Subgroups — #11748705

Order 960.

380 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#685	C₂	1	1	yes	maximal, characteristic
3	C₂×#6669	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#6665	C₂	1	1	yes	maximal, characteristic
5	C₂×#6665	C₂	1	1	yes	maximal, characteristic
6	#91115	C₂	1	1	yes	maximal, characteristic, complemented
7	#90994	C₂	1	1	yes	maximal, characteristic, complemented
8	#65229	—	3	1	no	maximal, Hall
9	C₂³×C₃×D₅	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₅	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₅	C₂²	1	1	yes	characteristic
13	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
16	C₄×C₅×Dic₃	—	2	2	no	—
17	C₃×#685	—	2	2	no	—
18	#6669	—	2	2	no	—
19	#6665	—	2	2	no	—
20	#6665	C₄	1	4	yes	complemented
21	#4944	—	5	1	no	maximal, Hall
22	C₂×C₄²×C₅	—	3	1	no	—
23	C₂×#685	—	3	1	no	—
24	C₂×#685	—	3	1	no	—
25	C₂×#685	D₃	1	1	yes	characteristic
26	C₂×#683	—	3	1	no	—
27	#3963	—	3	1	no	—
28	#3963	—	3	1	no	—
29	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
30	C₂²×C₃×D₅	—	2	1	no	—
31	C₂²×C₃×D₅	—	2	1	no	—
32	C₂²×C₃×D₅	—	2	1	no	—
33	C₂²×C₃×D₅	—	2	1	no	—
34	C₂²×C₃×D₅	—	2	1	no	—
35	C₂²×C₃×D₅	—	2	1	no	—
36	C₂²×C₃×D₅	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₄×C₅	—	2	1	no	—
38	C₂×C₃×C₄×C₅	—	2	1	no	—
39	C₂×C₃×C₄×C₅	D₄	1	2	yes	—
40	C₂×Dic₁₅	—	2	1	no	—
41	C₂×Dic₁₅	—	2	1	no	—
42	C₂×Dic₁₅	—	2	1	no	—
43	C₂×Dic₁₅	—	2	1	no	—
44	C₂×Dic₁₅	—	2	1	no	—
45	C₂×Dic₁₅	C₂×C₄	1	2	yes	—
46	C₂×C₅×Dic₃	—	2	1	no	—
47	C₂×C₅×Dic₃	—	2	1	no	—
48	C₂×C₅×Dic₃	—	2	1	no	—
49	C₂×C₅×Dic₃	—	2	1	no	—
50	C₂×C₅×Dic₃	C₂×C₄	1	2	yes	—
51	C₂×C₅×Dic₃	D₄	1	2	yes	—
52	C₂×C₃×Dic₅	—	2	1	no	—
53	C₂×C₃×Dic₅	—	2	1	no	—
54	C₂×C₃×Dic₅	—	2	1	no	—
55	C₂×C₃×C₂²⋊C₄	—	5	1	no	—
56	C₂×#217	—	5	1	no	—
57	C₂×#217	—	5	1	no	—
58	C₂×Dic₃⋊C₄	—	5	1	no	—
59	C₂×C₄×Dic₃	D₅	1	1	yes	characteristic, complemented
60	C₃⋊#95	—	5	1	no	—
61	C₃⋊#95	—	5	1	no	—
62	C₂³×D₅	C₂×D₃	1	1	yes	characteristic
63	C₂²×C₄×C₅	—	3	1	no	—
64	C₂²×C₄×C₅	—	3	1	no	—
65	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
66	C₂²×Dic₅	—	3	1	no	—
67	C₂²×Dic₅	—	3	1	no	—
68	C₂²×Dic₅	C₂×D₃	1	1	yes	characteristic
69	C₄²×C₅	—	6	2	no	—
70	#685	—	2	2	no	—
71	#685	—	3	4	no	—
72	#685	—	6	2	no	—
73	#683	—	6	2	no	—
74	#388	—	15	1	no	Sylow
