Subgroups — #41003133

Order 26944.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212365	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₄₂₁	C₂	1	1	yes	maximal, characteristic
3	#27212300	C₂	1	1	yes	maximal, characteristic
4	#27212299	C₂	1	1	yes	maximal, characteristic
5	#27212295	C₂	1	1	yes	maximal, characteristic
6	#27212290	C₂	1	1	yes	maximal, characteristic
7	#103×C₄₂₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₈₄₂	—	2	1	no	—
9	C₂×#16809890	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₄₂₁	—	2	1	no	—
11	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
13	#20269750	—	2	1	no	—
14	C₄×Dic₄₂₁	—	2	1	no	—
15	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₄₂₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₄₂₁	C₂²	1	1	yes	characteristic
19	Dic₈₄₂	—	4	2	no	—
20	C₂×Dic₄₂₁	—	2	1	no	—
21	C₂×Dic₄₂₁	—	2	1	no	—
22	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
23	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
24	#16809890	—	2	1	no	—
25	Q₈×C₄₂₁	—	2	1	no	—
26	Q₈×C₄₂₁	Q₈	1	2	yes	complemented
27	C₂×C₄×C₄₂₁	—	2	1	no	—
28	C₂×C₄×C₄₂₁	—	2	1	no	—
29	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
30	C₈×C₄₂₁	—	2	1	no	—
31	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
32	C₄×C₄₂₁	—	2	2	no	—
33	C₄×C₄₂₁	—	4	1	no	—
34	C₄×C₄₂₁	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₄₂₁	#41	1	1	yes	characteristic
36	Dic₄₂₁	—	2	1	no	—
37	Dic₄₂₁	—	4	1	no	—
38	Dic₄₂₁	—	4	1	no	—
39	Dic₄₂₁	Q₁₆	1	2	yes	—
40	C₂×C₄₂₁	C₂²⋊Q₈	1	1	yes	characteristic
41	C₂×C₄₂₁	C₂×Q₁₆	1	1	yes	characteristic
42	C₂×C₄₂₁	AGL(1,ℤ/8)	1	1	yes	characteristic
43	C₄₂₁	#477	1	1	yes	characteristic, complemented, Sylow
44	#477	—	421	1	no	maximal, Sylow
45	C₄⋊Q₈	—	421	1	no	—
46	C₄×Q₈	—	421	1	no	—
47	#107	—	421	1	no	—
48	#107	—	421	1	no	—
49	C₄⋊C₈	—	421	1	no	—
50	#103	—	421	1	no	—
51	#103	D₄₂₁	1	1	yes	characteristic
52	C₂×Q₈	—	842	1	no	—
53	C₂×Q₈	C₂×D₄₂₁	1	1	yes	characteristic
54	C₂×C₈	—	421	1	no	—
55	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
56	C₄⋊C₄	—	842	1	no	—
57	C₄⋊C₄	—	842	1	no	—
58	C₄⋊C₄	—	421	1	no	—
59	C₄⋊C₄	—	421	1	no	—
60	C₄⋊C₄	C₂×D₄₂₁	1	1	yes	characteristic
61	C₄²	—	842	1	no	—
62	C₄²	—	421	1	no	—
63	Q₈	—	2	1	no	—
64	Q₈	—	1684	2	no	—
65	Q₈	Dic₈₄₂	1	2	yes	complemented
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	842	1	no	—
69	C₂×C₄	—	842	1	no	—
70	C₂×C₄	—	421	1	no	—
71	C₂×C₄	—	421	1	no	—
72	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	842	1	no	—
75	C₂²	D₄×D₄₂₁	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	842	1	no	—
79	C₄	—	1684	1	no	—
80	C₄	—	1684	1	no	—
81	C₄	—	421	2	no	—
82	C₄	C₂×Dic₈₄₂	1	1	yes	characteristic
83	C₄	#20269770	1	1	yes	characteristic
84	C₂	#27212343	1	1	yes	characteristic
85	C₂	Q₁₆×D₄₂₁	1	1	yes	characteristic
86	C₂	#27212393	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₂₁
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#103×C₄₂₁	C₂
socle, prime radical	C₂²×C₄₂₁	C₂×D₄
hypercenter	#103	D₄₂₁
hypocenter	C₄₂₁	#477
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
26. Q₈×C₄₂₁	Q₈	semidirect
43. C₄₂₁	#477	semidirect
65. Q₈	Dic₈₄₂	semidirect

Maximal subgroups

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

subgroup	index
#27212365	2
Q₈×Dic₄₂₁	2
#27212300	2
#27212299	2
#27212295	2
#27212290	2
#103×C₄₂₁	2
#477	421