Subgroups — #11753429

Order 960.

560 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	C₂×D₅×Dic₆	C₂	1	2	yes	maximal, complemented
2	#91777	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#888	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4039	C₂	1	1	yes	maximal, characteristic
5	#91468	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#1204	C₂	1	1	yes	maximal, characteristic, complemented
7	#91434	C₂	1	1	yes	maximal, characteristic
8	#91431	C₂	1	2	yes	maximal, complemented
9	Dic₃×D₂₀	C₂	1	1	yes	maximal, characteristic
10	#91421	C₂	1	2	yes	maximal, complemented
11	#91349	C₂	1	2	yes	maximal, complemented
12	#66061	—	3	1	no	maximal, Hall
13	C₂×Dic₃₀	C₂²	1	2	yes	—
14	C₂×C₅×Dic₆	C₂²	1	2	yes	—
15	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
17	C₂×#1204	C₂²	1	2	yes	—
18	C₂×#1204	C₂²	1	1	yes	characteristic
19	C₂×D₅×Dic₃	C₂²	1	2	yes	—
20	C₂×D₅×Dic₃	C₂²	1	2	yes	—
21	D₅×Dic₆	—	2	8	no	—
22	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
24	#6712	C₂²	1	2	yes	—
25	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
26	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
28	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
29	C₃×#685	C₂²	1	2	yes	—
30	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
31	Dic₃⋊Dic₅	C₂²	1	1	yes	characteristic
32	#6669	C₂²	1	1	yes	characteristic
33	#6668	C₂²	1	2	yes	—
34	#6668	C₂²	1	1	yes	characteristic
35	#6665	C₂²	1	2	yes	—
36	#6665	C₂²	1	2	yes	—
37	D₄×Dic₆	—	5	1	no	maximal, Hall
38	C₂×Q₈×D₅	—	3	2	no	—
39	C₅×C₄⋊Q₈	—	3	1	no	—
40	#4092	—	3	2	no	—
41	#4042	—	3	2	no	—
42	#4042	—	3	2	no	—
43	#4039	—	3	1	no	—
44	#4039	D₃	1	1	yes	characteristic
45	#4034	—	3	1	no	—
46	#4034	—	3	1	no	—
47	C₄×D₂₀	—	3	1	no	—
48	C₄×Dic₁₀	—	3	1	no	—
49	C₂²×C₃×D₅	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₁₅	C₂³	1	2	yes	—
54	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
55	Dic₃₀	—	2	4	no	—
56	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₅×Dic₃	C₂³	1	2	yes	—
58	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₅×Dic₆	—	2	4	no	—
60	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
61	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
62	C₃×D₂₀	Q₈	1	4	yes	complemented
63	C₃×C₄×D₅	—	2	4	no	—
64	#1204	—	2	4	no	—
65	#1204	D₄	1	4	yes	complemented
66	D₅×Dic₃	—	2	4	no	—
67	D₅×Dic₃	—	2	4	no	—
68	C₂²×Dic₆	—	5	2	no	—
69	C₃×C₄×D₄	—	5	1	no	—
