Subgroups — #11756314

Order 960.

584 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6819	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#91819	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊Dic₁₅	C₂	1	2	yes	maximal, complemented
5	#91786	C₂	1	2	yes	maximal, complemented
6	#91775	C₂	1	4	yes	maximal, complemented
7	C₄×D₆₀	C₂	1	1	yes	maximal, characteristic, complemented
8	#91758	C₂	1	1	yes	maximal, characteristic, complemented
9	#65889	—	3	1	no	maximal, Hall
10	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
12	C₂×#1220	C₂²	1	4	yes	—
13	C₂²×Dic₁₅	C₂²	1	4	yes	—
14	#6819	—	2	8	no	—
15	C₂×D₆₀	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₅	C₂²	1	2	yes	—
17	C₂×Dic₃₀	C₂²	1	2	yes	—
18	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	#6714	C₂²	1	2	yes	—
22	#6714	C₂²	1	4	yes	—
23	C₄⋊Dic₁₅	C₂²	1	2	yes	—
24	C₄⋊Dic₁₅	C₂²	1	2	yes	—
25	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₁₅	C₂²	1	4	yes	—
27	#5555	—	5	1	no	maximal, Hall
28	C₂×#711	—	3	2	no	—
29	C₄×C₅×D₄	D₃	1	1	yes	characteristic, complemented
30	#4076	—	3	2	no	—
31	C₂×C₄⋊Dic₅	—	3	2	no	—
32	#4043	—	3	2	no	—
33	#4032	—	3	4	no	—
34	C₄×D₂₀	—	3	1	no	—
35	#4015	—	3	1	no	—
36	C₂³×C₃×C₅	C₂³	1	2	yes	—
37	C₂²×D₁₅	C₂³	1	2	yes	—
38	C₃×C₅×D₄	D₄	1	4	yes	complemented
39	C₂×C₃×C₄×C₅	—	2	4	no	—
40	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
42	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
44	#1220	—	2	8	no	—
45	C₂×Dic₁₅	—	2	8	no	—
46	C₂×Dic₁₅	C₂³	1	4	yes	—
47	C₂×Dic₁₅	C₂³	1	2	yes	—
48	D₆₀	—	2	2	no	—
49	C₄×D₁₅	—	2	4	no	—
50	Dic₃₀	—	2	4	no	—
51	C₂×#237	—	5	2	no	—
52	C₃×C₄×D₄	D₅	1	1	yes	characteristic, complemented
53	#930	—	5	2	no	—
54	C₂×C₄⋊Dic₃	—	5	2	no	—
55	#897	—	5	2	no	—
56	#886	—	5	4	no	—
57	C₄×D₁₂	—	5	1	no	—
58	#869	—	5	1	no	—
59	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×D₃	1	2	yes	—
61	C₂×C₂²⋊D₅	—	3	4	no	—
62	C₂²×Dic₅	—	3	4	no	—
63	#711	—	6	8	no	—
64	C₂×D₂₀	—	3	1	no	—
65	C₂×C₄×D₅	—	3	2	no	—
66	C₂×Dic₁₀	—	3	2	no	—
67	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
68	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
69	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
70	#685	—	3	2	no	—
71	#685	—	3	4	no	—
72	C₄⋊Dic₅	—	3	2	no	—
73	C₄⋊Dic₅	—	3	2	no	—
74	C₄⋊Dic₅	—	3	1	no	—
75	C₄⋊Dic₅	—	3	4	no	—
76	#547	—	15	1	no	Sylow
77	C₂²×C₃×C₅	—	2	2	no	—
78	C₂²×C₃×C₅	—	2	2	no	—
79	C₂²×C₃×C₅	C₂×D₄	1	4	yes	—
80	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×D₁₅	—	2	2	no	—
82	C₂×D₁₅	—	2	2	no	—
83	C₂×D₁₅	—	2	2	no	—
84	C₃×C₄×C₅	—	2	2	no	—
85	C₃×C₄×C₅	—	2	1	no	—
