Subgroups — #11752667

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×#6771	C₂	1	1	yes	maximal, characteristic
2	C₄²×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#872	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4018	C₂	1	1	yes	maximal, characteristic, complemented
5	#91442	C₂	1	2	yes	maximal, complemented
6	#91406	C₂	1	2	yes	maximal, complemented
7	C₄×D₃×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	#91396	C₂	1	2	yes	maximal, complemented
9	C₄×D₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#91386	C₂	1	1	yes	maximal, characteristic, complemented
11	#91325	C₂	1	2	yes	maximal, complemented
12	#65856	—	3	1	no	maximal, Hall
13	C₂×C₄×D₁₅	C₂²	1	2	yes	—
14	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
17	C₂×#1204	C₂²	1	1	yes	characteristic
18	C₂×#1203	C₂²	1	1	yes	characteristic
19	C₂×#1202	C₂²	1	1	yes	characteristic
20	C₂×#1201	C₂²	1	1	yes	characteristic
21	C₂×D₃×Dic₅	C₂²	1	2	yes	—
22	C₂×D₅×Dic₃	C₂²	1	2	yes	—
23	#6771	C₄	1	8	yes	complemented
24	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₅	C₂²	1	2	yes	—
27	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
28	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
29	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
30	C₃×#685	C₂²	1	2	yes	—
31	C₃×#683	C₂²	1	2	yes	—
32	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
33	#6669	C₂²	1	2	yes	—
34	#6667	C₂²	1	2	yes	—
35	Dic₃×Dic₅	C₂²	1	2	yes	—
36	Dic₃×Dic₅	C₂²	1	2	yes	—
37	#5522	—	5	1	no	maximal, Hall
38	C₂×#709	—	3	1	no	—
39	C₅×#117	—	3	1	no	—
40	C₂×C₄×Dic₅	—	3	1	no	—
41	#4039	—	3	2	no	—
42	#4033	—	3	2	no	—
43	#4027	—	3	2	no	—
44	#4027	—	3	2	no	—
45	#4018	—	3	1	no	—
46	#4018	D₃	1	1	yes	characteristic, complemented
47	C₄²×D₅	—	3	1	no	—
48	C₄²×D₅	—	3	1	no	—
49	C₂²×D₁₅	C₂³	1	1	yes	characteristic
50	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
51	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₅	C₂³	1	2	yes	—
56	C₄×D₁₅	—	2	4	no	—
57	C₄×D₁₅	C₂×C₄	1	4	yes	complemented
58	C₂×C₅×Dic₃	C₂³	1	2	yes	—
59	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₄×C₅×D₃	C₂×C₄	1	4	yes	complemented
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	#1204	C₂×C₄	1	4	yes	complemented
65	#1203	C₂×C₄	1	4	yes	complemented
66	#1202	C₂×C₄	1	4	yes	complemented
67	#1201	C₂×C₄	1	4	yes	complemented
68	D₃×Dic₅	—	2	4	no	—
69	D₅×Dic₃	—	2	4	no	—
70	C₂×#235	—	5	1	no	—
71	C₃×#117	—	5	1	no	—
72	C₂×C₄×Dic₃	—	5	1	no	—
73	#893	—	5	2	no	—
74	#887	—	5	2	no	—
75	#67⋊C₄	—	5	2	no	—
76	#67⋊C₄	—	5	2	no	—
