Subgroups — #41002634

Order 26944.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212353	C₂	1	1	yes	maximal, characteristic
2	#27212281	C₂	1	1	yes	maximal, characteristic
3	#99×C₄₂₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₈₄₂	C₄	1	1	yes	characteristic, complemented
5	C₂²⋊Dic₄₂₁	—	2	1	no	—
6	C₂²⋊Dic₄₂₁	—	2	1	no	—
7	C₂²⋊Dic₄₂₁	—	2	1	no	—
8	C₄×Dic₄₂₁	C₄	1	1	yes	characteristic, complemented
9	C₂×D₄×C₄₂₁	C₂²	1	1	yes	characteristic
10	C₂²⋊C₄×C₄₂₁	—	2	1	no	—
11	Dic₈₄₂	—	4	1	no	—
12	C₂×Dic₄₂₁	—	2	1	no	—
13	C₂×Dic₄₂₁	—	2	1	no	—
14	C₂×Dic₄₂₁	—	4	1	no	—
15	C₂³×C₄₂₁	D₄	1	1	yes	characteristic
16	C₂³×C₄₂₁	D₄	1	1	yes	characteristic
17	D₄×C₄₂₁	—	4	1	no	—
18	C₂×C₄×C₄₂₁	—	4	1	no	—
19	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄₂₁	—	2	1	no	—
21	C₂²×C₄₂₁	—	2	1	no	—
22	C₂²×C₄₂₁	—	4	1	no	—
23	C₂²×C₄₂₁	—	4	1	no	—
24	C₂²×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₄×C₄₂₁	—	2	1	no	—
26	C₄×C₄₂₁	—	8	1	no	—
27	Dic₄₂₁	—	4	1	no	—
28	Dic₄₂₁	—	4	1	no	—
29	Dic₄₂₁	—	8	1	no	—
30	C₂×C₄₂₁	—	2	1	no	—
31	C₂×C₄₂₁	—	4	1	no	—
32	C₂×C₄₂₁	—	4	1	no	—
33	C₂×C₄₂₁	#99	1	1	yes	characteristic
34	C₄₂₁	#354	1	1	yes	characteristic, complemented, Sylow
35	#354	—	421	1	no	maximal, Sylow
36	#124	—	421	1	no	—
37	#99	—	421	1	no	—
38	#99	D₄₂₁	1	1	yes	characteristic
39	C₂×Q₈	—	421	1	no	—
40	C₂×D₄	C₂×D₄₂₁	1	1	yes	characteristic
41	C₂²⋊C₄	—	2	1	no	—
42	C₂²⋊C₄	—	842	1	no	—
43	C₂²⋊C₄	—	842	1	no	—
44	C₂²⋊C₄	—	842	1	no	—
45	C₄²	—	421	1	no	—
46	C₂³	C₂²⋊D₄₂₁	1	1	yes	characteristic
47	C₂³	D₁₆₈₄	1	1	yes	characteristic
48	Q₈	—	1684	1	no	—
49	D₄	—	4	1	no	—
50	C₂×C₄	—	4	1	no	—
51	C₂×C₄	—	842	1	no	—
52	C₂×C₄	—	842	1	no	—
53	C₂×C₄	—	1684	1	no	—
54	C₂×C₄	C₄×D₄₂₁	1	1	yes	characteristic
55	C₂²	—	2	1	no	—
56	C₂²	—	2	1	no	—
57	C₂²	—	4	1	no	—
58	C₂²	—	4	1	no	—
59	C₂²	#20269748	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	—	8	1	no	—
62	C₄	—	3368	1	no	—
63	C₄	—	1684	1	no	—
64	C₄	—	1684	1	no	—
65	C₂	—	2	1	no	—
66	C₂	—	4	1	no	—
67	C₂	—	4	1	no	—
68	C₂	#27212280	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27212280
commutator	C₂×C₄×C₄₂₁	C₂×C₄
Frattini	C₂×D₄	C₂×D₄₂₁
Fitting	#99×C₄₂₁	C₂
socle	C₂×C₄₂₁	#99
prime radical, cosocle	C₂×D₄×C₄₂₁	C₂²
hypercenter	#99	D₄₂₁
hypocenter	C₄₂₁	#354
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂²×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #99

Complements

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

N	H ≅ G/N	type
4. C₂×Dic₈₄₂	C₄	semidirect
8. C₄×Dic₄₂₁	C₄	semidirect
34. C₄₂₁	#354	semidirect

Maximal subgroups

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

subgroup	index
#27212353	2
#27212281	2
#99×C₄₂₁	2
#354	421