Subgroups — #27213316

Order 13504.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20270022	C₂	1	1	yes	maximal, characteristic, complemented
2	#20270005	C₂	1	2	yes	maximal, complemented
3	#20269973	C₂	1	1	yes	maximal, characteristic, complemented
4	#20269903	C₂	1	2	yes	maximal, complemented
5	#100×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#16809962	C₂²	1	2	yes	—
7	#16809961	—	2	2	no	—
8	C₂×D₈₄₄	C₂²	1	1	yes	characteristic, complemented
9	C₂×C₂²⋊D₂₁₁	—	2	2	no	—
10	C₂×C₂²⋊D₂₁₁	—	2	2	no	—
11	D₄×D₂₁₁	—	2	2	no	—
12	#16809951	—	2	2	no	—
13	D₁₆₈₈	—	2	2	no	—
14	C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic, complemented
15	C₂×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
16	M₁₆×C₂₁₁	C₂²	1	2	yes	complemented
17	C₂²×D₂₁₁	—	2	2	no	—
18	C₂²×D₂₁₁	—	2	1	no	—
19	Dic₄₂₂	D₄	1	2	yes	complemented
20	D₈₄₄	—	2	2	no	—
21	D₈₄₄	D₄	1	2	yes	complemented
22	C₂²⋊D₂₁₁	—	2	4	no	—
23	C₂²⋊D₂₁₁	—	2	2	no	—
24	C₂²⋊D₂₁₁	—	4	4	no	—
25	C₂×Dic₂₁₁	—	2	2	no	—
26	C₂×Dic₂₁₁	—	2	1	no	—
27	C₄×D₂₁₁	—	2	2	no	—
28	C₂³×C₂₁₁	D₄	1	2	yes	—
29	D₄×C₂₁₁	—	2	2	no	—
30	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
31	C₈×C₂₁₁	—	2	2	no	—
32	C₂²×C₂₁₁	—	2	2	no	—
33	C₂²×C₂₁₁	—	4	2	no	—
34	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
35	C₂×D₂₁₁	—	2	2	no	—
36	C₂×D₂₁₁	—	4	2	no	—
37	C₂×D₂₁₁	—	4	1	no	—
38	C₂×D₂₁₁	—	8	2	no	—
39	C₄×C₂₁₁	C₂×D₄	1	2	yes	—
40	Dic₂₁₁	—	2	2	no	—
41	Dic₂₁₁	—	2	2	no	—
42	C₂×C₂₁₁	—	2	1	no	—
43	C₂×C₂₁₁	—	4	2	no	—
44	C₂×C₂₁₁	#120	1	1	yes	characteristic
45	D₂₁₁	—	4	2	no	—
46	D₂₁₁	—	8	1	no	—
47	C₂₁₁	#453	1	1	yes	characteristic, complemented, Sylow
48	#453	—	211	1	no	maximal, Sylow
49	H(5,𝔽₂)	—	211	1	no	—
50	AGL(1,ℤ/8)	—	211	2	no	—
51	#127	—	211	1	no	—
52	#104	—	211	2	no	—
53	#100	D₂₁₁	1	1	yes	characteristic, complemented
54	#41	—	422	2	no	—
55	#41	—	211	2	no	—
56	C₂×D₄	—	422	2	no	—
57	C₂×D₄	—	422	2	no	—
58	C₂×D₄	—	422	2	no	—
59	C₂×D₄	—	211	1	no	—
60	C₂×D₄	C₂×D₂₁₁	1	1	yes	characteristic
61	SD₁₆	—	422	2	no	—
62	D₈	—	422	2	no	—
63	M₁₆	C₂×D₂₁₁	1	2	yes	complemented
64	C₄²	—	211	1	no	—
65	C₂³	—	422	2	no	—
66	C₂³	—	422	1	no	—
67	C₂³	D₈₄₄	1	2	yes	—
68	Q₈	—	211	2	no	—
69	D₄	—	2	2	no	—
70	D₄	—	844	4	no	—
71	D₄	—	422	4	no	—
72	D₄	—	422	2	no	—
73	D₄	—	422	2	no	—
74	D₄	—	211	2	no	—
75	C₂×C₄	—	422	2	no	—
76	C₂×C₄	—	422	1	no	—
77	C₂×C₄	—	422	2	no	—
78	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	4	2	no	—
82	C₂²	—	844	2	no	—
83	C₂²	—	844	1	no	—
84	C₂²	—	1688	2	no	—
85	C₂²	—	422	2	no	—
86	C₂²	C₂×D₈₄₄	1	1	yes	characteristic
87	C₄	—	422	2	no	—
88	C₄	—	422	2	no	—
89	C₄	D₄×D₂₁₁	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	844	2	no	—
93	C₂	—	1688	1	no	—
94	C₂	#20269947	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269947
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#100×C₂₁₁	C₂
socle	C₂×C₂₁₁	#120
hypercenter	#100	D₂₁₁
hypocenter	C₂₁₁	#453
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

Complements

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

N	H ≅ G/N	type
1. #20270022	C₂	semidirect
2. #20270005	C₂	semidirect
3. #20269973	C₂	semidirect
4. #20269903	C₂	semidirect
5. #100×C₂₁₁	C₂	semidirect
8. C₂×D₈₄₄	C₂²	semidirect
14. C₄×Dic₂₁₁	C₂²	semidirect
16. M₁₆×C₂₁₁	C₂²	semidirect
19. Dic₄₂₂	D₄	semidirect
21. D₈₄₄	D₄	semidirect
47. C₂₁₁	#453	semidirect
53. #100	D₂₁₁	semidirect
63. M₁₆	C₂×D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20270022	2
#20270005	2
#20269973	2
#20269903	2
#100×C₂₁₁	2
#453	211