Subgroups — #41002600

Order 26944.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269755	C₂	1	1	yes	maximal, characteristic
2	#27212325	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×C₄₂₁	C₂	1	1	yes	maximal, characteristic
4	#20269755	—	2	1	no	—
5	#20269755	C₄	1	2	yes	complemented
6	C₂×#16809890	—	2	1	no	—
7	C₄⋊Dic₄₂₁	C₄	1	2	yes	complemented
8	C₂²⋊Dic₄₂₁	—	2	1	no	—
9	C₄×Dic₄₂₁	—	2	1	no	—
10	C₂²×C₄×C₄₂₁	—	2	1	no	—
11	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
12	C₄²×C₄₂₁	—	2	1	no	—
13	C₄²×C₄₂₁	C₄	1	2	yes	complemented
14	C₂×Dic₄₂₁	—	2	2	no	—
15	#16809890	—	2	2	no	—
16	C₂³×C₄₂₁	D₄	1	1	yes	characteristic
17	C₂×C₄×C₄₂₁	—	2	4	no	—
18	C₂×C₄×C₄₂₁	—	2	2	no	—
19	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₄₂₁	Q₈	1	1	yes	characteristic
25	C₂²×C₄₂₁	—	2	2	no	—
26	C₂²×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₄₂₁	C₄⋊C₄	1	1	yes	characteristic
29	C₄×C₄₂₁	—	2	4	no	—
30	C₄×C₄₂₁	C₄²	1	1	yes	characteristic
31	C₄×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₄₂₁	C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₄₂₁	C₄⋊C₄	1	1	yes	characteristic
34	Dic₄₂₁	—	4	2	no	—
35	C₂×C₄₂₁	—	2	1	no	—
36	C₂×C₄₂₁	—	2	1	no	—
37	C₂×C₄₂₁	#95	1	1	yes	characteristic
38	C₂×C₄₂₁	#104	1	2	yes	—
39	C₄₂₁	#339	1	1	yes	characteristic, complemented, Sylow
40	#339	—	421	1	no	maximal, Sylow
41	C₂×M₁₆	—	421	1	no	—
42	#117	—	421	1	no	—
43	C₂×C₄²	D₄₂₁	1	1	yes	characteristic
44	C₂²×C₄	—	2	1	no	—
45	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
46	M₁₆	—	842	1	no	—
47	M₁₆	—	421	2	no	—
48	C₂×C₈	—	842	1	no	—
49	C₄⋊C₄	—	421	2	no	—
50	C₂²⋊C₄	—	842	1	no	—
51	C₄²	—	2	1	no	—
52	C₄²	—	842	1	no	—
53	C₄²	Dic₄₂₁	1	2	yes	complemented
54	C₂³	C₂²⋊D₄₂₁	1	1	yes	characteristic
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	842	2	no	—
58	C₂×C₄	C₄×D₄₂₁	1	1	yes	characteristic
59	C₂×C₄	C₄×D₄₂₁	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₄₂₁	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₄₂₁	1	1	yes	characteristic
62	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
63	C₂×C₄	Dic₈₄₂	1	1	yes	characteristic
64	C₈	—	842	2	no	—
65	C₂²	—	2	2	no	—
66	C₂²	#20269748	1	1	yes	characteristic
67	C₂²	C₂²⋊Dic₄₂₁	1	1	yes	characteristic
68	C₂²	#20269750	1	1	yes	characteristic
69	C₄	—	2	4	no	—
70	C₄	—	1684	2	no	—
71	C₄	C₄×Dic₄₂₁	1	1	yes	characteristic
72	C₄	#20269748	1	1	yes	characteristic
73	C₄	#20269750	1	1	yes	characteristic
74	C₄	C₄⋊Dic₄₂₁	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	2	1	no	—
77	C₂	#27212273	1	1	yes	characteristic
78	C₂	#27212294	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₁₆₈₄
commutator	C₄×C₄₂₁	C₄²
Frattini	C₂²×C₄	C₂×D₄₂₁
Fitting	C₂×C₄²×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂²⋊C₄
prime radical	C₂³×C₄₂₁	D₄
cosocle	C₂²×C₄×C₄₂₁	C₂²
hypercenter	C₂×C₄²	D₄₂₁
hypocenter	C₄₂₁	#339
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
5. #20269755	C₄	semidirect
7. C₄⋊Dic₄₂₁	C₄	semidirect
13. C₄²×C₄₂₁	C₄	semidirect
39. C₄₂₁	#339	semidirect
53. C₄²	Dic₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269755	2
#27212325	2
C₂×C₄²×C₄₂₁	2
#339	421