Subgroups — #41002905

Order 26944.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄₂₁	C₂	1	2	yes	maximal, complemented
2	C₄×D₁₆₈₄	C₂	1	1	yes	maximal, characteristic
3	#27212278	C₂	1	2	yes	maximal, complemented
4	C₄×#16809890	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆₈₄	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄₂₁	C₂²	1	2	yes	—
8	C₂×#16809890	C₂²	1	1	yes	characteristic
9	C₂×#16809890	C₂²	1	1	yes	characteristic
10	C₈×D₄₂₁	—	2	4	no	—
11	C₄⋊Dic₄₂₁	C₄	1	1	yes	characteristic
12	#20269748	C₄	1	2	yes	—
13	C₂×C₈×C₄₂₁	C₂²	1	2	yes	—
14	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
15	C₂²×D₄₂₁	C₂×C₄	1	2	yes	—
16	D₁₆₈₄	C₈	1	4	yes	complemented
17	C₂×Dic₄₂₁	C₂×C₄	1	2	yes	—
18	C₄×D₄₂₁	—	2	2	no	—
19	C₄×D₄₂₁	—	2	2	no	—
20	#16809890	—	2	1	no	—
21	#16809890	D₄	1	2	yes	complemented
22	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
25	C₈×C₄₂₁	—	2	2	no	—
26	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₄₂₁	—	2	2	no	—
28	C₂×D₄₂₁	—	2	2	no	—
29	C₂×D₄₂₁	C₂×C₈	1	4	yes	complemented
30	C₄×C₄₂₁	—	2	1	no	—
31	C₄×C₄₂₁	C₂×C₈	1	2	yes	complemented
32	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
33	C₄×C₄₂₁	#41	1	1	yes	characteristic
34	Dic₄₂₁	—	2	2	no	—
35	C₂×C₄₂₁	C₄×D₄	1	1	yes	characteristic
36	C₂×C₄₂₁	C₂²×C₈	1	1	yes	characteristic
37	C₂×C₄₂₁	#131	1	1	yes	characteristic
38	D₄₂₁	—	2	4	no	—
39	C₄₂₁	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
40	C₈×D₄	—	421	1	no	maximal, Sylow
41	C₂²×C₈	—	421	2	no	—
42	C₄×D₄	—	421	1	no	—
43	C₄⋊C₈	D₄₂₁	1	1	yes	characteristic, complemented
44	C₂²⋊C₈	—	421	2	no	—
45	C₄×C₈	—	421	1	no	—
46	C₂×D₄	—	421	1	no	—
47	C₂²×C₄	—	421	2	no	—
48	C₂×C₈	—	842	4	no	—
49	C₂×C₈	—	421	1	no	—
50	C₂×C₈	—	421	1	no	—
51	C₂×C₈	C₂×D₄₂₁	1	2	yes	—
52	C₄⋊C₄	—	421	1	no	—
53	C₂²⋊C₄	—	421	2	no	—
54	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
55	C₂³	—	421	2	no	—
56	D₄	—	421	4	no	—
57	C₂×C₄	—	842	2	no	—
58	C₂×C₄	—	842	2	no	—
59	C₂×C₄	—	421	2	no	—
60	C₂×C₄	C₄×D₄₂₁	1	1	yes	characteristic
61	C₂×C₄	C₄×D₄₂₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₈	—	842	1	no	—
65	C₈	—	421	2	no	—
66	C₂²	—	842	2	no	—
67	C₂²	—	842	2	no	—
68	C₂²	—	421	4	no	—
69	C₂²	C₂×C₄×D₄₂₁	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	842	2	no	—
72	C₄	C₈×D₄₂₁	1	2	yes	complemented
73	C₄	D₄×D₄₂₁	1	1	yes	characteristic
74	C₄	#20269772	1	1	yes	characteristic
75	C₂	—	842	4	no	—
76	C₂	#27212328	1	1	yes	characteristic
77	C₂	C₂×C₈×D₄₂₁	1	1	yes	characteristic
78	C₂	#27212374	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₂₁
commutator	C₂×C₄₂₁	C₂²×C₈
Fitting	C₄⋊C₈×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂²×C₄
prime radical	C₂×D₁₆₈₄	C₄
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	C₄⋊C₈	D₄₂₁
hypocenter	C₄₂₁	C₈×D₄
LCS	G ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×D₄₂₁	C₂	semidirect
3. #27212278	C₂	semidirect
4. C₄×#16809890	C₂	semidirect
5. C₄⋊C₈×C₄₂₁	C₂	semidirect
16. D₁₆₈₄	C₈	semidirect
21. #16809890	D₄	semidirect
29. C₂×D₄₂₁	C₂×C₈	semidirect
31. C₄×C₄₂₁	C₂×C₈	semidirect
39. C₄₂₁	C₈×D₄	semidirect
43. C₄⋊C₈	D₄₂₁	semidirect
72. C₄	C₈×D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄₂₁	2
C₄×D₁₆₈₄	2
#27212278	2
C₄×#16809890	2
C₄⋊C₈×C₄₂₁	2
C₈×D₄	421