70	D₄×Dic₃	—	5	1	no	—
71	D₄×Dic₃	—	5	1	no	—
72	#924	—	5	2	no	—
73	#888	—	5	1	no	—
74	#888	D₅	1	1	yes	characteristic, complemented
75	#878	—	5	2	no	—
76	#878	—	5	2	no	—
77	#869	—	5	1	no	—
78	C₄×Dic₆	—	5	1	no	—
79	C₂×C₅×Q₈	—	3	2	no	—
80	Q₈×D₅	—	6	8	no	—
81	C₂×D₂₀	C₂×D₃	1	1	yes	characteristic
82	C₂×C₄×D₅	—	3	2	no	—
83	C₂×C₄×D₅	—	3	2	no	—
84	C₂×C₄×D₅	C₂×D₃	1	2	yes	—
85	C₂×Dic₁₀	—	3	1	no	—
86	C₂×Dic₁₀	—	3	2	no	—
87	C₂×Dic₁₀	—	3	2	no	—
88	C₅×C₄⋊C₄	—	3	2	no	—
89	C₅×C₄⋊C₄	—	3	1	no	—
90	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
91	C₄²×C₅	—	3	1	no	—
92	#685	—	3	2	no	—
93	#685	—	3	2	no	—
94	#685	C₂×D₃	1	2	yes	—
95	C₄⋊Dic₅	—	3	1	no	—
96	C₄⋊Dic₅	—	3	1	no	—
97	#683	—	3	2	no	—
98	#683	—	3	1	no	—
99	#683	—	3	2	no	—
100	#683	—	3	1	no	—
101	C₄×Dic₅	—	3	1	no	—
102	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
103	D₄×Q₈	—	15	1	no	Sylow
104	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
105	C₂×C₃×D₅	—	2	2	no	—
106	C₂×C₃×D₅	—	2	2	no	—
107	C₂×C₃×D₅	C₂×Q₈	1	4	yes	complemented
108	C₃×C₄×C₅	—	2	2	no	—
109	C₃×C₄×C₅	C₂×Q₈	1	2	yes	complemented
110	Dic₁₅	—	2	2	no	—
111	Dic₁₅	—	2	1	no	—
112	Dic₁₅	C₂×D₄	1	2	yes	—
113	C₃×Dic₅	—	2	1	no	—
114	C₃×Dic₅	C₂×D₄	1	2	yes	—
115	C₅×Dic₃	—	2	2	no	—
116	C₅×Dic₃	—	2	1	no	—
117	C₅×Dic₃	C₂×D₄	1	2	yes	—
118	C₂×C₃×D₄	—	5	1	no	—
119	C₂²×C₃×C₄	—	5	2	no	—
120	C₂²×Dic₃	—	5	2	no	—
121	C₂²×Dic₃	—	5	2	no	—
122	C₂×Dic₆	—	5	1	no	—
123	C₂×Dic₆	—	5	2	no	—
124	C₂×Dic₆	—	5	2	no	—
125	C₂×Dic₆	—	10	8	no	—
126	C₂×Dic₆	C₂×D₅	1	2	yes	—
127	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
128	C₃×C₂²⋊C₄	—	5	2	no	—
129	C₃×C₄²	—	5	1	no	—
130	#217	—	5	2	no	—
131	#217	—	5	2	no	—
132	C₄⋊Dic₃	—	5	1	no	—
133	C₄⋊Dic₃	—	5	1	no	—
134	C₄⋊Dic₃	—	5	2	no	—
135	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
136	Dic₃⋊C₄	—	5	2	no	—
137	Dic₃⋊C₄	—	5	1	no	—
138	Dic₃⋊C₄	—	5	1	no	—
139	Dic₃⋊C₄	C₂×D₅	1	2	yes	—
140	C₄×Dic₃	—	5	1	no	—
141	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
142	C₂²×D₅	C₂²×D₃	1	2	yes	—
143	C₅×Q₈	—	6	4	no	—
144	C₂×C₄×C₅	—	3	1	no	—
145	C₂×C₄×C₅	—	3	1	no	—
146	C₂×C₄×C₅	—	3	2	no	—
147	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
148	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
149	C₂×Dic₅	—	3	1	no	—
150	C₂×Dic₅	—	3	1	no	—
151	C₂×Dic₅	—	3	2	no	—
152	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
153	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
