Subgroups — #41002797

Order 26944.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269754	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₄₂₁	C₂	1	1	yes	maximal, characteristic
3	#27212323	C₂	1	1	yes	maximal, characteristic
4	#27212276	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#16809890	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#97×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
9	#20269754	C₄	1	4	yes	complemented
10	C₂×#16809890	C₂²	1	1	yes	characteristic
11	C₂×#16809890	C₂²	1	1	yes	characteristic
12	C₈×D₄₂₁	C₄	1	4	yes	complemented
13	#20269750	C₄	1	2	yes	—
14	#20269748	C₄	1	2	yes	—
15	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
18	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
19	C₂²×D₄₂₁	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₄₂₁	C₂×C₄	1	2	yes	—
21	C₂×Dic₄₂₁	C₂×C₄	1	1	yes	characteristic
22	C₄×D₄₂₁	C₂×C₄	1	2	yes	—
23	C₄×D₄₂₁	C₂×C₄	1	2	yes	—
24	#16809890	C₂×C₄	1	2	yes	—
25	#16809890	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₄₂₁	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
28	C₈×C₄₂₁	C₂×C₄	1	2	yes	—
29	C₈×C₄₂₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
31	C₂×D₄₂₁	—	2	2	no	—
32	C₂×D₄₂₁	C₄²	1	2	yes	—
33	C₄×C₄₂₁	—	2	2	no	—
34	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
36	Dic₄₂₁	—	2	2	no	—
37	Dic₄₂₁	C₄²	1	2	yes	—
38	C₂×C₄₂₁	C₂×C₄²	1	1	yes	characteristic
39	C₂×C₄₂₁	#131	1	2	yes	—
40	D₄₂₁	—	2	2	no	—
41	C₄₂₁	#405	1	1	yes	characteristic, complemented, Sylow
42	#405	—	421	1	no	maximal, Sylow
43	C₂×M₁₆	—	421	1	no	—
44	C₂²×C₈	—	421	1	no	—
45	#117	—	421	1	no	—
46	#97	—	421	1	no	—
47	#97	D₄₂₁	1	1	yes	characteristic, complemented
48	C₄×C₈	—	421	1	no	—
49	C₄×C₈	—	421	1	no	—
50	C₂²×C₄	—	421	1	no	—
51	M₁₆	—	421	4	no	—
52	C₂×C₈	—	421	1	no	—
53	C₂×C₈	—	421	1	no	—
54	C₂×C₈	—	421	4	no	—
55	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
56	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
57	C₄⋊C₄	—	421	2	no	—
58	C₂²⋊C₄	—	421	2	no	—
59	C₄²	—	421	1	no	—
60	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
61	C₂³	—	421	1	no	—
62	C₂×C₄	—	421	2	no	—
63	C₂×C₄	—	421	1	no	—
64	C₂×C₄	—	421	2	no	—
65	C₂×C₄	—	421	2	no	—
66	C₂×C₄	C₄×D₄₂₁	1	2	yes	—
67	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
68	C₈	—	421	2	no	—
69	C₈	—	421	2	no	—
70	C₈	C₄×D₄₂₁	1	2	yes	—
71	C₈	C₄×D₄₂₁	1	2	yes	—
72	C₂²	—	842	2	no	—
73	C₂²	—	421	2	no	—
74	C₂²	C₂×C₄×D₄₂₁	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	842	2	no	—
77	C₄	—	421	2	no	—
78	C₄	C₂×C₄×D₄₂₁	1	1	yes	characteristic
79	C₄	C₂×C₄×D₄₂₁	1	1	yes	characteristic
80	C₂	—	842	2	no	—
81	C₂	C₄²×D₄₂₁	1	1	yes	characteristic
82	C₂	#27212374	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₂₁
commutator	C₂×C₄₂₁	C₂×C₄²
Fitting	#97×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂²×C₄
prime radical	C₂²×D₄₂₁	C₂×C₄
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	#97	D₄₂₁
hypocenter	C₄₂₁	#405
LCS	G ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
4. #27212276	C₂	semidirect
5. C₄×#16809890	C₂	semidirect
6. C₈×Dic₄₂₁	C₂	semidirect
7. #97×C₄₂₁	C₂	semidirect
9. #20269754	C₄	semidirect
12. C₈×D₄₂₁	C₄	semidirect
41. C₄₂₁	#405	semidirect
47. #97	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269754	2
C₂×C₈×D₄₂₁	2
#27212323	2
#27212276	2
C₄×#16809890	2
C₈×Dic₄₂₁	2
#97×C₄₂₁	2
#405	421