Subgroups — #11752700

Order 960.

472 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#91764	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
4	#91458	C₂	1	1	yes	maximal, characteristic
5	#91455	C₂	1	2	yes	maximal, complemented
6	#91444	C₂	1	2	yes	maximal, complemented
7	#91428	C₂	1	1	yes	maximal, characteristic, complemented
8	#91364	C₂	1	1	yes	maximal, characteristic, complemented
9	#91356	C₂	1	2	yes	maximal, complemented
10	#91348	C₂	1	2	yes	maximal, complemented
11	#91340	C₂	1	1	yes	maximal, characteristic, complemented
12	#65919	—	3	1	no	maximal, Hall
13	C₂×D₆₀	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
15	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
16	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
18	C₂×#1202	C₂²	1	2	yes	—
19	C₂×#1202	C₂²	1	2	yes	—
20	C₂×#1200	C₂²	1	2	yes	—
21	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
22	#6714	C₂²	1	2	yes	—
23	#6714	C₂²	1	2	yes	—
24	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
25	C₅×Dic₃⋊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	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	#6669	C₂²	1	2	yes	—
32	#6667	C₂²	1	2	yes	—
33	#6665	C₂²	1	2	yes	—
34	#6665	C₂²	1	2	yes	—
35	Dic₃×Dic₅	C₂²	1	2	yes	—
36	#5559	—	5	1	no	maximal, Hall
37	C₄×C₅×Q₈	—	3	1	no	—
38	#4093	—	3	2	no	—
39	#4040	—	3	2	no	—
40	#4040	—	3	2	no	—
41	#4039	—	3	2	no	—
42	#4021	—	3	1	no	—
43	#4021	—	3	1	no	—
44	#4020	—	3	1	no	—
45	C₄×D₂₀	—	3	1	no	—
46	C₄×D₂₀	D₃	1	1	yes	characteristic, complemented
47	C₄×Dic₁₀	—	3	1	no	—
48	C₂²×D₁₅	C₂³	1	2	yes	—
49	C₂²×C₃×D₅	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₁₅	C₂³	1	2	yes	—
54	D₆₀	—	2	2	no	—
55	Dic₃₀	—	2	2	no	—
56	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₅×Dic₆	—	2	2	no	—
61	C₂×C₃×Dic₅	C₂³	1	2	yes	—
62	C₃×D₂₀	—	2	2	no	—
63	C₃×C₄×D₅	—	4	2	no	—
64	#1202	—	2	4	no	—
65	#1202	—	4	2	no	—
66	#1200	—	4	2	no	—
67	C₃×C₄×D₄	—	5	1	no	—
68	#940	—	5	1	no	—
69	#936	—	5	1	no	—
70	#929	—	5	2	no	—
71	C₄×#67	—	5	2	no	—
72	#893	—	5	1	no	—
73	#887	—	5	1	no	—
74	#885	—	5	2	no	—
75	#883	—	5	2	no	—
76	#875	—	5	1	no	—
77	C₄×Dic₆	D₅	1	1	yes	characteristic, complemented
78	C₂×C₅×Q₈	—	3	1	no	—
79	C₂×D₂₀	—	3	2	no	—
80	C₂×D₂₀	—	3	1	no	—
81	C₂×D₂₀	—	3	2	no	—
82	C₂×D₂₀	C₂×D₃	1	1	yes	characteristic
83	C₂×C₄×D₅	—	3	2	no	—
84	C₂×C₄×D₅	C₂×D₃	1	2	yes	—
85	C₂×Dic₁₀	—	3	1	no	—
86	C₅×C₄⋊C₄	—	3	1	no	—
87	C₅×C₄⋊C₄	—	3	1	no	—
88	C₅×C₄⋊C₄	—	3	1	no	—
89	C₄²×C₅	—	3	1	no	—
90	C₄²×C₅	—	3	1	no	—
91	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
92	#685	—	3	2	no	—
93	#685	—	3	2	no	—
94	#685	—	3	2	no	—
95	#685	—	3	2	no	—
96	#685	—	3	2	no	—
97	#685	C₂×D₃	1	2	yes	—
98	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
99	#683	—	3	2	no	—
100	C₄×Dic₅	—	3	2	no	—
101	#559	—	15	1	no	Sylow
102	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
103	C₂×D₁₅	—	2	2	no	—
104	C₂×D₁₅	—	2	2	no	—
105	C₂×D₁₅	—	2	2	no	—
106	C₂×C₃×D₅	—	2	2	no	—
107	C₂×C₃×D₅	—	2	2	no	—
