Subgroups — #41002934

Order 26944.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₆₈₄	C₂	1	1	yes	maximal, characteristic
2	#27212331	C₂	1	2	yes	maximal
3	#27212289	C₂	1	2	yes	maximal
4	C₈×Dic₄₂₁	C₂	1	1	yes	maximal, characteristic
5	#107×C₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₄₂	C₂²	1	2	yes	—
7	Dic₁₆₈₄	C₄	1	4	yes	complemented
8	C₂×#16809890	C₂²	1	1	yes	characteristic
9	#20269750	—	2	2	no	—
10	C₄×Dic₄₂₁	—	2	2	no	—
11	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₄₂₁	C₂²	1	2	yes	—
14	Dic₈₄₂	—	2	2	no	—
15	Dic₈₄₂	C₂×C₄	1	4	yes	—
16	C₂×Dic₄₂₁	—	2	2	no	—
17	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
18	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
19	#16809890	—	2	1	no	—
20	C₂×C₄×C₄₂₁	—	2	2	no	—
21	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
22	C₈×C₄₂₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
24	C₄×C₄₂₁	—	4	2	no	—
25	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
26	C₄×C₄₂₁	#41	1	1	yes	characteristic
27	Dic₄₂₁	—	2	4	no	—
28	Dic₄₂₁	—	2	1	no	—
29	Dic₄₂₁	Q₁₆	1	2	yes	complemented
30	C₂×C₄₂₁	C₄×D₄	1	1	yes	characteristic
31	C₂×C₄₂₁	C₂×Q₁₆	1	1	yes	characteristic
32	C₂×C₄₂₁	#135	1	1	yes	characteristic
33	C₄₂₁	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×Q₁₆	—	421	1	no	maximal, Sylow
35	C₂×Q₁₆	—	421	1	no	—
36	C₄×Q₈	—	421	2	no	—
37	#107	D₄₂₁	1	1	yes	characteristic
38	#103	—	421	2	no	—
39	C₄×C₈	—	421	1	no	—
40	C₂×Q₈	—	421	2	no	—
41	Q₁₆	—	421	4	no	—
42	C₂×C₈	—	421	1	no	—
43	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
44	C₄⋊C₄	—	842	2	no	—
45	C₄⋊C₄	C₂×D₄₂₁	1	2	yes	—
46	C₄²	—	842	2	no	—
47	C₄²	—	421	1	no	—
48	Q₈	—	842	2	no	—
49	Q₈	—	421	4	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	842	2	no	—
52	C₂×C₄	—	421	1	no	—
53	C₂×C₄	—	421	1	no	—
54	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
55	C₈	—	842	1	no	—
56	C₈	C₄×D₄₂₁	1	2	yes	—
57	C₂²	D₄×D₄₂₁	1	1	yes	characteristic
58	C₄	—	4	2	no	—
59	C₄	—	842	4	no	—
60	C₄	—	842	1	no	—
61	C₄	—	421	2	no	—
62	C₄	C₂×C₄×D₄₂₁	1	1	yes	characteristic
63	C₄	#20269772	1	1	yes	characteristic
64	C₂	#27212328	1	1	yes	characteristic
65	C₂	Q₁₆×D₄₂₁	1	1	yes	characteristic
66	C₂	#27212385	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₂₁
commutator	C₄×C₄₂₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#107×C₄₂₁	C₂
socle, prime radical	C₂²×C₄₂₁	C₂×D₄
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	#107	D₄₂₁
hypocenter	C₄₂₁	C₄×Q₁₆
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
7. Dic₁₆₈₄	C₄	semidirect
29. Dic₄₂₁	Q₁₆	semidirect
33. C₄₂₁	C₄×Q₁₆	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
#27212331	2
#27212289	2
C₈×Dic₄₂₁	2
#107×C₄₂₁	2
C₄×Q₁₆	421