Subgroups — #11753402

Order 960.

620 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	C₂×#6772	C₂	1	1	yes	maximal, characteristic
2	#91776	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#893	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4038	C₂	1	1	yes	maximal, characteristic, complemented
5	#91442	C₂	1	2	yes	maximal, complemented
6	#91441	C₂	1	2	yes	maximal, complemented
7	#91435	C₂	1	1	yes	maximal, characteristic, complemented
8	#91398	C₂	1	2	yes	maximal, complemented
9	C₄×D₃×Dic₅	C₂	1	2	yes	maximal, complemented
10	C₄×D₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#91340	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#711	—	3	1	no	maximal, Hall
13	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
14	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
16	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
17	C₂×#1201	C₂²	1	2	yes	—
18	C₂×D₃×Dic₅	C₂²	1	2	yes	—
19	C₂×D₃×Dic₅	C₂²	1	2	yes	—
20	C₂×D₅×Dic₃	C₂²	1	1	yes	characteristic
21	C₂×D₅×Dic₃	C₂²	1	1	yes	characteristic
22	#6772	C₄	1	8	yes	complemented
23	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	#6712	C₂²	1	2	yes	—
25	C₄×Dic₁₅	C₂²	1	2	yes	—
26	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
27	C₅×(C₂×D₃)⋊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	C₃×C₄×Dic₅	C₂²	1	2	yes	—
32	#6669	C₂²	1	2	yes	—
33	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
34	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
35	Dic₃×Dic₅	C₂²	1	2	yes	—
36	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
37	#5522	—	5	1	no	maximal, Hall
38	C₂×#711	—	3	1	no	—
39	C₄×C₅×D₄	—	3	1	no	—
40	D₄×Dic₅	—	3	1	no	—
41	C₄×C₂²⋊D₅	—	3	2	no	—
42	C₂×C₄×Dic₅	—	3	2	no	—
43	#4038	D₃	1	1	yes	characteristic, complemented
44	#4033	—	3	1	no	—
45	#4027	—	3	2	no	—
46	#4023	—	3	2	no	—
47	C₄²×D₅	—	3	1	no	—
48	C₄×Dic₁₀	—	3	1	no	—
49	C₂²×C₅×D₃	C₂³	1	2	yes	—
50	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
54	C₂×Dic₁₅	C₂³	1	2	yes	—
55	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
56	Dic₃₀	C₂×C₄	1	4	yes	complemented
57	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₅×D₁₂	C₂×C₄	1	4	yes	complemented
60	C₄×C₅×D₃	—	2	4	no	—
61	C₂×C₃×Dic₅	C₂³	1	2	yes	—
62	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
63	C₃×C₄×D₅	C₂×C₄	1	4	yes	complemented
64	#1201	C₂×C₄	1	8	yes	complemented
65	D₃×Dic₅	—	2	4	no	—
66	D₃×Dic₅	C₂×C₄	1	8	yes	complemented
67	D₅×Dic₃	—	2	2	no	—
68	D₅×Dic₃	—	2	2	no	—
69	C₂×#235	—	5	1	no	—
70	C₃×#117	—	5	1	no	—
71	C₂×C₄×Dic₃	—	5	1	no	—
72	#893	—	5	1	no	—
73	#893	D₅	1	1	yes	characteristic, complemented
