Subgroups — #11751112

Order 960.

248 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#6710	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#91114	C₂	1	1	yes	maximal, characteristic, complemented
5	#91114	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊(C₃⋊#168)	C₂	1	1	yes	maximal, characteristic, complemented
7	#91087	C₂	1	1	yes	maximal, characteristic, complemented
8	#65300	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×D₄	C₄	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₃×C₅×C₂²⋊C₄	C₄	1	1	yes	characteristic
14	C₃×C₅×C₂²⋊C₄	C₄	1	1	yes	characteristic
15	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	#6710	—	2	2	no	—
17	C₂×C₃⋊#168	—	2	2	no	—
18	C₂×C₃⋊#168	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊#168	C₂²	1	1	yes	characteristic
20	C₂×C₃⋊#168	C₂²	1	1	yes	characteristic
21	C₂×C₃⋊#168	C₂²	1	1	yes	characteristic
22	#5011	—	5	1	no	maximal, Hall
23	C₄×C₅×D₄	D₃	1	1	yes	characteristic
24	C₂×#681	—	3	1	no	—
25	C₂²×#168	—	3	1	no	—
26	C₂²⋊#168	—	3	1	no	—
27	C₂²⋊#168	—	3	1	no	—
28	C₄⋊#168	—	3	1	no	—
29	#3935	—	3	1	no	—
30	C₂³×C₃×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₃×C₅	C₂×C₄	1	1	yes	characteristic
32	C₃×C₅×D₄	—	2	2	no	—
33	C₂×C₃×C₄×C₅	—	2	1	no	—
34	C₂×C₃×C₄×C₅	—	2	1	no	—
35	C₂×C₃×C₄×C₅	—	2	1	no	—
36	C₂×C₃×C₄×C₅	—	2	1	no	—
37	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₃⋊#168	—	2	1	no	—
43	C₃⋊#168	—	2	1	no	—
44	C₃⋊#168	—	2	1	no	—
45	C₃⋊#168	D₄	1	2	yes	complemented
46	C₃×C₄×D₄	D₅	1	1	yes	characteristic
47	C₂×#208	—	5	1	no	—
48	C₂²×C₃⋊C₈	—	5	1	no	—
49	#830	—	5	1	no	—
50	#830	—	5	1	no	—
51	C₄⋊(C₃⋊C₈)	—	5	1	no	—
52	#803	—	5	1	no	—
53	C₂×C₅×D₄	Dic₃	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
56	C₅×C₄⋊C₄	Dic₃	1	1	yes	characteristic
57	C₅×C₂²⋊C₄	Dic₃	1	1	yes	characteristic
58	C₅×C₂²⋊C₄	Dic₃	1	1	yes	characteristic
59	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
60	#681	—	6	2	no	—
61	C₂×#168	—	3	1	no	—
62	C₂×#168	—	3	1	no	—
63	C₂×#168	—	3	1	no	—
64	C₂×#168	—	3	1	no	—
65	C₂×#168	—	6	2	no	—
66	#435	—	15	1	no	Sylow
67	C₂²×C₃×C₅	—	2	1	no	—
68	C₂²×C₃×C₅	—	2	1	no	—
69	C₂²×C₃×C₅	—	2	1	no	—
70	C₂²×C₃×C₅	—	2	1	no	—
71	C₂²×C₃×C₅	—	2	1	no	—
72	C₂²×C₃×C₅	M₁₆	1	2	yes	complemented
73	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₃×C₄×C₅	—	2	1	no	—
75	C₃×C₄×C₅	—	2	1	no	—
76	C₃×C₄×C₅	—	2	1	no	—
77	C₃×C₄×C₅	—	2	1	no	—
78	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₃×C₄×C₅	#41	1	1	yes	characteristic
80	C₂×C₃×D₄	Dic₅	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
83	C₃×C₄⋊C₄	Dic₅	1	1	yes	characteristic
84	C₃×C₂²⋊C₄	Dic₅	1	1	yes	characteristic
85	C₃×C₂²⋊C₄	Dic₅	1	1	yes	characteristic
86	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
87	#208	—	10	2	no	—
88	C₂×C₃⋊C₈	—	5	1	no	—
89	C₂×C₃⋊C₈	—	5	1	no	—
90	C₂×C₃⋊C₈	—	5	1	no	—
91	C₂×C₃⋊C₈	—	5	1	no	—
92	C₂×C₃⋊C₈	—	10	2	no	—
93	C₂³×C₅	C₂×Dic₃	1	1	yes	characteristic
94	C₂³×C₅	C₂×Dic₃	1	1	yes	characteristic
95	C₅×D₄	—	2	2	no	—
96	C₂×C₄×C₅	—	2	1	no	—
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	2	1	no	—
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
105	#168	—	3	2	no	—
106	#168	—	6	1	no	—
107	#168	—	6	1	no	—
108	#168	—	6	1	no	—
109	C₂×M₁₆	—	15	1	no	—
110	C₂²×C₈	—	15	1	no	—
111	C₄×D₄	D₁₅	1	1	yes	characteristic
112	C₄⋊C₈	—	15	1	no	—
113	C₂²⋊C₈	—	15	1	no	—
114	C₂²⋊C₈	—	15	1	no	—
115	#97	—	15	1	no	—
116	C₂×C₃×C₅	—	2	2	no	—
117	C₂×C₃×C₅	—	4	1	no	—
118	C₂×C₃×C₅	C₄×D₄	1	1	yes	characteristic
119	C₂×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₃×C₅	#131	1	1	yes	characteristic