75	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₃×C₅	#41	1	1	yes	characteristic
79	C₂²×C₃×C₅	#41	1	1	yes	characteristic
80	C₂²×C₃×C₅	#41	1	1	yes	characteristic
81	C₂²×C₃×C₅	#41	1	1	yes	characteristic
82	C₂×C₃×D₅	—	2	2	no	—
83	C₂×C₃×D₅	—	2	2	no	—
84	C₂×C₃×D₅	—	2	2	no	—
85	C₂×C₃×D₅	—	4	1	no	—
86	C₂×C₃×D₅	—	4	1	no	—
87	C₂×C₃×D₅	—	4	1	no	—
88	C₂×C₃×D₅	—	4	1	no	—
89	C₃×C₄×C₅	—	2	2	no	—
90	Dic₁₅	—	2	2	no	—
91	Dic₁₅	—	4	1	no	—
92	C₃×Dic₅	—	4	1	no	—
93	C₅×Dic₃	—	2	2	no	—
94	C₅×Dic₃	—	2	2	no	—
95	C₂⁴×C₃	—	5	1	no	—
96	C₂²×C₃×C₄	—	5	1	no	—
97	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
98	C₂²×Dic₃	—	5	1	no	—
99	C₂²×Dic₃	—	5	1	no	—
100	C₂²×Dic₃	C₂×D₅	1	1	yes	characteristic
101	C₂²×Dic₃	C₂×D₅	1	1	yes	characteristic
102	C₃×C₂²⋊C₄	—	10	2	no	—
103	#217	—	5	4	no	—
104	#217	—	10	2	no	—
105	Dic₃⋊C₄	—	10	2	no	—
106	C₄×Dic₃	—	2	2	no	—
107	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
108	C₂²×D₅	—	2	1	no	—
109	C₂²×D₅	—	2	1	no	—
110	C₂²×D₅	—	2	1	no	—
111	C₂²×D₅	—	2	1	no	—
112	C₂²×D₅	—	2	1	no	—
113	C₂²×D₅	—	2	1	no	—
114	C₂²×D₅	C₄×D₃	1	2	yes	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	3	2	no	—
118	C₂×C₄×C₅	—	3	2	no	—
119	C₂×C₄×C₅	—	6	1	no	—
120	C₂×C₄×C₅	—	6	1	no	—
121	C₂×C₄×C₅	—	6	1	no	—
122	C₂×C₄×C₅	—	6	1	no	—
123	C₂×C₄×C₅	#67	1	2	yes	—
124	C₂×Dic₅	—	2	1	no	—
125	C₂×Dic₅	—	2	1	no	—
126	C₂×Dic₅	—	2	1	no	—
127	C₂×Dic₅	—	3	2	no	—
128	C₂×Dic₅	—	6	1	no	—
129	C₂×Dic₅	—	6	1	no	—
130	C₂×Dic₅	—	6	1	no	—
131	C₂×Dic₅	—	6	1	no	—
132	C₂×Dic₅	—	6	1	no	—
133	C₂×C₄⋊C₄	—	15	1	no	—
134	C₂×C₂²⋊C₄	—	5	1	no	—
135	C₂×C₂²⋊C₄	—	15	1	no	—
136	C₂×C₂²⋊C₄	—	15	1	no	—
137	C₂×C₄²	—	3	1	no	—
138	#95	—	15	1	no	—
139	#95	—	15	1	no	—
140	C₂×C₃×C₅	#117	1	1	yes	characteristic
141	C₂×C₃×C₅	C₄×D₄	1	1	yes	characteristic
142	C₂×C₃×C₅	C₄×D₄	1	1	yes	characteristic
143	C₂×C₃×C₅	#121	1	1	yes	characteristic
144	C₂×C₃×C₅	#123	1	1	yes	characteristic
145	C₂×C₃×C₅	#124	1	1	yes	characteristic
146	C₂×C₃×C₅	#126	1	1	yes	characteristic
147	C₃×D₅	—	4	2	no	—
148	C₂³×C₃	—	5	2	no	—
149	C₂³×C₃	—	10	1	no	—
150	C₂³×C₃	—	10	1	no	—
151	C₂³×C₃	—	10	1	no	—
152	C₂³×C₃	—	10	1	no	—
153	C₂³×C₃	—	10	1	no	—
154	C₂³×C₃	—	10	1	no	—
155	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	10	1	no	—
159	C₂×C₃×C₄	—	10	1	no	—
160	C₂×C₃×C₄	—	10	1	no	—