77	#872	—	5	1	no	—
78	#872	D₅	1	1	yes	characteristic, complemented
79	C₄²×D₃	—	5	1	no	—
80	C₄²×D₃	—	5	1	no	—
81	C₂²×C₄×C₅	—	3	1	no	—
82	C₂×C₂²⋊D₅	—	3	1	no	—
83	C₂×C₂²⋊D₅	—	3	1	no	—
84	C₂²×Dic₅	—	3	2	no	—
85	#709	—	3	8	no	—
86	C₂×D₂₀	—	3	1	no	—
87	C₂×C₄×D₅	—	3	1	no	—
88	C₂×C₄×D₅	—	3	2	no	—
89	C₂×C₄×D₅	—	3	2	no	—
90	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
91	C₂×Dic₁₀	—	3	1	no	—
92	C₅×C₄⋊C₄	—	3	2	no	—
93	C₅×C₂²⋊C₄	—	3	2	no	—
94	C₄²×C₅	—	3	1	no	—
95	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
96	#685	—	3	2	no	—
97	#685	C₂×D₃	1	2	yes	—
98	#683	—	3	2	no	—
99	#683	C₂×D₃	1	2	yes	—
100	C₄×Dic₅	—	3	1	no	—
101	C₄×Dic₅	—	3	2	no	—
102	C₄×Dic₅	—	3	2	no	—
103	C₄×Dic₅	—	3	2	no	—
104	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
105	C₄×#41	—	15	1	no	Sylow
106	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
107	C₂×D₁₅	—	2	2	no	—
108	C₂×D₁₅	C₂²×C₄	1	2	yes	—
109	C₂×C₅×D₃	—	2	2	no	—
110	C₂×C₅×D₃	C₂²×C₄	1	2	yes	—
111	C₂×C₃×D₅	—	2	2	no	—
112	C₂×C₃×D₅	C₂²×C₄	1	2	yes	—
113	C₃×C₄×C₅	—	2	2	no	—
114	C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
115	Dic₁₅	C₂²×C₄	1	2	yes	—
116	Dic₁₅	#41	1	4	yes	complemented
117	C₃×Dic₅	—	2	2	no	—
118	C₃×Dic₅	C₂²×C₄	1	2	yes	—
119	C₅×Dic₃	—	2	2	no	—
120	C₅×Dic₃	C₂²×C₄	1	2	yes	—
121	C₂²×C₃×C₄	—	5	1	no	—
122	C₂×#67	—	5	1	no	—
123	C₂×#67	—	5	1	no	—
124	C₂²×Dic₃	—	5	2	no	—
125	#235	—	5	8	no	—
126	C₂×D₁₂	—	5	1	no	—
127	C₂×C₄×D₃	—	5	1	no	—
128	C₂×C₄×D₃	—	5	2	no	—
129	C₂×C₄×D₃	—	5	2	no	—
130	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
131	C₂×Dic₆	—	5	1	no	—
132	C₃×C₄⋊C₄	—	5	2	no	—
133	C₃×C₂²⋊C₄	—	5	2	no	—
134	C₃×C₄²	—	5	1	no	—
135	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
136	(C₂×D₃)⋊C₄	—	5	2	no	—
137	(C₂×D₃)⋊C₄	C₂×D₅	1	2	yes	—
138	Dic₃⋊C₄	—	5	2	no	—
139	Dic₃⋊C₄	C₂×D₅	1	2	yes	—
140	C₄×Dic₃	—	5	1	no	—
141	C₄×Dic₃	—	5	2	no	—
142	C₄×Dic₃	—	5	2	no	—
143	C₄×Dic₃	—	5	2	no	—
144	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
145	C₂³×C₅	—	3	1	no	—
146	C₂²×D₅	—	3	1	no	—
147	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
148	C₂×C₄×C₅	—	3	1	no	—
149	C₂×C₄×C₅	—	3	4	no	—
150	C₂×C₄×C₅	—	3	2	no	—
151	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
152	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
153	C₂²⋊D₅	—	3	4	no	—
154	C₂²⋊D₅	—	3	4	no	—
155	C₂×Dic₅	—	3	2	no	—
156	C₂×Dic₅	—	3	1	no	—
157	C₂×Dic₅	—	6	4	no	—
158	C₂×Dic₅	C₂²×D₃	1	2	yes	—
159	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
160	D₂₀	—	3	4	no	—
161	C₄×D₅	—	3	4	no	—
162	C₄×D₅	—	6	4	no	—
