Subgroups — #41002876

Order 26944.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16809890	C₂	1	1	yes	maximal, characteristic
2	#27212325	C₂	1	1	yes	maximal, characteristic
3	#27212299	C₂	1	2	yes	maximal, complemented
4	#27212297	C₂	1	2	yes	maximal, complemented
5	#117×C₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809890	C₄	1	4	yes	complemented
7	C₂×#16809890	C₂²	1	1	yes	characteristic
8	C₂×#16809890	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
10	C₂²⋊Dic₄₂₁	—	2	1	no	—
11	C₄×Dic₄₂₁	—	2	1	no	—
12	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₄₂₁	C₂²	1	2	yes	—
14	C₂²⋊C₄×C₄₂₁	—	2	1	no	—
15	C₄²×C₄₂₁	—	2	1	no	—
16	C₂×Dic₄₂₁	—	2	2	no	—
17	#16809890	C₂×C₄	1	2	yes	—
18	#16809890	C₂×C₄	1	2	yes	—
19	C₂³×C₄₂₁	D₄	1	1	yes	characteristic
20	C₂×C₄×C₄₂₁	—	2	2	no	—
21	C₂×C₄×C₄₂₁	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₂₁	Q₈	1	2	yes	—
24	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄₂₁	—	2	2	no	—
26	C₂²×C₄₂₁	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
28	C₄×C₄₂₁	—	4	2	no	—
29	C₄×C₄₂₁	C₄⋊C₄	1	2	yes	—
30	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₄₂₁	C₂×Q₈	1	1	yes	characteristic
32	Dic₄₂₁	—	4	2	no	—
33	C₂×C₄₂₁	—	2	2	no	—
34	C₂×C₄₂₁	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄₂₁	#135	1	2	yes	—
36	C₄₂₁	#427	1	1	yes	characteristic, complemented, Sylow
37	#427	—	421	1	no	maximal, Sylow
38	C₂²×C₈	—	421	1	no	—
39	#117	—	421	1	no	—
40	#117	D₄₂₁	1	1	yes	characteristic
41	#107	—	421	2	no	—
42	#106	—	421	2	no	—
43	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
44	C₂×C₈	—	421	4	no	—
45	C₂×C₈	—	421	1	no	—
46	C₂×C₈	—	421	1	no	—
47	C₄⋊C₄	—	421	2	no	—
48	C₄⋊C₄	C₂×D₄₂₁	1	2	yes	—
49	C₂²⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	—	842	1	no	—
51	C₄²	—	2	1	no	—
52	C₄²	—	842	1	no	—
53	C₂³	C₂²⋊D₄₂₁	1	1	yes	characteristic
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	842	2	no	—
56	C₂×C₄	C₄×D₄₂₁	1	2	yes	—
57	C₂×C₄	C₂²⋊D₄₂₁	1	1	yes	characteristic
58	C₂×C₄	Dic₈₄₂	1	2	yes	—
59	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
60	C₈	—	421	2	no	—
61	C₈	—	421	2	no	—
62	C₂²	—	2	2	no	—
63	C₂²	#20269750	1	2	yes	—
64	C₂²	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
65	C₄	—	4	2	no	—
66	C₄	—	1684	2	no	—
67	C₄	#20269750	1	2	yes	—
68	C₄	C₂×C₄×D₄₂₁	1	1	yes	characteristic
69	C₄	C₂×Dic₈₄₂	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	C₂×#20269750	1	1	yes	characteristic
72	C₂	#27212386	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₄₂₁
commutator	C₄×C₄₂₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#117×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
prime radical	C₂³×C₄₂₁	D₄
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	#117	D₄₂₁
hypocenter	C₄₂₁	#427
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #27212299	C₂	semidirect
4. #27212297	C₂	semidirect
6. C₂×#16809890	C₄	semidirect
36. C₄₂₁	#427	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#16809890	2
#27212325	2
#27212299	2
#27212297	2
#117×C₄₂₁	2
#427	421