Subgroups — #41002962

Order 26944.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212373	C₂	1	2	yes	maximal, complemented
2	#27212302	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212293	C₂	1	2	yes	maximal, complemented
4	#27212276	C₂	1	1	yes	maximal, characteristic, complemented
5	#135×C₄₂₁	C₂	1	1	yes	maximal, characteristic
6	#20269755	—	2	2	no	—
7	#20269755	C₂²	1	2	yes	—
8	C₂×#16809890	—	2	2	no	—
9	C₂×#16809890	C₂²	1	1	yes	characteristic
10	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
11	#41×C₄₂₁	C₂²	1	2	yes	—
12	Q₁₆×C₄₂₁	C₄	1	1	yes	characteristic, complemented
13	SD₁₆×C₄₂₁	C₄	1	2	yes	complemented
14	D₈×C₄₂₁	C₄	1	1	yes	characteristic, complemented
15	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₂₁	—	2	1	no	—
17	#16809890	—	2	2	no	—
18	#16809890	D₄	1	2	yes	—
19	Q₈×C₄₂₁	C₂×C₄	1	2	yes	—
20	D₄×C₄₂₁	—	2	2	no	—
21	D₄×C₄₂₁	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₄₂₁	—	2	2	no	—
23	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
24	C₈×C₄₂₁	C₂×C₄	1	1	yes	characteristic
25	C₈×C₄₂₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄₂₁	—	2	2	no	—
27	C₂²×C₄₂₁	#41	1	1	yes	characteristic
28	C₄×C₄₂₁	—	2	2	no	—
29	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
31	Dic₄₂₁	—	4	1	no	—
32	C₂×C₄₂₁	—	2	1	no	—
33	C₂×C₄₂₁	—	4	2	no	—
34	C₂×C₄₂₁	C₄×D₄	1	1	yes	characteristic
35	C₄₂₁	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	421	1	no	maximal, Sylow
37	#135	D₄₂₁	1	1	yes	characteristic
38	#131	—	421	2	no	—
39	#108	—	421	1	no	—
40	#104	—	421	2	no	—
41	#97	—	421	1	no	—
42	#41	C₂×D₄₂₁	1	2	yes	—
43	Q₁₆	Dic₄₂₁	1	1	yes	characteristic, complemented
44	SD₁₆	Dic₄₂₁	1	2	yes	complemented
45	D₈	Dic₄₂₁	1	1	yes	characteristic, complemented
46	M₁₆	—	842	2	no	—
47	M₁₆	—	421	2	no	—
48	C₂×C₈	—	842	2	no	—
49	C₂×C₈	—	421	1	no	—
50	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
51	C₄²	—	421	1	no	—
52	Q₈	C₂×Dic₄₂₁	1	2	yes	—
53	D₄	—	2	2	no	—
54	D₄	C₂×Dic₄₂₁	1	2	yes	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	842	1	no	—
57	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
58	C₈	—	842	2	no	—
59	C₈	—	421	2	no	—
60	C₈	C₂×Dic₄₂₁	1	1	yes	characteristic
61	C₈	C₂×Dic₄₂₁	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	#20269770	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	1684	1	no	—
66	C₄	C₂²×Dic₄₂₁	1	1	yes	characteristic
67	C₄	D₄×D₄₂₁	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	2	no	—
70	C₂	D₄×Dic₄₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₄₂₁
commutator	C₄×C₄₂₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting, prime radical	#135×C₄₂₁	C₂
socle	C₂×C₄₂₁	C₄×D₄
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	#135	D₄₂₁
hypocenter	C₄₂₁	#444
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #27212373	C₂	semidirect
2. #27212302	C₂	semidirect
3. #27212293	C₂	semidirect
4. #27212276	C₂	semidirect
12. Q₁₆×C₄₂₁	C₄	semidirect
13. SD₁₆×C₄₂₁	C₄	semidirect
14. D₈×C₄₂₁	C₄	semidirect
35. C₄₂₁	#444	semidirect
43. Q₁₆	Dic₄₂₁	semidirect
44. SD₁₆	Dic₄₂₁	semidirect
45. D₈	Dic₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212373	2
#27212302	2
#27212293	2
#27212276	2
#135×C₄₂₁	2
#444	421