Subgroups — #27156894

Order 12416.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690860	C₂	1	2	yes	maximal
2	C₁₆×Dic₉₇	C₂	1	1	yes	maximal, characteristic
3	#15690860	—	2	2	no	—
4	C₂×#13095486	C₂²	1	1	yes	characteristic
5	C₈×Dic₉₇	C₄	1	1	yes	characteristic
6	C₂×C₁₆×C₉₇	C₄	1	1	yes	characteristic
7	C₂×#11715895	C₈	1	1	yes	characteristic
8	C₄×Dic₉₇	C₈	1	1	yes	characteristic
9	#13095486	D₄	1	1	yes	characteristic
10	#13095486	Q₈	1	1	yes	characteristic
11	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₉₇	—	2	1	no	—
13	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
14	C₂×Dic₉₇	C₁₆	1	1	yes	characteristic
15	C₂×Dic₉₇	C₁₆	1	1	yes	characteristic
16	C₈×C₉₇	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₉₇	M₁₆	1	1	yes	characteristic
18	#11715895	—	2	1	no	—
19	C₂²×C₉₇	C₂×C₁₆	1	1	yes	characteristic
20	C₄×C₉₇	C₄⋊C₈	1	1	yes	characteristic
21	C₄×C₉₇	M₃₂	1	1	yes	characteristic
22	Dic₉₇	—	2	1	no	—
23	Dic₉₇	C₃₂	1	2	yes	complemented
24	C₂×C₉₇	C₄⋊C₁₆	1	1	yes	characteristic
25	C₂×C₉₇	C₂×C₃₂	1	1	yes	characteristic
26	C₂×C₉₇	M₆₄	1	1	yes	characteristic
27	C₄⋊C₃₂	—	97	1	no	maximal, Sylow
28	C₉₇	C₄⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₃₂	—	97	2	no	—
30	C₄×C₁₆	—	97	1	no	—
31	C₂×C₁₆	—	97	1	no	—
32	C₂×C₁₆	#87604	1	1	yes	characteristic
33	C₄×C₈	—	97	1	no	—
34	C₃₂	—	194	2	no	—
35	C₂×C₈	—	97	1	no	—
36	C₂×C₈	C₂×#87604	1	1	yes	characteristic
37	C₄²	—	97	1	no	—
38	C₁₆	—	2	1	no	—
39	C₁₆	—	97	1	no	—
40	C₁₆	—	97	1	no	—
41	C₂×C₄	—	97	1	no	—
42	C₂×C₄	—	97	1	no	—
43	C₂×C₄	#13095519	1	1	yes	characteristic
44	C₈	—	194	1	no	—
45	C₈	C₄⋊#87604	1	1	yes	characteristic
46	C₈	#13095520	1	1	yes	characteristic
47	C₂²	#15690883	1	1	yes	characteristic
48	C₄	—	97	2	no	—
49	C₄	—	194	1	no	—
50	C₄	#15690877	1	1	yes	characteristic
51	C₄	#15690885	1	1	yes	characteristic
52	C₂	#20261656	1	1	yes	characteristic
53	C₂	#20261666	1	1	yes	characteristic
54	C₂	#20261668	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×#87604
commutator	C₂×C₉₇	C₂×C₃₂
Fitting	C₂×C₁₆×C₉₇	C₄
socle, prime radical	C₂²×C₉₇	C₂×C₁₆
cosocle	C₂×#13095486	C₂²
hypercenter	C₂×C₁₆	#87604
hypocenter	C₉₇	C₄⋊C₃₂
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
23. Dic₉₇	C₃₂	semidirect
28. C₉₇	C₄⋊C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690860	2
C₁₆×Dic₉₇	2
C₄⋊C₃₂	97