Subgroups — #27158796

Order 12416.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261824	C₂	1	2	yes	maximal
2	#20261622	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11715906	—	2	2	no	—
4	#15690883	C₄	1	1	yes	characteristic
5	C₈×#87604	C₂²	1	1	yes	characteristic
6	C₁₆×D₉₇	C₄	1	1	yes	characteristic
7	C₂×#11715906	—	4	2	no	—
8	#13095519	Q₈	1	1	yes	characteristic
9	C₄×#87604	D₄	1	1	yes	characteristic
10	#13095514	—	2	1	no	—
11	C₈×D₉₇	C₂×C₄	1	1	yes	characteristic
12	#13095486	C₈	1	1	yes	characteristic, complemented
13	C₁₆×C₉₇	C₈	1	1	yes	characteristic, complemented
14	C₂×#87604	SD₁₆	1	2	yes	—
15	#11715906	—	8	2	no	—
16	C₄×D₉₇	C₄⋊C₄	1	1	yes	characteristic
17	#11715897	—	2	1	no	—
18	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
19	#11715895	M₁₆	1	1	yes	characteristic
20	C₂×D₉₇	#106	1	1	yes	characteristic
21	#87604	—	2	2	no	—
22	C₄×C₉₇	C₄⋊C₈	1	1	yes	characteristic
23	Dic₉₇	#108	1	1	yes	characteristic
24	C₂×C₉₇	#334	1	1	yes	characteristic
25	D₉₇	#365	1	2	yes	—
26	#1368	—	97	1	no	maximal, Sylow
27	C₉₇	#1368	1	1	yes	characteristic, complemented, Sylow
28	#346	—	97	1	no	—
29	#334	—	97	2	no	—
30	C₂×C₁₆	—	97	1	no	—
31	C₂×C₁₆	—	97	1	no	—
32	C₄⋊C₈	—	194	2	no	—
33	C₄×C₈	—	97	1	no	—
34	C₂×C₈	—	97	1	no	—
35	C₂×C₈	—	97	1	no	—
36	C₂×C₈	—	388	2	no	—
37	C₄²	—	97	1	no	—
38	C₁₆	—	97	1	no	—
39	C₁₆	—	194	1	no	—
40	C₁₆	#11715906	1	1	yes	characteristic, complemented
41	C₂×C₄	—	97	1	no	—
42	C₂×C₄	—	97	2	no	—
43	C₈	—	776	2	no	—
44	C₈	—	97	1	no	—
45	C₈	—	194	1	no	—
46	C₈	C₂×#11715906	1	1	yes	characteristic
47	C₂²	—	97	1	no	—
48	C₄	—	97	1	no	—
49	C₄	—	194	2	no	—
50	C₄	C₄⋊#11715906	1	1	yes	characteristic
51	C₂	—	97	2	no	—
52	C₂	#20261824	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261824
commutator	C₈×C₉₇	C₂×C₈
Frattini	C₈	C₂×#11715906
Fitting	C₁₆×C₉₇	C₈
socle	C₂×C₉₇	#334
prime radical	C₂×D₉₇	#106
cosocle	C₈×#87604	C₂²
hypercenter	C₁₆	#11715906
hypocenter	C₉₇	#1368
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
12. #13095486	C₈	semidirect
13. C₁₆×C₉₇	C₈	semidirect
27. C₉₇	#1368	semidirect
40. C₁₆	#11715906	semidirect

Maximal subgroups

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

subgroup	index
#20261824	2
#20261622	2
#1368	97