Subgroups — #11753527

Order 960.

456 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#91779	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#890	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4039	C₂	1	1	yes	maximal, characteristic
4	#91468	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#1204	C₂	1	1	yes	maximal, characteristic, complemented
6	#91434	C₂	1	1	yes	maximal, characteristic
7	#91430	C₂	1	2	yes	maximal, complemented
8	Dic₃×D₂₀	C₂	1	1	yes	maximal, characteristic
9	#91422	C₂	1	2	yes	maximal, complemented
10	D₅×C₄⋊Dic₃	C₂	1	2	yes	maximal, complemented
11	#91349	C₂	1	2	yes	maximal, complemented
12	#66024	—	3	1	no	maximal, Hall
13	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
15	C₂×#1204	C₂²	1	1	yes	characteristic
16	C₂×#1204	C₂²	1	2	yes	—
17	C₂×D₅×Dic₃	C₂²	1	2	yes	—
18	C₂×D₅×Dic₃	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₄⋊Dic₁₅	C₂²	1	2	yes	—
21	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
22	#6712	C₂²	1	2	yes	—
23	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
24	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
25	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
26	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
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	Dic₃⋊Dic₅	C₂²	1	1	yes	characteristic
31	#6669	C₂²	1	1	yes	characteristic
32	#6668	C₂²	1	1	yes	characteristic
33	#6668	C₂²	1	2	yes	—
34	#6665	C₂²	1	2	yes	—
35	#6665	C₂²	1	2	yes	—
36	#5543	—	5	1	no	maximal, Hall
37	C₅×#125	—	3	1	no	—
38	#4043	—	3	2	no	—
39	#4042	—	3	2	no	—
40	#4042	—	3	2	no	—
41	#4039	—	3	1	no	—
42	#4039	D₃	1	1	yes	characteristic
43	D₅×C₄⋊C₄	—	3	2	no	—
44	#4036	—	3	1	no	—
45	#4034	—	3	1	no	—
46	C₄×D₂₀	—	3	1	no	—
47	C₄×Dic₁₀	—	3	1	no	—
48	C₂²×C₃×D₅	C₂³	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
52	C₂×Dic₁₅	C₂³	1	2	yes	—
53	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
54	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
55	C₂×C₅×Dic₃	C₂³	1	2	yes	—
56	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
58	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
59	C₃×D₂₀	Q₈	1	4	yes	complemented
60	C₃×C₄×D₅	—	2	4	no	—
61	#1204	—	2	2	no	—
62	#1204	—	4	2	no	—
63	D₅×Dic₃	—	4	2	no	—
64	D₅×Dic₃	—	4	2	no	—
65	C₃×C₄×D₄	—	5	1	no	—
66	D₄×Dic₃	—	5	1	no	—
67	D₄×Dic₃	—	5	1	no	—
68	C₂×C₄⋊Dic₃	—	5	2	no	—
69	#924	—	5	2	no	—
70	#890	—	5	1	no	—
71	#890	D₅	1	1	yes	characteristic, complemented
72	#878	—	5	2	no	—
73	#878	—	5	2	no	—
74	#869	—	5	1	no	—
75	C₄×Dic₆	—	5	1	no	—
76	C₂×D₂₀	C₂×D₃	1	1	yes	characteristic
77	C₂×C₄×D₅	—	3	2	no	—
78	C₂×C₄×D₅	—	3	2	no	—
79	C₂×C₄×D₅	C₂×D₃	1	2	yes	—
80	C₂×Dic₁₀	—	3	1	no	—
81	C₂×Dic₁₀	—	3	2	no	—
82	C₅×C₄⋊C₄	—	3	2	no	—
83	C₅×C₄⋊C₄	—	3	1	no	—
84	C₅×C₄⋊C₄	—	3	2	no	—
85	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
86	C₄²×C₅	—	3	1	no	—
87	#685	—	3	2	no	—
88	#685	—	3	2	no	—
89	#685	C₂×D₃	1	2	yes	—
90	C₄⋊Dic₅	—	3	1	no	—
91	C₄⋊Dic₅	—	3	2	no	—
92	C₄⋊Dic₅	—	3	1	no	—
93	#683	—	3	2	no	—
