Subgroups — #41003131

Order 26944.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212366	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₁₆₈₄	C₂	1	1	yes	maximal, characteristic
3	#27212298	C₂	1	2	yes	maximal, complemented
4	#27212288	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₈₄₂	—	2	1	no	—
7	C₂×D₁₆₈₄	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄₂₁	—	2	1	no	—
9	C₄⋊Dic₄₂₁	—	2	1	no	—
10	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
11	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
12	#20269748	—	2	1	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	—	4	2	no	—
17	D₁₆₈₄	—	2	1	no	—
18	D₁₆₈₄	Q₈	1	2	yes	complemented
19	C₂×Dic₄₂₁	—	2	1	no	—
20	C₂×Dic₄₂₁	—	2	2	no	—
21	C₄×D₄₂₁	—	4	2	no	—
22	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
25	C₈×C₄₂₁	—	2	2	no	—
26	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
27	C₂×D₄₂₁	—	2	2	no	—
28	C₂×D₄₂₁	—	4	1	no	—
29	C₂×D₄₂₁	—	4	1	no	—
30	C₄×C₄₂₁	—	2	1	no	—
31	C₄×C₄₂₁	SD₁₆	1	2	yes	—
32	C₄×C₄₂₁	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₄₂₁	#41	1	1	yes	characteristic
34	Dic₄₂₁	—	4	1	no	—
35	Dic₄₂₁	—	4	2	no	—
36	C₂×C₄₂₁	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₄₂₁	C₂×SD₁₆	1	1	yes	characteristic
38	C₂×C₄₂₁	AGL(1,ℤ/8)	1	1	yes	characteristic
39	D₄₂₁	—	4	2	no	—
40	C₄₂₁	#476	1	1	yes	characteristic, complemented, Sylow
41	#476	—	421	1	no	maximal, Sylow
42	C₄⋊Q₈	—	421	1	no	—
43	C₄×D₄	—	421	1	no	—
44	#106	—	421	2	no	—
45	C₄⋊C₈	D₄₂₁	1	1	yes	characteristic, complemented
46	#102	—	421	2	no	—
47	C₂×Q₈	—	842	1	no	—
48	C₂×D₄	—	421	1	no	—
49	C₂²×C₄	—	842	1	no	—
50	C₂×C₈	C₂×D₄₂₁	1	2	yes	—
51	C₄⋊C₄	—	842	1	no	—
52	C₄⋊C₄	—	421	2	no	—
53	C₄⋊C₄	—	421	1	no	—
54	C₂²⋊C₄	—	842	1	no	—
55	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
56	C₂³	—	842	1	no	—
57	Q₈	—	1684	2	no	—
58	D₄	—	842	1	no	—
59	D₄	—	421	2	no	—
60	C₂×C₄	—	842	1	no	—
61	C₂×C₄	—	842	2	no	—
62	C₂×C₄	—	1684	2	no	—
63	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
64	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₂²	—	842	2	no	—
68	C₂²	—	1684	1	no	—
69	C₂²	—	1684	1	no	—
70	C₂²	C₂×D₁₆₈₄	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	1684	1	no	—
73	C₄	—	1684	2	no	—
74	C₄	#20269760	1	2	yes	—
75	C₄	Q₈×D₄₂₁	1	1	yes	characteristic
76	C₄	#20269770	1	1	yes	characteristic
77	C₂	—	1684	2	no	—
78	C₂	#27212346	1	1	yes	characteristic
79	C₂	C₂×#20269760	1	1	yes	characteristic
80	C₂	#27212394	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	C₂²×C₄₂₁	C₂×D₄
prime radical	C₂×D₁₆₈₄	C₂²
hypercenter	C₄⋊C₈	D₄₂₁
hypocenter	C₄₂₁	#476
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
1. #27212366	C₂	semidirect
3. #27212298	C₂	semidirect
5. C₄⋊C₈×C₄₂₁	C₂	semidirect
18. D₁₆₈₄	Q₈	semidirect
40. C₄₂₁	#476	semidirect
45. C₄⋊C₈	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212366	2
C₄×D₁₆₈₄	2
#27212298	2
#27212288	2
C₄⋊C₈×C₄₂₁	2
#476	421