Group #38742456 — subgroups

Subgroups — #38742456

Order 25216.

296 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₂×C₈×Dic₁₉₇	C₂	1	2	yes	maximal
2	C₄²×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic
3	#27160834	C₂	1	1	yes	maximal, characteristic
4	#27160833	C₂	1	2	yes	maximal
5	#336×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13096802	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₉₇	C₄	1	2	yes	—
8	C₂×C₄×Dic₁₉₇	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₉₇	C₄	1	8	yes	complemented
12	C₂²×C₈×C₁₉₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₉₇	C₂×C₄	1	2	yes	—
15	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
17	C₂×#13096802	—	2	2	no	—
18	C₂×#13096802	—	2	2	no	—
19	C₂×#13096802	C₂×C₄	1	4	yes	—
20	C₄×Dic₁₉₇	—	2	4	no	—
21	C₄×Dic₁₉₇	—	2	1	no	—
22	C₄×Dic₁₉₇	—	2	1	no	—
23	C₄×Dic₁₉₇	C₈	1	8	yes	complemented
24	C₄×Dic₁₉₇	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₉₇	—	2	2	no	—
28	C₂×C₈×C₁₉₇	—	2	2	no	—
29	C₂×C₈×C₁₉₇	C₂×C₄	1	4	yes	—
30	C₄²×C₁₉₇	—	2	2	no	—
31	C₂×Dic₁₉₇	—	2	2	no	—
32	C₂×Dic₁₉₇	—	2	2	no	—
33	C₂×Dic₁₉₇	C₄²	1	2	yes	—
34	C₂×Dic₁₉₇	C₄²	1	2	yes	—
35	C₂×Dic₁₉₇	C₂×C₈	1	4	yes	—
36	C₂×Dic₁₉₇	C₂×C₈	1	4	yes	—
37	C₂×Dic₁₉₇	M₁₆	1	4	yes	—
38	#13096802	—	2	4	no	—
39	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₇	—	2	2	no	—
41	C₂×C₄×C₁₉₇	—	2	2	no	—
42	C₂×C₄×C₁₉₇	C₂×C₈	1	4	yes	—
43	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₁₉₇	#41	1	2	yes	—
45	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
46	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
47	C₈×C₁₉₇	—	2	4	no	—
48	C₂²×C₁₉₇	C₂×C₄²	1	1	yes	characteristic
49	C₂²×C₁₉₇	#117	1	1	yes	characteristic
50	C₂²×C₁₉₇	#117	1	1	yes	characteristic
51	C₂²×C₁₉₇	C₂²×C₈	1	2	yes	—
52	C₂²×C₁₉₇	C₂×M₁₆	1	2	yes	—
53	C₄×C₁₉₇	—	2	4	no	—
54	C₄×C₁₉₇	#117	1	2	yes	—
55	C₄×C₁₉₇	#117	1	2	yes	—
56	Dic₁₉₇	—	2	4	no	—
57	Dic₁₉₇	C₄×C₈	1	4	yes	—
58	Dic₁₉₇	#97	1	4	yes	—
