Subgroups — #41003171

Order 26944.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212345	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₄₂₁	C₂	1	1	yes	maximal, characteristic
3	#27212298	C₂	1	2	yes	maximal, complemented
4	#27212291	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
7	C₂²×Dic₄₂₁	—	2	1	no	—
8	C₄⋊Dic₄₂₁	—	2	1	no	—
9	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
10	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
11	#20269750	—	2	1	no	—
12	C₂²⋊Dic₄₂₁	—	2	1	no	—
13	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₄₂₁	C₂²	1	2	yes	—
15	Dic₈₄₂	—	2	2	no	—
16	C₂×Dic₄₂₁	—	2	1	no	—
17	C₂×Dic₄₂₁	—	2	1	no	—
18	C₂×Dic₄₂₁	—	2	2	no	—
19	C₂×Dic₄₂₁	—	4	2	no	—
20	C₂³×C₄₂₁	D₄	1	1	yes	characteristic
21	C₂×C₄×C₄₂₁	—	2	2	no	—
22	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
24	C₈×C₄₂₁	—	2	2	no	—
25	C₂²×C₄₂₁	—	2	1	no	—
26	C₂²×C₄₂₁	—	2	1	no	—
27	C₂²×C₄₂₁	SD₁₆	1	2	yes	—
28	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
29	C₄×C₄₂₁	—	2	1	no	—
30	C₄×C₄₂₁	#41	1	2	yes	—
31	Dic₄₂₁	—	4	1	no	—
32	Dic₄₂₁	—	4	2	no	—
33	Dic₄₂₁	—	4	1	no	—
34	C₂×C₄₂₁	—	2	2	no	—
35	C₂×C₄₂₁	#123	1	1	yes	characteristic
36	C₂×C₄₂₁	C₂×SD₁₆	1	1	yes	characteristic
37	C₂×C₄₂₁	#137	1	1	yes	characteristic
38	C₄₂₁	#483	1	1	yes	characteristic, complemented, Sylow
39	#483	—	421	1	no	maximal, Sylow
40	C₂²⋊Q₈	—	421	1	no	—
41	C₂×C₄⋊C₄	—	421	1	no	—
42	#106	—	421	2	no	—
43	#103	—	421	2	no	—
44	C₂²⋊C₈	D₄₂₁	1	1	yes	characteristic
45	C₂×Q₈	—	421	1	no	—
46	C₂²×C₄	—	842	1	no	—
47	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
48	C₂×C₈	C₂×D₄₂₁	1	2	yes	—
49	C₄⋊C₄	—	842	1	no	—
50	C₄⋊C₄	—	842	1	no	—
51	C₄⋊C₄	—	421	2	no	—
52	C₄⋊C₄	—	421	1	no	—
53	C₂²⋊C₄	—	842	1	no	—
54	C₂³	D₁₆₈₄	1	1	yes	characteristic
55	Q₈	—	842	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	842	1	no	—
58	C₂×C₄	—	842	1	no	—
59	C₂×C₄	—	842	2	no	—
60	C₂×C₄	—	1684	2	no	—
61	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	#20269760	1	2	yes	—
67	C₂²	C₂×D₁₆₈₄	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	1684	1	no	—
70	C₄	—	1684	2	no	—
71	C₄	—	1684	1	no	—
72	C₄	#20269770	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#27212350	1	1	yes	characteristic
75	C₂	C₂×#20269760	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	C₂²⋊C₈×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
prime radical	C₂³×C₄₂₁	D₄
hypercenter	C₂²⋊C₈	D₄₂₁
hypocenter	C₄₂₁	#483
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. #27212298	C₂	semidirect
4. #27212291	C₂	semidirect
38. C₄₂₁	#483	semidirect

Maximal subgroups

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

subgroup	index
#27212345	2
C₂×C₄⋊Dic₄₂₁	2
#27212298	2
#27212291	2
C₂²⋊C₈×C₄₂₁	2
#483	421