Subgroups — #41003144

Order 26944.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₈₄₂	C₂	1	1	yes	maximal, characteristic
2	#27212299	C₂	1	1	yes	maximal, characteristic
3	#27212297	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#16809890	C₂	1	1	yes	maximal, characteristic
5	#27212289	C₂	1	1	yes	maximal, characteristic
6	#27212289	C₂	1	1	yes	maximal, characteristic
7	#125×C₄₂₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
9	C₂×#16809890	C₂²	1	1	yes	characteristic
10	C₂×#16809890	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
12	#20269750	—	2	1	no	—
13	C₄×Dic₄₂₁	—	2	1	no	—
14	C₄⋊C₄×C₄₂₁	—	2	1	no	—
15	C₄⋊C₄×C₄₂₁	—	2	1	no	—
16	C₄⋊C₄×C₄₂₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₄₂₁	C₂²	1	1	yes	characteristic
18	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
19	Dic₈₄₂	—	2	1	no	—
20	Dic₈₄₂	Q₈	1	2	yes	—
21	C₂×Dic₄₂₁	—	2	1	no	—
22	C₂×Dic₄₂₁	—	2	1	no	—
23	#16809890	—	2	1	no	—
24	#16809890	—	2	1	no	—
25	C₂×C₄×C₄₂₁	—	2	1	no	—
26	C₂×C₄×C₄₂₁	—	2	1	no	—
27	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
28	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
29	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
30	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
31	C₄×C₄₂₁	—	2	1	no	—
32	C₄×C₄₂₁	—	2	1	no	—
33	C₄×C₄₂₁	—	4	1	no	—
34	C₄×C₄₂₁	—	4	1	no	—
35	C₄×C₄₂₁	C₂×Q₈	1	1	yes	characteristic
36	C₄×C₄₂₁	#41	1	1	yes	characteristic
37	Dic₄₂₁	—	2	2	no	—
38	Dic₄₂₁	—	4	1	no	—
39	C₂×C₄₂₁	C₂²⋊Q₈	1	1	yes	characteristic
40	C₂×C₄₂₁	#135	1	1	yes	characteristic
41	C₂×C₄₂₁	#137	1	1	yes	characteristic
42	C₄₂₁	#479	1	1	yes	characteristic, complemented, Sylow
43	#479	—	421	1	no	maximal, Sylow
44	#125	D₄₂₁	1	1	yes	characteristic
45	C₄×Q₈	—	421	1	no	—
46	#107	—	421	1	no	—
47	#106	—	421	1	no	—
48	C₄⋊C₈	—	421	1	no	—
49	#103	—	421	1	no	—
50	#103	—	421	1	no	—
51	C₂×Q₈	—	421	1	no	—
52	C₂×C₈	—	421	1	no	—
53	C₂×C₈	—	421	1	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	2	1	no	—
56	C₄⋊C₄	—	842	1	no	—
57	C₄⋊C₄	—	421	1	no	—
58	C₄⋊C₄	C₂×D₄₂₁	1	1	yes	characteristic
59	C₄⋊C₄	C₂×D₄₂₁	1	1	yes	characteristic
60	C₄²	—	842	1	no	—
61	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
62	Q₈	—	842	1	no	—
63	Q₈	—	421	2	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	842	1	no	—
67	C₂×C₄	—	842	1	no	—
68	C₂×C₄	C₂²⋊D₄₂₁	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₄₂₁	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
71	C₈	—	842	1	no	—
72	C₈	—	842	1	no	—
73	C₂²	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	842	2	no	—
79	C₄	—	1684	1	no	—
80	C₄	Q₈×D₄₂₁	1	1	yes	characteristic
81	C₄	#20269772	1	1	yes	characteristic
82	C₂	#27212342	1	1	yes	characteristic
83	C₂	#27212386	1	1	yes	characteristic
84	C₂	#27212396	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₂₁
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#125×C₄₂₁	C₂
socle, prime radical	C₂²×C₄₂₁	C₂×D₄
hypercenter	#125	D₄₂₁
hypocenter	C₄₂₁	#479
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
42. C₄₂₁	#479	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₈₄₂	2
#27212299	2
#27212297	2
C₄⋊#16809890	2
#27212289	2
#125×C₄₂₁	2
#479	421