94	#683	—	3	1	no	—
95	#683	—	3	2	no	—
96	#683	—	3	1	no	—
97	C₄×Dic₅	—	3	1	no	—
98	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
99	#554	—	15	1	no	Sylow
100	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
101	C₂×C₃×D₅	—	2	2	no	—
102	C₂×C₃×D₅	—	2	2	no	—
103	C₂×C₃×D₅	C₂×Q₈	1	4	yes	—
104	C₃×C₄×C₅	—	2	2	no	—
105	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
106	Dic₁₅	—	2	1	no	—
107	Dic₁₅	—	2	1	no	—
108	Dic₁₅	—	2	2	no	—
109	C₃×Dic₅	—	2	1	no	—
110	C₃×Dic₅	#41	1	2	yes	—
111	C₅×Dic₃	—	2	1	no	—
112	C₅×Dic₃	—	2	1	no	—
113	C₅×Dic₃	—	2	2	no	—
114	C₂×C₃×D₄	—	5	1	no	—
115	C₂²×C₃×C₄	—	5	2	no	—
116	C₂²×Dic₃	—	5	2	no	—
117	C₂²×Dic₃	—	5	2	no	—
118	C₂×Dic₆	—	5	1	no	—
119	C₂×Dic₆	—	5	2	no	—
120	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
121	C₃×C₂²⋊C₄	—	5	2	no	—
122	C₃×C₄²	—	5	1	no	—
123	#217	—	5	2	no	—
124	#217	—	5	2	no	—
125	C₄⋊Dic₃	—	5	1	no	—
126	C₄⋊Dic₃	—	5	2	no	—
127	C₄⋊Dic₃	—	5	1	no	—
128	C₄⋊Dic₃	—	5	2	no	—
129	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
130	C₄⋊Dic₃	C₂×D₅	1	2	yes	—
131	Dic₃⋊C₄	—	5	2	no	—
132	Dic₃⋊C₄	—	5	1	no	—
133	Dic₃⋊C₄	—	5	1	no	—
134	Dic₃⋊C₄	C₂×D₅	1	2	yes	—
135	C₄×Dic₃	—	5	1	no	—
136	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
137	C₂²×D₅	C₂²×D₃	1	2	yes	—
138	C₂×C₄×C₅	—	3	1	no	—
139	C₂×C₄×C₅	—	3	2	no	—
140	C₂×C₄×C₅	—	3	1	no	—
141	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
142	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
143	C₂×Dic₅	—	3	1	no	—
144	C₂×Dic₅	—	3	1	no	—
145	C₂×Dic₅	—	3	2	no	—
146	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
147	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
148	D₂₀	Dic₆	1	4	yes	complemented
149	C₄×D₅	—	2	4	no	—
150	C₄×D₅	—	12	2	no	—
151	C₄×D₅	—	12	2	no	—
152	Dic₁₀	—	6	2	no	—
153	Dic₁₀	—	12	2	no	—
154	C₄⋊Q₈	—	15	1	no	—
155	#125	—	3	1	no	—
156	#125	—	15	1	no	—
157	C₂²⋊Q₈	—	15	2	no	—
158	C₂²⋊Q₈	—	15	2	no	—
159	C₂²⋊Q₈	—	15	2	no	—
160	C₄×Q₈	—	15	1	no	—
161	C₄×D₄	—	5	1	no	—
162	C₄×D₄	—	15	1	no	—
163	C₄×D₄	—	15	1	no	—
164	C₂×C₄⋊C₄	—	15	2	no	—
165	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
166	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
167	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
168	C₃×D₅	—	2	4	no	—
169	C₂³×C₃	—	5	2	no	—
170	C₃×D₄	—	5	4	no	—
171	C₂×C₃×C₄	—	5	1	no	—
172	C₂×C₃×C₄	—	5	1	no	—
173	C₂×C₃×C₄	—	10	4	no	—
174	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
175	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
176	C₂×Dic₃	—	5	1	no	—
177	C₂×Dic₃	—	5	1	no	—
178	C₂×Dic₃	—	5	2	no	—
179	C₂×Dic₃	—	20	2	no	—
180	C₂×Dic₃	—	20	2	no	—
181	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
182	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
183	C₂×Dic₃	C₂²×D₅	1	2	yes	—
184	Dic₆	—	10	2	no	—
185	Dic₆	—	20	2	no	—
186	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
187	C₂×D₅	—	2	2	no	—
188	C₂×D₅	—	2	2	no	—
189	C₂×D₅	C₂×Dic₆	1	4	yes	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	6	1	no	—
