Subgroups — #11748899

Order 960.

468 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₅×Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#685	C₂	1	1	yes	maximal, characteristic
4	C₂×#6668	C₂	1	1	yes	maximal, characteristic, complemented
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	#65278	—	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	C₂×D₅×Dic₃	—	2	2	no	—
17	C₂×D₅×Dic₃	—	2	2	no	—
18	C₂×D₅×Dic₃	—	2	2	no	—
19	C₅×C₄⋊Dic₃	—	2	2	no	—
20	C₃×#685	C₄	1	4	yes	complemented
21	#6668	—	2	2	no	—
22	#6665	—	2	2	no	—
23	#4950	—	5	1	no	maximal, Hall
24	C₂²×C₄×D₅	—	3	1	no	—
25	C₂×C₅×C₄⋊C₄	—	3	1	no	—
26	C₂×#685	—	3	1	no	—
27	C₂×#685	D₃	1	1	yes	characteristic
28	C₂×#683	—	3	1	no	—
29	#3963	—	3	1	no	—
30	#3963	—	3	1	no	—
31	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
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	C₂²×C₃×D₅	D₄	1	2	yes	—
38	C₂²×C₃×D₅	Q₈	1	2	yes	—
39	C₂×C₃×C₄×C₅	—	2	1	no	—
40	C₂×C₃×C₄×C₅	—	2	1	no	—
41	C₂×C₃×C₄×C₅	C₂×C₄	1	2	yes	complemented
42	C₂×Dic₁₅	—	2	1	no	—
43	C₂×Dic₁₅	—	2	1	no	—
44	C₂×Dic₁₅	—	2	1	no	—
45	C₂×Dic₁₅	—	2	1	no	—
46	C₂×Dic₁₅	—	2	1	no	—
47	C₂×Dic₁₅	D₄	1	2	yes	—
48	C₂×C₅×Dic₃	—	2	1	no	—
49	C₂×C₅×Dic₃	—	2	1	no	—
50	C₂×C₅×Dic₃	—	2	1	no	—
51	C₂×C₅×Dic₃	—	2	1	no	—
52	C₂×C₅×Dic₃	—	2	1	no	—
53	C₂×C₅×Dic₃	D₄	1	2	yes	—
54	C₂×C₃×Dic₅	—	2	1	no	—
55	C₂×C₃×Dic₅	—	2	1	no	—
56	C₂×C₃×Dic₅	C₂×C₄	1	2	yes	complemented
57	D₅×Dic₃	—	2	8	no	—
58	C₂³×Dic₃	—	5	1	no	—
59	C₂×C₃×C₂²⋊C₄	—	5	1	no	—
60	C₂×#217	—	5	1	no	—
61	C₂×C₄⋊Dic₃	—	5	1	no	—
62	C₂×C₄⋊Dic₃	D₅	1	1	yes	characteristic, complemented
63	C₃⋊#95	—	5	1	no	—
64	C₃⋊#95	—	5	1	no	—
65	C₂³×D₅	C₂×D₃	1	1	yes	characteristic
66	C₂²×C₄×C₅	—	3	1	no	—
67	C₂²×C₄×C₅	—	3	1	no	—
68	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
69	C₂²×Dic₅	—	3	1	no	—
70	C₂²×Dic₅	—	3	1	no	—
71	C₂²×Dic₅	C₂×D₃	1	1	yes	characteristic
72	C₂×C₄×D₅	—	6	2	no	—
73	C₂×C₄×D₅	—	6	2	no	—
74	C₂×C₄×D₅	—	6	2	no	—
75	C₅×C₄⋊C₄	—	6	2	no	—
76	#685	—	6	2	no	—
77	#685	Dic₃	1	4	yes	complemented
78	#683	—	6	2	no	—
79	#385	—	15	1	no	Sylow
80	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₃×C₅	#41	1	1	yes	characteristic
86	C₂²×C₃×C₅	#41	1	1	yes	characteristic
87	C₂×C₃×D₅	—	2	2	no	—
88	C₂×C₃×D₅	—	2	2	no	—
89	C₂×C₃×D₅	—	2	2	no	—
90	C₂×C₃×D₅	—	2	2	no	—
91	C₂×C₃×D₅	—	2	2	no	—
92	C₂×C₃×D₅	—	2	2	no	—