163	C₄×D₅	—	6	4	no	—
164	C₄×D₅	C₄×D₃	1	4	yes	complemented
165	Dic₁₀	—	3	4	no	—
166	C₂×#41	—	15	1	no	—
167	C₄×Q₈	—	15	2	no	—
168	C₄×D₄	—	15	2	no	—
169	C₄×D₄	—	15	2	no	—
170	C₄×D₄	—	15	2	no	—
171	#117	—	3	1	no	—
172	#117	—	5	1	no	—
173	#117	—	15	1	no	—
174	C₂×C₄²	—	15	1	no	—
175	C₂×C₄²	—	15	1	no	—
176	C₂×C₄²	—	15	1	no	—
177	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₃×C₅	C₂×#41	1	2	yes	—
179	D₁₅	—	2	2	no	—
180	C₃×D₅	—	2	2	no	—
181	C₅×D₃	—	2	2	no	—
182	C₂³×C₃	—	5	1	no	—
183	C₂²×D₃	—	5	1	no	—
184	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
185	C₂×C₃×C₄	—	5	1	no	—
186	C₂×C₃×C₄	—	5	4	no	—
187	C₂×C₃×C₄	—	5	2	no	—
188	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
189	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
190	#67	—	5	4	no	—
191	#67	—	5	4	no	—
192	C₂×Dic₃	—	5	2	no	—
193	C₂×Dic₃	—	5	1	no	—
194	C₂×Dic₃	—	10	4	no	—
195	C₂×Dic₃	C₂²×D₅	1	2	yes	—
196	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
197	D₁₂	—	5	4	no	—
198	C₄×D₃	—	5	4	no	—
199	C₄×D₃	—	10	4	no	—
200	C₄×D₃	—	10	4	no	—
201	C₄×D₃	C₄×D₅	1	4	yes	complemented
202	Dic₆	—	5	4	no	—
203	C₂²×C₅	—	3	2	no	—
204	C₂²×C₅	—	6	2	no	—
205	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
206	C₂×D₅	—	2	2	no	—
207	C₂×D₅	—	3	2	no	—
208	C₂×D₅	—	6	2	no	—
209	C₂×D₅	C₂×C₄×D₃	1	2	yes	—
210	C₄×C₅	—	2	2	no	—
211	C₄×C₅	—	3	2	no	—
212	C₄×C₅	—	6	2	no	—
213	C₄×C₅	C₂×C₄×D₃	1	2	yes	—
214	Dic₅	—	2	2	no	—
215	Dic₅	—	3	2	no	—
216	Dic₅	—	3	4	no	—
217	Dic₅	C₂×C₄×D₃	1	2	yes	—
218	#41	—	15	8	no	—
219	C₂×Q₈	—	15	1	no	—
220	C₂×D₄	—	15	1	no	—
221	C₂×D₄	—	15	1	no	—
222	C₂×D₄	—	15	1	no	—
223	C₂²×C₄	—	3	1	no	—
224	C₂²×C₄	—	5	1	no	—
225	C₂²×C₄	—	15	2	no	—
226	C₂²×C₄	—	15	2	no	—
227	C₂²×C₄	—	15	1	no	—
228	C₂²×C₄	—	15	2	no	—
229	C₄⋊C₄	—	3	2	no	—
230	C₄⋊C₄	—	5	2	no	—
231	C₄⋊C₄	—	15	2	no	—
232	C₂²⋊C₄	—	3	2	no	—
233	C₂²⋊C₄	—	5	2	no	—
234	C₂²⋊C₄	—	15	2	no	—
235	C₄²	—	3	1	no	—
236	C₄²	—	5	1	no	—
237	C₄²	—	15	2	no	—
238	C₄²	—	15	2	no	—
239	C₄²	—	15	2	no	—
240	C₄²	—	15	1	no	—
241	C₄²	D₃×D₅	1	1	yes	characteristic
242	C₃×C₅	C₄×#41	1	1	yes	characteristic, complemented, Hall
243	C₂²×C₃	—	5	2	no	—
244	C₂²×C₃	—	10	2	no	—
245	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
246	C₂×D₃	—	2	2	no	—
247	C₂×D₃	—	5	2	no	—
248	C₂×D₃	—	10	2	no	—
249	C₂×D₃	C₂×C₄×D₅	1	2	yes	—
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	—	5	2	no	—
252	C₃×C₄	—	10	2	no	—
253	C₃×C₄	C₂×C₄×D₅	1	2	yes	—
254	Dic₃	—	2	2	no	—
255	Dic₃	—	5	2	no	—
256	Dic₃	—	5	4	no	—
257	Dic₃	C₂×C₄×D₅	1	2	yes	—
258	C₂×C₅	—	6	2	no	—