74	#887	—	5	1	no	—
75	#887	—	5	1	no	—
76	#67⋊C₄	—	5	2	no	—
77	#67⋊C₄	—	5	2	no	—
78	#872	—	5	2	no	—
79	C₄²×D₃	—	5	2	no	—
80	C₂×C₅×D₄	—	3	1	no	—
81	C₂²×C₄×C₅	—	3	2	no	—
82	C₂×C₂²⋊D₅	—	3	2	no	—
83	C₂²×Dic₅	—	3	2	no	—
84	C₂²×Dic₅	—	3	2	no	—
85	#711	—	3	8	no	—
86	C₂×C₄×D₅	—	3	1	no	—
87	C₂×C₄×D₅	—	3	1	no	—
88	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
89	C₂×Dic₁₀	—	3	1	no	—
90	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
91	C₅×C₂²⋊C₄	—	3	2	no	—
92	C₄²×C₅	—	3	1	no	—
93	C₂²⋊Dic₅	—	3	2	no	—
94	#685	C₂×D₃	1	2	yes	—
95	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
96	#683	—	3	2	no	—
97	#683	—	3	2	no	—
98	C₄×Dic₅	—	3	1	no	—
99	C₄×Dic₅	—	3	2	no	—
100	C₄×Dic₅	—	3	2	no	—
101	C₄×Dic₅	—	3	1	no	—
102	C₄×Dic₅	—	3	1	no	—
103	C₄×Dic₅	C₂×D₃	1	2	yes	—
104	C₄×#41	—	15	1	no	Sylow
105	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
106	C₂×C₅×D₃	—	2	2	no	—
107	C₂×C₅×D₃	—	2	2	no	—
108	C₂×C₅×D₃	C₂²×C₄	1	4	yes	—
109	C₂×C₃×D₅	—	2	1	no	—
110	C₂×C₃×D₅	—	2	1	no	—
111	C₂×C₃×D₅	C₂²×C₄	1	2	yes	—
112	C₃×C₄×C₅	—	2	2	no	—
113	C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
114	Dic₁₅	—	2	1	no	—
115	Dic₁₅	C₂²×C₄	1	4	yes	—
116	Dic₁₅	#41	1	2	yes	complemented
117	C₃×Dic₅	—	2	2	no	—
118	C₃×Dic₅	C₂²×C₄	1	2	yes	—
119	C₅×Dic₃	—	2	1	no	—
120	C₅×Dic₃	#41	1	2	yes	complemented
121	C₂²×C₃×C₄	—	5	1	no	—
122	C₂×#67	—	5	2	no	—
123	C₂²×Dic₃	—	5	1	no	—
124	C₂²×Dic₃	—	5	1	no	—
125	#235	—	5	8	no	—
126	C₂×D₁₂	C₂×D₅	1	1	yes	characteristic
127	C₂×C₄×D₃	—	5	2	no	—
128	C₂×C₄×D₃	—	5	2	no	—
129	C₂×C₄×D₃	C₂×D₅	1	2	yes	—
130	C₂×Dic₆	—	5	1	no	—
131	C₃×C₄⋊C₄	—	5	1	no	—
132	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
133	C₃×C₂²⋊C₄	—	5	2	no	—
134	C₃×C₄²	—	5	2	no	—
135	(C₂×D₃)⋊C₄	—	5	2	no	—
136	(C₂×D₃)⋊C₄	C₂×D₅	1	2	yes	—
137	Dic₃⋊C₄	—	5	2	no	—
138	Dic₃⋊C₄	—	5	2	no	—
139	C₄×Dic₃	—	5	2	no	—
140	C₄×Dic₃	—	5	1	no	—
141	C₄×Dic₃	—	5	1	no	—
142	C₄×Dic₃	—	5	2	no	—
143	C₄×Dic₃	—	5	1	no	—
144	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
145	C₂³×C₅	—	3	2	no	—
146	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
147	C₅×D₄	—	3	4	no	—
148	C₂×C₄×C₅	—	3	1	no	—
149	C₂×C₄×C₅	—	3	1	no	—
150	C₂×C₄×C₅	—	6	4	no	—
151	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
152	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
153	C₂²⋊D₅	—	3	8	no	—
154	C₂×Dic₅	—	3	1	no	—
155	C₂×Dic₅	—	3	2	no	—
156	C₂×Dic₅	—	3	8	no	—
157	C₂×Dic₅	—	3	1	no	—
158	C₂×Dic₅	—	6	4	no	—
159	C₂×Dic₅	C₂²×D₃	1	2	yes	—
160	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
161	C₄×D₅	—	6	2	no	—
162	C₄×D₅	—	6	2	no	—
163	C₄×D₅	C₄×D₃	1	4	yes	complemented
