Subgroups — #41003107

Order 26944.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212392	C₂	1	2	yes	maximal, complemented
2	#27212372	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212302	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212283	C₂	1	2	yes	maximal, complemented
5	C₂×D₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#20269771	C₂²	1	1	yes	characteristic
7	#20269759	—	2	2	no	—
8	#20269756	—	2	2	no	—
9	#20269755	C₂²	1	2	yes	—
10	#20269755	C₂²	1	1	yes	characteristic
11	#20269754	—	2	1	no	—
12	C₈×D₄₂₁	—	2	1	no	—
13	C₂×D₄×C₄₂₁	C₂²	1	2	yes	—
14	D₈×C₄₂₁	—	2	1	no	—
15	D₈×C₄₂₁	—	2	1	no	—
16	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
17	Dic₈₄₂	D₄	1	1	yes	characteristic
18	D₁₆₈₄	D₄	1	1	yes	characteristic
19	C₂²⋊D₄₂₁	—	2	1	no	—
20	C₄×D₄₂₁	—	2	1	no	—
21	#16809890	—	2	1	no	—
22	#16809890	—	2	2	no	—
23	C₂³×C₄₂₁	—	2	2	no	—
24	D₄×C₄₂₁	—	2	2	no	—
25	D₄×C₄₂₁	—	2	2	no	—
26	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
27	C₈×C₄₂₁	D₄	1	1	yes	characteristic
28	C₈×C₄₂₁	D₄	1	1	yes	characteristic
29	C₂²×C₄₂₁	—	4	2	no	—
30	C₂²×C₄₂₁	—	8	2	no	—
31	C₂²×C₄₂₁	#41	1	1	yes	characteristic
32	C₂×D₄₂₁	—	2	1	no	—
33	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
34	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
35	Dic₄₂₁	—	2	1	no	—
36	C₂×C₄₂₁	—	2	1	no	—
37	C₂×C₄₂₁	—	8	2	no	—
38	C₂×C₄₂₁	#121	1	1	yes	characteristic
39	D₄₂₁	—	4	1	no	—
40	C₄₂₁	#472	1	1	yes	characteristic, complemented, Sylow
41	#472	—	421	1	no	maximal, Sylow
42	AGL(1,ℤ/8)	—	421	2	no	—
43	C₂×D₈	D₄₂₁	1	1	yes	characteristic, complemented
44	#131	—	421	1	no	—
45	#108	—	421	1	no	—
46	#100	—	421	2	no	—
47	#41	—	421	1	no	—
48	C₂×D₄	C₂×D₄₂₁	1	2	yes	—
49	SD₁₆	—	842	2	no	—
50	D₈	—	2	1	no	—
51	D₈	—	2	1	no	—
52	D₈	—	842	2	no	—
53	M₁₆	—	842	1	no	—
54	M₁₆	—	421	1	no	—
55	M₁₆	—	421	2	no	—
56	C₂×C₈	—	842	1	no	—
57	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
58	C₂³	—	2	2	no	—
59	Q₈	—	421	1	no	—
60	D₄	—	2	2	no	—
61	D₄	—	2	2	no	—
62	D₄	—	842	1	no	—
63	D₄	—	421	1	no	—
64	C₂×C₄	—	842	1	no	—
65	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
66	C₈	—	842	2	no	—
67	C₈	—	842	1	no	—
68	C₈	C₂²⋊D₄₂₁	1	1	yes	characteristic
69	C₈	C₂²⋊D₄₂₁	1	1	yes	characteristic
70	C₂²	—	4	2	no	—
71	C₂²	—	8	2	no	—
72	C₂²	—	842	1	no	—
73	C₂²	#20269770	1	1	yes	characteristic
74	C₄	—	842	1	no	—
75	C₄	D₄×D₄₂₁	1	1	yes	characteristic
76	C₄	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	8	2	no	—
79	C₂	—	1684	1	no	—
80	C₂	#27212337	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27212337
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	C₂×D₈×C₄₂₁	C₂
socle	C₂×C₄₂₁	#121
hypercenter	C₂×D₈	D₄₂₁
hypocenter	C₄₂₁	#472
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #27212392	C₂	semidirect
2. #27212372	C₂	semidirect
3. #27212302	C₂	semidirect
4. #27212283	C₂	semidirect
5. C₂×D₈×C₄₂₁	C₂	semidirect
40. C₄₂₁	#472	semidirect
43. C₂×D₈	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212392	2
#27212372	2
#27212302	2
#27212283	2
C₂×D₈×C₄₂₁	2
#472	421