Subgroups — #11753611

Order 960.

444 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#91777	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#890	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4038	C₂	1	1	yes	maximal, characteristic
4	#91470	C₂	1	1	yes	maximal, characteristic, complemented
5	#91431	C₂	1	2	yes	maximal
6	#91430	C₂	1	2	yes	maximal
7	C₄×D₅×Dic₃	C₂	1	1	yes	maximal, characteristic
8	#91381	C₂	1	1	yes	maximal, characteristic, complemented
9	#91358	C₂	1	1	yes	maximal, characteristic
10	#91333	C₂	1	2	yes	maximal, complemented
11	#91323	C₂	1	2	yes	maximal, complemented
12	#66022	—	3	1	no	maximal, Hall
13	C₂×Dic₃₀	C₂²	1	2	yes	—
14	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
15	C₂×#1204	C₂²	1	2	yes	—
16	C₂×D₅×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×D₅×Dic₃	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
20	#6712	C₂²	1	2	yes	—
21	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
23	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
25	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
26	C₃×#685	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
28	C₃×C₄×Dic₅	C₂²	1	2	yes	—
29	#6669	C₂²	1	2	yes	—
30	#6669	C₂²	1	2	yes	—
31	#6668	C₂²	1	2	yes	—
32	#6665	C₂²	1	2	yes	—
33	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
34	Dic₃×Dic₅	C₂²	1	2	yes	—
35	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
36	#5517	—	5	1	no	maximal, Hall
37	C₅×#125	—	3	1	no	—
38	#4042	—	3	2	no	—
39	#4042	—	3	2	no	—
40	#4038	—	3	1	no	—
41	#4038	D₃	1	1	yes	characteristic
42	#4034	—	3	1	no	—
43	#4034	—	3	1	no	—
44	#4033	—	3	2	no	—
45	#4033	—	3	2	no	—
46	C₄²×D₅	—	3	1	no	—
47	#4016	—	3	1	no	—
48	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂×Dic₁₅	C₂³	1	2	yes	—
52	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
54	Dic₃₀	—	2	4	no	—
55	C₂×C₅×Dic₃	C₂³	1	2	yes	—
56	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₃×Dic₅	C₂³	1	2	yes	—
59	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
60	C₃×C₄×D₅	Q₈	1	4	yes	complemented
61	#1204	—	2	4	no	—
62	D₅×Dic₃	—	2	2	no	—
63	D₅×Dic₃	—	2	2	no	—
64	C₃×#117	—	5	1	no	—
65	#926	—	5	1	no	—
66	C₂×C₄×Dic₃	—	5	1	no	—
67	#890	—	5	1	no	—
68	#890	D₅	1	1	yes	characteristic, complemented
69	#888	—	5	1	no	—
70	#888	—	5	1	no	—
71	#878	—	5	2	no	—
72	#878	—	5	2	no	—
73	C₄×Dic₆	—	5	2	no	—
74	C₄×Dic₆	—	5	2	no	—
75	C₂×C₄×D₅	—	3	1	no	—
76	C₂×C₄×D₅	—	3	1	no	—
77	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
78	C₂×Dic₁₀	—	3	2	no	—
79	C₂×Dic₁₀	—	3	2	no	—
80	C₅×C₄⋊C₄	—	3	2	no	—
81	C₅×C₄⋊C₄	—	3	2	no	—
82	C₅×C₄⋊C₄	—	3	1	no	—
83	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
84	C₄²×C₅	—	3	1	no	—
85	#685	—	3	2	no	—
86	#685	C₂×D₃	1	2	yes	—
87	C₄⋊Dic₅	—	3	1	no	—
88	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
89	#683	—	3	2	no	—
90	#683	—	3	2	no	—
91	#683	—	3	2	no	—
92	#683	—	3	2	no	—
93	C₄×Dic₅	—	3	1	no	—
94	C₄×Dic₅	—	3	1	no	—