93	C₂×C₃×D₅	C₄⋊C₄	1	4	yes	complemented
94	C₃×C₄×C₅	—	2	2	no	—
95	Dic₁₅	—	2	2	no	—
96	Dic₁₅	—	4	1	no	—
97	C₃×Dic₅	—	2	2	no	—
98	C₅×Dic₃	—	2	2	no	—
99	C₅×Dic₃	—	4	1	no	—
100	C₂⁴×C₃	—	5	1	no	—
101	C₂²×C₃×C₄	—	5	1	no	—
102	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
103	C₂²×Dic₃	—	5	1	no	—
104	C₂²×Dic₃	—	5	1	no	—
105	C₂²×Dic₃	—	10	2	no	—
106	C₂²×Dic₃	—	10	2	no	—
107	C₂²×Dic₃	—	10	2	no	—
108	C₂²×Dic₃	C₂×D₅	1	1	yes	characteristic
109	C₂²×Dic₃	C₂×D₅	1	1	yes	characteristic
110	C₃×C₂²⋊C₄	—	5	4	no	—
111	#217	—	10	2	no	—
112	C₄⋊Dic₃	—	2	2	no	—
113	C₄⋊Dic₃	—	10	2	no	—
114	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
115	C₂²×D₅	—	2	1	no	—
116	C₂²×D₅	—	2	1	no	—
117	C₂²×D₅	—	2	1	no	—
118	C₂²×D₅	—	2	1	no	—
119	C₂²×D₅	Dic₆	1	2	yes	—
120	C₂²×D₅	D₁₂	1	2	yes	—
121	C₂²×D₅	C₂×Dic₃	1	2	yes	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	3	2	no	—
125	C₂×C₄×C₅	—	6	1	no	—
126	C₂×C₄×C₅	—	6	1	no	—
127	C₂×C₄×C₅	—	6	1	no	—
128	C₂×C₄×C₅	—	6	1	no	—
129	C₂×C₄×C₅	—	6	1	no	—
130	C₂×C₄×C₅	C₂×Dic₃	1	2	yes	complemented
131	C₂×Dic₅	—	2	1	no	—
132	C₂×Dic₅	—	2	1	no	—
133	C₂×Dic₅	—	3	2	no	—
134	C₂×Dic₅	—	6	1	no	—
135	C₂×Dic₅	—	6	1	no	—
136	C₂×Dic₅	—	6	1	no	—
137	C₂×Dic₅	—	6	1	no	—
138	C₂×Dic₅	—	6	1	no	—
139	C₂×Dic₅	C₂×Dic₃	1	2	yes	complemented
140	C₄×D₅	—	6	8	no	—
141	C₂³×C₄	—	15	1	no	—
142	C₂×C₄⋊C₄	—	3	1	no	—
143	C₂×C₄⋊C₄	—	15	1	no	—
144	C₂×C₂²⋊C₄	—	5	1	no	—
145	C₂×C₂²⋊C₄	—	15	1	no	—
146	#95	—	15	1	no	—
147	#95	—	15	1	no	—
148	C₂×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₃×C₅	C₄×D₄	1	1	yes	characteristic
150	C₂×C₃×C₅	C₄×D₄	1	1	yes	characteristic
151	C₂×C₃×C₅	#120	1	1	yes	characteristic
152	C₂×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
153	C₂×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
154	C₂×C₃×C₅	#123	1	1	yes	characteristic
155	C₃×D₅	—	2	4	no	—
156	C₂³×C₃	—	5	2	no	—
157	C₂³×C₃	—	5	2	no	—
158	C₂³×C₃	—	5	2	no	—
159	C₂³×C₃	—	10	1	no	—
160	C₂³×C₃	—	10	1	no	—
161	C₂³×C₃	—	10	1	no	—
162	C₂³×C₃	—	10	1	no	—
163	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	5	2	no	—
167	C₂×C₃×C₄	—	10	1	no	—
168	C₂×C₃×C₄	—	10	1	no	—
169	C₂×C₃×C₄	C₄×D₅	1	2	yes	complemented
170	C₂×Dic₃	—	2	1	no	—
171	C₂×Dic₃	—	2	1	no	—
172	C₂×Dic₃	—	2	1	no	—
173	C₂×Dic₃	—	2	1	no	—
174	C₂×Dic₃	—	2	1	no	—
175	C₂×Dic₃	—	5	2	no	—
176	C₂×Dic₃	—	10	8	no	—
