Subgroups — #41002887

Order 26944.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269754	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₄₂₁	C₂	1	2	yes	maximal
3	#27212302	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212301	C₂	1	2	yes	maximal, complemented
5	#108×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
7	#20269755	C₂²	1	2	yes	—
8	#20269754	—	2	2	no	—
9	#20269754	C₄	1	4	yes	—
10	C₂×#16809890	C₂²	1	1	yes	characteristic
11	C₈×D₄₂₁	—	2	2	no	—
12	M₁₆×C₄₂₁	C₂²	1	2	yes	—
13	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
14	C₂²×D₄₂₁	Q₈	1	1	yes	characteristic
15	C₂×Dic₄₂₁	Q₈	1	1	yes	characteristic
16	C₄×D₄₂₁	C₂×C₄	1	2	yes	—
17	C₄×D₄₂₁	D₄	1	2	yes	—
18	#16809890	—	2	2	no	—
19	#16809890	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
21	C₈×C₄₂₁	—	2	2	no	—
22	C₈×C₄₂₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄₂₁	C₂×Q₈	1	1	yes	characteristic
24	C₂×D₄₂₁	—	4	1	no	—
25	C₂×D₄₂₁	C₄⋊C₄	1	2	yes	—
26	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
28	Dic₄₂₁	C₄⋊C₄	1	2	yes	—
29	C₂×C₄₂₁	—	2	1	no	—
30	C₂×C₄₂₁	C₂×C₄⋊C₄	1	1	yes	characteristic
31	D₄₂₁	—	2	2	no	—
32	C₄₂₁	#430	1	1	yes	characteristic, complemented, Sylow
33	#430	—	421	1	no	maximal, Sylow
34	C₂×M₁₆	—	421	2	no	—
35	C₂×M₁₆	—	421	1	no	—
36	#108	—	421	2	no	—
37	#108	—	421	1	no	—
38	#108	D₄₂₁	1	1	yes	characteristic, complemented
39	C₂²×C₄	—	421	1	no	—
40	M₁₆	—	842	2	no	—
41	M₁₆	—	421	2	no	—
42	M₁₆	—	421	4	no	—
43	M₁₆	C₂×D₄₂₁	1	2	yes	—
44	C₂×C₈	—	842	2	no	—
45	C₂×C₈	—	421	1	no	—
46	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
47	C₂³	—	421	1	no	—
48	C₂×C₄	—	421	1	no	—
49	C₂×C₄	—	421	2	no	—
50	C₂×C₄	—	421	2	no	—
51	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
52	C₈	—	2	2	no	—
53	C₈	—	842	2	no	—
54	C₈	—	421	2	no	—
55	C₈	C₄×D₄₂₁	1	2	yes	—
56	C₂²	—	1684	1	no	—
57	C₂²	—	421	2	no	—
58	C₂²	Q₈×D₄₂₁	1	1	yes	characteristic
59	C₄	—	421	2	no	—
60	C₄	C₂×C₄×D₄₂₁	1	1	yes	characteristic
61	C₄	D₄×D₄₂₁	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	842	2	no	—
64	C₂	C₄⋊C₄×D₄₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₄₂₁
commutator	C₄×C₄₂₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#108×C₄₂₁	C₂
socle	C₂×C₄₂₁	C₂×C₄⋊C₄
prime radical	C₂²×D₄₂₁	Q₈
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	#108	D₄₂₁
hypocenter	C₄₂₁	#430
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
3. #27212302	C₂	semidirect
4. #27212301	C₂	semidirect
5. #108×C₄₂₁	C₂	semidirect
32. C₄₂₁	#430	semidirect
38. #108	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269754	2
M₁₆×D₄₂₁	2
#27212302	2
#27212301	2
#108×C₄₂₁	2
#430	421