161	C₂×C₃×C₄	C₂²⋊D₅	1	2	yes	—
162	C₂×Dic₃	—	2	1	no	—
163	C₂×Dic₃	—	2	1	no	—
164	C₂×Dic₃	—	2	1	no	—
165	C₂×Dic₃	—	2	1	no	—
166	C₂×Dic₃	—	5	2	no	—
167	C₂×Dic₃	—	10	1	no	—
168	C₂×Dic₃	—	10	1	no	—
169	C₂×Dic₃	—	10	1	no	—
170	C₂×Dic₃	—	10	1	no	—
171	C₂×Dic₃	—	10	1	no	—
172	C₂×Dic₃	C₄×D₅	1	2	yes	—
173	C₂×Dic₃	D₂₀	1	2	yes	—
174	C₂²×C₅	C₂×C₄×D₃	1	1	yes	characteristic
175	C₂²×C₅	#235	1	1	yes	characteristic
176	C₂²×C₅	D₃×D₄	1	1	yes	characteristic
177	C₂²×C₅	#237	1	1	yes	characteristic
178	C₂²×C₅	#237	1	1	yes	characteristic
179	C₂²×C₅	#237	1	1	yes	characteristic
180	C₂²×C₅	C₂×#67	1	1	yes	characteristic
181	C₂×D₅	—	2	2	no	—
182	C₂×D₅	—	2	2	no	—
183	C₂×D₅	—	2	2	no	—
184	C₂×D₅	—	4	1	no	—
185	C₂×D₅	—	4	1	no	—
186	C₂×D₅	—	4	1	no	—
187	C₂×D₅	—	4	1	no	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	6	2	no	—
190	C₄×C₅	—	6	2	no	—
191	Dic₅	—	4	1	no	—
192	Dic₅	—	6	2	no	—
193	Dic₅	—	12	1	no	—
194	C₂⁴	—	5	1	no	—
195	C₂²×C₄	—	3	1	no	—
196	C₂²×C₄	—	3	1	no	—
197	C₂²×C₄	—	5	1	no	—
198	C₂²×C₄	—	15	1	no	—
199	C₂²×C₄	—	15	1	no	—
200	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
201	C₄⋊C₄	—	30	2	no	—
202	C₂²⋊C₄	—	10	2	no	—
203	C₂²⋊C₄	—	15	4	no	—
204	C₂²⋊C₄	—	30	2	no	—
205	C₄²	—	6	2	no	—
206	C₃×C₅	#388	1	1	yes	characteristic, complemented, Hall
207	C₂²×C₃	—	10	2	no	—
208	C₂²×C₃	—	10	2	no	—
209	C₂²×C₃	—	10	2	no	—
210	C₂²×C₃	—	20	1	no	—
211	C₂²×C₃	—	20	1	no	—
212	C₂²×C₃	—	20	1	no	—
213	C₂²×C₃	—	20	1	no	—
214	C₂²×C₃	C₂×C₄×D₅	1	1	yes	characteristic
215	C₂²×C₃	C₂×D₂₀	1	1	yes	characteristic
216	C₂²×C₃	#709	1	1	yes	characteristic
217	C₂²×C₃	#709	1	1	yes	characteristic
218	C₂²×C₃	#709	1	1	yes	characteristic
219	C₂²×C₃	#709	1	1	yes	characteristic
220	C₂²×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
221	C₃×C₄	—	2	2	no	—
222	C₃×C₄	—	20	1	no	—
223	Dic₃	—	2	2	no	—
224	Dic₃	—	2	2	no	—
225	Dic₃	—	10	2	no	—
226	Dic₃	—	20	1	no	—
227	C₂×C₅	#877	1	1	yes	characteristic
228	C₂×C₅	#879	1	1	yes	characteristic
229	C₂×C₅	#67⋊C₄	1	1	yes	characteristic
230	C₂×C₅	#885	1	1	yes	characteristic
231	C₂×C₅	C₄×#67	1	1	yes	characteristic
232	C₂×C₅	#935	1	1	yes	characteristic
233	C₂×C₅	#938	1	1	yes	characteristic
234	D₅	—	4	2	no	—
235	C₂³	—	5	2	no	—
236	C₂³	—	10	1	no	—
237	C₂³	—	10	1	no	—
238	C₂³	—	10	1	no	—
239	C₂³	—	10	1	no	—
240	C₂³	—	10	1	no	—
