Subgroups — #41003234

Order 26944.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₆₈₄	C₂	1	2	yes	maximal, complemented
2	C₂×#20269760	C₂	1	2	yes	maximal
3	#27212366	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212354	C₂	1	1	yes	maximal, characteristic
5	#97×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
7	C₂×Dic₈₄₂	C₂²	1	2	yes	—
8	C₂×D₁₆₈₄	C₂²	1	1	yes	characteristic
9	Dic₁₆₈₄	—	2	4	no	—
10	#20269760	—	2	4	no	—
11	C₄⋊Dic₄₂₁	—	2	2	no	—
12	#20269748	—	2	2	no	—
13	C₂×C₈×C₄₂₁	C₂²	1	2	yes	—
14	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
15	C₂²×D₄₂₁	—	2	1	no	—
16	Dic₈₄₂	—	2	2	no	—
17	Dic₈₄₂	—	2	2	no	—
18	Dic₈₄₂	—	2	2	no	—
19	D₁₆₈₄	—	2	2	no	—
20	C₂×Dic₄₂₁	—	2	1	no	—
21	C₂×Dic₄₂₁	—	2	2	no	—
22	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
23	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
24	C₈×C₄₂₁	D₄	1	4	yes	—
25	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
26	C₂×D₄₂₁	—	4	1	no	—
27	C₂×D₄₂₁	—	4	2	no	—
28	C₄×C₄₂₁	—	2	2	no	—
29	C₄×C₄₂₁	C₂×D₄	1	2	yes	—
30	Dic₄₂₁	—	4	1	no	—
31	Dic₄₂₁	—	4	2	no	—
32	C₂×C₄₂₁	#127	1	1	yes	characteristic
33	C₂×C₄₂₁	#137	1	2	yes	—
34	D₄₂₁	—	8	1	no	—
35	C₄₂₁	#497	1	1	yes	characteristic, complemented, Sylow
36	#497	—	421	1	no	maximal, Sylow
37	C₂×Q₁₆	—	421	2	no	—
38	C₂×SD₁₆	—	421	2	no	—
39	C₄⋊Q₈	—	421	1	no	—
40	#124	—	421	1	no	—
41	#97	D₄₂₁	1	1	yes	characteristic, complemented
42	C₂×Q₈	—	421	1	no	—
43	C₂×Q₈	—	421	2	no	—
44	C₂×D₄	—	421	1	no	—
45	Q₁₆	—	842	4	no	—
46	SD₁₆	—	842	4	no	—
47	C₂×C₈	C₂×D₄₂₁	1	2	yes	—
48	C₄⋊C₄	—	842	2	no	—
49	C₂²⋊C₄	—	842	2	no	—
50	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
51	C₂³	—	842	1	no	—
52	Q₈	—	842	2	no	—
53	Q₈	—	842	2	no	—
54	Q₈	—	842	2	no	—
55	D₄	—	842	2	no	—
56	C₂×C₄	—	842	1	no	—
57	C₂×C₄	—	842	2	no	—
58	C₂×C₄	D₁₆₈₄	1	2	yes	—
59	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
60	C₈	D₁₆₈₄	1	4	yes	—
61	C₂²	—	1684	1	no	—
62	C₂²	—	1684	2	no	—
63	C₂²	C₂×D₁₆₈₄	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	1684	1	no	—
66	C₄	—	1684	2	no	—
67	C₄	C₂×D₁₆₈₄	1	2	yes	—
68	C₂	—	3368	1	no	—
69	C₂	#27212363	1	1	yes	characteristic
70	C₂	#27212399	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₆₈₄
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#97×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
prime radical	C₂×D₁₆₈₄	C₂²
hypercenter	#97	D₄₂₁
hypocenter	C₄₂₁	#497
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₆₈₄	C₂	semidirect
3. #27212366	C₂	semidirect
5. #97×C₄₂₁	C₂	semidirect
35. C₄₂₁	#497	semidirect
41. #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×Dic₁₆₈₄	2
C₂×#20269760	2
#27212366	2
#27212354	2
#97×C₄₂₁	2
#497	421