154	D₂₀	Dic₆	1	4	yes	complemented
155	C₄×D₅	—	2	4	no	—
156	C₄×D₅	—	6	4	no	—
157	C₄×D₅	—	6	4	no	—
158	Dic₁₀	—	3	4	no	—
159	Dic₁₀	—	6	4	no	—
160	Dic₁₀	—	6	4	no	—
161	C₂²×Q₈	—	15	2	no	—
162	C₄⋊Q₈	—	3	1	no	—
163	C₄⋊Q₈	—	15	1	no	—
164	C₄⋊Q₈	—	15	1	no	—
165	C₂²⋊Q₈	—	15	2	no	—
166	C₂²⋊Q₈	—	15	2	no	—
167	C₂²⋊Q₈	—	15	2	no	—
168	C₄×Q₈	—	15	1	no	—
169	C₄×D₄	—	5	1	no	—
170	C₄×D₄	—	15	1	no	—
171	C₄×D₄	—	15	1	no	—
172	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
174	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
175	C₃×D₅	—	2	4	no	—
176	C₂³×C₃	—	5	2	no	—
177	C₃×D₄	—	5	4	no	—
178	C₂×C₃×C₄	—	5	1	no	—
179	C₂×C₃×C₄	—	5	1	no	—
180	C₂×C₃×C₄	—	10	4	no	—
181	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
182	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
183	C₂×Dic₃	—	5	1	no	—
184	C₂×Dic₃	—	5	1	no	—
185	C₂×Dic₃	—	5	2	no	—
186	C₂×Dic₃	—	10	4	no	—
187	C₂×Dic₃	—	10	4	no	—
188	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
189	C₂×Dic₃	C₂²×D₅	1	2	yes	—
190	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
191	Dic₆	—	2	4	no	—
192	Dic₆	—	5	4	no	—
193	Dic₆	—	10	4	no	—
194	Dic₆	—	10	4	no	—
195	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
196	C₂×D₅	—	2	2	no	—
197	C₂×D₅	—	2	2	no	—
198	C₂×D₅	C₂×Dic₆	1	4	yes	complemented
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	3	2	no	—
201	C₄×C₅	—	6	2	no	—
202	C₄×C₅	—	6	1	no	—
203	C₄×C₅	C₂×Dic₆	1	2	yes	complemented
204	Dic₅	—	2	1	no	—
205	Dic₅	—	3	2	no	—
206	Dic₅	—	6	2	no	—
207	Dic₅	—	6	1	no	—
208	Dic₅	D₃×D₄	1	2	yes	—
209	C₂×Q₈	—	3	2	no	—
210	C₂×Q₈	—	15	1	no	—
211	C₂×Q₈	—	15	2	no	—
212	C₂×Q₈	—	15	2	no	—
213	C₂×Q₈	—	30	8	no	—
214	C₂×D₄	—	5	1	no	—
215	C₂²×C₄	—	5	2	no	—
216	C₂²×C₄	—	15	2	no	—
217	C₂²×C₄	—	15	2	no	—
218	C₄⋊C₄	—	3	1	no	—
219	C₄⋊C₄	—	3	2	no	—
220	C₄⋊C₄	—	15	1	no	—
221	C₄⋊C₄	—	15	1	no	—
222	C₄⋊C₄	—	15	1	no	—
223	C₄⋊C₄	—	15	2	no	—
224	C₄⋊C₄	—	15	1	no	—
225	C₄⋊C₄	—	15	2	no	—
226	C₄⋊C₄	D₃×D₅	1	1	yes	characteristic
227	C₂²⋊C₄	—	5	2	no	—
228	C₂²⋊C₄	—	15	2	no	—
229	C₂²⋊C₄	—	15	2	no	—
230	C₄²	—	3	1	no	—
231	C₄²	—	5	1	no	—
232	C₄²	—	15	1	no	—
233	C₃×C₅	D₄×Q₈	1	1	yes	characteristic, complemented, Hall
234	C₂²×C₃	—	5	4	no	—
235	C₂²×C₃	—	10	2	no	—
236	C₂²×C₃	—	10	2	no	—
237	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
238	C₃×C₄	—	2	2	no	—
239	C₃×C₄	—	5	2	no	—
240	C₃×C₄	—	10	1	no	—
241	C₃×C₄	Q₈×D₅	1	2	yes	complemented
242	Dic₃	—	2	2	no	—
