Subgroups — #41003073

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	1	yes	maximal, characteristic, complemented
2	#27212337	C₂	1	2	yes	maximal, complemented
3	#27212300	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212287	C₂	1	2	yes	maximal, complemented
5	C₂×D₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊D₄₂₁	—	2	2	no	—
7	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
8	C₂×#16809890	C₂²	1	1	yes	characteristic
9	C₈×D₄₂₁	—	2	2	no	—
10	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
11	C₂²⋊Dic₄₂₁	—	2	2	no	—
12	C₂×D₄×C₄₂₁	C₂²	1	2	yes	—
13	D₈×C₄₂₁	—	2	2	no	—
14	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
15	C₂²×D₄₂₁	D₄	1	1	yes	characteristic
16	C₂²⋊D₄₂₁	—	4	4	no	—
17	C₂×Dic₄₂₁	—	2	2	no	—
18	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
19	C₄×D₄₂₁	—	2	1	no	—
20	C₄×D₄₂₁	—	2	1	no	—
21	#16809890	—	2	1	no	—
22	C₂³×C₄₂₁	—	2	2	no	—
23	D₄×C₄₂₁	—	2	2	no	—
24	D₄×C₄₂₁	—	2	2	no	—
25	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
26	C₈×C₄₂₁	D₄	1	2	yes	complemented
27	C₂²×C₄₂₁	—	4	2	no	—
28	C₂²×C₄₂₁	—	4	2	no	—
29	C₂²×C₄₂₁	—	4	2	no	—
30	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
31	C₂×D₄₂₁	—	2	1	no	—
32	C₂×D₄₂₁	—	2	1	no	—
33	C₂×D₄₂₁	D₈	1	2	yes	complemented
34	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
35	C₄×C₄₂₁	#41	1	1	yes	characteristic
36	Dic₄₂₁	—	2	1	no	—
37	Dic₄₂₁	—	4	2	no	—
38	C₂×C₄₂₁	—	8	2	no	—
39	C₂×C₄₂₁	#121	1	1	yes	characteristic
40	C₂×C₄₂₁	C₂×D₈	1	1	yes	characteristic
41	C₂×C₄₂₁	#135	1	1	yes	characteristic
42	D₄₂₁	—	2	2	no	—
43	C₄₂₁	#466	1	1	yes	characteristic, complemented, Sylow
44	#466	—	421	1	no	maximal, Sylow
45	C₂×D₈	D₄₂₁	1	1	yes	characteristic, complemented
46	C₂²×C₈	—	421	1	no	—
47	#121	—	421	2	no	—
48	#107	—	421	1	no	—
49	#102	—	421	2	no	—
50	C₂×D₄	—	842	2	no	—
51	C₂×D₄	C₂×D₄₂₁	1	2	yes	—
52	C₂²×C₄	—	421	1	no	—
53	D₈	—	2	2	no	—
54	C₂×C₈	—	842	2	no	—
55	C₂×C₈	—	421	1	no	—
56	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
57	C₄⋊C₄	—	421	2	no	—
58	C₂²⋊C₄	—	842	2	no	—
59	C₂³	—	2	2	no	—
60	C₂³	—	421	1	no	—
61	D₄	—	2	2	no	—
62	D₄	—	2	2	no	—
63	D₄	—	1684	4	no	—
64	C₂×C₄	—	842	1	no	—
65	C₂×C₄	—	842	1	no	—
66	C₂×C₄	—	842	2	no	—
67	C₂×C₄	—	421	1	no	—
68	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
69	C₈	—	842	1	no	—
70	C₈	C₂²⋊D₄₂₁	1	2	yes	complemented
71	C₂²	—	4	2	no	—
72	C₂²	—	4	2	no	—
73	C₂²	—	4	2	no	—
74	C₂²	—	842	1	no	—
75	C₂²	—	842	1	no	—
76	C₂²	—	421	2	no	—
77	C₂²	D₄×D₄₂₁	1	1	yes	characteristic
78	C₄	—	842	1	no	—
79	C₄	—	1684	2	no	—
80	C₄	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
81	C₄	#20269770	1	1	yes	characteristic
82	C₂	—	8	2	no	—
83	C₂	—	842	2	no	—
84	C₂	#27212337	1	1	yes	characteristic
85	C₂	D₈×D₄₂₁	1	1	yes	characteristic
86	C₂	#27212385	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₂₁
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	C₂×D₈×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
hypercenter	C₂×D₈	D₄₂₁
hypocenter	C₄₂₁	#466
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×D₄₂₁	C₂	semidirect
2. #27212337	C₂	semidirect
3. #27212300	C₂	semidirect
4. #27212287	C₂	semidirect
5. C₂×D₈×C₄₂₁	C₂	semidirect
26. C₈×C₄₂₁	D₄	semidirect
33. C₂×D₄₂₁	D₈	semidirect
43. C₄₂₁	#466	semidirect
45. C₂×D₈	D₄₂₁	semidirect
70. C₈	C₂²⋊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
#27212337	2
#27212300	2
#27212287	2
C₂×D₈×C₄₂₁	2
#466	421