Subgroups — #41003094

Order 26944.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₃₆₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#20269754	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212341	C₂	1	2	yes	maximal, complemented
4	#27212286	C₂	1	2	yes	maximal, complemented
5	#106×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆₈₄	—	2	2	no	—
7	C₂×D₁₆₈₄	C₂²	1	2	yes	—
8	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
9	D₃₃₆₈	—	2	2	no	—
10	#20269754	—	2	2	no	—
11	C₂×#16809890	C₂²	1	1	yes	characteristic
12	#20269748	—	2	2	no	—
13	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₄₂₁	C₂²	1	2	yes	—
15	C₂²×D₄₂₁	—	2	2	no	—
16	C₂²×D₄₂₁	D₄	1	1	yes	characteristic
17	D₁₆₈₄	—	2	2	no	—
18	D₁₆₈₄	—	2	2	no	—
19	D₁₆₈₄	—	4	4	no	—
20	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
21	C₄×D₄₂₁	—	2	1	no	—
22	C₄×D₄₂₁	—	2	1	no	—
23	#16809890	—	2	1	no	—
24	C₂×C₄×C₄₂₁	—	2	2	no	—
25	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
26	C₈×C₄₂₁	D₄	1	2	yes	complemented
27	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
28	C₂×D₄₂₁	—	2	1	no	—
29	C₂×D₄₂₁	—	2	2	no	—
30	C₂×D₄₂₁	—	4	2	no	—
31	C₂×D₄₂₁	—	4	2	no	—
32	C₂×D₄₂₁	—	4	2	no	—
33	C₄×C₄₂₁	—	4	2	no	—
34	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
35	C₄×C₄₂₁	#41	1	1	yes	characteristic
36	Dic₄₂₁	—	2	1	no	—
37	C₂×C₄₂₁	#121	1	1	yes	characteristic
38	C₂×C₄₂₁	AGL(1,ℤ/8)	1	2	yes	—
39	D₄₂₁	—	4	1	no	—
40	D₄₂₁	—	8	2	no	—
41	C₄₂₁	#469	1	1	yes	characteristic, complemented, Sylow
42	#469	—	421	1	no	maximal, Sylow
43	C₂×D₈	—	421	1	no	—
44	C₂×M₁₆	—	421	1	no	—
45	#121	—	421	2	no	—
46	#106	D₄₂₁	1	1	yes	characteristic, complemented
47	#102	—	421	2	no	—
48	C₂×D₄	—	842	2	no	—
49	C₂×D₄	—	421	2	no	—
50	C₂²×C₄	—	421	1	no	—
51	D₈	—	842	2	no	—
52	M₁₆	—	842	2	no	—
53	C₂×C₈	—	421	1	no	—
54	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
55	C₄⋊C₄	C₂×D₄₂₁	1	2	yes	—
56	C₂²⋊C₄	—	842	2	no	—
57	C₂³	—	842	2	no	—
58	C₂³	—	421	1	no	—
59	D₄	—	842	2	no	—
60	D₄	—	842	2	no	—
61	D₄	—	1684	4	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	842	1	no	—
64	C₂×C₄	—	842	1	no	—
65	C₂×C₄	—	421	1	no	—
66	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
67	C₈	—	842	1	no	—
68	C₈	D₁₆₈₄	1	2	yes	complemented
69	C₂²	—	842	1	no	—
70	C₂²	—	842	2	no	—
71	C₂²	—	1684	2	no	—
72	C₂²	—	1684	2	no	—
73	C₂²	—	1684	2	no	—
74	C₂²	D₄×D₄₂₁	1	1	yes	characteristic
75	C₄	—	4	2	no	—
76	C₄	—	842	1	no	—
77	C₄	C₂×D₁₆₈₄	1	1	yes	characteristic
78	C₄	#20269772	1	1	yes	characteristic
79	C₂	—	3368	2	no	—
80	C₂	—	1684	1	no	—
81	C₂	#27212341	1	1	yes	characteristic
82	C₂	#27212393	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₄₂₁
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#106×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
hypercenter	#106	D₄₂₁
hypocenter	C₄₂₁	#469
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
1. C₂×D₃₃₆₈	C₂	semidirect
2. C₂×#20269754	C₂	semidirect
3. #27212341	C₂	semidirect
4. #27212286	C₂	semidirect
5. #106×C₄₂₁	C₂	semidirect
26. C₈×C₄₂₁	D₄	semidirect
41. C₄₂₁	#469	semidirect
46. #106	D₄₂₁	semidirect
68. C₈	D₁₆₈₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₃₆₈	2
C₂×#20269754	2
#27212341	2
#27212286	2
#106×C₄₂₁	2
#469	421