Subgroups — #13070525

Order 1408.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₁⋊C₁₆)	C₂	1	2	yes	maximal
3	C₄⋊(C₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#10622099	C₂	1	2	yes	maximal
5	C₄×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₄×Q₈×C₁₁	C₄	1	1	yes	characteristic
7	C₁₁×C₄⋊C₈	C₄	1	2	yes	—
8	C₁₁×C₄⋊C₈	C₄	1	1	yes	characteristic
9	C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₁₁	C₂²	1	2	yes	—
11	C₂×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₁₁⋊C₁₆	C₂²	1	2	yes	—
13	C₂×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₁₁	C₈	1	1	yes	characteristic
15	C₂×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₁	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
18	C₁₁×C₄⋊C₄	C₈	1	2	yes	—
19	C₁₁×C₄⋊C₄	C₈	1	1	yes	characteristic
20	C₄²×C₁₁	C₂×C₄	1	2	yes	—
21	C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₁₁⋊C₁₆	—	2	1	no	—
23	C₁₁⋊C₁₆	—	2	2	no	—
24	C₁₁⋊C₁₆	Q₈	1	2	yes	complemented
25	#2183	—	11	1	no	maximal, Sylow
26	Q₈×C₁₁	—	2	2	no	—
27	C₂×C₄×C₁₁	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₁	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₁₁	—	2	1	no	—
33	C₈×C₁₁	—	2	2	no	—
34	C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
35	C₈×C₁₁	#41	1	1	yes	characteristic
36	C₈×Q₈	D₁₁	1	1	yes	characteristic
37	C₄⋊C₁₆	—	11	1	no	—
38	C₄⋊C₁₆	—	11	2	no	—
39	#346	—	11	2	no	—
40	C₄×C₁₆	—	11	1	no	—
41	C₂²×C₁₁	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₁	—	2	2	no	—
43	C₄×C₁₁	—	2	2	no	—
44	C₄×C₁₁	—	2	1	no	—
45	C₄×C₁₁	M₃₂	1	2	yes	—
46	C₄×C₁₁	C₄×Q₈	1	1	yes	characteristic
47	C₄×C₁₁	#131	1	1	yes	characteristic
48	C₄×Q₈	Dic₁₁	1	1	yes	characteristic
49	C₂×C₁₆	—	11	1	no	—
50	C₂×C₁₆	—	11	1	no	—
51	C₂×C₁₆	—	11	2	no	—
52	C₄⋊C₈	Dic₁₁	1	2	yes	—
53	C₄⋊C₈	Dic₁₁	1	1	yes	characteristic
54	C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
55	C₄×C₈	C₂×D₁₁	1	2	yes	—
56	C₂×C₁₁	C₈×Q₈	1	1	yes	characteristic
57	C₂×C₁₁	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₁₁	#504	1	1	yes	characteristic
59	C₂×Q₈	C₁₁⋊C₈	1	1	yes	characteristic
60	C₂×C₈	C₂×Dic₁₁	1	1	yes	characteristic
61	C₂×C₈	C₂×Dic₁₁	1	2	yes	—
62	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
63	C₄⋊C₄	C₁₁⋊C₈	1	2	yes	—
64	C₄⋊C₄	C₁₁⋊C₈	1	1	yes	characteristic
65	C₄²	C₂×Dic₁₁	1	2	yes	—
66	C₄²	C₂×Dic₁₁	1	1	yes	characteristic
67	C₁₆	—	11	2	no	—
68	C₁₆	—	22	1	no	—
69	C₁₆	—	22	2	no	—
70	C₁₁	#2183	1	1	yes	characteristic, complemented, Sylow
71	Q₈	—	2	2	no	—
72	C₂×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
73	C₂×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
74	C₂×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
75	C₂×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
76	C₂×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	2	2	no	—
79	C₈	Q₈×D₁₁	1	1	yes	characteristic
80	C₈	#4343	1	1	yes	characteristic
81	C₂²	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	2	1	no	—
84	C₄	—	2	2	no	—
85	C₄	#66876	1	2	yes	—
86	C₄	Q₈×Dic₁₁	1	1	yes	characteristic
87	C₄	#67001	1	1	yes	characteristic
88	C₂	Q₈×C₁₁⋊C₈	1	1	yes	characteristic
89	C₂	C₂×#66876	1	1	yes	characteristic
90	C₂	#10622753	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₁
commutator, socle, prime radical	C₂²×C₁₁	C₂²×C₈
Fitting	C₈×Q₈×C₁₁	C₂
cosocle	C₂×C₈×C₁₁	C₂³
hypercenter	C₈×Q₈	D₁₁
hypocenter	C₁₁	#2183
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
24. C₁₁⋊C₁₆	Q₈	semidirect
70. C₁₁	#2183	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈×C₁₁	2
C₄⋊(C₁₁⋊C₁₆)	2
#10622099	2
C₄×C₁₁⋊C₁₆	2
#2183	11