241	C₂³	—	10	1	no	—
242	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	3	2	no	—
246	C₂×C₄	—	3	2	no	—
247	C₂×C₄	—	6	1	no	—
248	C₂×C₄	—	6	1	no	—
249	C₂×C₄	—	6	1	no	—
250	C₂×C₄	—	6	1	no	—
251	C₂×C₄	—	10	1	no	—
252	C₂×C₄	—	10	1	no	—
253	C₂×C₄	—	10	1	no	—
254	C₂×C₄	—	15	2	no	—
255	C₂×C₄	—	30	1	no	—
256	C₂×C₄	—	30	1	no	—
257	C₂×C₄	—	30	1	no	—
258	C₂×C₄	—	30	1	no	—
259	C₂×C₄	—	30	1	no	—
260	C₂×C₄	#1201	1	2	yes	—
261	C₂×C₃	—	20	2	no	—
262	C₂×C₃	#4018	1	1	yes	characteristic
263	C₂×C₃	C₄×D₂₀	1	1	yes	characteristic
264	C₂×C₃	#4021	1	1	yes	characteristic
265	C₂×C₃	#4022	1	1	yes	characteristic
266	C₂×C₃	C₄×C₂²⋊D₅	1	1	yes	characteristic
267	C₂×C₃	#4075	1	1	yes	characteristic
268	C₂×C₃	#4076	1	1	yes	characteristic
269	C₅	#4944	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	10	2	no	—
271	C₂²	—	10	2	no	—
272	C₂²	—	10	2	no	—
273	C₂²	—	20	1	no	—
274	C₂²	—	20	1	no	—
275	C₂²	—	20	1	no	—
276	C₂²	—	20	1	no	—
277	C₂²	#6765	1	1	yes	characteristic
278	C₂²	#6771	1	1	yes	characteristic
279	C₂²	C₄×D₃×D₅	1	1	yes	characteristic
280	C₂²	D₃×D₂₀	1	1	yes	characteristic
281	C₂²	#6779	1	1	yes	characteristic
282	C₂²	#6779	1	1	yes	characteristic
283	C₂²	C₂×#1201	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	6	2	no	—
286	C₄	—	6	2	no	—
287	C₄	—	20	1	no	—
288	C₄	—	30	2	no	—
289	C₄	—	60	1	no	—
290	C₃	#65229	1	1	yes	characteristic, complemented, Sylow
291	C₂	—	20	2	no	—
292	C₂	#91353	1	1	yes	characteristic
293	C₂	#91363	1	1	yes	characteristic
294	C₂	#91393	1	1	yes	characteristic
295	C₂	#91395	1	1	yes	characteristic
296	C₂	C₄×#1201	1	1	yes	characteristic
297	C₂	#91456	1	1	yes	characteristic
298	C₂	#91536	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂×D₃×D₅
commutator	C₂²×C₃×C₅	C₂²×C₄
Fitting	C₂²×C₃×C₄×C₅	C₂²
socle, cosocle	C₂³×C₃×C₅	C₂³
prime radical	C₂³×C₃×D₅	C₂²
hypercenter	C₂²×C₄	D₃×D₅
hypocenter	C₃×C₅	#388
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×C₅×Dic₃	C₂	semidirect
3. C₂×#6669	C₂	semidirect
6. #91115	C₂	semidirect
7. #90994	C₂	semidirect
20. #6665	C₄	semidirect
59. C₂×C₄×Dic₃	D₅	semidirect
206. C₃×C₅	#388	semidirect
269. C₅	#4944	semidirect
290. C₃	#65229	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×Dic₃	2
C₂×C₃×#685	2
C₂×#6669	2
C₂×#6665	2
#91115	2
#90994	2
#65229	3
#4944	5