95	C₄×Dic₅	—	3	1	no	—
96	C₄×Dic₅	—	3	2	no	—
97	C₄×Dic₅	C₂×D₃	1	2	yes	—
98	#533	—	15	1	no	Sylow
99	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
100	C₂×C₃×D₅	—	2	1	no	—
101	C₂×C₃×D₅	—	2	1	no	—
102	C₂×C₃×D₅	C₂×Q₈	1	2	yes	—
103	C₃×C₄×C₅	—	2	2	no	—
104	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
105	Dic₁₅	—	2	2	no	—
106	Dic₁₅	#41	1	2	yes	—
107	Dic₁₅	#41	1	2	yes	—
108	C₃×Dic₅	—	2	2	no	—
109	C₃×Dic₅	C₂×Q₈	1	2	yes	—
110	C₅×Dic₃	—	2	1	no	—
111	C₅×Dic₃	—	2	1	no	—
112	C₅×Dic₃	—	2	2	no	—
113	C₂²×C₃×C₄	—	5	1	no	—
114	C₂²×Dic₃	—	5	1	no	—
115	C₂²×Dic₃	—	5	1	no	—
116	C₂×Dic₆	—	5	2	no	—
117	C₂×Dic₆	—	5	2	no	—
118	C₃×C₄⋊C₄	—	5	1	no	—
119	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
120	C₃×C₂²⋊C₄	—	5	2	no	—
121	C₃×C₄²	—	5	2	no	—
122	#217	—	5	2	no	—
123	C₄⋊Dic₃	—	5	1	no	—
124	C₄⋊Dic₃	—	5	2	no	—
125	C₄⋊Dic₃	C₂×D₅	1	2	yes	—
126	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
127	Dic₃⋊C₄	—	5	2	no	—
128	Dic₃⋊C₄	—	5	2	no	—
129	Dic₃⋊C₄	—	5	2	no	—
130	Dic₃⋊C₄	C₂×D₅	1	2	yes	—
131	C₄×Dic₃	—	5	1	no	—
132	C₄×Dic₃	—	5	1	no	—
133	C₄×Dic₃	—	5	1	no	—
134	C₄×Dic₃	—	5	2	no	—
135	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
136	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
137	C₂×C₄×C₅	—	3	1	no	—
138	C₂×C₄×C₅	—	3	1	no	—
139	C₂×C₄×C₅	—	3	2	no	—
140	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
141	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
142	C₂×Dic₅	—	3	1	no	—
143	C₂×Dic₅	—	3	1	no	—
144	C₂×Dic₅	—	3	2	no	—
145	C₂×Dic₅	C₂²×D₃	1	2	yes	—
146	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
147	C₄×D₅	—	6	2	no	—
148	C₄×D₅	—	6	2	no	—
149	C₄×D₅	Dic₆	1	4	yes	complemented
150	Dic₁₀	—	6	4	no	—
151	Dic₁₀	—	6	4	no	—
152	C₄⋊Q₈	—	15	1	no	—
153	C₄⋊Q₈	—	15	1	no	—
154	#125	—	3	1	no	—
155	#125	—	15	1	no	—
156	C₂²⋊Q₈	—	15	2	no	—
157	C₂²⋊Q₈	—	15	2	no	—
158	C₄×Q₈	—	15	2	no	—
159	C₄×Q₈	—	15	2	no	—
160	#117	—	5	1	no	—
161	#117	—	15	1	no	—
162	C₂×C₄²	—	15	1	no	—
163	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
165	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
166	C₃×D₅	—	2	2	no	—
167	C₂³×C₃	—	5	1	no	—
168	C₂×C₃×C₄	—	5	1	no	—
169	C₂×C₃×C₄	—	5	4	no	—
170	C₂×C₃×C₄	—	5	2	no	—
171	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
172	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
173	C₂×Dic₃	—	5	1	no	—
174	C₂×Dic₃	—	5	1	no	—
175	C₂×Dic₃	—	5	2	no	—
176	C₂×Dic₃	—	10	2	no	—
177	C₂×Dic₃	—	10	2	no	—
178	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
179	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
180	C₂×Dic₃	C₂²×D₅	1	2	yes	—
181	Dic₆	—	10	4	no	—
182	Dic₆	—	10	4	no	—
183	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
184	C₂×D₅	—	2	1	no	—
185	C₂×D₅	—	2	1	no	—
186	C₂×D₅	C₂×Dic₆	1	2	yes	—
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	6	2	no	—
189	C₄×C₅	—	6	1	no	—
190	C₄×C₅	—	6	1	no	—
191	C₄×C₅	C₂×Dic₆	1	2	yes	—
192	Dic₅	—	2	2	no	—
193	Dic₅	—	3	2	no	—