108	C₂×C₃×D₅	—	2	2	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	2	1	no	—
111	C₃×C₄×C₅	#41	1	2	yes	—
112	Dic₁₅	—	2	2	no	—
113	C₃×Dic₅	—	2	2	no	—
114	C₅×Dic₃	—	2	1	no	—
115	C₅×Dic₃	—	2	1	no	—
116	C₅×Dic₃	—	2	1	no	—
117	C₅×Dic₃	#41	1	2	yes	—
118	C₂×C₃×D₄	—	5	1	no	—
119	C₂²×C₃×C₄	—	5	2	no	—
120	C₂×#67	—	5	2	no	—
121	C₂×#67	—	5	2	no	—
122	C₂×D₁₂	—	5	1	no	—
123	C₂×C₄×D₃	—	5	2	no	—
124	C₂×Dic₆	—	5	1	no	—
125	C₂×Dic₆	C₂×D₅	1	1	yes	characteristic
126	C₃×C₄⋊C₄	—	5	1	no	—
127	C₃×C₂²⋊C₄	—	5	2	no	—
128	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
129	#217	—	5	2	no	—
130	#217	—	5	2	no	—
131	(C₂×D₃)⋊C₄	—	5	2	no	—
132	(C₂×D₃)⋊C₄	—	5	2	no	—
133	(C₂×D₃)⋊C₄	—	5	2	no	—
134	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
135	Dic₃⋊C₄	—	5	2	no	—
136	Dic₃⋊C₄	C₂×D₅	1	1	yes	characteristic
137	Dic₃⋊C₄	C₂×D₅	1	1	yes	characteristic
138	C₄×Dic₃	—	5	2	no	—
139	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
140	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
141	C₂²×D₅	—	3	2	no	—
142	C₂²×D₅	C₂²×D₃	1	2	yes	—
143	C₅×Q₈	—	6	2	no	—
144	C₂×C₄×C₅	—	3	1	no	—
145	C₂×C₄×C₅	—	3	1	no	—
146	C₂×C₄×C₅	—	3	1	no	—
147	C₂×C₄×C₅	—	3	1	no	—
148	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
151	C₂×Dic₅	—	3	2	no	—
152	C₂×Dic₅	C₂²×D₃	1	2	yes	—
153	D₂₀	—	2	2	no	—
154	D₂₀	—	6	4	no	—
155	D₂₀	—	6	2	no	—
156	D₂₀	—	12	2	no	—
157	C₄×D₅	—	4	2	no	—
158	C₄×D₅	—	12	2	no	—
159	Dic₁₀	—	6	2	no	—
160	#127	—	15	1	no	—
161	#124	—	15	1	no	—
162	#124	—	15	2	no	—
163	#124	—	15	1	no	—
164	#123	—	15	2	no	—
165	#123	—	15	2	no	—
166	C₄×Q₈	—	3	1	no	—
167	C₄×Q₈	—	15	1	no	—
168	C₄×D₄	—	5	1	no	—
169	C₄×D₄	—	15	1	no	—
170	C₄×D₄	—	15	2	no	—
171	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
172	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
173	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
174	D₁₅	—	4	2	no	—
175	C₃×D₅	—	4	2	no	—
176	C₂³×C₃	—	5	2	no	—
177	C₂²×D₃	—	5	2	no	—
178	C₃×D₄	—	10	2	no	—
179	C₂×C₃×C₄	—	5	2	no	—
180	C₂×C₃×C₄	—	20	2	no	—
181	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
184	#67	—	10	4	no	—
185	#67	—	20	2	no	—
186	C₂×Dic₃	—	5	2	no	—
187	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
188	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
189	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
190	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
191	D₁₂	—	10	2	no	—
192	C₄×D₃	—	20	2	no	—
193	Dic₆	—	2	2	no	—
194	Dic₆	—	10	2	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	—	2	2	no	—
199	C₂×D₅	—	6	2	no	—
200	C₂×D₅	—	6	2	no	—
201	C₂×D₅	—	6	2	no	—
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	2	1	no	—
204	C₄×C₅	—	3	2	no	—
205	C₄×C₅	—	6	1	no	—
206	C₄×C₅	—	6	1	no	—
207	C₄×C₅	—	6	1	no	—
208	C₄×C₅	#235	1	2	yes	—
209	Dic₅	—	2	2	no	—
210	Dic₅	—	6	2	no	—
211	C₂×Q₈	—	3	1	no	—
212	C₂×Q₈	—	15	1	no	—
213	C₂×D₄	—	5	1	no	—
214	C₂×D₄	—	15	2	no	—
215	C₂×D₄	—	15	2	no	—
216	C₂×D₄	—	15	1	no	—
217	C₂²×C₄	—	5	2	no	—
218	C₂²×C₄	—	15	2	no	—
