Subgroups — #27139936

Order 12416.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260561	C₂	1	2	yes	maximal
2	C₂×C₃₂×C₉₇	C₂	1	1	yes	maximal, characteristic
3	#15690744	—	2	2	no	—
4	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
5	C₃₂×C₉₇	C₄	1	2	yes	complemented
6	C₄⋊Dic₉₇	—	4	2	no	—
7	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
8	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
9	C₁₆×C₉₇	Q₈	1	1	yes	characteristic
10	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
11	C₂×Dic₉₇	—	8	2	no	—
12	C₈×C₉₇	C₄⋊C₄	1	1	yes	characteristic
13	C₈×C₉₇	Q₁₆	1	1	yes	characteristic
14	C₂²×C₉₇	D₁₆	1	1	yes	characteristic
15	C₄×C₉₇	#107	1	1	yes	characteristic
16	C₄×C₉₇	Q₃₂	1	1	yes	characteristic
17	Dic₉₇	—	16	2	no	—
18	C₂×C₉₇	#366	1	1	yes	characteristic
19	C₂×C₉₇	D₃₂	1	1	yes	characteristic
20	C₂×C₉₇	Q₆₄	1	1	yes	characteristic
21	#1423	—	97	1	no	maximal, Sylow
22	C₉₇	#1423	1	1	yes	characteristic, complemented, Sylow
23	C₂×C₃₂	D₉₇	1	1	yes	characteristic
24	#366	—	97	2	no	—
25	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
26	#107	—	194	2	no	—
27	C₃₂	Dic₉₇	1	2	yes	complemented
28	C₂×C₈	D₃₈₈	1	1	yes	characteristic
29	C₄⋊C₄	—	388	2	no	—
30	C₁₆	Dic₁₉₄	1	1	yes	characteristic
31	C₁₆	C₂×Dic₉₇	1	1	yes	characteristic
32	C₂×C₄	—	776	2	no	—
33	C₂×C₄	D₇₇₆	1	1	yes	characteristic
34	C₈	C₄⋊Dic₉₇	1	1	yes	characteristic
35	C₈	Dic₃₈₈	1	1	yes	characteristic
36	C₂²	D₁₅₅₂	1	1	yes	characteristic
37	C₄	—	1552	2	no	—
38	C₄	#15690744	1	1	yes	characteristic
39	C₄	Dic₇₇₆	1	1	yes	characteristic
40	C₂	#20260561	1	1	yes	characteristic
41	C₂	D₃₁₀₄	1	1	yes	characteristic
42	C₂	Dic₁₅₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₅₅₂
commutator	C₁₆×C₉₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₉₇
Fitting	C₂×C₃₂×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	D₁₆
cosocle	C₂×C₁₆×C₉₇	C₂²
hypercenter	C₂×C₃₂	D₉₇
hypocenter	C₉₇	#1423
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
5. C₃₂×C₉₇	C₄	semidirect
22. C₉₇	#1423	semidirect
27. C₃₂	Dic₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20260561	2
C₂×C₃₂×C₉₇	2
#1423	97