164	Dic₁₀	—	3	4	no	—
165	C₂×#41	—	15	1	no	—
166	C₄×Q₈	—	15	1	no	—
167	C₄×Q₈	—	15	1	no	—
168	C₄×D₄	—	3	1	no	—
169	C₄×D₄	—	15	2	no	—
170	C₄×D₄	—	15	2	no	—
171	C₄×D₄	—	15	1	no	—
172	#117	—	5	1	no	—
173	#117	—	15	2	no	—
174	C₂×C₄²	—	15	1	no	—
175	C₂×C₄²	—	15	2	no	—
176	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
177	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
178	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
179	C₃×D₅	—	2	2	no	—
180	C₅×D₃	—	2	4	no	—
181	C₂³×C₃	—	5	1	no	—
182	C₂²×D₃	C₂²×D₅	1	2	yes	—
183	C₂×C₃×C₄	—	5	1	no	—
184	C₂×C₃×C₄	—	5	4	no	—
185	C₂×C₃×C₄	—	5	2	no	—
186	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
187	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
188	#67	—	5	8	no	—
189	C₂×Dic₃	—	5	1	no	—
190	C₂×Dic₃	—	5	2	no	—
191	C₂×Dic₃	—	5	1	no	—
192	C₂×Dic₃	—	10	2	no	—
193	C₂×Dic₃	—	10	2	no	—
194	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
195	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
196	D₁₂	C₄×D₅	1	4	yes	complemented
197	C₄×D₃	—	2	4	no	—
198	C₄×D₃	—	5	8	no	—
199	C₄×D₃	—	10	4	no	—
200	Dic₆	—	5	4	no	—
201	C₂²×C₅	—	3	4	no	—
202	C₂²×C₅	—	6	2	no	—
203	C₂²×C₅	—	6	2	no	—
204	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
205	C₂×D₅	—	2	1	no	—
206	C₂×D₅	—	2	1	no	—
207	C₂×D₅	C₂×C₄×D₃	1	2	yes	—
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	3	2	no	—
210	C₄×C₅	—	6	1	no	—
211	C₄×C₅	C₂×C₄×D₃	1	2	yes	—
212	Dic₅	—	2	2	no	—
213	Dic₅	—	3	4	no	—
214	Dic₅	—	3	2	no	—
215	Dic₅	—	6	1	no	—
216	Dic₅	C₂×C₄×D₃	1	2	yes	—
217	#41	—	15	8	no	—
218	C₂×Q₈	—	15	1	no	—
219	C₂×D₄	—	3	1	no	—
220	C₂×D₄	—	15	2	no	—
221	C₂²×C₄	—	3	2	no	—
222	C₂²×C₄	—	5	1	no	—
223	C₂²×C₄	—	15	1	no	—
224	C₂²×C₄	—	15	1	no	—
225	C₂²×C₄	—	15	2	no	—
226	C₂²×C₄	—	15	2	no	—
227	C₄⋊C₄	—	5	1	no	—
228	C₄⋊C₄	—	15	2	no	—
229	C₄⋊C₄	—	15	2	no	—
230	C₄⋊C₄	D₃×D₅	1	1	yes	characteristic
231	C₂²⋊C₄	—	3	2	no	—
232	C₂²⋊C₄	—	5	2	no	—
233	C₂²⋊C₄	—	15	2	no	—
234	C₄²	—	3	1	no	—
235	C₄²	—	5	2	no	—
236	C₄²	—	15	1	no	—
237	C₄²	—	15	1	no	—
238	C₄²	—	15	2	no	—
239	C₄²	—	15	2	no	—
240	C₄²	—	15	1	no	—
241	C₃×C₅	C₄×#41	1	1	yes	characteristic, complemented, Hall
242	C₂²×C₃	—	5	2	no	—
243	C₂²×C₃	—	10	1	no	—
244	C₂²×C₃	—	10	1	no	—
245	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
246	C₂×D₃	—	2	2	no	—
247	C₂×D₃	—	2	2	no	—
248	C₂×D₃	C₂×C₄×D₅	1	4	yes	—
249	C₃×C₄	—	2	2	no	—
250	C₃×C₄	—	5	2	no	—
251	C₃×C₄	—	10	2	no	—
252	C₃×C₄	C₂×C₄×D₅	1	2	yes	—
253	Dic₃	—	2	1	no	—
254	Dic₃	—	5	4	no	—
255	Dic₃	—	5	2	no	—
256	Dic₃	—	10	1	no	—
257	Dic₃	#711	1	2	yes	complemented
258	C₂×C₅	—	6	4	no	—
