Group #27144922 — subgroups

Subgroups — #27144922

Order 12416.

408 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260634	C₂	1	4	yes	maximal
2	#20260621	C₂	1	4	yes	maximal
3	#20260621	C₂	1	4	yes	maximal
4	#20260601	C₂	1	2	yes	maximal
5	C₂×#125×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₉₇	C₂²	1	4	yes	—
8	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
9	C₂×C₄×Dic₉₇	C₂²	1	4	yes	—
10	#15690717	C₂²	1	4	yes	—
11	#15690717	C₂²	1	4	yes	—
12	#15690717	C₂²	1	4	yes	—
13	#15690717	C₂²	1	4	yes	—
14	#125×C₉₇	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₉₇	C₂²	1	4	yes	—
17	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
18	C₂²×Dic₉₇	C₂³	1	4	yes	—
19	C₂²×Dic₉₇	C₂³	1	4	yes	—
20	C₄⋊Dic₉₇	—	4	2	no	—
21	C₄⋊Dic₉₇	—	4	4	no	—
22	C₄⋊Dic₉₇	—	4	2	no	—
23	C₄×Dic₉₇	—	4	4	no	—
24	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
25	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
26	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₉₇	C₂×C₄	1	8	yes	—
28	C₄⋊C₄×C₉₇	C₂×C₄	1	16	yes	—
29	C₄²×C₉₇	C₂×C₄	1	4	yes	—
30	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
31	C₂×C₄×C₉₇	—	2	4	no	—
32	C₂×C₄×C₉₇	—	2	2	no	—
33	C₂×C₄×C₉₇	—	2	4	no	—
34	C₂×C₄×C₉₇	—	2	2	no	—
35	C₂×C₄×C₉₇	—	2	1	no	—
36	C₂×C₄×C₉₇	—	2	1	no	—
37	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
39	C₂×C₄×C₉₇	C₂²×C₄	1	8	yes	—
40	C₂×Dic₉₇	—	2	4	no	—
41	C₂×Dic₉₇	—	2	4	no	—
42	C₂×Dic₉₇	—	2	4	no	—
43	C₂×Dic₉₇	—	2	4	no	—
44	C₂×Dic₉₇	—	2	4	no	—
45	C₂×Dic₉₇	—	2	4	no	—
46	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
47	C₂²×C₉₇	H(5,𝔽₂)	1	2	yes	—
48	C₂²×C₉₇	2₋¹⁺⁴	1	2	yes	—
49	C₂²×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
50	C₂²×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
51	C₄×C₉₇	—	2	2	no	—
52	C₄×C₉₇	—	2	8	no	—
53	C₄×C₉₇	—	2	4	no	—
54	Dic₉₇	—	4	4	no	—
55	Dic₉₇	—	4	4	no	—
56	C₂×C₉₇	#519	1	1	yes	characteristic
57	C₂×C₉₇	#520	1	2	yes	—
58	C₂×C₉₇	#562	1	1	yes	characteristic
59	C₂×C₉₇	#563	1	2	yes	—
60	C₂×C₉₇	#564	1	1	yes	characteristic
61	#2382	—	97	1	no	maximal, Sylow
62	C₉₇	#2382	1	1	yes	characteristic, complemented, Sylow
63	C₂×#125	D₉₇	1	1	yes	characteristic
64	#389	—	97	4	no	—
65	#387	—	97	4	no	—
66	#387	—	97	4	no	—
67	#382	—	97	2	no	—
68	#125	Dic₉₇	1	8	yes	complemented
69	C₂×C₄⋊C₄	—	97	2	no	—
70	C₂×C₄⋊C₄	—	97	2	no	—
71	C₂×C₄⋊C₄	—	97	4	no	—
72	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
73	C₂×C₄⋊C₄	C₂×D₉₇	1	4	yes	—
74	C₂×C₄²	—	97	4	no	—
75	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
76	#95	—	97	4	no	—
77	#95	—	97	4	no	—
78	#95	—	97	4	no	—
79	#95	—	97	4	no	—
80	C₂²×C₄	—	97	4	no	—
81	C₂²×C₄	—	97	4	no	—
82	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
83	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
84	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
85	C₄⋊C₄	—	388	4	no	—
86	C₄⋊C₄	—	388	2	no	—
87	C₄⋊C₄	—	388	2	no	—
88	C₄⋊C₄	C₂×Dic₉₇	1	8	yes	—
89	C₄⋊C₄	C₂×Dic₉₇	1	16	yes	—
90	C₄²	—	388	4	no	—
91	C₄²	C₂×Dic₉₇	1	4	yes	—
92	C₂³	C₂³×D₉₇	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	194	4	no	—
100	C₂×C₄	—	194	4	no	—
101	C₂×C₄	—	194	4	no	—
102	C₂×C₄	—	194	4	no	—
103	C₂×C₄	—	194	4	no	—
104	C₂×C₄	—	194	4	no	—
105	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
106	C₂×C₄	C₂²×Dic₉₇	1	4	yes	—
107	C₂×C₄	C₂²×Dic₉₇	1	8	yes	—
108	C₂²	C₂³×Dic₉₇	1	1	yes	characteristic
109	C₂²	#15690856	1	2	yes	—
110	C₂²	#15690857	1	2	yes	—
111	C₂²	#15690858	1	1	yes	characteristic
112	C₂²	#15690858	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	2	8	no	—
115	C₄	—	2	4	no	—
116	C₄	—	388	4	no	—
117	C₄	—	388	4	no	—
118	C₂	#20261213	1	1	yes	characteristic
119	C₂	#20261218	1	2	yes	—
120	C₂	#20261483	1	1	yes	characteristic
121	C₂	#20261489	1	2	yes	—
122	C₂	C₉₇⋊#564	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260634

