Subgroups — #41003216

Order 26944.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269756	C₂	1	4	yes	maximal, complemented
2	#27212363	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#16809890	C₂	1	1	yes	maximal, characteristic, complemented
4	#127×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₆₈₄	—	2	2	no	—
6	C₂×D₁₆₈₄	C₂²	1	2	yes	—
7	#20269756	—	2	8	no	—
8	C₂×#16809890	C₂²	1	2	yes	—
9	C₂×D₄×C₄₂₁	—	2	2	no	—
10	C₂×D₄×C₄₂₁	C₂²	1	2	yes	—
11	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
12	C₂²×D₄₂₁	—	2	2	no	—
13	D₁₆₈₄	—	2	4	no	—
14	D₁₆₈₄	—	4	4	no	—
15	#16809890	D₄	1	4	yes	complemented
16	C₂³×C₄₂₁	—	2	2	no	—
17	D₄×C₄₂₁	—	2	4	no	—
18	D₄×C₄₂₁	—	4	4	no	—
19	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
20	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
21	C₂²×C₄₂₁	—	4	2	no	—
22	C₂²×C₄₂₁	—	4	4	no	—
23	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
24	C₂×D₄₂₁	—	4	2	no	—
25	C₂×D₄₂₁	—	4	4	no	—
26	C₄×C₄₂₁	D₈	1	4	yes	complemented
27	C₄×C₄₂₁	C₂×D₄	1	2	yes	—
28	C₂×C₄₂₁	—	8	2	no	—
29	C₂×C₄₂₁	#127	1	1	yes	characteristic
30	C₂×C₄₂₁	C₂×D₈	1	2	yes	—
31	D₄₂₁	—	8	2	no	—
32	C₄₂₁	#493	1	1	yes	characteristic, complemented, Sylow
33	#493	—	421	1	no	maximal, Sylow
34	C₂×D₈	—	421	4	no	—
35	#127	—	421	1	no	—
36	#127	D₄₂₁	1	1	yes	characteristic, complemented
37	C₄×C₈	—	421	1	no	—
38	C₂×D₄	—	2	2	no	—
39	C₂×D₄	—	842	2	no	—
40	C₂×D₄	—	421	2	no	—
41	C₂×D₄	C₂×D₄₂₁	1	2	yes	—
42	D₈	—	842	8	no	—
43	C₂×C₈	—	421	2	no	—
44	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
45	C₂³	—	2	2	no	—
46	C₂³	—	842	2	no	—
47	D₄	—	2	4	no	—
48	D₄	—	4	4	no	—
49	D₄	—	842	4	no	—
50	D₄	—	1684	4	no	—
51	C₂×C₄	C₂²⋊D₄₂₁	1	2	yes	—
52	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
53	C₈	—	421	4	no	—
54	C₂²	—	4	2	no	—
55	C₂²	—	4	4	no	—
56	C₂²	—	1684	2	no	—
57	C₂²	—	1684	4	no	—
58	C₂²	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
59	C₄	#20269756	1	4	yes	complemented
60	C₄	D₄×D₄₂₁	1	2	yes	—
61	C₂	—	8	2	no	—
62	C₂	—	3368	2	no	—
63	C₂	#27212362	1	1	yes	characteristic
64	C₂	C₂×#20269756	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₂₁
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#127×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
hypercenter	#127	D₄₂₁
hypocenter	C₄₂₁	#493
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×#20269756	C₂	semidirect
2. #27212363	C₂	semidirect
3. C₄×#16809890	C₂	semidirect
4. #127×C₄₂₁	C₂	semidirect
15. #16809890	D₄	semidirect
26. C₄×C₄₂₁	D₈	semidirect
32. C₄₂₁	#493	semidirect
36. #127	D₄₂₁	semidirect
59. C₄	#20269756	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269756	2
#27212363	2
C₄×#16809890	2
#127×C₄₂₁	2
#493	421