219	C₄⋊C₄	—	3	1	no	—
220	C₄⋊C₄	—	3	1	no	—
221	C₄⋊C₄	—	3	1	no	—
222	C₄⋊C₄	—	5	1	no	—
223	C₄⋊C₄	—	15	2	no	—
224	C₂²⋊C₄	—	5	2	no	—
225	C₂²⋊C₄	—	15	2	no	—
226	C₂²⋊C₄	—	15	2	no	—
227	C₂²⋊C₄	—	15	2	no	—
228	C₂²⋊C₄	—	15	2	no	—
229	C₂²⋊C₄	—	15	2	no	—
230	C₄²	—	3	1	no	—
231	C₄²	—	3	1	no	—
232	C₄²	—	15	2	no	—
233	C₄²	D₃×D₅	1	1	yes	characteristic
234	C₃×C₅	#559	1	1	yes	characteristic, complemented, Hall
235	C₂²×C₃	—	10	2	no	—
236	C₂²×C₃	—	10	2	no	—
237	C₂²×C₃	—	10	2	no	—
238	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
239	C₂×D₃	—	10	2	no	—
240	C₂×D₃	—	10	2	no	—
241	C₂×D₃	—	10	2	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	10	2	no	—
245	C₃×C₄	#713	1	2	yes	—
246	Dic₃	—	2	1	no	—
247	Dic₃	—	2	1	no	—
248	Dic₃	—	2	1	no	—
249	Dic₃	—	10	2	no	—
250	Dic₃	#709	1	2	yes	—
251	C₂×C₅	C₂×#235	1	1	yes	characteristic
252	C₂×C₅	#1004	1	1	yes	characteristic
253	C₂×C₅	D₃×#41	1	1	yes	characteristic
254	D₅	—	4	2	no	—
255	D₅	—	12	2	no	—
256	C₂³	—	5	2	no	—
257	C₂³	—	15	2	no	—
258	Q₈	—	6	2	no	—
259	Q₈	—	30	2	no	—
260	D₄	—	10	2	no	—
261	D₄	—	30	4	no	—
262	D₄	—	30	2	no	—
263	D₄	—	60	2	no	—
264	C₂×C₄	—	3	1	no	—
265	C₂×C₄	—	3	1	no	—
266	C₂×C₄	—	3	1	no	—
267	C₂×C₄	—	3	1	no	—
268	C₂×C₄	—	5	2	no	—
269	C₂×C₄	—	15	2	no	—
270	C₂×C₄	—	20	2	no	—
271	C₂×C₄	—	60	2	no	—
272	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
273	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
274	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
275	C₂×C₃	—	20	2	no	—
276	C₂×C₃	C₂×#709	1	1	yes	characteristic
277	C₂×C₃	C₂×#713	1	1	yes	characteristic
278	C₂×C₃	#4149	1	1	yes	characteristic
279	D₃	—	20	2	no	—
280	C₅	#5559	1	1	yes	characteristic, complemented, Sylow
281	C₂²	—	10	2	no	—
282	C₂²	—	10	2	no	—
283	C₂²	—	10	2	no	—
284	C₂²	—	30	2	no	—
285	C₂²	—	30	2	no	—
286	C₂²	—	30	2	no	—
287	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	3	2	no	—
291	C₄	—	6	1	no	—
292	C₄	—	6	1	no	—
293	C₄	—	6	1	no	—
294	C₄	—	10	2	no	—
295	C₄	—	30	2	no	—
296	C₄	#6773	1	2	yes	—
297	C₃	#65919	1	1	yes	characteristic, complemented, Sylow
298	C₂	—	20	2	no	—
299	C₂	—	60	2	no	—
300	C₂	C₂×#6773	1	1	yes	characteristic
301	C₂	D₃×#709	1	1	yes	characteristic
302	C₂	#92017	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₄²×C₅	C₂²
prime radical	#91458	C₂
hypercenter	C₄²	D₃×D₅
hypocenter	C₃×C₅	#559
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. #91764	C₂	semidirect
2. C₄×C₅×Dic₆	C₂	semidirect
3. C₃×C₄×D₂₀	C₂	semidirect
5. #91455	C₂	semidirect
6. #91444	C₂	semidirect
7. #91428	C₂	semidirect
8. #91364	C₂	semidirect
9. #91356	C₂	semidirect
10. #91348	C₂	semidirect
11. #91340	C₂	semidirect
46. C₄×D₂₀	D₃	semidirect
77. C₄×Dic₆	D₅	semidirect
234. C₃×C₅	#559	semidirect
280. C₅	#5559	semidirect
297. C₃	#65919	semidirect

Maximal subgroups

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

subgroup	index
#91764	2
C₄×C₅×Dic₆	2
C₃×C₄×D₂₀	2
#91458	2
#91455	2
#91444	2
#91428	2
#91364	2
#91356	2
#91348	2
#91340	2
#65919	3
#5559	5