Subgroups — #41001579

Order 26848.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27211955	C₂	1	2	yes	maximal, complemented
2	Dic₃₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#27211944	C₂	1	2	yes	maximal, complemented
4	D₆₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×C₈₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₆₇₈	C₂²	1	2	yes	—
7	D₃₃₅₆	C₂²	1	2	yes	—
8	C₂²⋊D₈₃₉	—	2	2	no	—
9	C₄×D₈₃₉	—	2	2	no	—
10	C₂×C₄×C₈₃₉	C₂²	1	1	yes	characteristic
11	C₈×C₈₃₉	C₂²	1	1	yes	characteristic
12	C₈×C₈₃₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈₃₉	D₄	1	1	yes	characteristic
14	C₂×D₈₃₉	—	2	2	no	—
15	C₄×C₈₃₉	D₄	1	1	yes	characteristic
16	C₄×C₈₃₉	C₂³	1	1	yes	characteristic
17	Dic₈₃₉	—	2	2	no	—
18	C₂×C₈₃₉	—	2	1	no	—
19	C₂×C₈₃₉	C₂×D₄	1	1	yes	characteristic
20	D₈₃₉	—	4	2	no	—
21	C₈₃₉	#135	1	1	yes	characteristic, complemented, Sylow
22	#135	—	839	1	no	maximal, Sylow
23	#41	—	839	2	no	—
24	Q₁₆	—	839	1	no	—
25	SD₁₆	—	839	2	no	—
26	D₈	—	839	1	no	—
27	C₂×C₈	D₈₃₉	1	1	yes	characteristic, complemented
28	Q₈	—	839	2	no	—
29	D₄	—	839	2	no	—
30	D₄	—	1678	2	no	—
31	C₂×C₄	—	1678	2	no	—
32	C₂×C₄	C₂×D₈₃₉	1	1	yes	characteristic
33	C₈	C₂×D₈₃₉	1	1	yes	characteristic
34	C₈	C₂×D₈₃₉	1	1	yes	characteristic
35	C₂²	—	1678	2	no	—
36	C₂²	D₃₃₅₆	1	1	yes	characteristic
37	C₄	—	1678	2	no	—
38	C₄	D₃₃₅₆	1	1	yes	characteristic
39	C₄	C₂²×D₈₃₉	1	1	yes	characteristic
40	C₂	—	2	1	no	—
41	C₂	—	3356	2	no	—
42	C₂	C₂×D₃₃₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₃₃₅₆
commutator, cosocle	C₄×C₈₃₉	C₂³
Frattini	C₄	C₂²×D₈₃₉
Fitting	C₂×C₈×C₈₃₉	C₂
socle	C₂×C₈₃₉	C₂×D₄
hypercenter	C₂×C₈	D₈₃₉
hypocenter	C₈₃₉	#135
LCS	G ▷ C₄×C₈₃₉ ▷ C₂×C₈₃₉ ▷ C₈₃₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #27211955	C₂	semidirect
2. Dic₃₃₅₆	C₂	semidirect
3. #27211944	C₂	semidirect
4. D₆₇₁₂	C₂	semidirect
5. C₂×C₈×C₈₃₉	C₂	semidirect
21. C₈₃₉	#135	semidirect
27. C₂×C₈	D₈₃₉	semidirect

Maximal subgroups

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

subgroup	index
#27211955	2
Dic₃₃₅₆	2
#27211944	2
D₆₇₁₂	2
C₂×C₈×C₈₃₉	2
#135	839