259	C₂×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
260	C₂×C₅	D₃×#41	1	2	yes	—
261	D₅	—	2	2	no	—
262	D₅	—	6	2	no	—
263	C₂³	—	3	1	no	—
264	C₂³	—	5	1	no	—
265	C₂³	—	15	1	no	—
266	Q₈	—	15	4	no	—
267	D₄	—	15	4	no	—
268	D₄	—	15	4	no	—
269	D₄	—	15	4	no	—
270	C₂×C₄	—	3	1	no	—
271	C₂×C₄	—	3	4	no	—
272	C₂×C₄	—	3	2	no	—
273	C₂×C₄	—	5	1	no	—
274	C₂×C₄	—	5	4	no	—
275	C₂×C₄	—	5	2	no	—
276	C₂×C₄	—	15	2	no	—
277	C₂×C₄	—	15	1	no	—
278	C₂×C₄	—	15	4	no	—
279	C₂×C₄	—	30	4	no	—
280	C₂×C₄	—	30	4	no	—
281	C₂×C₄	—	30	4	no	—
282	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
283	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
284	C₂×C₃	—	10	2	no	—
285	C₂×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
286	C₂×C₃	D₅×#41	1	2	yes	—
287	D₃	—	2	2	no	—
288	D₃	—	10	2	no	—
289	C₅	#5522	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	3	2	no	—
291	C₂²	—	5	2	no	—
292	C₂²	—	6	2	no	—
293	C₂²	—	10	2	no	—
294	C₂²	—	15	2	no	—
295	C₂²	—	30	2	no	—
296	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	3	2	no	—
299	C₄	—	5	2	no	—
300	C₄	—	6	2	no	—
301	C₄	—	10	2	no	—
302	C₄	—	15	2	no	—
303	C₄	—	15	4	no	—
304	C₄	C₄×D₃×D₅	1	2	yes	—
305	C₃	#65856	1	1	yes	characteristic, complemented, Sylow
306	C₂	—	6	2	no	—
307	C₂	—	10	2	no	—
308	C₂	—	30	2	no	—
309	C₂	C₂×C₄×D₃×D₅	1	1	yes	characteristic
310	C₂	#92016	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₅
commutator	C₂×C₃×C₅	C₂³×C₄
Frattini	C₂²	C₂²×D₃×D₅
Fitting	C₃×C₄²×C₅	C₂²
socle, cosocle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×#6771	C₂
hypercenter	C₄²	D₃×D₅
hypocenter	C₃×C₅	C₄×#41
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄²×D₁₅	C₂	semidirect
3. C₅×#872	C₂	semidirect
4. C₃×#4018	C₂	semidirect
5. #91442	C₂	semidirect
6. #91406	C₂	semidirect
7. C₄×D₃×Dic₅	C₂	semidirect
8. #91396	C₂	semidirect
9. C₄×D₅×Dic₃	C₂	semidirect
10. #91386	C₂	semidirect
11. #91325	C₂	semidirect
23. #6771	C₄	semidirect
46. #4018	D₃	semidirect
57. C₄×D₁₅	C₂×C₄	semidirect
60. C₄×C₅×D₃	C₂×C₄	semidirect
63. C₃×C₄×D₅	C₂×C₄	semidirect
64. #1204	C₂×C₄	semidirect
65. #1203	C₂×C₄	semidirect
66. #1202	C₂×C₄	semidirect
67. #1201	C₂×C₄	semidirect
78. #872	D₅	semidirect
116. Dic₁₅	#41	semidirect
164. C₄×D₅	C₄×D₃	semidirect
201. C₄×D₃	C₄×D₅	semidirect
242. C₃×C₅	C₄×#41	semidirect
289. C₅	#5522	semidirect
305. C₃	#65856	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6771	2
C₄²×D₁₅	2
C₅×#872	2
C₃×#4018	2
#91442	2
#91406	2
C₄×D₃×Dic₅	2
#91396	2
C₄×D₅×Dic₃	2
#91386	2
#91325	2
#65856	3
#5522	5