Subgroups — #41003248

Order 26944.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212365	C₂	1	1	yes	maximal, characteristic
2	#27212357	C₂	1	1	yes	maximal, characteristic
3	#27212299	C₂	1	1	yes	maximal, characteristic
4	#27212298	C₂	1	1	yes	maximal, characteristic
5	#27212297	C₂	1	1	yes	maximal, characteristic
6	#27212276	C₂	1	1	yes	maximal, characteristic
7	#107×C₄₂₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₈₄₂	—	2	1	no	—
9	C₂×#16809890	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₄₂₁	—	2	1	no	—
11	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
13	#20269750	—	2	1	no	—
14	#20269750	—	2	1	no	—
15	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₄₂₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₄₂₁	C₂²	1	1	yes	characteristic
19	Dic₈₄₂	—	4	2	no	—
20	C₂×Dic₄₂₁	—	2	1	no	—
21	C₂×Dic₄₂₁	—	2	1	no	—
22	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
23	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
24	#16809890	Q₈	1	2	yes	complemented
25	C₂×C₄×C₄₂₁	—	2	1	no	—
26	C₂×C₄×C₄₂₁	—	2	1	no	—
27	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
28	C₈×C₄₂₁	Q₈	1	2	yes	complemented
29	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
30	C₄×C₄₂₁	—	4	1	no	—
31	C₄×C₄₂₁	—	4	1	no	—
32	C₄×C₄₂₁	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₄₂₁	C₂×Q₈	1	1	yes	characteristic
34	Dic₄₂₁	—	2	1	no	—
35	Dic₄₂₁	—	2	1	no	—
36	Dic₄₂₁	—	4	1	no	—
37	Dic₄₂₁	—	4	1	no	—
38	C₂×C₄₂₁	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₄₂₁	AGL(1,ℤ/8)	1	1	yes	characteristic
40	C₂×C₄₂₁	#137	1	1	yes	characteristic
41	C₄₂₁	#501	1	1	yes	characteristic, complemented, Sylow
42	#501	—	421	1	no	maximal, Sylow
43	C₄⋊Q₈	—	421	1	no	—
44	#125	—	421	1	no	—
45	#107	—	421	1	no	—
46	#107	D₄₂₁	1	1	yes	characteristic
47	#106	—	421	1	no	—
48	#106	—	421	1	no	—
49	#97	—	421	1	no	—
50	C₂×Q₈	—	842	1	no	—
51	C₂×C₈	—	421	1	no	—
52	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
53	C₄⋊C₄	—	842	1	no	—
54	C₄⋊C₄	—	842	1	no	—
55	C₄⋊C₄	—	842	1	no	—
56	C₄⋊C₄	—	421	1	no	—
57	C₄⋊C₄	—	421	1	no	—
58	C₄⋊C₄	C₂×D₄₂₁	1	1	yes	characteristic
59	C₄⋊C₄	C₂×D₄₂₁	1	1	yes	characteristic
60	C₄²	—	421	1	no	—
61	Q₈	—	1684	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	842	1	no	—
65	C₂×C₄	—	842	1	no	—
66	C₂×C₄	—	421	1	no	—
67	C₂×C₄	—	421	1	no	—
68	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
69	C₈	—	421	2	no	—
70	C₈	Dic₈₄₂	1	2	yes	complemented
71	C₂²	D₄×D₄₂₁	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	842	1	no	—
75	C₄	—	842	1	no	—
76	C₄	—	1684	1	no	—
77	C₄	—	1684	1	no	—
78	C₄	Q₈×D₄₂₁	1	1	yes	characteristic
79	C₄	C₂×Dic₈₄₂	1	1	yes	characteristic
80	C₂	#27212365	1	1	yes	characteristic
81	C₂	#27212390	1	1	yes	characteristic
82	C₂	#27212398	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₂₁
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#107×C₄₂₁	C₂
socle, prime radical	C₂²×C₄₂₁	C₂×D₄
hypercenter	#107	D₄₂₁
hypocenter	C₄₂₁	#501
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
24. #16809890	Q₈	semidirect
28. C₈×C₄₂₁	Q₈	semidirect
41. C₄₂₁	#501	semidirect
70. C₈	Dic₈₄₂	semidirect

Maximal subgroups

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

subgroup	index
#27212365	2
#27212357	2
#27212299	2
#27212298	2
#27212297	2
#27212276	2
#107×C₄₂₁	2
#501	421