243	Dic₃	—	2	1	no	—
244	Dic₃	—	5	2	no	—
245	Dic₃	—	10	2	no	—
246	Dic₃	—	10	1	no	—
247	Dic₃	D₄×D₅	1	2	yes	—
248	C₂×C₅	C₂²×Dic₆	1	1	yes	characteristic
249	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
250	C₂×C₅	#1010	1	1	yes	characteristic
251	D₅	—	2	4	no	—
252	C₂³	—	5	2	no	—
253	Q₈	—	6	4	no	—
254	Q₈	—	15	4	no	—
255	Q₈	—	30	4	no	—
256	Q₈	—	30	4	no	—
257	D₄	—	5	4	no	—
258	C₂×C₄	—	3	1	no	—
259	C₂×C₄	—	3	1	no	—
260	C₂×C₄	—	3	2	no	—
261	C₂×C₄	—	5	1	no	—
262	C₂×C₄	—	5	1	no	—
263	C₂×C₄	—	10	4	no	—
264	C₂×C₄	—	15	1	no	—
265	C₂×C₄	—	15	1	no	—
266	C₂×C₄	—	15	2	no	—
267	C₂×C₄	—	30	4	no	—
268	C₂×C₄	—	30	4	no	—
269	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
270	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
271	C₂×C₃	—	10	4	no	—
272	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
273	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
274	C₂×C₃	#4147	1	1	yes	characteristic
275	C₅	D₄×Dic₆	1	1	yes	characteristic, complemented, Sylow
276	C₂²	—	5	4	no	—
277	C₂²	—	10	2	no	—
278	C₂²	—	10	2	no	—
279	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	3	2	no	—
282	C₄	—	5	2	no	—
283	C₄	—	6	2	no	—
284	C₄	—	6	1	no	—
285	C₄	—	10	1	no	—
286	C₄	—	15	2	no	—
287	C₄	—	30	2	no	—
288	C₄	—	30	1	no	—
289	C₄	D₅×Dic₆	1	2	yes	complemented
290	C₃	#66061	1	1	yes	characteristic, complemented, Sylow
291	C₂	—	10	4	no	—
292	C₂	C₂×D₅×Dic₆	1	1	yes	characteristic
293	C₂	D₃×D₄×D₅	1	1	yes	characteristic
294	C₂	#92028	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₅×C₄⋊C₄	C₂²
prime radical	C₂×C₃×D₂₀	C₂²
hypercenter	C₄⋊C₄	D₃×D₅
hypocenter	C₃×C₅	D₄×Q₈
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₅×Dic₆	C₂	semidirect
2. #91777	C₂	semidirect
3. C₅×#888	C₂	semidirect
5. #91468	C₂	semidirect
6. C₄×#1204	C₂	semidirect
8. #91431	C₂	semidirect
10. #91421	C₂	semidirect
11. #91349	C₂	semidirect
62. C₃×D₂₀	Q₈	semidirect
65. #1204	D₄	semidirect
74. #888	D₅	semidirect
107. C₂×C₃×D₅	C₂×Q₈	semidirect
109. C₃×C₄×C₅	C₂×Q₈	semidirect
154. D₂₀	Dic₆	semidirect
198. C₂×D₅	C₂×Dic₆	semidirect
203. C₄×C₅	C₂×Dic₆	semidirect
233. C₃×C₅	D₄×Q₈	semidirect
241. C₃×C₄	Q₈×D₅	semidirect
275. C₅	D₄×Dic₆	semidirect
289. C₄	D₅×Dic₆	semidirect
290. C₃	#66061	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅×Dic₆	2
#91777	2
C₅×#888	2
C₃×#4039	2
#91468	2
C₄×#1204	2
#91434	2
#91431	2
Dic₃×D₂₀	2
#91421	2
#91349	2
#66061	3
D₄×Dic₆	5