177	C₂×Dic₃	—	10	1	no	—
178	C₂×Dic₃	—	10	1	no	—
179	C₂×Dic₃	—	10	1	no	—
180	C₂×Dic₃	—	10	1	no	—
181	C₂×Dic₃	—	10	1	no	—
182	C₂×Dic₃	C₂²⋊D₅	1	2	yes	—
183	C₂²×C₅	C₂×Dic₆	1	1	yes	characteristic
184	C₂²×C₅	C₂×D₁₂	1	1	yes	characteristic
185	C₂²×C₅	D₃×D₄	1	1	yes	characteristic
186	C₂²×C₅	D₃×D₄	1	1	yes	characteristic
187	C₂²×C₅	#237	1	1	yes	characteristic
188	C₂²×C₅	#237	1	1	yes	characteristic
189	C₂²×C₅	C₂²×Dic₃	1	1	yes	characteristic
190	C₂×D₅	—	2	2	no	—
191	C₂×D₅	—	2	2	no	—
192	C₂×D₅	—	2	2	no	—
193	C₂×D₅	—	2	2	no	—
194	C₂×D₅	—	2	2	no	—
195	C₂×D₅	—	2	2	no	—
196	C₂×D₅	C₄⋊Dic₃	1	4	yes	complemented
197	C₄×C₅	—	2	2	no	—
198	C₄×C₅	—	6	2	no	—
199	C₄×C₅	—	12	1	no	—
200	Dic₅	—	2	2	no	—
201	Dic₅	—	6	2	no	—
202	Dic₅	—	12	1	no	—
203	C₂⁴	—	5	1	no	—
204	C₂²×C₄	—	3	1	no	—
205	C₂²×C₄	—	3	1	no	—
206	C₂²×C₄	—	5	1	no	—
207	C₂²×C₄	—	15	1	no	—
208	C₂²×C₄	—	15	1	no	—
209	C₂²×C₄	—	30	2	no	—
210	C₂²×C₄	—	30	2	no	—
211	C₂²×C₄	—	30	2	no	—
212	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
213	C₄⋊C₄	—	6	2	no	—
214	C₄⋊C₄	—	30	2	no	—
215	C₂²⋊C₄	—	5	4	no	—
216	C₂²⋊C₄	—	30	2	no	—
217	C₃×C₅	#385	1	1	yes	characteristic, complemented, Hall
218	C₂²×C₃	—	5	4	no	—
219	C₂²×C₃	—	10	2	no	—
220	C₂²×C₃	—	10	2	no	—
221	C₂²×C₃	—	10	2	no	—
222	C₂²×C₃	—	10	2	no	—
223	C₂²×C₃	—	10	2	no	—
224	C₂²×C₃	—	10	2	no	—
225	C₂²×C₃	C₂×C₄×D₅	1	1	yes	characteristic
226	C₂²×C₃	#709	1	1	yes	characteristic
227	C₂²×C₃	D₄×D₅	1	1	yes	characteristic
228	C₂²×C₃	D₄×D₅	1	1	yes	characteristic
229	C₂²×C₃	Q₈×D₅	1	1	yes	characteristic
230	C₂²×C₃	#713	1	1	yes	characteristic
231	C₂²×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	10	2	no	—
234	Dic₃	—	2	2	no	—
235	Dic₃	—	4	1	no	—
236	Dic₃	—	10	2	no	—
237	Dic₃	—	20	1	no	—
238	C₂×C₅	#878	1	1	yes	characteristic
239	C₂×C₅	#878	1	1	yes	characteristic
240	C₂×C₅	#882	1	1	yes	characteristic
241	C₂×C₅	#886	1	1	yes	characteristic
242	C₂×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
243	C₂×C₅	D₄×Dic₃	1	1	yes	characteristic
244	C₂×C₅	D₄×Dic₃	1	1	yes	characteristic
245	D₅	—	2	4	no	—
246	C₂³	—	5	2	no	—
247	C₂³	—	5	2	no	—
248	C₂³	—	5	2	no	—
249	C₂³	—	10	1	no	—
250	C₂³	—	10	1	no	—
251	C₂³	—	10	1	no	—
252	C₂³	—	10	1	no	—
253	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	3	2	no	—
257	C₂×C₄	—	5	2	no	—
258	C₂×C₄	—	6	1	no	—
259	C₂×C₄	—	6	1	no	—
260	C₂×C₄	—	6	1	no	—
