Subgroups — #41003051

Order 26944.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₆₈₄	C₂	1	2	yes	maximal
2	#27212366	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₈₄₂	C₂	1	1	yes	maximal, characteristic
4	#27212291	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₄₂	C₂²	1	2	yes	—
7	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
8	Dic₁₆₈₄	—	2	4	no	—
9	C₄⋊Dic₄₂₁	—	2	1	no	—
10	C₄⋊Dic₄₂₁	—	2	1	no	—
11	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
12	#20269750	—	2	1	no	—
13	C₄×Dic₄₂₁	—	2	1	no	—
14	C₂×C₈×C₄₂₁	C₂²	1	2	yes	—
15	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
16	Dic₈₄₂	—	2	1	no	—
17	Dic₈₄₂	—	2	2	no	—
18	Dic₈₄₂	—	2	2	no	—
19	Dic₈₄₂	D₄	1	2	yes	—
20	C₂×Dic₄₂₁	—	2	1	no	—
21	C₂×Dic₄₂₁	—	2	1	no	—
22	C₂×Dic₄₂₁	—	2	2	no	—
23	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
25	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
26	C₈×C₄₂₁	—	2	2	no	—
27	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
28	C₄×C₄₂₁	—	2	1	no	—
29	C₄×C₄₂₁	Q₁₆	1	2	yes	complemented
30	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
31	C₄×C₄₂₁	#41	1	1	yes	characteristic
32	Dic₄₂₁	—	2	2	no	—
33	Dic₄₂₁	—	4	2	no	—
34	Dic₄₂₁	—	4	1	no	—
35	C₂×C₄₂₁	#121	1	1	yes	characteristic
36	C₂×C₄₂₁	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₄₂₁	#137	1	1	yes	characteristic
38	C₄₂₁	#462	1	1	yes	characteristic, complemented, Sylow
39	#462	—	421	1	no	maximal, Sylow
40	C₂×Q₁₆	—	421	2	no	—
41	C₄⋊Q₈	—	421	1	no	—
42	C₄×Q₈	—	421	1	no	—
43	C₄⋊C₈	D₄₂₁	1	1	yes	characteristic
44	#103	—	421	2	no	—
45	C₂×Q₈	—	421	2	no	—
46	C₂×Q₈	—	421	1	no	—
47	Q₁₆	—	842	4	no	—
48	C₂×C₈	C₂×D₄₂₁	1	2	yes	—
49	C₄⋊C₄	—	842	1	no	—
50	C₄⋊C₄	—	842	1	no	—
51	C₄⋊C₄	—	842	1	no	—
52	C₄⋊C₄	—	421	1	no	—
53	C₄²	—	842	1	no	—
54	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
55	Q₈	—	842	1	no	—
56	Q₈	—	842	2	no	—
57	Q₈	—	842	2	no	—
58	Q₈	—	421	2	no	—
59	C₂×C₄	—	842	1	no	—
60	C₂×C₄	—	842	1	no	—
61	C₂×C₄	—	842	2	no	—
62	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
63	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₂²	C₂×D₁₆₈₄	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	842	2	no	—
69	C₄	—	1684	2	no	—
70	C₄	—	1684	1	no	—
71	C₄	Dic₁₆₈₄	1	2	yes	complemented
72	C₄	D₄×D₄₂₁	1	1	yes	characteristic
73	C₄	#20269772	1	1	yes	characteristic
74	C₂	#27212341	1	1	yes	characteristic
75	C₂	C₂×Dic₁₆₈₄	1	1	yes	characteristic
76	C₂	#27212399	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₆₈₄
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	C₄⋊C₈×C₄₂₁	C₂
socle, prime radical	C₂²×C₄₂₁	C₂×D₄
hypercenter	C₄⋊C₈	D₄₂₁
hypocenter	C₄₂₁	#462
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
29. C₄×C₄₂₁	Q₁₆	semidirect
38. C₄₂₁	#462	semidirect
71. C₄	Dic₁₆₈₄	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
#27212366	2
C₄×Dic₈₄₂	2
#27212291	2
C₄⋊C₈×C₄₂₁	2
#462	421