Subgroups — #40996522

Order 26464.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27210322	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₈₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#27210311	C₂	1	1	yes	maximal, characteristic, complemented
4	#27210310	C₂	1	1	yes	maximal, characteristic, complemented
5	#27210308	C₂	1	1	yes	maximal, characteristic, complemented
6	#27210304	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×C₈₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₁₆₅₄	—	2	1	no	—
9	Dic₁₆₅₄	C₂²	1	1	yes	characteristic
10	D₃₃₀₈	—	2	1	no	—
11	D₃₃₀₈	C₂²	1	1	yes	characteristic
12	C₄×D₈₂₇	—	2	1	no	—
13	C₄×D₈₂₇	—	2	1	no	—
14	C₄×D₈₂₇	C₂²	1	1	yes	characteristic
15	C₈₂₇⋊C₈	C₂²	1	1	yes	characteristic
16	Q₈×C₈₂₇	C₂²	1	1	yes	characteristic
17	Q₈×C₈₂₇	C₂²	1	1	yes	characteristic
18	C₈×C₈₂₇	C₂²	1	1	yes	characteristic
19	C₂×D₈₂₇	—	2	1	no	—
20	C₂×D₈₂₇	D₄	1	1	yes	characteristic
21	C₄×C₈₂₇	—	2	1	no	—
22	C₄×C₈₂₇	—	2	1	no	—
23	C₄×C₈₂₇	C₂³	1	1	yes	characteristic
24	Dic₈₂₇	—	2	1	no	—
25	Dic₈₂₇	D₄	1	1	yes	characteristic
26	C₂×C₈₂₇	C₂×D₄	1	1	yes	characteristic
27	D₈₂₇	—	2	1	no	—
28	D₈₂₇	—	4	1	no	—
29	C₈₂₇	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	827	1	no	maximal, Sylow
31	#41	—	827	1	no	—
32	C₂×Q₈	—	827	1	no	—
33	Q₁₆	—	827	1	no	—
34	Q₁₆	D₈₂₇	1	1	yes	characteristic, complemented
35	SD₁₆	—	827	1	no	—
36	SD₁₆	—	827	1	no	—
37	M₁₆	—	827	1	no	—
38	Q₈	—	827	1	no	—
39	Q₈	—	1654	1	no	—
40	Q₈	C₂×D₈₂₇	1	1	yes	characteristic
41	Q₈	C₂×D₈₂₇	1	1	yes	characteristic
42	D₄	—	827	1	no	—
43	D₄	—	1654	1	no	—
44	C₂×C₄	—	827	1	no	—
45	C₂×C₄	—	1654	1	no	—
46	C₂×C₄	—	1654	1	no	—
47	C₈	—	827	1	no	—
48	C₈	C₂×D₈₂₇	1	1	yes	characteristic
49	C₂²	—	827	1	no	—
50	C₂²	—	1654	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	827	1	no	—
54	C₄	—	1654	1	no	—
55	C₄	C₂²×D₈₂₇	1	1	yes	characteristic
56	C₂	—	1654	1	no	—
57	C₂	—	3308	1	no	—
58	C₂	D₄×D₈₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₈₂₇
commutator, cosocle	C₄×C₈₂₇	C₂³
Frattini	C₄	C₂²×D₈₂₇
Fitting	Q₁₆×C₈₂₇	C₂
socle	C₂×C₈₂₇	C₂×D₄
prime radical	#27210322	C₂
hypercenter	Q₁₆	D₈₂₇
hypocenter	C₈₂₇	#137
LCS	G ▷ C₄×C₈₂₇ ▷ C₂×C₈₂₇ ▷ C₈₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₈₂₇	C₂	semidirect
3. #27210311	C₂	semidirect
4. #27210310	C₂	semidirect
5. #27210308	C₂	semidirect
6. #27210304	C₂	semidirect
7. Q₁₆×C₈₂₇	C₂	semidirect
29. C₈₂₇	#137	semidirect
34. Q₁₆	D₈₂₇	semidirect

Maximal subgroups

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

subgroup	index
#27210322	2
Q₈×D₈₂₇	2
#27210311	2
#27210310	2
#27210308	2
#27210304	2
Q₁₆×C₈₂₇	2
#137	827