259	C₂×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
260	C₂×C₅	D₃×#41	1	1	yes	characteristic
261	C₂×C₅	D₃×#41	1	1	yes	characteristic
262	D₅	—	2	2	no	—
263	C₂³	—	3	2	no	—
264	C₂³	—	5	1	no	—
265	Q₈	—	15	4	no	—
266	D₄	—	3	4	no	—
267	D₄	—	15	8	no	—
268	C₂×C₄	—	3	1	no	—
269	C₂×C₄	—	3	1	no	—
270	C₂×C₄	—	5	1	no	—
271	C₂×C₄	—	5	4	no	—
272	C₂×C₄	—	5	2	no	—
273	C₂×C₄	—	6	4	no	—
274	C₂×C₄	—	15	8	no	—
275	C₂×C₄	—	15	1	no	—
276	C₂×C₄	—	15	2	no	—
277	C₂×C₄	—	15	1	no	—
278	C₂×C₄	—	30	2	no	—
279	C₂×C₄	—	30	2	no	—
280	C₂×C₄	—	30	4	no	—
281	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
282	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
283	C₂×C₃	—	10	2	no	—
284	C₂×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
285	C₂×C₃	C₂×#711	1	1	yes	characteristic
286	C₂×C₃	D₅×#41	1	1	yes	characteristic
287	D₃	—	2	4	no	—
288	C₅	#5522	1	1	yes	characteristic, complemented, Sylow
289	C₂²	—	3	4	no	—
290	C₂²	—	5	2	no	—
291	C₂²	—	6	2	no	—
292	C₂²	—	6	2	no	—
293	C₂²	—	10	1	no	—
294	C₂²	—	10	1	no	—
295	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	3	2	no	—
298	C₄	—	5	2	no	—
299	C₄	—	6	1	no	—
300	C₄	—	10	2	no	—
301	C₄	—	15	4	no	—
302	C₄	—	15	2	no	—
303	C₄	—	30	1	no	—
304	C₄	C₄×D₃×D₅	1	2	yes	—
305	C₃	C₄×#711	1	1	yes	characteristic, complemented, Sylow
306	C₂	—	6	4	no	—
307	C₂	—	10	2	no	—
308	C₂	C₂×C₄×D₃×D₅	1	1	yes	characteristic
309	C₂	D₃×#711	1	1	yes	characteristic
310	C₂	#92034	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator	C₂×C₃×C₅	C₂³×C₄
Fitting	C₃×C₅×C₄⋊C₄	C₂²
socle, cosocle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×#6772	C₂
hypercenter	C₄⋊C₄	D₃×D₅
hypocenter	C₃×C₅	C₄×#41
LCS	G ▷ C₂×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
2. #91776	C₂	semidirect
3. C₅×#893	C₂	semidirect
4. C₃×#4038	C₂	semidirect
5. #91442	C₂	semidirect
6. #91441	C₂	semidirect
7. #91435	C₂	semidirect
8. #91398	C₂	semidirect
9. C₄×D₃×Dic₅	C₂	semidirect
10. C₄×D₅×Dic₃	C₂	semidirect
11. #91340	C₂	semidirect
22. #6772	C₄	semidirect
43. #4038	D₃	semidirect
56. Dic₃₀	C₂×C₄	semidirect
59. C₅×D₁₂	C₂×C₄	semidirect
63. C₃×C₄×D₅	C₂×C₄	semidirect
64. #1201	C₂×C₄	semidirect
66. D₃×Dic₅	C₂×C₄	semidirect
73. #893	D₅	semidirect
116. Dic₁₅	#41	semidirect
120. C₅×Dic₃	#41	semidirect
163. C₄×D₅	C₄×D₃	semidirect
196. D₁₂	C₄×D₅	semidirect
241. C₃×C₅	C₄×#41	semidirect
257. Dic₃	#711	semidirect
288. C₅	#5522	semidirect
305. C₃	C₄×#711	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6772	2
#91776	2
C₅×#893	2
C₃×#4038	2
#91442	2
#91441	2
#91435	2
#91398	2
C₄×D₃×Dic₅	2
C₄×D₅×Dic₃	2
#91340	2
C₄×#711	3
#5522	5