121	C₂³×C₃	C₂×Dic₅	1	1	yes	characteristic
122	C₂³×C₃	C₂×Dic₅	1	1	yes	characteristic
123	C₃×D₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
133	C₃⋊C₈	—	5	2	no	—
134	C₃⋊C₈	—	10	1	no	—
135	C₃⋊C₈	—	10	1	no	—
136	C₃⋊C₈	—	10	1	no	—
137	C₂²×C₅	—	2	1	no	—
138	C₂²×C₅	—	2	1	no	—
139	C₂²×C₅	—	2	1	no	—
140	C₂²×C₅	—	2	1	no	—
141	C₂²×C₅	—	2	1	no	—
142	C₂²×C₅	#208	1	2	yes	complemented
143	C₂²×C₅	C₂²×Dic₃	1	1	yes	characteristic
144	C₄×C₅	—	2	1	no	—
145	C₄×C₅	—	2	1	no	—
146	C₄×C₅	—	2	1	no	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	D₃×D₄	1	1	yes	characteristic
149	C₄×C₅	#237	1	1	yes	characteristic
150	C₂×D₄	Dic₁₅	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₁₅	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₁₅	1	1	yes	characteristic
153	M₁₆	—	30	2	no	—
154	C₂×C₈	—	15	1	no	—
155	C₂×C₈	—	15	1	no	—
156	C₂×C₈	—	15	1	no	—
157	C₂×C₈	—	15	1	no	—
158	C₂×C₈	—	30	2	no	—
159	C₄⋊C₄	Dic₁₅	1	1	yes	characteristic
160	C₂²⋊C₄	Dic₁₅	1	1	yes	characteristic
161	C₂²⋊C₄	Dic₁₅	1	1	yes	characteristic
162	C₄²	C₂×D₁₅	1	1	yes	characteristic
163	C₃×C₅	#435	1	1	yes	characteristic, complemented, Hall
164	C₂²×C₃	—	2	1	no	—
165	C₂²×C₃	—	2	1	no	—
166	C₂²×C₃	—	2	1	no	—
167	C₂²×C₃	—	2	1	no	—
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	#681	1	2	yes	complemented
170	C₂²×C₃	C₂²×Dic₅	1	1	yes	characteristic
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	—	2	1	no	—
175	C₃×C₄	D₄×D₅	1	1	yes	characteristic
176	C₃×C₄	#711	1	1	yes	characteristic
177	C₂×C₅	—	2	2	no	—
178	C₂×C₅	—	4	1	no	—
179	C₂×C₅	C₂×#208	1	1	yes	characteristic
180	C₂×C₅	D₄×Dic₃	1	1	yes	characteristic
181	C₂×C₅	#948	1	1	yes	characteristic
182	C₂³	C₂×Dic₁₅	1	1	yes	characteristic
183	C₂³	C₂×Dic₁₅	1	1	yes	characteristic
184	D₄	—	2	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	C₂×Dic₁₅	1	1	yes	characteristic
190	C₂×C₄	C₂×Dic₁₅	1	1	yes	characteristic
191	C₂×C₄	C₂×Dic₁₅	1	1	yes	characteristic
192	C₂×C₄	C₂×Dic₁₅	1	1	yes	characteristic
193	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
194	C₈	—	15	2	no	—
195	C₈	—	30	1	no	—
196	C₈	—	30	1	no	—
197	C₈	—	30	1	no	—
198	C₂×C₃	—	2	2	no	—
199	C₂×C₃	—	4	1	no	—
200	C₂×C₃	C₂×#681	1	1	yes	characteristic
201	C₂×C₃	D₄×Dic₅	1	1	yes	characteristic
202	C₂×C₃	#4094	1	1	yes	characteristic
203	C₅	#5011	1	1	yes	characteristic, complemented, Sylow
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	#6710	1	2	yes	complemented
210	C₂²	C₂²×Dic₁₅	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	D₄×D₁₅	1	1	yes	characteristic
216	C₄	#6819	1	1	yes	characteristic
217	C₃	#65300	1	1	yes	characteristic, complemented, Sylow
218	C₂	—	2	2	no	—
219	C₂	—	4	1	no	—
220	C₂	C₂×#6710	1	1	yes	characteristic
221	C₂	D₄×Dic₁₅	1	1	yes	characteristic
222	C₂	#91837	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₅
commutator, socle	C₂²×C₃×C₅	C₂²×C₄
Fitting	C₃×C₄×C₅×D₄	C₂
prime radical	C₂×C₃×C₅×D₄	C₄
cosocle	C₂×C₃×C₄×C₅	C₂³
hypercenter	C₄×D₄	D₁₅
hypocenter	C₃×C₅	#435
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#6710	C₂	semidirect
3. C₂²×C₃⋊#168	C₂	semidirect
4. #91114	C₂	semidirect
5. #91114	C₂	semidirect
6. C₄⋊(C₃⋊#168)	C₂	semidirect
7. #91087	C₂	semidirect
45. C₃⋊#168	D₄	semidirect
72. C₂²×C₃×C₅	M₁₆	semidirect
142. C₂²×C₅	#208	semidirect
163. C₃×C₅	#435	semidirect
169. C₂²×C₃	#681	semidirect
203. C₅	#5011	semidirect
209. C₂²	#6710	semidirect
217. C₃	#65300	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×D₄	2
C₂×#6710	2
C₂²×C₃⋊#168	2
#91114	2
C₄⋊(C₃⋊#168)	2
#91087	2
#65300	3
#5011	5