C₂

yes

maximal

#20260621

C₂

yes

maximal

#20260621

C₂

yes

maximal

#20260601

C₂

yes

maximal

C₂×#125×C₉₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×C₄×Dic₉₇

C₂²

yes

—

#15690717

C₂²

yes

—

#15690717

C₂²

yes

—

#15690717

C₂²

yes

—

#15690717

C₂²

yes

—

#125×C₉₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

C₂³

yes

—

C₂²×Dic₉₇

C₂³

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄×Dic₉₇

—

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

C₂×C₄

yes

—

C₂³×C₉₇

C₂⁴

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂²×C₉₇

C₂³×C₄

yes

characteristic

C₂²×C₉₇

H(5,𝔽₂)

yes

—

C₂²×C₉₇

2₋¹⁺⁴

yes

—

C₂²×C₉₇

2₋¹⁺⁴

yes

characteristic

C₂²×C₉₇

2₋¹⁺⁴

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#519

yes

characteristic

C₂×C₉₇

#520

yes

—

C₂×C₉₇

#562

yes

characteristic

C₂×C₉₇

#563

yes

—

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×#125

D₉₇

yes

—

—

—

—

Dic₉₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉₇

yes

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

—

C₄⋊C₄

—

388

—

C₄⋊C₄

—

388

—

C₄⋊C₄

—

388

—

C₄⋊C₄

C₂×Dic₉₇

yes

—

C₄⋊C₄

C₂×Dic₉₇

yes

—

C₄²

—

388

—

C₄²

C₂×Dic₉₇

yes

—

C₂³

C₂³×D₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

194

—

100

C₂×C₄

—

194

—

101

C₂×C₄

—

194

—

102

C₂×C₄

—

194

—

103

C₂×C₄

—

194

—

104

C₂×C₄

—

194

—

105

C₂×C₄

C₂²×Dic₉₇

yes

—

106

C₂×C₄

C₂²×Dic₉₇

yes

—

107

C₂×C₄

C₂²×Dic₉₇

yes

—

108

C₂²

C₂³×Dic₉₇

yes

characteristic

109

C₂²

#15690856

yes

—

110

C₂²

#15690857

yes

—

111

C₂²

#15690858

yes

characteristic

112

C₂²

#15690858

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

388

—

117

C₄

—

388

—

118

C₂

#20261213

yes

characteristic

119

C₂

#20261218

yes

—

120

C₂

#20261483

yes

characteristic

121

C₂

#20261489

yes

—

122

C₂

C₉₇⋊#564

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator	C₂²×C₉₇	C₂³×C₄
Fitting	C₂×#125×C₉₇	C₂
socle, prime radical, cosocle	C₂³×C₉₇	C₂⁴
hypercenter	C₂×#125	D₉₇
hypocenter	C₉₇	#2382
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₉₇

commutator

C₂²×C₉₇

C₂³×C₄

Fitting

C₂×#125×C₉₇

C₂

socle, prime radical, cosocle

C₂³×C₉₇

C₂⁴

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
14. #125×C₉₇	C₄	semidirect
62. C₉₇	#2382	semidirect
68. #125	Dic₉₇	semidirect

H ≅ G/N

type

14. #125×C₉₇

C₄

semidirect

62. C₉₇

#2382

semidirect

68. #125

Dic₉₇

semidirect

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

subgroup	index
#20260634	2
#20260621	2
#20260601	2
C₂×#125×C₉₇	2
#2382	97

subgroup

index

C₂×#125×C₉₇

Subgroups — #27144922

Conjugacy classes of subgroups

Complements

Maximal subgroups