Subgroups — #27150128

Order 12416.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260626	C₂	1	7	yes	maximal
2	#20260619	C₂	1	7	yes	maximal, complemented
3	C₄³×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13095490	C₂²	1	7	yes	—
5	C₂×#13095488	C₂²	1	7	yes	—
6	#15690717	C₂²	1	14	yes	—
7	C₂×C₄²×C₉₇	C₂²	1	7	yes	—
8	C₂³×D₉₇	C₂³	1	1	yes	characteristic
9	C₂²×Dic₉₇	C₂³	1	7	yes	—
10	#13095490	—	2	14	no	—
11	#13095488	—	2	14	no	—
12	C₂²×C₄×C₉₇	C₂³	1	7	yes	—
13	C₄²×C₉₇	—	2	14	no	—
14	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
15	C₂²×D₉₇	—	2	7	no	—
16	C₂×C₄×C₉₇	—	2	14	no	—
17	C₂×C₄×C₉₇	#41	1	14	yes	—
18	C₂×Dic₉₇	—	2	7	no	—
19	C₂×Dic₉₇	—	2	14	no	—
20	C₂²×C₉₇	C₂×#41	1	7	yes	—
21	C₂×D₉₇	—	4	7	no	—
22	C₄×C₉₇	—	2	14	no	—
23	Dic₉₇	—	4	7	no	—
24	C₂×C₉₇	#529	1	7	yes	—
25	D₉₇	—	8	1	no	—
26	#2866	—	97	1	no	maximal, Sylow
27	C₉₇	#2866	1	1	yes	characteristic, complemented, Sylow
28	#388	—	97	7	no	—
29	#387	—	97	7	no	—
30	C₄³	D₉₇	1	1	yes	characteristic, complemented
31	C₂×C₄⋊C₄	—	97	7	no	—
32	C₂×C₂²⋊C₄	—	97	7	no	—
33	C₂×C₄²	C₂×D₉₇	1	7	yes	—
34	#95	—	97	14	no	—
35	C₂⁴	—	97	1	no	—
36	C₂²×C₄	—	97	7	no	—
37	C₂²×C₄	C₂²×D₉₇	1	7	yes	—
38	C₄⋊C₄	—	194	14	no	—
39	C₂²⋊C₄	—	194	14	no	—
40	C₄²	—	2	14	no	—
41	C₂³	—	194	7	no	—
42	C₂³	C₂³×D₉₇	1	1	yes	characteristic
43	C₂×C₄	—	2	14	no	—
44	C₂×C₄	—	194	7	no	—
45	C₂×C₄	—	194	14	no	—
46	C₂×C₄	#13095511	1	14	yes	—
47	C₂²	—	388	7	no	—
48	C₂²	C₂×#13095511	1	7	yes	—
49	C₄	—	2	14	no	—
50	C₄	—	388	7	no	—
51	C₂	—	776	1	no	—
52	C₂	#20261269	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator, socle, cosocle	C₂³×C₉₇	C₂⁴
Fitting	C₄³×C₉₇	C₂
prime radical	C₂³×D₉₇	C₂³
hypercenter	C₄³	D₉₇
hypocenter	C₉₇	#2866
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
2. #20260619	C₂	semidirect
3. C₄³×C₉₇	C₂	semidirect
27. C₉₇	#2866	semidirect
30. C₄³	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20260626	2
#20260619	2
C₄³×C₉₇	2
#2866	97