	C₄⋊Dic₃	—	5	1	no	—
138	C₄⋊Dic₃	—	5	2	no	—
139	Dic₃⋊C₄	—	5	2	no	—
140	Dic₃⋊C₄	C₂×D₅	1	1	yes	characteristic
141	C₄×Dic₃	—	5	2	no	—
142	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
143	C₂³×C₅	—	3	1	no	—
144	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
145	C₂²×D₅	C₂²×D₃	1	2	yes	—
146	C₅×D₄	—	6	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	3	1	no	—
149	C₂×C₄×C₅	—	3	1	no	—
150	C₂×C₄×C₅	—	3	1	no	—
151	C₂×C₄×C₅	—	6	2	no	—
152	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
154	C₂²⋊D₅	—	2	4	no	—
155	C₂²⋊D₅	—	12	2	no	—
156	C₂×Dic₅	—	3	2	no	—
157	C₂×Dic₅	—	3	2	no	—
158	C₂×Dic₅	—	12	2	no	—
159	C₂×Dic₅	C₂²×D₃	1	2	yes	—
160	D₂₀	—	2	2	no	—
161	C₄×D₅	—	12	2	no	—
162	#126	—	15	2	no	—
163	#125	—	15	1	no	—
164	#123	—	15	2	no	—
165	#121	—	5	1	no	—
166	#121	—	15	1	no	—
167	#121	—	15	2	no	—
168	C₄×D₄	—	3	1	no	—
169	C₄×D₄	—	15	1	no	—
170	C₄×D₄	—	15	2	no	—
171	#117	—	15	1	no	—
172	C₂×C₄⋊C₄	—	15	1	no	—
173	C₂×C₃×C₅	—	2	2	no	—
174	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
175	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
176	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
177	C₃×D₅	—	4	2	no	—
178	C₅×D₃	—	4	1	no	—
179	C₂³×C₃	—	5	2	no	—
180	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
181	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
182	C₃×D₄	—	10	4	no	—
183	C₃×D₄	—	10	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	5	2	no	—
186	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
188	#67	—	2	2	no	—
189	#67	—	20	2	no	—
190	C₂×Dic₃	—	5	2	no	—
191	C₂×Dic₃	—	5	2	no	—
192	C₂×Dic₃	—	20	2	no	—
193	C₂×Dic₃	—	20	2	no	—
194	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
195	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
196	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
197	C₄×D₃	—	2	2	no	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	—	2	1	no	—
200	C₂²×C₅	—	6	1	no	—
201	C₂²×C₅	—	6	1	no	—
202	C₂²×C₅	—	6	1	no	—
203	C₂²×C₅	#237	1	2	yes	—
204	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
205	C₂×D₅	—	2	2	no	—
206	C₂×D₅	—	2	2	no	—
207	C₂×D₅	—	2	2	no	—
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	6	1	no	—
210	C₄×C₅	—	6	1	no	—
211	C₄×C₅	—	6	1	no	—
212	C₄×C₅	#237	1	2	yes	—
213	Dic₅	—	2	2	no	—
214	Dic₅	—	6	2	no	—
215	Dic₅	—	6	2	no	—
216	C₂×D₄	—	3	1	no	—
217	C₂×D₄	—	5	2	no	—
218	C₂×D₄	—	5	1	no	—
219	C₂×D₄	—	15	2	no	—
220	C₂²×C₄	—	3	1	no	—
221	C₂²×C₄	—	15	2	no	—
222	C₂²×C₄	—	15	2	no	—
223	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
224	C₄⋊C₄	—	3	1	no	—
225	C₄⋊C₄	—	5	1	no	—
226	C₄⋊C₄	—	15	2	no	—
227	C₄⋊C₄	—	15	2	no	—
228	C₄⋊C₄	—	15	1	no	—
229	C₄⋊C₄	—	15	1	no	—
230	C₄⋊C₄	—	15	2	no	—
231	C₂²⋊C₄	—	3	1	no	—
232	C₂²⋊C₄	—	3	1	no	—
233	C₂²⋊C₄	—	5	2	no	—
234	C₂²⋊C₄	—	15	2	no	—
235	C₂²⋊C₄	—	15	2	no	—
236	C₂²⋊C₄	—	15	2	no	—