86	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
87	C₃×C₄×C₅	#41	1	2	yes	complemented
88	Dic₁₅	—	2	4	no	—
89	Dic₁₅	—	2	2	no	—
90	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₂×D₅	1	2	yes	—
92	C₂×#67	—	5	4	no	—
93	C₂²×Dic₃	—	5	4	no	—
94	#237	—	10	8	no	—
95	C₂×D₁₂	—	5	1	no	—
96	C₂×C₄×D₃	—	5	2	no	—
97	C₂×Dic₆	—	5	2	no	—
98	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
99	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
100	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
101	(C₂×D₃)⋊C₄	—	5	2	no	—
102	(C₂×D₃)⋊C₄	—	5	4	no	—
103	C₄⋊Dic₃	—	5	2	no	—
104	C₄⋊Dic₃	—	5	2	no	—
105	C₄⋊Dic₃	—	5	1	no	—
106	C₄⋊Dic₃	—	5	4	no	—
107	C₂³×C₅	C₂²×D₃	1	2	yes	—
108	C₂²×D₅	—	3	2	no	—
109	C₅×D₄	D₁₂	1	4	yes	complemented
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
113	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
115	C₂²⋊D₅	—	6	8	no	—
116	C₂×Dic₅	—	3	4	no	—
117	C₂×Dic₅	—	3	2	no	—
118	C₂×Dic₅	—	6	8	no	—
119	D₂₀	—	6	2	no	—
120	C₄×D₅	—	6	4	no	—
121	Dic₁₀	—	6	4	no	—
122	C₂×#41	—	15	2	no	—
123	C₄⋊Q₈	—	15	1	no	—
124	#123	—	15	4	no	—
125	C₂²⋊Q₈	—	15	2	no	—
126	#121	—	15	2	no	—
127	C₄×D₄	—	15	1	no	—
128	C₄×D₄	D₁₅	1	1	yes	characteristic, complemented
129	C₂×C₄⋊C₄	—	15	2	no	—
130	C₂×C₃×C₅	—	2	4	no	—
131	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
133	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
134	D₁₅	—	4	2	no	—
135	C₂³×C₃	C₂²×D₅	1	2	yes	—
136	C₂²×D₃	—	5	2	no	—
137	C₃×D₄	D₂₀	1	4	yes	complemented
138	C₂×C₃×C₄	—	2	4	no	—
139	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
141	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
143	#67	—	10	8	no	—
144	C₂×Dic₃	—	5	4	no	—
145	C₂×Dic₃	—	5	2	no	—
146	C₂×Dic₃	—	10	8	no	—
147	D₁₂	—	10	2	no	—
148	C₄×D₃	—	10	4	no	—
149	Dic₆	—	10	4	no	—
150	C₂²×C₅	—	2	2	no	—
151	C₂²×C₅	—	2	2	no	—
152	C₂²×C₅	C₂×D₁₂	1	4	yes	—
153	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
154	C₂×D₅	—	6	2	no	—
155	C₂×D₅	—	6	2	no	—
156	C₂×D₅	—	6	2	no	—
157	C₄×C₅	—	2	2	no	—
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	C₂×D₁₂	1	2	yes	—
160	C₄×C₅	#237	1	2	yes	complemented
161	Dic₅	—	6	4	no	—
162	Dic₅	—	6	2	no	—
163	#41	—	30	8	no	—
164	C₂×Q₈	—	15	2	no	—
165	C₂×D₄	—	15	4	no	—
166	C₂×D₄	—	15	1	no	—
167	C₂×D₄	C₂×D₁₅	1	1	yes	characteristic
168	C₂²×C₄	—	15	4	no	—
169	C₂²×C₄	—	15	2	no	—
170	C₂²×C₄	C₂×D₁₅	1	2	yes	—
171	C₄⋊C₄	—	15	1	no	—
172	C₄⋊C₄	—	15	2	no	—
173	C₄⋊C₄	—	15	4	no	—
174	C₄⋊C₄	—	15	2	no	—
175	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
176	C₂²⋊C₄	—	15	2	no	—
177	C₂²⋊C₄	—	15	4	no	—
178	C₂²⋊C₄	C₂×D₁₅	1	2	yes	—
179	C₄²	C₂×D₁₅	1	1	yes	characteristic