192	C₄×C₅	—	6	2	no	—
193	C₄×C₅	—	6	1	no	—
194	C₄×C₅	C₂×Dic₆	1	2	yes	—
195	Dic₅	—	2	1	no	—
196	Dic₅	—	6	1	no	—
197	Dic₅	—	6	2	no	—
198	Dic₅	—	6	1	no	—
199	Dic₅	#237	1	2	yes	—
200	C₂×Q₈	—	15	1	no	—
201	C₂×Q₈	—	15	2	no	—
202	C₂×D₄	—	5	1	no	—
203	C₂²×C₄	—	5	2	no	—
204	C₂²×C₄	—	15	2	no	—
205	C₂²×C₄	—	15	2	no	—
206	C₄⋊C₄	—	3	2	no	—
207	C₄⋊C₄	—	3	1	no	—
208	C₄⋊C₄	—	3	2	no	—
209	C₄⋊C₄	—	15	1	no	—
210	C₄⋊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	1	no	—
215	C₄⋊C₄	—	15	2	no	—
216	C₄⋊C₄	D₃×D₅	1	1	yes	characteristic
217	C₂²⋊C₄	—	5	2	no	—
218	C₂²⋊C₄	—	15	2	no	—
219	C₂²⋊C₄	—	15	2	no	—
220	C₄²	—	3	1	no	—
221	C₄²	—	5	1	no	—
222	C₄²	—	15	1	no	—
223	C₃×C₅	#554	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	—	5	4	no	—
225	C₂²×C₃	—	10	2	no	—
226	C₂²×C₃	—	10	2	no	—
227	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	—	5	2	no	—
230	C₃×C₄	—	10	1	no	—
231	C₃×C₄	Q₈×D₅	1	2	yes	—
232	Dic₃	—	2	1	no	—
233	Dic₃	—	2	2	no	—
234	Dic₃	—	2	1	no	—
235	Dic₃	—	10	1	no	—
236	Dic₃	—	10	2	no	—
237	Dic₃	—	10	1	no	—
238	C₂×C₅	C₂²×Dic₆	1	1	yes	characteristic
239	C₂×C₅	C₂×#237	1	1	yes	characteristic
240	C₂×C₅	#1009	1	1	yes	characteristic
241	D₅	—	2	4	no	—
242	C₂³	—	5	2	no	—
243	Q₈	—	30	2	no	—
244	Q₈	—	60	2	no	—
245	D₄	—	5	4	no	—
246	C₂×C₄	—	3	1	no	—
247	C₂×C₄	—	3	1	no	—
248	C₂×C₄	—	3	2	no	—
249	C₂×C₄	—	5	1	no	—
250	C₂×C₄	—	5	1	no	—
251	C₂×C₄	—	10	4	no	—
252	C₂×C₄	—	15	1	no	—
253	C₂×C₄	—	15	1	no	—
254	C₂×C₄	—	15	2	no	—
255	C₂×C₄	—	60	2	no	—
256	C₂×C₄	—	60	2	no	—
257	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
258	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
259	C₂×C₃	—	10	4	no	—
260	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
261	C₂×C₃	D₅×#41	1	1	yes	characteristic
262	C₂×C₃	#4149	1	1	yes	characteristic
263	C₅	#5543	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	5	4	no	—
265	C₂²	—	10	2	no	—
266	C₂²	—	10	2	no	—
267	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	5	2	no	—
270	C₄	—	6	1	no	—
271	C₄	—	6	1	no	—
272	C₄	—	6	2	no	—
273	C₄	—	10	1	no	—
274	C₄	—	30	1	no	—
275	C₄	—	30	1	no	—
276	C₄	—	30	2	no	—
277	C₄	D₅×Dic₆	1	2	yes	—
278	C₃	#66024	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	10	4	no	—
280	C₂	C₂×D₅×Dic₆	1	1	yes	characteristic
281	C₂	D₅×#237	1	1	yes	characteristic
282	C₂	#92035	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	#554
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. #91779	C₂	semidirect
2. C₅×#890	C₂	semidirect
4. #91468	C₂	semidirect
5. C₄×#1204	C₂	semidirect
7. #91430	C₂	semidirect
9. #91422	C₂	semidirect
10. D₅×C₄⋊Dic₃	C₂	semidirect
11. #91349	C₂	semidirect
59. C₃×D₂₀	Q₈	semidirect
71. #890	D₅	semidirect
148. D₂₀	Dic₆	semidirect
223. C₃×C₅	#554	semidirect
263. C₅	#5543	semidirect
278. C₃	#66024	semidirect

Maximal subgroups

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

subgroup	index
#91779	2
C₅×#890	2
C₃×#4039	2
#91468	2
C₄×#1204	2
#91434	2
#91430	2
Dic₃×D₂₀	2
#91422	2
D₅×C₄⋊Dic₃	2
#91349	2
#66024	3
#5543	5