Subgroups — #41003475

Order 26944.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×D₄₂₁	C₂	1	4	yes	maximal, complemented
2	#27212337	C₂	1	4	yes	maximal, complemented
3	#27212334	C₂	1	4	yes	maximal, complemented
4	C₄×D₁₆₈₄	C₂	1	2	yes	maximal, complemented
5	#127×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄₂₁	C₂²	1	4	yes	complemented
7	C₂×D₁₆₈₄	C₂²	1	2	yes	—
8	C₂×C₂²⋊D₄₂₁	C₂²	1	8	yes	complemented
9	D₄×D₄₂₁	—	2	16	no	—
10	C₂×C₄×D₄₂₁	C₂²	1	4	yes	complemented
11	C₄⋊Dic₄₂₁	C₂²	1	2	yes	complemented
12	C₂²⋊Dic₄₂₁	C₂²	1	4	yes	complemented
13	#20269748	C₂²	1	4	yes	complemented
14	C₂×D₄×C₄₂₁	C₂²	1	4	yes	complemented
15	C₂×D₄×C₄₂₁	C₂²	1	2	yes	complemented
16	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic, complemented
17	C₂²×D₄₂₁	—	2	16	no	—
18	C₂²×D₄₂₁	—	4	4	no	—
19	C₂²×D₄₂₁	C₂³	1	4	yes	—
20	D₁₆₈₄	D₄	1	8	yes	complemented
21	C₂²⋊D₄₂₁	—	2	16	no	—
22	C₂×Dic₄₂₁	C₂³	1	4	yes	—
23	C₄×D₄₂₁	—	2	8	no	—
24	C₂³×C₄₂₁	C₂³	1	4	yes	—
25	D₄×C₄₂₁	—	2	8	no	—
26	D₄×C₄₂₁	—	4	2	no	—
27	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
29	C₂²×C₄₂₁	—	2	8	no	—
30	C₂²×C₄₂₁	—	2	4	no	—
31	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
32	C₂×D₄₂₁	—	2	4	no	—
33	C₂×D₄₂₁	—	2	4	no	—
34	C₂×D₄₂₁	—	4	16	no	—
35	C₂×D₄₂₁	C₂×D₄	1	8	yes	complemented
36	C₄×C₄₂₁	—	2	1	no	—
37	C₄×C₄₂₁	C₂×D₄	1	4	yes	complemented
38	Dic₄₂₁	—	2	4	no	—
39	C₂×C₄₂₁	—	4	4	no	—
40	C₂×C₄₂₁	C₂²×D₄	1	2	yes	—
41	C₂×C₄₂₁	H(5,𝔽₂)	1	1	yes	characteristic
42	D₄₂₁	—	2	8	no	—
43	C₄₂₁	D₄²	1	1	yes	characteristic, complemented, Sylow
44	D₄²	—	421	1	no	maximal, Sylow
45	C₂²×D₄	—	421	4	no	—
46	#127	D₄₂₁	1	1	yes	characteristic, complemented
47	#121	—	421	4	no	—
48	#120	—	421	4	no	—
49	C₄×D₄	—	421	2	no	—
50	C₂⁴	—	421	4	no	—
51	C₂×D₄	—	842	16	no	—
52	C₂×D₄	—	421	8	no	—
53	C₂×D₄	—	421	2	no	—
54	C₂×D₄	C₂×D₄₂₁	1	4	yes	complemented
55	C₂×D₄	C₂×D₄₂₁	1	2	yes	complemented
56	C₂²×C₄	—	421	4	no	—
57	C₄⋊C₄	—	421	2	no	—
58	C₂²⋊C₄	—	421	4	no	—
59	C₂²⋊C₄	—	421	4	no	—
60	C₄²	C₂×D₄₂₁	1	1	yes	characteristic, complemented
61	C₂³	—	842	16	no	—
62	C₂³	—	1684	4	no	—
63	C₂³	—	421	4	no	—
64	C₂³	C₂²×D₄₂₁	1	4	yes	—
65	D₄	—	2	8	no	—
66	D₄	—	4	2	no	—
67	D₄	—	842	16	no	—
68	D₄	—	421	8	no	—
69	C₂×C₄	—	842	8	no	—
70	C₂×C₄	—	421	4	no	—
71	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
73	C₂²	—	2	8	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	842	4	no	—
76	C₂²	—	842	4	no	—
77	C₂²	—	1684	16	no	—
78	C₂²	—	421	8	no	—
79	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	842	4	no	—
82	C₄	D₄×D₄₂₁	1	4	yes	complemented
83	C₂	—	4	4	no	—
84	C₂	—	842	8	no	—
85	C₂	C₂×D₄×D₄₂₁	1	2	yes	—
86	C₂	#27212411	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₂₁
commutator, socle, cosocle	C₂²×C₄₂₁	C₂⁴
Fitting	#127×C₄₂₁	C₂
hypercenter	#127	D₄₂₁
hypocenter	C₄₂₁	D₄²
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×D₄₂₁	C₂	semidirect
2. #27212337	C₂	semidirect
3. #27212334	C₂	semidirect
4. C₄×D₁₆₈₄	C₂	semidirect
5. #127×C₄₂₁	C₂	semidirect
6. C₂³×D₄₂₁	C₂²	semidirect
8. C₂×C₂²⋊D₄₂₁	C₂²	semidirect
10. C₂×C₄×D₄₂₁	C₂²	semidirect
11. C₄⋊Dic₄₂₁	C₂²	semidirect
12. C₂²⋊Dic₄₂₁	C₂²	semidirect
13. #20269748	C₂²	semidirect
14. C₂×D₄×C₄₂₁	C₂²	semidirect
15. C₂×D₄×C₄₂₁	C₂²	semidirect
16. C₄²×C₄₂₁	C₂²	semidirect
20. D₁₆₈₄	D₄	semidirect
35. C₂×D₄₂₁	C₂×D₄	semidirect
37. C₄×C₄₂₁	C₂×D₄	semidirect
43. C₄₂₁	D₄²	semidirect
46. #127	D₄₂₁	semidirect
54. C₂×D₄	C₂×D₄₂₁	semidirect
55. C₂×D₄	C₂×D₄₂₁	semidirect
60. C₄²	C₂×D₄₂₁	semidirect
82. C₄	D₄×D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₄₂₁	2
#27212337	2
#27212334	2
C₄×D₁₆₈₄	2
#127×C₄₂₁	2
D₄²	421