Group #38742856 — subgroups

Subgroups — #38742856

Order 25216.

196 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 85 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	#27160925	C₂	1	1	yes	maximal, characteristic
2	#27160834	C₂	1	1	yes	maximal, characteristic
3	#27160833	C₂	1	2	yes	maximal
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	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
9	#20263233	C₄	1	2	yes	—
10	#20263233	C₄	1	2	yes	—
11	C₂²×C₈×C₁₉₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₁₉₇	C₂×C₄	1	2	yes	—
16	C₂×#13096802	—	2	2	no	—
17	C₂×#13096802	—	2	2	no	—
18	C₂×#13096802	—	2	2	no	—
19	C₄×Dic₁₉₇	—	2	1	no	—
20	C₄×Dic₁₉₇	—	2	2	no	—
21	C₄×Dic₁₉₇	—	2	1	no	—
22	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₁₉₇	—	2	2	no	—
25	C₂×C₈×C₁₉₇	—	2	2	no	—
26	C₂×C₈×C₁₉₇	—	2	2	no	—
27	C₄²×C₁₉₇	—	2	2	no	—
28	C₂×Dic₁₉₇	—	2	1	no	—
29	C₂×Dic₁₉₇	—	2	1	no	—
30	C₂×Dic₁₉₇	—	2	2	no	—
31	C₂×Dic₁₉₇	—	2	2	no	—
32	C₂×Dic₁₉₇	—	2	2	no	—
33	C₂×Dic₁₉₇	—	2	2	no	—
34	C₂×Dic₁₉₇	—	2	2	no	—
35	#13096802	—	4	2	no	—
36	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₉₇	—	2	2	no	—
38	C₂×C₄×C₁₉₇	—	2	2	no	—
39	C₂×C₄×C₁₉₇	—	2	2	no	—
40	C₂×C₄×C₁₉₇	#41	1	2	yes	—
41	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₉₇	#41	1	2	yes	—
43	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
44	C₈×C₁₉₇	—	4	2	no	—
45	C₂²×C₁₉₇	#117	1	1	yes	characteristic
46	C₂²×C₁₉₇	#117	1	1	yes	characteristic
47	C₂²×C₁₉₇	#117	1	1	yes	characteristic
48	C₂²×C₁₉₇	#131	1	2	yes	—
49	C₂²×C₁₉₇	#131	1	2	yes	—
50	C₄×C₁₉₇	—	4	2	no	—
51	C₄×C₁₉₇	#126	1	2	yes	—
52	C₄×C₁₉₇	#126	1	2	yes	—
53	Dic₁₉₇	—	4	1	no	—
54	Dic₁₉₇	—	4	2	no	—
55	Dic₁₉₇	—	4	1	no	—
56	C₂×C₁₉₇	#383	1	1	yes	characteristic
57	C₂×C₁₉₇	#433	1	2	yes	—
58	C₂×C₁₉₇	#433	1	1	yes	characteristic
59	C₂×C₁₉₇	#433	1	2	yes	—
60	C₂×C₁₉₇	#433	1	1	yes	characteristic
61	C₁₉₇	#1840	1	1	yes	characteristic, complemented, Sylow
62	#1840	—	197	1	no	maximal, Sylow
63	#376	—	197	1	no	—
64	#336	—	197	2	no	—
65	#336	—	197	2	no	—
66	#336	—	197	1	no	—
67	#336	D₁₉₇	1	1	yes	characteristic
68	C₂²×C₈	—	197	2	no	—
69	C₂²×C₈	C₂×D₁₉₇	1	2	yes	—
70	C₂×C₄²	—	197	1	no	—
71	C₂×C₄²	—	197	1	no	—
72	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
73	#95	—	197	2	no	—
74	#95	—	197	2	no	—
75	C₂²×C₄	—	197	1	no	—
76	C₂²×C₄	—	197	1	no	—
77	C₂²×C₄	—	197	2	no	—
78	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
79	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	394	2	no	—
84	C₂×C₈	—	394	2	no	—
85	C₂×C₈	—	394	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	—	394	1	no	—
88	C₄²	—	394	2	no	—
89	C₄²	—	394	1	no	—
90	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	394	1	no	—
95	C₂×C₄	—	394	1	no	—
96	C₂×C₄	—	394	2	no	—
97	C₂×C₄	—	394	2	no	—
98	C₂×C₄	—	394	2	no	—
99	C₂×C₄	—	394	2	no	—
100	C₂×C₄	—	394	2	no	—
101	C₂×C₄	#15691357	1	1	yes	characteristic
102	C₂×C₄	#15691357	1	2	yes	—
103	C₂×C₄	#15691358	1	2	yes	—
104	C₂×C₄	#15691359	1	1	yes	characteristic
105	C₈	—	4	2	no	—
106	C₈	—	788	2	no	—
107	C₂²	#20263282	1	1	yes	characteristic
108	C₂²	#20263283	1	1	yes	characteristic
109	C₂²	#20263284	1	1	yes	characteristic
110	C₂²	#20263332	1	2	yes	—
111	C₂²	#20263334	1	2	yes	—
112	C₄	—	4	2	no	—
113	C₄	—	788	1	no	—
114	C₄	—	788	2	no	—
115	C₄	—	788	1	no	—
116	C₄	#20263319	1	2	yes	—
117	C₄	#20263321	1	2	yes	—
118	C₂	#27160946	1	1	yes	characteristic
119	C₂	#27161137	1	2	yes	—
120	C₂	#27161138	1	1	yes	characteristic
121	C₂	#27161139	1	2	yes	—
122	C₂	#27161141	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27160925

