Subgroups — #27139942

Order 12416.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260565	C₂	1	1	yes	maximal, characteristic
2	#20260563	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₉₇	C₂	1	1	yes	maximal, characteristic
4	#15690746	—	2	1	no	—
5	#15690744	—	2	1	no	—
6	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
7	C₃₂×C₉₇	C₄	1	2	yes	complemented
8	#13095495	—	4	1	no	—
9	C₄⋊Dic₉₇	—	4	1	no	—
10	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
11	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₉₇	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
14	C₂×Dic₉₇	—	8	1	no	—
15	C₈×C₉₇	C₄⋊C₄	1	1	yes	characteristic
16	C₈×C₉₇	Q₁₆	1	1	yes	characteristic
17	#11715895	—	8	1	no	—
18	C₂²×C₉₇	SD₃₂	1	1	yes	characteristic
19	C₄×C₉₇	#107	1	1	yes	characteristic
20	C₄×C₉₇	SD₃₂	1	1	yes	characteristic
21	Dic₉₇	—	16	1	no	—
22	C₂×C₉₇	—	2	1	no	—
23	C₂×C₉₇	#367	1	1	yes	characteristic
24	#1426	—	97	1	no	maximal, Sylow
25	C₉₇	#1426	1	1	yes	characteristic, complemented, Sylow
26	M₆₄	D₉₇	1	1	yes	characteristic
27	#368	—	97	1	no	—
28	#367	—	97	1	no	—
29	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
30	#108	—	194	1	no	—
31	#107	—	194	1	no	—
32	C₃₂	Dic₉₇	1	2	yes	complemented
33	M₁₆	—	388	1	no	—
34	C₂×C₈	D₃₈₈	1	1	yes	characteristic
35	C₄⋊C₄	—	388	1	no	—
36	C₁₆	Dic₁₉₄	1	1	yes	characteristic
37	C₁₆	C₂×Dic₉₇	1	1	yes	characteristic
38	C₂×C₄	—	776	1	no	—
39	C₂×C₄	D₇₇₆	1	1	yes	characteristic
40	C₈	—	776	1	no	—
41	C₈	C₄⋊Dic₉₇	1	1	yes	characteristic
42	C₈	Dic₃₈₈	1	1	yes	characteristic
43	C₂²	#15690755	1	1	yes	characteristic
44	C₄	—	1552	1	no	—
45	C₄	#15690744	1	1	yes	characteristic
46	C₄	#15690755	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#20260563	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260563
commutator	C₁₆×C₉₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₉₇
Fitting	M₆₄×C₉₇	C₂
socle	C₂×C₉₇	#367
prime radical	C₂²×C₉₇	SD₃₂
cosocle	C₂×C₁₆×C₉₇	C₂²
hypercenter	M₆₄	D₉₇
hypocenter	C₉₇	#1426
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
7. C₃₂×C₉₇	C₄	semidirect
25. C₉₇	#1426	semidirect
32. C₃₂	Dic₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20260565	2
#20260563	2
M₆₄×C₉₇	2
#1426	97