Subgroups — #12008317

Order 1280.

290 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#10600537	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#672	C₂	1	1	yes	maximal, characteristic
4	#10597929	C₂	1	2	yes	maximal, complemented
5	#10597928	C₂	1	2	yes	maximal, complemented
6	C₅×#431	C₄	1	1	yes	characteristic
7	C₄×C₅×M₁₆	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂²×#672	C₄	1	2	yes	—
10	C₅×#322	C₄	1	2	yes	—
11	C₅×C₈²	C₂²	1	2	yes	—
12	C₂²⋊#672	C₄	1	2	yes	—
13	C₄⋊#672	C₄	1	2	yes	—
14	C₄×#672	C₄	1	2	yes	—
15	C₄×#672	C₂²	1	1	yes	characteristic
16	C₄×#672	C₂²	1	2	yes	—
17	C₄×#672	C₂²	1	1	yes	characteristic
18	#11408	—	5	1	no	maximal, Sylow
19	C₂×C₅×M₁₆	C₈	1	2	yes	—
20	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₅×C₄⋊C₈	C₈	1	2	yes	—
24	C₅×C₂²⋊C₈	C₈	1	2	yes	—
25	C₅×#97	C₈	1	2	yes	—
26	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₂×#672	—	2	4	no	—
33	C₂×#672	C₂×C₄	1	4	yes	—
34	C₂×#672	C₂×C₄	1	2	yes	—
35	C₂×#672	C₂×C₄	1	2	yes	—
36	#2162	—	5	1	no	—
37	C₂×C₄×C₁₆	—	5	1	no	—
38	C₈×M₁₆	D₅	1	1	yes	characteristic
39	#1312	—	5	2	no	—
40	#1311	—	5	2	no	—
41	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
44	C₅×M₁₆	—	2	4	no	—
45	C₂×C₅×C₈	—	2	2	no	—
46	C₂×C₅×C₈	—	2	2	no	—
47	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
48	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
50	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
51	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₅	C₄²	1	1	yes	characteristic
54	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
56	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
57	#672	—	2	2	no	—
58	#672	—	2	4	no	—
59	#672	M₁₆	1	4	yes	complemented
60	C₂²×C₁₆	—	5	2	no	—
61	#431	Dic₅	1	1	yes	characteristic
62	C₄×M₁₆	Dic₅	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
64	C₄⋊C₁₆	—	5	2	no	—
65	C₂²⋊C₁₆	—	5	2	no	—
66	C₄×C₁₆	—	5	2	no	—
67	C₄×C₁₆	—	5	1	no	—
68	C₄×C₁₆	—	5	2	no	—
69	C₄×C₁₆	—	5	1	no	—
70	#322	Dic₅	1	2	yes	—
71	C₈²	C₂×D₅	1	2	yes	—
72	C₂³×C₅	C₄×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅	—	2	1	no	—
74	C₂×C₄×C₅	—	2	1	no	—
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
78	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
79	C₂×C₄×C₅	M₃₂	1	4	yes	—
80	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
83	C₅×C₈	—	2	1	no	—
84	C₅×C₈	—	2	1	no	—
85	C₅×C₈	—	2	4	no	—
86	C₅×C₈	—	2	4	no	—
87	C₅×C₈	C₂×M₁₆	1	2	yes	—
88	C₅×C₈	#131	1	2	yes	—
89	C₂×M₁₆	#168	1	2	yes	—
90	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
91	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
92	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
93	C₂×C₁₆	—	5	2	no	—
94	C₂×C₁₆	—	5	2	no	—
95	C₂×C₁₆	—	5	4	no	—
96	C₂×C₁₆	—	10	4	no	—
97	C₄⋊C₈	#168	1	2	yes	—
98	C₂²⋊C₈	#168	1	2	yes	—
99	#97	#168	1	2	yes	—
100	C₄×C₈	C₄×D₅	1	1	yes	characteristic
101	C₄×C₈	C₄×D₅	1	1	yes	characteristic
102	C₄×C₈	C₂×Dic₅	1	2	yes	—
103	C₄×C₈	C₂×Dic₅	1	2	yes	—
104	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
105	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
106	C₂²×C₅	—	2	1	no	—
107	C₂²×C₅	—	2	1	no	—
108	C₂²×C₅	#346	1	2	yes	—
109	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
110	C₄×C₅	—	2	1	no	—
111	C₄×C₅	—	2	1	no	—
112	C₄×C₅	#346	1	2	yes	—
113	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
114	C₄×C₅	#405	1	1	yes	characteristic
115	C₄×C₅	C₂×M₃₂	1	2	yes	—
116	C₄×C₅	#504	1	2	yes	—
117	C₂²×C₄	C₂×#168	1	1	yes	characteristic
118	C₂²×C₄	C₂×#168	1	1	yes	characteristic
119	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
120	M₁₆	—	2	4	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	C₂×#168	1	4	yes	—
124	C₂×C₈	C₂×#168	1	4	yes	—
125	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
126	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
127	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
128	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
129	C₄²	C₂×#168	1	1	yes	characteristic
130	C₄²	C₂×#168	1	1	yes	characteristic
131	C₄²	C₄×Dic₅	1	1	yes	characteristic
132	C₄²	C₂²×Dic₅	1	1	yes	characteristic
133	C₁₆	—	5	4	no	—
134	C₁₆	—	10	2	no	—
135	C₁₆	—	10	4	no	—
136	C₂×C₅	—	2	2	no	—
137	C₂×C₅	C₈×M₁₆	1	1	yes	characteristic
138	C₂×C₅	C₂×#346	1	1	yes	characteristic
139	C₂×C₅	#2108	1	1	yes	characteristic
140	C₂³	C₄×#168	1	1	yes	characteristic
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	C₄×#168	1	1	yes	characteristic
145	C₂×C₄	C₄×#168	1	1	yes	characteristic
146	C₂×C₄	C₄×#168	1	1	yes	characteristic
147	C₂×C₄	#3944	1	4	yes	—
148	C₂×C₄	C₂²×#168	1	1	yes	characteristic
149	C₂×C₄	C₂²×#168	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
151	C₈	—	2	4	no	—
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	2	4	no	—
155	C₈	D₅×M₁₆	1	2	yes	—
156	C₈	#4053	1	2	yes	—
157	C₅	#11408	1	1	yes	characteristic, complemented, Sylow
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#64681	1	2	yes	—
161	C₂²	C₂×C₄×#168	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	#64681	1	2	yes	—
165	C₄	C₂×#3944	1	2	yes	—
166	C₄	M₁₆×Dic₅	1	1	yes	characteristic
167	C₄	#65411	1	1	yes	characteristic
168	C₄	#65429	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	C₂×#64681	1	1	yes	characteristic
171	C₂	M₁₆×#168	1	1	yes	characteristic
172	C₂	#10600674	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₅
commutator, socle	C₂²×C₅	C₂×C₄×C₈
Fitting	C₅×C₈×M₁₆	C₂
prime radical	C₂³×C₅	C₄×C₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	C₈×M₁₆	D₅
hypocenter	C₅	#11408
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
4. #10597929	C₂	semidirect
5. #10597928	C₂	semidirect
59. #672	M₁₆	semidirect
157. C₅	#11408	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₈×M₁₆	2
#10600537	2
C₂×C₄×#672	2
#10597929	2
#10597928	2
#11408	5