Subgroups — #27213318

Order 13504.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20270025	C₂	1	1	yes	maximal, characteristic, complemented
2	#20270003	C₂	1	2	yes	maximal, complemented
3	#20269905	C₂	1	2	yes	maximal, complemented
4	#20269896	C₂	1	1	yes	maximal, characteristic, complemented
5	#124×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#16809963	—	2	2	no	—
7	#16809962	—	2	2	no	—
8	#16809962	C₂²	1	2	yes	—
9	Q₈×D₂₁₁	—	2	2	no	—
10	#16809950	—	2	2	no	—
11	#16809947	—	2	2	no	—
12	#16809946	C₂²	1	2	yes	—
13	C₂×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
16	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
17	Dic₄₂₂	—	2	2	no	—
18	Dic₄₂₂	D₄	1	2	yes	—
19	D₈₄₄	—	2	2	no	—
20	D₈₄₄	D₄	1	2	yes	—
21	C₂²⋊D₂₁₁	—	2	2	no	—
22	C₄×D₂₁₁	—	2	4	no	—
23	C₄×D₂₁₁	—	2	2	no	—
24	C₂₁₁⋊C₈	—	2	2	no	—
25	C₂³×C₂₁₁	—	2	1	no	—
26	Q₈×C₂₁₁	—	2	2	no	—
27	D₄×C₂₁₁	—	2	2	no	—
28	C₂×C₄×C₂₁₁	—	2	1	no	—
29	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
30	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
31	C₂²×C₂₁₁	—	4	2	no	—
32	C₂²×C₂₁₁	—	4	1	no	—
33	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
34	C₂×D₂₁₁	—	2	2	no	—
35	C₄×C₂₁₁	—	2	2	no	—
36	C₄×C₂₁₁	—	4	1	no	—
37	C₄×C₂₁₁	C₂×D₄	1	2	yes	—
38	Dic₂₁₁	—	2	2	no	—
39	C₂×C₂₁₁	—	2	1	no	—
40	C₂×C₂₁₁	—	8	1	no	—
41	C₂×C₂₁₁	#120	1	1	yes	characteristic
42	D₂₁₁	—	4	2	no	—
43	C₂₁₁	#454	1	1	yes	characteristic, complemented, Sylow
44	#454	—	211	1	no	maximal, Sylow
45	2₋¹⁺⁴	—	211	1	no	—
46	AGL(1,ℤ/8)	—	211	2	no	—
47	#124	D₂₁₁	1	1	yes	characteristic, complemented
48	#104	—	211	2	no	—
49	#101	—	211	1	no	—
50	#41	—	422	2	no	—
51	#41	—	422	2	no	—
52	#41	—	211	2	no	—
53	C₂×Q₈	—	422	2	no	—
54	C₂×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
55	C₂×D₄	C₂×D₂₁₁	1	1	yes	characteristic
56	SD₁₆	—	422	2	no	—
57	D₈	—	422	2	no	—
58	M₁₆	—	211	2	no	—
59	C₂²⋊C₄	—	2	2	no	—
60	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
61	C₂³	—	2	1	no	—
62	Q₈	—	2	2	no	—
63	Q₈	—	422	2	no	—
64	Q₈	—	211	2	no	—
65	D₄	—	2	2	no	—
66	D₄	—	422	2	no	—
67	D₄	—	422	2	no	—
68	D₄	—	211	2	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	422	4	no	—
71	C₂×C₄	—	422	2	no	—
72	C₂×C₄	C₂²⋊D₂₁₁	1	2	yes	—
73	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
74	C₈	—	422	2	no	—
75	C₂²	—	4	2	no	—
76	C₂²	—	4	1	no	—
77	C₂²	—	422	2	no	—
78	C₂²	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	4	1	no	—
81	C₄	—	422	2	no	—
82	C₄	D₄×D₂₁₁	1	2	yes	—
83	C₂	—	2	1	no	—
84	C₂	—	8	1	no	—
85	C₂	—	844	2	no	—
86	C₂	#20269945	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269945
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#124×C₂₁₁	C₂
socle	C₂×C₂₁₁	#120
hypercenter	#124	D₂₁₁
hypocenter	C₂₁₁	#454
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #124

Complements

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

N	H ≅ G/N	type
1. #20270025	C₂	semidirect
2. #20270003	C₂	semidirect
3. #20269905	C₂	semidirect
4. #20269896	C₂	semidirect
5. #124×C₂₁₁	C₂	semidirect
43. C₂₁₁	#454	semidirect
47. #124	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20270025	2
#20270003	2
#20269905	2
#20269896	2
#124×C₂₁₁	2
#454	211