Subgroups — #27156902

Order 12416.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261664	C₂	1	1	yes	maximal, characteristic
2	#20261662	C₂	1	1	yes	maximal, characteristic
3	#20260568	C₂	1	1	yes	maximal, characteristic
4	#15690881	—	2	1	no	—
5	#15690879	—	2	1	no	—
6	C₁₆×D₉₇	C₂²	1	1	yes	characteristic
7	#15690716	C₄	1	1	yes	characteristic, complemented
8	C₃₂×C₉₇	C₄	1	1	yes	characteristic, complemented
9	#13095521	—	4	1	no	—
10	C₄⋊#87604	—	4	1	no	—
11	C₈×D₉₇	D₄	1	1	yes	characteristic
12	#13095486	Q₈	1	1	yes	characteristic
13	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
14	C₂×#87604	—	8	1	no	—
15	C₄×D₉₇	D₈	1	1	yes	characteristic
16	#11715897	—	8	1	no	—
17	C₈×C₉₇	C₄⋊C₄	1	1	yes	characteristic
18	#11715895	Q₁₆	1	1	yes	characteristic
19	C₂×D₉₇	SD₃₂	1	1	yes	characteristic
20	#87604	—	16	1	no	—
21	C₄×C₉₇	#107	1	1	yes	characteristic
22	Dic₉₇	SD₃₂	1	1	yes	characteristic
23	C₂×C₉₇	#367	1	1	yes	characteristic
24	D₉₇	—	2	1	no	—
25	#1426	—	97	1	no	maximal, Sylow
26	C₉₇	#1426	1	1	yes	characteristic, complemented, Sylow
27	M₆₄	—	97	1	no	—
28	#368	—	97	1	no	—
29	#367	—	97	1	no	—
30	C₂×C₁₆	—	97	1	no	—
31	#108	—	194	1	no	—
32	#107	—	194	1	no	—
33	C₃₂	—	97	1	no	—
34	C₃₂	#87604	1	1	yes	characteristic, complemented
35	M₁₆	—	388	1	no	—
36	C₂×C₈	—	97	1	no	—
37	C₄⋊C₄	—	388	1	no	—
38	C₁₆	—	97	1	no	—
39	C₁₆	C₂×#87604	1	1	yes	characteristic
40	C₂×C₄	—	776	1	no	—
41	C₂×C₄	—	97	1	no	—
42	C₈	—	776	1	no	—
43	C₈	—	97	1	no	—
44	C₈	C₄⋊#87604	1	1	yes	characteristic
45	C₂²	—	97	1	no	—
46	C₄	—	1552	1	no	—
47	C₄	—	97	1	no	—
48	C₄	#15690879	1	1	yes	characteristic
49	C₂	—	194	1	no	—
50	C₂	#20261662	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261662
commutator	C₁₆×C₉₇	C₂×C₄
Frattini	C₁₆	C₂×#87604
Fitting	C₃₂×C₉₇	C₄
socle	C₂×C₉₇	#367
prime radical	C₂×D₉₇	SD₃₂
cosocle	C₁₆×D₉₇	C₂²
hypercenter	C₃₂	#87604
hypocenter	C₉₇	#1426
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

Complements

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

N	H ≅ G/N	type
7. #15690716	C₄	semidirect
8. C₃₂×C₉₇	C₄	semidirect
26. C₉₇	#1426	semidirect
34. C₃₂	#87604	semidirect

Maximal subgroups

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

subgroup	index
#20261664	2
#20261662	2
#20260568	2
#1426	97