Subgroups — #27213396

Order 13504.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16809945	C₂	1	1	yes	maximal, characteristic, complemented
2	#20269948	C₂	1	2	yes	maximal, complemented
3	#20269909	C₂	1	1	yes	maximal, characteristic, complemented
4	#20269898	C₂	1	2	yes	maximal, complemented
5	C₂×D₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊D₂₁₁	—	2	2	no	—
7	C₂×C₄×D₂₁₁	C₂²	1	1	yes	characteristic
8	#16809945	—	2	2	no	—
9	C₂×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
11	C₂²⋊Dic₂₁₁	—	2	2	no	—
12	C₂×D₄×C₂₁₁	C₂²	1	2	yes	—
13	D₈×C₂₁₁	—	2	2	no	—
14	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂²×D₂₁₁	D₄	1	1	yes	characteristic
16	C₂²⋊D₂₁₁	—	4	4	no	—
17	C₂×Dic₂₁₁	—	2	2	no	—
18	C₂×Dic₂₁₁	D₄	1	1	yes	characteristic
19	C₄×D₂₁₁	—	2	1	no	—
20	C₄×D₂₁₁	—	2	1	no	—
21	C₂₁₁⋊C₈	—	2	1	no	—
22	C₂³×C₂₁₁	—	2	2	no	—
23	D₄×C₂₁₁	—	2	2	no	—
24	D₄×C₂₁₁	—	2	2	no	—
25	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
26	C₈×C₂₁₁	D₄	1	2	yes	complemented
27	C₂²×C₂₁₁	—	4	2	no	—
28	C₂²×C₂₁₁	—	4	2	no	—
29	C₂²×C₂₁₁	—	4	2	no	—
30	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
31	C₂×D₂₁₁	—	2	1	no	—
32	C₂×D₂₁₁	—	2	2	no	—
33	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
34	C₄×C₂₁₁	#41	1	1	yes	characteristic
35	Dic₂₁₁	—	2	1	no	—
36	Dic₂₁₁	—	4	2	no	—
37	C₂×C₂₁₁	—	8	2	no	—
38	C₂×C₂₁₁	#121	1	1	yes	characteristic
39	C₂×C₂₁₁	AGL(1,ℤ/8)	1	2	yes	—
40	D₂₁₁	—	4	1	no	—
41	C₂₁₁	#469	1	1	yes	characteristic, complemented, Sylow
42	#469	—	211	1	no	maximal, Sylow
43	C₂×D₈	D₂₁₁	1	1	yes	characteristic, complemented
44	C₂×M₁₆	—	211	1	no	—
45	#121	—	211	2	no	—
46	#106	—	211	1	no	—
47	#102	—	211	2	no	—
48	C₂×D₄	—	422	2	no	—
49	C₂×D₄	C₂×D₂₁₁	1	2	yes	—
50	C₂²×C₄	—	211	1	no	—
51	D₈	—	2	2	no	—
52	M₁₆	—	422	2	no	—
53	C₂×C₈	—	211	1	no	—
54	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
55	C₄⋊C₄	—	211	2	no	—
56	C₂²⋊C₄	—	422	2	no	—
57	C₂³	—	2	2	no	—
58	C₂³	—	211	1	no	—
59	D₄	—	2	2	no	—
60	D₄	—	2	2	no	—
61	D₄	—	844	4	no	—
62	C₂×C₄	—	422	2	no	—
63	C₂×C₄	—	422	1	no	—
64	C₂×C₄	—	422	1	no	—
65	C₂×C₄	—	211	1	no	—
66	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
67	C₈	—	422	1	no	—
68	C₈	C₂²⋊D₂₁₁	1	2	yes	complemented
69	C₂²	—	4	2	no	—
70	C₂²	—	4	2	no	—
71	C₂²	—	4	2	no	—
72	C₂²	—	422	1	no	—
73	C₂²	—	422	2	no	—
74	C₂²	D₄×D₂₁₁	1	1	yes	characteristic
75	C₄	—	844	2	no	—
76	C₄	—	422	1	no	—
77	C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
78	C₄	#16809961	1	1	yes	characteristic
79	C₂	—	8	2	no	—
80	C₂	—	844	1	no	—
81	C₂	#20269948	1	1	yes	characteristic
82	C₂	#20270001	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₁₁
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	C₂×D₈×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×D₄
hypercenter	C₂×D₈	D₂₁₁
hypocenter	C₂₁₁	#469
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16809945	C₂	semidirect
2. #20269948	C₂	semidirect
3. #20269909	C₂	semidirect
4. #20269898	C₂	semidirect
5. C₂×D₈×C₂₁₁	C₂	semidirect
26. C₈×C₂₁₁	D₄	semidirect
41. C₂₁₁	#469	semidirect
43. C₂×D₈	D₂₁₁	semidirect
68. C₈	C₂²⋊D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16809945	2
#20269948	2
#20269909	2
#20269898	2
C₂×D₈×C₂₁₁	2
#469	211