237	C₄²	—	3	1	no	—
238	C₄²	—	15	2	no	—
239	C₃×C₅	#553	1	1	yes	characteristic, complemented, Hall
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	10	2	no	—
243	C₂²×C₃	—	10	2	no	—
244	C₂²×C₃	—	10	2	no	—
245	C₂²×C₃	#709	1	2	yes	—
246	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
247	C₂×D₃	—	2	1	no	—
248	C₂×D₃	—	2	1	no	—
249	C₂×D₃	—	2	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	10	2	no	—
252	C₃×C₄	#711	1	2	yes	—
253	Dic₃	—	2	1	no	—
254	Dic₃	—	2	1	no	—
255	Dic₃	—	2	1	no	—
256	Dic₃	—	10	2	no	—
257	Dic₃	—	10	2	no	—
258	C₂×C₅	—	2	2	no	—
259	C₂×C₅	—	12	1	no	—
260	C₂×C₅	C₂×#237	1	1	yes	characteristic
261	C₂×C₅	C₂×#237	1	1	yes	characteristic
262	C₂×C₅	#1009	1	1	yes	characteristic
263	D₅	—	4	2	no	—
264	C₂³	—	3	1	no	—
265	C₂³	—	5	2	no	—
266	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
267	D₄	—	6	2	no	—
268	D₄	—	10	4	no	—
269	D₄	—	10	2	no	—
270	D₄	—	60	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	3	1	no	—
273	C₂×C₄	—	3	1	no	—
274	C₂×C₄	—	3	1	no	—
275	C₂×C₄	—	5	2	no	—
276	C₂×C₄	—	6	2	no	—
277	C₂×C₄	—	15	2	no	—
278	C₂×C₄	—	15	2	no	—
279	C₂×C₄	—	60	2	no	—
280	C₂×C₄	—	60	2	no	—
281	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
282	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
283	C₂×C₃	—	2	2	no	—
284	C₂×C₃	—	20	2	no	—
285	C₂×C₃	C₂×#709	1	1	yes	characteristic
286	C₂×C₃	C₂×#711	1	1	yes	characteristic
287	C₂×C₃	#4149	1	1	yes	characteristic
288	D₃	—	4	1	no	—
289	C₅	#5618	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	6	1	no	—
293	C₂²	—	6	1	no	—
294	C₂²	—	6	1	no	—
295	C₂²	—	10	2	no	—
296	C₂²	—	10	2	no	—
297	C₂²	—	10	2	no	—
298	C₂²	#6765	1	2	yes	—
299	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	6	1	no	—
302	C₄	—	6	1	no	—
303	C₄	—	6	1	no	—
304	C₄	—	10	2	no	—
305	C₄	—	30	2	no	—
306	C₄	—	30	2	no	—
307	C₄	#6780	1	2	yes	—
308	C₃	#65899	1	1	yes	characteristic, complemented, Sylow
309	C₂	—	2	2	no	—
310	C₂	—	12	1	no	—
311	C₂	—	20	2	no	—
312	C₂	C₂×#6765	1	1	yes	characteristic
313	C₂	#92019	1	1	yes	characteristic
314	C₂	C₂×#6780	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₂²×C₃×C₄×C₅	C₂²
hypercenter	C₂²×C₄	D₃×D₅
hypocenter	C₃×C₅	#553
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊Dic₁₅	C₂	semidirect
2. C₄×C₅×#67	C₂	semidirect
3. C₃×#4076	C₂	semidirect
4. #91571	C₂	semidirect
5. #91559	C₂	semidirect
6. #91456	C₂	semidirect
7. Dic₃×D₂₀	C₂	semidirect
8. #91414	C₂	semidirect
9. #91381	C₂	semidirect
10. #91362	C₂	semidirect
11. #91338	C₂	semidirect
40. #4076	D₃	semidirect
72. C₄×#67	D₅	semidirect
239. C₃×C₅	#553	semidirect
289. C₅	#5618	semidirect
308. C₃	#65899	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₁₅	2
C₄×C₅×#67	2
C₃×#4076	2
#91571	2
#91559	2
#91456	2
Dic₃×D₂₀	2
#91414	2
#91381	2
#91362	2
#91338	2
#65899	3
#5618	5