Subgroups — #41003274

Order 26944.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212385	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×D₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212311	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212310	C₂	1	1	yes	maximal, characteristic, complemented
5	#27212307	C₂	1	1	yes	maximal, characteristic, complemented
6	#27212305	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₆×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#20269770	—	2	1	no	—
9	D₄×D₄₂₁	—	2	1	no	—
10	Dic₁₆₈₄	C₂²	1	1	yes	characteristic, complemented
11	#20269759	—	2	1	no	—
12	D₃₃₆₈	C₂²	1	1	yes	characteristic
13	#20269756	—	2	1	no	—
14	C₈×D₄₂₁	C₂²	1	1	yes	characteristic
15	#20269746	C₂²	1	1	yes	characteristic, complemented
16	D₈×C₄₂₁	C₂²	1	1	yes	characteristic
17	D₈×C₄₂₁	C₂²	1	1	yes	characteristic, complemented
18	C₁₆×C₄₂₁	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₄₂₁	—	4	1	no	—
20	Dic₈₄₂	—	2	1	no	—
21	D₁₆₈₄	—	2	1	no	—
22	C₂²⋊D₄₂₁	—	4	1	no	—
23	C₂²⋊D₄₂₁	—	4	1	no	—
24	C₂×Dic₄₂₁	—	4	1	no	—
25	C₄×D₄₂₁	D₄	1	1	yes	characteristic
26	#16809890	D₄	1	1	yes	characteristic
27	D₄×C₄₂₁	—	2	1	no	—
28	D₄×C₄₂₁	—	2	1	no	—
29	C₈×C₄₂₁	C₂³	1	1	yes	characteristic
30	C₂²×C₄₂₁	—	4	1	no	—
31	C₂²×C₄₂₁	—	4	1	no	—
32	C₂×D₄₂₁	—	4	1	no	—
33	C₂×D₄₂₁	—	8	2	no	—
34	C₂×D₄₂₁	D₈	1	1	yes	characteristic
35	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
36	Dic₄₂₁	—	4	1	no	—
37	Dic₄₂₁	D₈	1	1	yes	characteristic
38	C₂×C₄₂₁	—	8	1	no	—
39	C₂×C₄₂₁	—	8	1	no	—
40	C₂×C₄₂₁	C₂×D₈	1	1	yes	characteristic
41	D₄₂₁	—	2	1	no	—
42	D₄₂₁	—	8	1	no	—
43	C₄₂₁	#509	1	1	yes	characteristic, complemented, Sylow
44	#509	—	421	1	no	maximal, Sylow
45	#135	—	421	1	no	—
46	C₂×D₈	—	421	1	no	—
47	SD₃₂	—	421	1	no	—
48	SD₃₂	—	421	1	no	—
49	D₁₆	—	421	1	no	—
50	D₁₆	D₄₂₁	1	1	yes	characteristic, complemented
51	M₃₂	—	421	1	no	—
52	#41	—	842	1	no	—
53	C₂×D₄	—	842	1	no	—
54	Q₁₆	—	421	1	no	—
55	SD₁₆	—	842	1	no	—
56	D₈	—	842	1	no	—
57	D₈	—	421	1	no	—
58	D₈	C₂×D₄₂₁	1	1	yes	characteristic
59	D₈	C₂×D₄₂₁	1	1	yes	characteristic, complemented
60	C₂×C₈	—	421	1	no	—
61	C₁₆	—	421	1	no	—
62	C₁₆	C₂×D₄₂₁	1	1	yes	characteristic, complemented
63	C₂³	—	1684	1	no	—
64	Q₈	—	842	1	no	—
65	D₄	—	2	1	no	—
66	D₄	—	2	1	no	—
67	D₄	—	842	1	no	—
68	D₄	—	1684	1	no	—
69	D₄	—	1684	1	no	—
70	C₂×C₄	—	1684	1	no	—
71	C₂×C₄	—	421	1	no	—
72	C₈	—	421	1	no	—
73	C₈	C₂²×D₄₂₁	1	1	yes	characteristic
74	C₂²	—	4	1	no	—
75	C₂²	—	4	1	no	—
76	C₂²	—	3368	2	no	—
77	C₂²	—	1684	1	no	—
78	C₂²	—	421	1	no	—
79	C₄	—	1684	1	no	—
80	C₄	—	421	1	no	—
81	C₄	D₄×D₄₂₁	1	1	yes	characteristic
82	C₂	—	8	1	no	—
83	C₂	—	8	1	no	—
84	C₂	—	3368	1	no	—
85	C₂	—	842	1	no	—
86	C₂	D₈×D₄₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₄₂₁
commutator, cosocle	C₈×C₄₂₁	C₂³
Frattini	C₈	C₂²×D₄₂₁
Fitting	D₁₆×C₄₂₁	C₂
socle	C₂×C₄₂₁	C₂×D₈
hypercenter	D₁₆	D₄₂₁
hypocenter	C₄₂₁	#509
LCS	G ▷ C₈×C₄₂₁ ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

Complements

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

N	H ≅ G/N	type
1. #27212385	C₂	semidirect
2. D₈×D₄₂₁	C₂	semidirect
3. #27212311	C₂	semidirect
4. #27212310	C₂	semidirect
5. #27212307	C₂	semidirect
6. #27212305	C₂	semidirect
7. D₁₆×C₄₂₁	C₂	semidirect
10. Dic₁₆₈₄	C₂²	semidirect
15. #20269746	C₂²	semidirect
17. D₈×C₄₂₁	C₂²	semidirect
18. C₁₆×C₄₂₁	C₂²	semidirect
43. C₄₂₁	#509	semidirect
50. D₁₆	D₄₂₁	semidirect
59. D₈	C₂×D₄₂₁	semidirect
62. C₁₆	C₂×D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212385	2
D₈×D₄₂₁	2
#27212311	2
#27212310	2
#27212307	2
#27212305	2
D₁₆×C₄₂₁	2
#509	421