180	C₃×C₅	#547	1	1	yes	characteristic, complemented, Hall
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	C₂×D₂₀	1	4	yes	—
184	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
185	C₂×D₃	—	10	2	no	—
186	C₂×D₃	—	10	2	no	—
187	C₂×D₃	—	10	2	no	—
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	C₂×D₂₀	1	2	yes	—
191	C₃×C₄	#711	1	2	yes	complemented
192	Dic₃	—	10	4	no	—
193	Dic₃	—	10	2	no	—
194	C₂×C₅	—	2	4	no	—
195	C₂×C₅	C₂²×D₁₂	1	1	yes	characteristic
196	C₂×C₅	C₂×#237	1	1	yes	characteristic
197	C₂×C₅	#1010	1	1	yes	characteristic
198	D₅	—	12	2	no	—
199	C₂³	—	15	2	no	—
200	C₂³	C₂²×D₁₅	1	2	yes	—
201	Q₈	—	30	4	no	—
202	D₄	—	30	8	no	—
203	D₄	—	30	2	no	—
204	D₄	D₆₀	1	4	yes	complemented
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	15	2	no	—
207	C₂×C₄	—	15	4	no	—
208	C₂×C₄	—	30	8	no	—
209	C₂×C₄	—	30	4	no	—
210	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
211	C₂×C₄	C₂²×D₁₅	1	2	yes	—
212	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
213	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
214	C₂×C₃	—	2	4	no	—
215	C₂×C₃	C₂²×D₂₀	1	1	yes	characteristic
216	C₂×C₃	C₂×#711	1	1	yes	characteristic
217	C₂×C₃	#4150	1	1	yes	characteristic
218	D₃	—	20	2	no	—
219	C₅	#5555	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	30	2	no	—
223	C₂²	—	30	2	no	—
224	C₂²	—	30	2	no	—
225	C₂²	C₂×D₆₀	1	4	yes	—
226	C₂²	C₂³×D₁₅	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	30	2	no	—
230	C₄	—	30	4	no	—
231	C₄	C₂×D₆₀	1	2	yes	—
232	C₄	#6819	1	2	yes	complemented
233	C₃	#65889	1	1	yes	characteristic, complemented, Sylow
234	C₂	—	2	4	no	—
235	C₂	—	60	2	no	—
236	C₂	C₂²×D₆₀	1	1	yes	characteristic
237	C₂	C₂×#6819	1	1	yes	characteristic
238	C₂	#92101	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₄×C₅×D₄	C₂
hypercenter	C₄×D₄	D₁₅
hypocenter	C₃×C₅	#547
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#6819	C₂	semidirect
2. C₃×C₄×C₅×D₄	C₂	semidirect
3. #91819	C₂	semidirect
4. C₂×C₄⋊Dic₁₅	C₂	semidirect
5. #91786	C₂	semidirect
6. #91775	C₂	semidirect
7. C₄×D₆₀	C₂	semidirect
8. #91758	C₂	semidirect
29. C₄×C₅×D₄	D₃	semidirect
38. C₃×C₅×D₄	D₄	semidirect
52. C₃×C₄×D₄	D₅	semidirect
87. C₃×C₄×C₅	#41	semidirect
109. C₅×D₄	D₁₂	semidirect
128. C₄×D₄	D₁₅	semidirect
137. C₃×D₄	D₂₀	semidirect
160. C₄×C₅	#237	semidirect
180. C₃×C₅	#547	semidirect
191. C₃×C₄	#711	semidirect
204. D₄	D₆₀	semidirect
219. C₅	#5555	semidirect
232. C₄	#6819	semidirect
233. C₃	#65889	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6819	2
C₃×C₄×C₅×D₄	2
#91819	2
C₂×C₄⋊Dic₁₅	2
#91786	2
#91775	2
C₄×D₆₀	2
#91758	2
#65889	3
#5555	5