Subgroups — #41003920

Order 26944.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212454	C₂	1	8	yes	maximal, complemented
2	D₄₂₁⋊M₁₆	C₂	1	6	yes	maximal, complemented
3	C₂×#20269772	C₂	1	1	yes	maximal, characteristic
4	#20269781	C₂²	1	12	yes	complemented
5	#20269780	C₂²	1	4	yes	—
6	D₄₂₁⋊C₈	C₂²	1	12	yes	complemented
7	#20269772	C₄	1	4	yes	—
8	#20269772	C₂²	1	4	yes	—
9	C₂×D₁₆₈₄	C₄	1	3	yes	—
10	C₂×C₄×D₄₂₁	C₂²	1	3	yes	—
11	C₂×Q₈×C₄₂₁	C₄	1	1	yes	characteristic
12	#16809897	C₂³	1	8	yes	—
13	C₂²×D₄₂₁	C₂×C₄	1	3	yes	—
14	D₁₆₈₄	C₂×C₄	1	12	yes	—
15	C₂×Dic₄₂₁	C₂³	1	1	yes	characteristic
16	C₄×D₄₂₁	C₂×C₄	1	6	yes	—
17	C₄×D₄₂₁	C₂³	1	6	yes	—
18	Q₈×C₄₂₁	C₂×C₄	1	4	yes	—
19	C₂×C₄×C₄₂₁	C₂×C₄	1	3	yes	—
20	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₄₂₁	—	4	3	no	—
22	C₂×D₄₂₁	C₂²×C₄	1	6	yes	—
23	C₄×C₄₂₁	C₂²×C₄	1	6	yes	—
24	Dic₄₂₁	C₂²×C₄	1	1	yes	characteristic
25	Dic₄₂₁	C₂⁴	1	1	yes	characteristic
26	C₂×C₄₂₁	—	2	1	no	—
27	C₂×C₄₂₁	C₂³×C₄	1	1	yes	characteristic
28	D₄₂₁	—	2	6	no	—
29	C₄₂₁	#568	1	1	yes	characteristic, complemented, Sylow
30	#568	—	421	1	no	maximal, Sylow
31	C₂×#41	—	421	1	no	—
32	#131	—	421	8	no	—
33	C₂×M₁₆	—	421	6	no	—
34	#41	—	421	4	no	—
35	#41	—	421	4	no	—
36	C₂×Q₈	#13132997	1	1	yes	characteristic
37	C₂×D₄	—	421	3	no	—
38	C₂²×C₄	—	421	3	no	—
39	M₁₆	—	421	4	no	—
40	M₁₆	—	421	12	no	—
41	C₂×C₈	—	421	12	no	—
42	C₂³	—	421	3	no	—
43	Q₈	C₂×#13132997	1	4	yes	—
44	D₄	—	421	12	no	—
45	C₂×C₄	—	421	1	no	—
46	C₂×C₄	—	421	6	no	—
47	C₂×C₄	—	421	6	no	—
48	C₂×C₄	C₂×#13132997	1	3	yes	—
49	C₈	—	421	8	no	—
50	C₂²	—	1684	3	no	—
51	C₂²	—	421	6	no	—
52	C₂²	C₂²×#13132997	1	1	yes	characteristic
53	C₄	—	421	1	no	—
54	C₄	—	421	1	no	—
55	C₄	C₂²×#13132997	1	6	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	842	6	no	—
58	C₂	C₂³×#13132997	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#13132997
commutator, socle	C₂×C₄₂₁	C₂³×C₄
Fitting	C₂×Q₈×C₄₂₁	C₄
prime radical	C₂×#20269772	C₂
cosocle	Dic₄₂₁	C₂⁴
hypercenter	C₂×Q₈	#13132997
hypocenter	C₄₂₁	#568
LCS	G ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #27212454	C₂	semidirect
2. D₄₂₁⋊M₁₆	C₂	semidirect
4. #20269781	C₂²	semidirect
6. D₄₂₁⋊C₈	C₂²	semidirect
29. C₄₂₁	#568	semidirect

Maximal subgroups

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

subgroup	index
#27212454	2
D₄₂₁⋊M₁₆	2
C₂×#20269772	2
#568	421