194	Dic₅	—	3	2	no	—
195	Dic₅	—	6	2	no	—
196	Dic₅	C₂×Dic₆	1	2	yes	—
197	C₂×Q₈	—	15	2	no	—
198	C₂×Q₈	—	15	2	no	—
199	C₂²×C₄	—	5	1	no	—
200	C₂²×C₄	—	15	1	no	—
201	C₂²×C₄	—	15	1	no	—
202	C₄⋊C₄	—	3	1	no	—
203	C₄⋊C₄	—	3	2	no	—
204	C₄⋊C₄	—	3	2	no	—
205	C₄⋊C₄	—	5	1	no	—
206	C₄⋊C₄	—	15	2	no	—
207	C₄⋊C₄	—	15	2	no	—
208	C₄⋊C₄	—	15	2	no	—
209	C₄⋊C₄	—	15	1	no	—
210	C₄⋊C₄	—	15	2	no	—
211	C₄⋊C₄	D₃×D₅	1	1	yes	characteristic
212	C₂²⋊C₄	—	5	2	no	—
213	C₂²⋊C₄	—	15	2	no	—
214	C₄²	—	3	1	no	—
215	C₄²	—	5	2	no	—
216	C₄²	—	15	1	no	—
217	C₄²	—	15	1	no	—
218	C₄²	—	15	2	no	—
219	C₄²	—	15	1	no	—
220	C₃×C₅	#533	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	—	5	2	no	—
222	C₂²×C₃	—	10	1	no	—
223	C₂²×C₃	—	10	1	no	—
224	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	5	2	no	—
227	C₃×C₄	—	10	2	no	—
228	C₃×C₄	Q₈×D₅	1	2	yes	—
229	Dic₃	—	2	2	no	—
230	Dic₃	—	2	1	no	—
231	Dic₃	—	2	1	no	—
232	Dic₃	—	5	2	no	—
233	Dic₃	—	5	2	no	—
234	Dic₃	—	10	2	no	—
235	C₂×C₅	C₂²×Dic₆	1	1	yes	characteristic
236	C₂×C₅	D₃×#41	1	1	yes	characteristic
237	C₂×C₅	D₃×#41	1	1	yes	characteristic
238	D₅	—	2	2	no	—
239	C₂³	—	5	1	no	—
240	Q₈	—	30	4	no	—
241	Q₈	—	30	4	no	—
242	C₂×C₄	—	3	1	no	—
243	C₂×C₄	—	3	1	no	—
244	C₂×C₄	—	3	2	no	—
245	C₂×C₄	—	5	1	no	—
246	C₂×C₄	—	5	4	no	—
247	C₂×C₄	—	5	2	no	—
248	C₂×C₄	—	15	1	no	—
249	C₂×C₄	—	15	1	no	—
250	C₂×C₄	—	15	2	no	—
251	C₂×C₄	—	30	2	no	—
252	C₂×C₄	—	30	2	no	—
253	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
254	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
255	C₂×C₃	—	10	2	no	—
256	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
257	C₂×C₃	D₅×#41	1	1	yes	characteristic
258	C₂×C₃	D₅×#41	1	1	yes	characteristic
259	C₅	#5517	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	5	2	no	—
261	C₂²	—	10	1	no	—
262	C₂²	—	10	1	no	—
263	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	5	2	no	—
266	C₄	—	6	1	no	—
267	C₄	—	6	1	no	—
268	C₄	—	6	2	no	—
269	C₄	—	10	2	no	—
270	C₄	—	15	2	no	—
271	C₄	—	15	2	no	—
272	C₄	—	30	2	no	—
273	C₄	D₅×Dic₆	1	2	yes	—
274	C₃	#66022	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	10	2	no	—
276	C₂	C₂×D₅×Dic₆	1	1	yes	characteristic
277	C₂	#92024	1	1	yes	characteristic
278	C₂	#92034	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₅×C₄⋊C₄	C₂²
prime radical	C₂²×C₃×D₅	C₂³
hypercenter	C₄⋊C₄	D₃×D₅
hypocenter	C₃×C₅	#533
LCS	G ▷ C₂²×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
1. #91777	C₂	semidirect
2. C₅×#890	C₂	semidirect
4. #91470	C₂	semidirect
8. #91381	C₂	semidirect
10. #91333	C₂	semidirect
11. #91323	C₂	semidirect
60. C₃×C₄×D₅	Q₈	semidirect
68. #890	D₅	semidirect
149. C₄×D₅	Dic₆	semidirect
220. C₃×C₅	#533	semidirect
259. C₅	#5517	semidirect
274. C₃	#66022	semidirect

Maximal subgroups

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

subgroup	index
#91777	2
C₅×#890	2
C₃×#4038	2
#91470	2
#91431	2
#91430	2
C₄×D₅×Dic₃	2
#91381	2
#91358	2
#91333	2
#91323	2
#66022	3
#5517	5