59	C₂×C₁₉₇	#376	1	1	yes	characteristic
60	C₂×C₁₉₇	C₂×C₄×C₈	1	1	yes	characteristic
61	C₂×C₁₉₇	C₂×#97	1	1	yes	characteristic
62	C₂×C₁₉₇	#431	1	2	yes	—
63	C₂×C₁₉₇	#431	1	2	yes	—
64	C₁₉₇	#1744	1	1	yes	characteristic, complemented, Sylow
65	#1744	—	197	1	no	maximal, Sylow
66	C₂×C₄×C₈	—	197	2	no	—
67	C₄³	—	197	1	no	—
68	#336	—	197	2	no	—
69	#336	—	197	1	no	—
70	#336	D₁₉₇	1	1	yes	characteristic
71	C₂²×C₈	—	197	2	no	—
72	C₂²×C₈	C₂×D₁₉₇	1	2	yes	—
73	C₂×C₄²	—	197	2	no	—
74	C₂×C₄²	—	197	1	no	—
75	C₂×C₄²	—	197	2	no	—
76	C₂×C₄²	—	197	1	no	—
77	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
78	C₄×C₈	—	197	8	no	—
79	C₂²×C₄	—	197	2	no	—
80	C₂²×C₄	—	197	1	no	—
81	C₂²×C₄	—	197	1	no	—
82	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	394	2	no	—
87	C₂×C₈	—	394	2	no	—
88	C₂×C₈	—	197	4	no	—
89	C₂×C₈	C₄×D₁₉₇	1	4	yes	—
90	C₄²	—	2	2	no	—
91	C₄²	—	394	4	no	—
92	C₄²	—	394	1	no	—
93	C₄²	—	394	1	no	—
94	C₄²	—	197	8	no	—
95	C₄²	—	197	4	no	—
96	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	394	2	no	—
100	C₂×C₄	—	394	2	no	—
101	C₂×C₄	—	197	4	no	—
102	C₂×C₄	—	197	2	no	—
103	C₂×C₄	—	197	2	no	—
104	C₂×C₄	—	197	4	no	—
105	C₂×C₄	—	197	4	no	—
106	C₂×C₄	C₈×D₁₉₇	1	4	yes	—
107	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
108	C₂×C₄	#15691357	1	1	yes	characteristic
109	C₂×C₄	#15691357	1	2	yes	—
110	C₂×C₄	#15691359	1	1	yes	characteristic
111	C₈	—	2	4	no	—
112	C₈	—	394	4	no	—
113	C₂²	C₄²×D₁₉₇	1	1	yes	characteristic
114	C₂²	#20263282	1	1	yes	characteristic
115	C₂²	#20263284	1	1	yes	characteristic
116	C₂²	C₂×C₈×D₁₉₇	1	2	yes	—
117	C₂²	M₁₆×D₁₉₇	1	2	yes	—
118	C₄	—	2	4	no	—
119	C₄	—	394	4	no	—
120	C₄	—	197	4	no	—
121	C₄	—	197	4	no	—
122	C₄	#20263282	1	2	yes	—
123	C₄	#20263284	1	2	yes	—
124	C₂	#27160924	1	1	yes	characteristic
125	C₂	C₄×C₈×D₁₉₇	1	1	yes	characteristic
126	C₂	#97×D₁₉₇	1	1	yes	characteristic
127	C₂	#27161129	1	2	yes	—
128	C₂	#27161131	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₉₇