C₂

yes

maximal, characteristic

#27160834

C₂

yes

maximal, characteristic

#27160833

C₂

yes

maximal

#27160833

C₂

yes

maximal

#336×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×#13096802

C₂²

yes

—

C₂×C₄×Dic₁₉₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₉₇

C₂²

yes

characteristic

#20263233

C₄

yes

—

#20263233

C₄

yes

—

C₂²×C₈×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂²×Dic₁₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₉₇

C₂×C₄

yes

—

C₂×#13096802

—

C₂×#13096802

—

C₂×#13096802

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₄²×C₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

#41

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

#41

yes

characteristic

C₈×C₁₉₇

—

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

yes

—

C₂²×C₁₉₇

#131

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#126

yes

—

C₄×C₁₉₇

#126

yes

—

Dic₁₉₇

—

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

#383

yes

characteristic

C₂×C₁₉₇

#433

yes

—

C₂×C₁₉₇

#433

yes

characteristic

C₂×C₁₉₇

#433

yes

—

C₂×C₁₉₇

#433

yes

characteristic

C₁₉₇

#1840

yes

characteristic, complemented, Sylow

#1840

—

197

maximal, Sylow

#376

—

197

—

#336

—

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

C₂×D₁₉₇

yes

characteristic

#95

—

197

—

#95

—

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

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₄²

—

C₄²

—

394

—

C₄²

—

394

—

C₄²

—

394

—

C₂³

C₂×C₄×D₁₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

394

—

C₂×C₄

—

394

—

C₂×C₄

—

394

—

C₂×C₄

—

394

—

C₂×C₄

—

394

—

C₂×C₄

—

394

—

100

C₂×C₄

—

394

—

101

C₂×C₄

#15691357

yes

characteristic

102

C₂×C₄

#15691357

yes

—

103

C₂×C₄

#15691358

yes

—

104

C₂×C₄

#15691359

yes

characteristic

105

C₈

—

106

C₈

—

788

—

107

C₂²

#20263282

yes

characteristic

108

C₂²

#20263283

yes

characteristic

109

C₂²

#20263284

yes

characteristic

110

C₂²

#20263332

yes

—

111

C₂²

#20263334

yes

—

112

C₄

—

113

C₄

—

788

—

114

C₄

—

788

—

115

C₄

—

788

—

116

C₄

#20263319

yes

—

117

C₄

#20263321

yes

—

118

C₂

#27160946

yes

characteristic

119

C₂

#27161137

yes

—

120

C₂

#27161138

yes

characteristic

121

C₂

#27161139

yes

—

122

C₂

#27161141

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₉₇
commutator, socle, prime radical	C₂³×C₁₉₇	C₂²×C₄
Fitting	#336×C₁₉₇	C₂
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	#336	D₁₉₇
hypocenter	C₁₉₇	#1840
LCS	G ▷ C₂³×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₉₇

commutator, socle, prime radical

C₂³×C₁₉₇

C₂²×C₄

Fitting

#336×C₁₉₇

C₂

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂³×C₁₉₇ ▷ C₁₉₇

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
61. C₁₉₇	#1840	semidirect

H ≅ G/N

type

61. C₁₉₇

#1840

semidirect

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

subgroup	index
#27160925	2
#27160834	2
#27160833	2
#336×C₁₉₇	2
#1840	197

subgroup

index

#336×C₁₉₇

197

Subgroups — #38742856

Conjugacy classes of subgroups

Complements

Maximal subgroups