261	C₂×C₄	—	6	1	no	—
262	C₂×C₄	—	6	1	no	—
263	C₂×C₄	—	10	1	no	—
264	C₂×C₄	—	10	1	no	—
265	C₂×C₄	—	15	2	no	—
266	C₂×C₄	—	30	8	no	—
267	C₂×C₄	—	30	1	no	—
268	C₂×C₄	—	30	1	no	—
269	C₂×C₄	—	30	1	no	—
270	C₂×C₄	—	30	1	no	—
271	C₂×C₄	—	30	1	no	—
272	C₂×C₄	D₅×Dic₃	1	2	yes	complemented
273	C₂×C₃	—	10	4	no	—
274	C₂×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
275	C₂×C₃	#4039	1	1	yes	characteristic
276	C₂×C₃	#4040	1	1	yes	characteristic
277	C₂×C₃	#4042	1	1	yes	characteristic
278	C₂×C₃	C₄×C₂²⋊D₅	1	1	yes	characteristic
279	C₂×C₃	#4083	1	1	yes	characteristic
280	C₂×C₃	#4092	1	1	yes	characteristic
281	C₅	#4950	1	1	yes	characteristic, complemented, Sylow
282	C₂²	—	5	4	no	—
283	C₂²	—	10	2	no	—
284	C₂²	—	10	2	no	—
285	C₂²	—	10	2	no	—
286	C₂²	—	10	2	no	—
287	C₂²	—	10	2	no	—
288	C₂²	—	10	2	no	—
289	C₂²	D₅×Dic₆	1	1	yes	characteristic
290	C₂²	#6766	1	1	yes	characteristic
291	C₂²	D₅×D₁₂	1	1	yes	characteristic
292	C₂²	#6777	1	1	yes	characteristic
293	C₂²	C₂×D₅×Dic₃	1	1	yes	characteristic
294	C₂²	#6779	1	1	yes	characteristic
295	C₂²	D₃×C₂²⋊D₅	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	6	2	no	—
298	C₄	—	10	2	no	—
299	C₄	—	12	1	no	—
300	C₄	—	30	2	no	—
301	C₄	—	60	1	no	—
302	C₃	#65278	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	10	4	no	—
304	C₂	D₅×C₄⋊Dic₃	1	1	yes	characteristic
305	C₂	#91413	1	1	yes	characteristic
306	C₂	#91421	1	1	yes	characteristic
307	C₂	#91430	1	1	yes	characteristic
308	C₂	#91434	1	1	yes	characteristic
309	C₂	#91475	1	1	yes	characteristic
310	C₂	Dic₃×C₂²⋊D₅	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	#385
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
2. C₂×C₅×C₄⋊Dic₃	C₂	semidirect
4. C₂×#6668	C₂	semidirect
6. #91115	C₂	semidirect
7. #90994	C₂	semidirect
20. C₃×#685	C₄	semidirect
41. C₂×C₃×C₄×C₅	C₂×C₄	semidirect
56. C₂×C₃×Dic₅	C₂×C₄	semidirect
62. C₂×C₄⋊Dic₃	D₅	semidirect
77. #685	Dic₃	semidirect
93. C₂×C₃×D₅	C₄⋊C₄	semidirect
130. C₂×C₄×C₅	C₂×Dic₃	semidirect
139. C₂×Dic₅	C₂×Dic₃	semidirect
169. C₂×C₃×C₄	C₄×D₅	semidirect
196. C₂×D₅	C₄⋊Dic₃	semidirect
217. C₃×C₅	#385	semidirect
272. C₂×C₄	D₅×Dic₃	semidirect
281. C₅	#4950	semidirect
302. C₃	#65278	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₅×Dic₃	2
C₂×C₅×C₄⋊Dic₃	2
C₂×C₃×#685	2
C₂×#6668	2
C₂×#6665	2
#91115	2
#90994	2
#65278	3
#4950	5