C₂

yes

maximal

C₄²×Dic₁₉₇

C₂

yes

maximal, characteristic

#27160834

C₂

yes

maximal, characteristic

#27160833

C₂

yes

maximal

#336×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×#13096802

C₂²

yes

—

C₂×C₄×Dic₁₉₇

C₄

yes

—

C₂×C₄×Dic₁₉₇

C₄

yes

—

C₂×C₄×Dic₁₉₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₉₇

C₂²

yes

characteristic

C₈×Dic₁₉₇

C₄

yes

complemented

C₂²×C₈×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂²×Dic₁₉₇

C₂×C₄

yes

—

C₂²×Dic₁₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₉₇

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

C₈

yes

complemented

C₄×Dic₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

C₂×C₄

yes

—

C₄²×C₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

C₄²

yes

—

C₂×Dic₁₉₇

C₄²

yes

—

C₂×Dic₁₉₇

C₂×C₈

yes

—

C₂×Dic₁₉₇

C₂×C₈

yes

—

C₂×Dic₁₉₇

M₁₆

yes

—

#13096802

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×C₈

yes

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

—

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

yes

C₂×C₄×C₁₉₇

yes

C₈×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₄²

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

C₂²×C₈

yes

—

C₂²×C₁₉₇

C₂×M₁₆

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#117

yes

—

C₄×C₁₉₇

#117

yes

—

Dic₁₉₇

—

Dic₁₉₇

C₄×C₈

yes

—

Dic₁₉₇

#97

yes

—

C₂×C₁₉₇

#376

yes

characteristic

C₂×C₁₉₇

C₂×C₄×C₈

yes

characteristic

C₂×C₁₉₇

C₂×#97

yes

characteristic

C₂×C₁₉₇

#431

yes

—

C₂×C₁₉₇

#431

yes

—

C₁₉₇

#1744

yes

characteristic, complemented, Sylow

#1744

—

197

maximal, Sylow

C₂×C₄×C₈

—

197

—

C₄³

—

197

—

#336

—

197

—

#336

—

197

—

#336

D₁₉₇

yes

characteristic

C₂²×C₈

—

197

—

C₂²×C₈

C₂×D₁₉₇

yes

—

C₂×C₄²

—

197

—

C₂×C₄²

—

197

—

C₂×C₄²

—

197

—

C₂×C₄²

—

197

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₄×C₈

—

197

—

C₂²×C₄

—

197

—

C₂²×C₄

—

197

—

C₂²×C₄

—

197

—

C₂²×C₄

C₄×D₁₉₇

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

197

—

C₂×C₈

C₄×D₁₉₇

yes

—

C₄²

—

C₄²

—

394

—

C₄²

—

394

—

C₄²

—

394

—

C₄²

—

197

—

C₄²

—

197

—

C₂³

C₂×C₄×D₁₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

394

—

100

C₂×C₄

—

394

—

101

C₂×C₄

—

197

—

102

C₂×C₄

—

197

—

103

C₂×C₄

—

197

—

104

C₂×C₄

—

197

—

105

C₂×C₄

—

197

—

106

C₂×C₄

C₈×D₁₉₇

yes

—

107

C₂×C₄

C₂×C₄×D₁₉₇

yes

—

108

C₂×C₄

#15691357

yes

characteristic

109

C₂×C₄

#15691357

yes

—

110

C₂×C₄

#15691359

yes

characteristic

111

C₈

—

112

C₈

—

394

—

113

C₂²

C₄²×D₁₉₇

yes

characteristic

114

C₂²

#20263282

yes

characteristic

115

C₂²

#20263284

yes

characteristic

116

C₂²

C₂×C₈×D₁₉₇

yes

—

117

C₂²

M₁₆×D₁₉₇

yes

—

118

C₄

—

119

C₄

—

394

—

120

C₄

—

197

—

121

C₄

—

197

—

122

C₄

#20263282

yes

—

123

C₄

#20263284

yes

—

124

C₂

#27160924

yes

characteristic

125

C₂

C₄×C₈×D₁₉₇

yes

characteristic

126

C₂

#97×D₁₉₇

yes

characteristic

127

C₂

#27161129

yes

—

128

C₂

#27161131

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₉₇
commutator	C₂×C₁₉₇	C₂×C₄×C₈
Fitting	#336×C₁₉₇	C₂
socle, prime radical	C₂³×C₁₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	#336	D₁₉₇
hypocenter	C₁₉₇	#1744
LCS	G ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₉₇

commutator

C₂×C₁₉₇

C₂×C₄×C₈

Fitting

#336×C₁₉₇

C₂

socle, prime radical

C₂³×C₁₉₇

C₂²×C₄

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂×C₁₉₇ ▷ C₁₉₇

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
11. C₈×Dic₁₉₇	C₄	semidirect
23. C₄×Dic₁₉₇	C₈	semidirect
64. C₁₉₇	#1744	semidirect

H ≅ G/N

type

11. C₈×Dic₁₉₇

C₄

semidirect

23. C₄×Dic₁₉₇

C₈

semidirect

64. C₁₉₇

#1744

semidirect

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

subgroup	index
C₂×C₈×Dic₁₉₇	2
C₄²×Dic₁₉₇	2
#27160834	2
#27160833	2
#336×C₁₉₇	2
#1744	197

subgroup

index

C₂×C₈×Dic₁₉₇

C₄²×Dic₁₉₇

#27160834

#27160833

#336×C₁₉₇

#1744

197

Subgroups — #38742456

Conjugacy classes of subgroups

Complements

Maximal subgroups