Subgroups — #27213781

Order 13504.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×D₂₁₁	C₂	1	4	yes	maximal, complemented
2	#20269948	C₂	1	4	yes	maximal, complemented
3	#20269945	C₂	1	4	yes	maximal, complemented
4	C₄×D₈₄₄	C₂	1	2	yes	maximal, complemented
5	#127×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₂₁₁	C₂²	1	4	yes	complemented
7	C₂×D₈₄₄	C₂²	1	2	yes	—
8	C₂×C₂²⋊D₂₁₁	C₂²	1	8	yes	complemented
9	D₄×D₂₁₁	—	2	16	no	—
10	C₂×C₄×D₂₁₁	C₂²	1	4	yes	complemented
11	C₄⋊Dic₂₁₁	C₂²	1	2	yes	complemented
12	C₂²⋊Dic₂₁₁	C₂²	1	4	yes	complemented
13	(C₂×D₂₁₁)⋊C₄	C₂²	1	4	yes	complemented
14	C₂×D₄×C₂₁₁	C₂²	1	4	yes	complemented
15	C₂×D₄×C₂₁₁	C₂²	1	2	yes	complemented
16	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic, complemented
17	C₂²×D₂₁₁	—	2	16	no	—
18	C₂²×D₂₁₁	—	4	4	no	—
19	C₂²×D₂₁₁	C₂³	1	4	yes	—
20	D₈₄₄	D₄	1	8	yes	complemented
21	C₂²⋊D₂₁₁	—	2	16	no	—
22	C₂×Dic₂₁₁	C₂³	1	4	yes	—
23	C₄×D₂₁₁	—	2	8	no	—
24	C₂³×C₂₁₁	C₂³	1	4	yes	—
25	D₄×C₂₁₁	—	2	8	no	—
26	D₄×C₂₁₁	—	4	2	no	—
27	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₂₁₁	C₂³	1	2	yes	—
29	C₂²×C₂₁₁	—	2	8	no	—
30	C₂²×C₂₁₁	—	2	4	no	—
31	C₂²×C₂₁₁	C₂⁴	1	1	yes	characteristic
32	C₂×D₂₁₁	—	2	4	no	—
33	C₂×D₂₁₁	—	2	4	no	—
34	C₂×D₂₁₁	—	4	16	no	—
35	C₂×D₂₁₁	C₂×D₄	1	8	yes	complemented
36	C₄×C₂₁₁	—	2	1	no	—
37	C₄×C₂₁₁	C₂×D₄	1	4	yes	complemented
38	Dic₂₁₁	—	2	4	no	—
39	C₂×C₂₁₁	—	4	4	no	—
40	C₂×C₂₁₁	C₂²×D₄	1	2	yes	—
41	C₂×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
42	D₂₁₁	—	2	8	no	—
43	C₂₁₁	D₄²	1	1	yes	characteristic, complemented, Sylow
44	D₄²	—	211	1	no	maximal, Sylow
45	C₂²×D₄	—	211	4	no	—
46	#127	D₂₁₁	1	1	yes	characteristic, complemented
47	#121	—	211	4	no	—
48	#120	—	211	4	no	—
49	C₄×D₄	—	211	2	no	—
50	C₂⁴	—	211	4	no	—
51	C₂×D₄	—	422	16	no	—
52	C₂×D₄	—	211	8	no	—
53	C₂×D₄	—	211	2	no	—
54	C₂×D₄	C₂×D₂₁₁	1	4	yes	complemented
55	C₂×D₄	C₂×D₂₁₁	1	2	yes	complemented
56	C₂²×C₄	—	211	4	no	—
57	C₄⋊C₄	—	211	2	no	—
58	C₂²⋊C₄	—	211	4	no	—
59	C₂²⋊C₄	—	211	4	no	—
60	C₄²	C₂×D₂₁₁	1	1	yes	characteristic, complemented
61	C₂³	—	844	4	no	—
62	C₂³	—	422	16	no	—
63	C₂³	—	211	4	no	—
64	C₂³	C₂²×D₂₁₁	1	4	yes	—
65	D₄	—	2	8	no	—
66	D₄	—	4	2	no	—
67	D₄	—	422	16	no	—
68	D₄	—	211	8	no	—
69	C₂×C₄	—	422	8	no	—
70	C₂×C₄	—	211	4	no	—
71	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₂₁₁	1	2	yes	—
73	C₂²	—	2	8	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	844	16	no	—
76	C₂²	—	422	4	no	—
77	C₂²	—	422	4	no	—
78	C₂²	—	211	8	no	—
79	C₂²	C₂³×D₂₁₁	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	422	4	no	—
82	C₄	D₄×D₂₁₁	1	4	yes	complemented
83	C₂	—	4	4	no	—
84	C₂	—	422	8	no	—
85	C₂	C₂×D₄×D₂₁₁	1	2	yes	—
86	C₂	#20270022	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁₁
commutator, socle, cosocle	C₂²×C₂₁₁	C₂⁴
Fitting	#127×C₂₁₁	C₂
hypercenter	#127	D₂₁₁
hypocenter	C₂₁₁	D₄²
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×D₂₁₁	C₂	semidirect
2. #20269948	C₂	semidirect
3. #20269945	C₂	semidirect
4. C₄×D₈₄₄	C₂	semidirect
5. #127×C₂₁₁	C₂	semidirect
6. C₂³×D₂₁₁	C₂²	semidirect
8. C₂×C₂²⋊D₂₁₁	C₂²	semidirect
10. C₂×C₄×D₂₁₁	C₂²	semidirect
11. C₄⋊Dic₂₁₁	C₂²	semidirect
12. C₂²⋊Dic₂₁₁	C₂²	semidirect
13. (C₂×D₂₁₁)⋊C₄	C₂²	semidirect
14. C₂×D₄×C₂₁₁	C₂²	semidirect
15. C₂×D₄×C₂₁₁	C₂²	semidirect
16. C₄²×C₂₁₁	C₂²	semidirect
20. D₈₄₄	D₄	semidirect
35. C₂×D₂₁₁	C₂×D₄	semidirect
37. C₄×C₂₁₁	C₂×D₄	semidirect
43. C₂₁₁	D₄²	semidirect
46. #127	D₂₁₁	semidirect
54. C₂×D₄	C₂×D₂₁₁	semidirect
55. C₂×D₄	C₂×D₂₁₁	semidirect
60. C₄²	C₂×D₂₁₁	semidirect
82. C₄	D₄×D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₂₁₁	2
#20269948	2
#20269945	2
C₄×D₈₄₄	2
#127×C₂₁₁	2
D₄²	211