Subgroups — #27156850

Order 12416.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690860	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₆×D₉₇	C₂	1	1	yes	maximal, characteristic
3	#15690860	—	2	2	no	—
4	C₂×C₈×D₉₇	C₄	1	1	yes	characteristic
5	C₂×#13095486	C₂²	1	1	yes	characteristic
6	C₁₆×D₉₇	—	2	2	no	—
7	C₂×C₁₆×C₉₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₉₇	C₈	1	1	yes	characteristic
9	C₂×#11715895	C₈	1	1	yes	characteristic
10	C₈×D₉₇	—	2	1	no	—
11	C₈×D₉₇	—	2	1	no	—
12	#13095486	D₄	1	2	yes	—
13	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
14	C₁₆×C₉₇	—	2	1	no	—
15	C₂²×D₉₇	C₁₆	1	1	yes	characteristic
16	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
17	C₂×Dic₉₇	C₁₆	1	1	yes	characteristic
18	C₄×D₉₇	—	2	1	no	—
19	C₄×D₉₇	—	2	1	no	—
20	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₈×C₉₇	M₁₆	1	1	yes	characteristic
22	#11715895	—	2	1	no	—
23	C₂²×C₉₇	C₂×C₁₆	1	1	yes	characteristic
24	C₂×D₉₇	—	2	1	no	—
25	C₂×D₉₇	—	2	1	no	—
26	C₂×D₉₇	C₃₂	1	2	yes	complemented
27	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₉₇	M₃₂	1	1	yes	characteristic
29	Dic₉₇	—	2	1	no	—
30	C₂×C₉₇	C₂²⋊C₁₆	1	1	yes	characteristic
31	C₂×C₉₇	C₂×C₃₂	1	1	yes	characteristic
32	C₂×C₉₇	M₆₄	1	1	yes	characteristic
33	D₉₇	—	2	2	no	—
34	C₂²⋊C₃₂	—	97	1	no	maximal, Sylow
35	C₉₇	C₂²⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
36	C₂²×C₁₆	—	97	1	no	—
37	C₂×C₃₂	—	97	2	no	—
38	C₂²×C₈	—	97	1	no	—
39	C₂×C₁₆	—	97	1	no	—
40	C₂×C₁₆	—	194	2	no	—
41	C₂×C₁₆	#87604	1	1	yes	characteristic
42	C₃₂	—	194	2	no	—
43	C₂²×C₄	—	97	1	no	—
44	C₂×C₈	—	97	1	no	—
45	C₂×C₈	—	194	1	no	—
46	C₂×C₈	—	194	1	no	—
47	C₂×C₈	C₂×#87604	1	1	yes	characteristic
48	C₁₆	—	2	1	no	—
49	C₁₆	—	97	2	no	—
50	C₂³	—	97	1	no	—
51	C₂×C₄	—	97	1	no	—
52	C₂×C₄	—	194	1	no	—
53	C₂×C₄	—	194	1	no	—
54	C₂×C₄	#13095519	1	1	yes	characteristic
55	C₈	—	194	1	no	—
56	C₈	C₂²⋊#87604	1	1	yes	characteristic
57	C₈	#13095521	1	1	yes	characteristic
58	C₂²	—	97	2	no	—
59	C₂²	—	194	1	no	—
60	C₂²	—	194	1	no	—
61	C₂²	#15690883	1	1	yes	characteristic
62	C₄	—	194	1	no	—
63	C₄	#15690867	1	1	yes	characteristic
64	C₄	#15690885	1	1	yes	characteristic
65	C₂	—	194	2	no	—
66	C₂	#20261630	1	1	yes	characteristic
67	C₂	#20261666	1	1	yes	characteristic
68	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	C₂²×C₉₇	C₂×C₁₆
prime radical	C₂²×D₉₇	C₁₆
cosocle	C₂×#13095486	C₂²
hypercenter	C₂×C₁₆	#87604
hypocenter	C₉₇	C₂²⋊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
1. C₂×#15690860	C₂	semidirect
26. C₂×D₉₇	C₃₂	semidirect
35. C₉₇	C₂²⋊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
C₂×C₁₆×D₉₇	2
C₂²⋊C₃₂	97