Subgroups — #27139871

Order 12416.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260569	C₂	1	1	yes	maximal, characteristic
2	#20260510	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095486	C₄	1	1	yes	characteristic
5	C₈×Dic₉₇	C₄	1	1	yes	characteristic
6	#15690716	C₄	1	2	yes	complemented
7	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
8	C₃₂×C₉₇	C₄	1	2	yes	complemented
9	C₂×#11715895	C₈	1	1	yes	characteristic
10	C₄×Dic₉₇	C₈	1	1	yes	characteristic
11	#13095486	C₈	1	2	yes	—
12	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
14	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
16	C₂×Dic₉₇	—	2	1	no	—
17	C₈×C₉₇	C₄²	1	1	yes	characteristic
18	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
19	#11715895	—	2	1	no	—
20	C₂²×C₉₇	M₃₂	1	1	yes	characteristic
21	C₄×C₉₇	C₄×C₈	1	1	yes	characteristic
22	C₄×C₉₇	M₃₂	1	1	yes	characteristic
23	Dic₉₇	—	4	1	no	—
24	C₂×C₉₇	—	2	1	no	—
25	C₂×C₉₇	#346	1	1	yes	characteristic
26	#1398	—	97	1	no	maximal, Sylow
27	C₉₇	#1398	1	1	yes	characteristic, complemented, Sylow
28	M₆₄	—	97	1	no	—
29	M₆₄	D₉₇	1	1	yes	characteristic
30	#346	—	97	1	no	—
31	C₂×C₁₆	—	97	1	no	—
32	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
33	C₄×C₈	—	97	1	no	—
34	C₃₂	—	97	2	no	—
35	C₃₂	Dic₉₇	1	2	yes	complemented
36	C₂×C₈	—	97	1	no	—
37	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
38	C₄²	—	97	1	no	—
39	C₁₆	—	97	2	no	—
40	C₁₆	C₄×D₉₇	1	1	yes	characteristic
41	C₁₆	C₂×Dic₉₇	1	1	yes	characteristic
42	C₂×C₄	—	194	1	no	—
43	C₂×C₄	C₈×D₉₇	1	1	yes	characteristic
44	C₈	—	194	1	no	—
45	C₈	C₄×Dic₉₇	1	1	yes	characteristic
46	C₈	C₈×D₉₇	1	1	yes	characteristic
47	C₂²	#15690749	1	1	yes	characteristic
48	C₄	—	388	1	no	—
49	C₄	C₈×Dic₉₇	1	1	yes	characteristic
50	C₄	#15690749	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	#20260510	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×Dic₉₇
commutator	C₄×C₉₇	C₄×C₈
Frattini	C₂×C₁₆	C₂×D₉₇
Fitting	M₆₄×C₉₇	C₂
socle	C₂×C₉₇	#346
prime radical	C₂²×C₉₇	M₃₂
cosocle	C₂×C₁₆×C₉₇	C₂²
hypercenter	M₆₄	D₉₇
hypocenter	C₉₇	#1398
LCS	G ▷ C₄×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₈ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
6. #15690716	C₄	semidirect
8. C₃₂×C₉₇	C₄	semidirect
27. C₉₇	#1398	semidirect
35. C₃₂	Dic₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20260569	2
#20260510	2
M₆₄×C₉₇	2
#1398	97