Group #38742792 — subgroups

Subgroups — #38742792

Order 25216.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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	#27161043	C₂	1	2	yes	maximal
2	#27160848	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄⋊C₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#15691342	C₂²	1	4	yes	—
5	C₂²⋊#13096802	—	2	2	no	—
6	C₂²⋊#13096802	—	2	2	no	—
7	C₂³×C₄×C₁₉₇	C₄	1	2	yes	—
8	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₁₉₇	—	2	2	no	—
10	C₂×C₄⋊C₄×C₁₉₇	—	2	2	no	—
11	C₂×C₄⋊C₄×C₁₉₇	C₄	1	2	yes	—
12	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
13	#15691342	—	2	8	no	—
14	C₂×#13096802	—	2	4	no	—
15	C₂⁴×C₁₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₉₇	—	2	4	no	—
17	C₂²×C₄×C₁₉₇	—	2	2	no	—
18	C₂²×C₄×C₁₉₇	—	2	2	no	—
19	C₂²×C₄×C₁₉₇	—	4	4	no	—
20	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₉₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
23	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
24	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₉₇	—	2	4	no	—
26	C₄⋊C₄×C₁₉₇	—	2	4	no	—
27	#13096802	—	4	4	no	—
28	C₂³×C₁₉₇	—	2	2	no	—
29	C₂³×C₁₉₇	—	2	2	no	—
30	C₂³×C₁₉₇	—	2	2	no	—
31	C₂³×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂³×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₉₇	—	2	2	no	—
35	C₂×C₄×C₁₉₇	—	2	4	no	—
36	C₂×C₄×C₁₉₇	—	2	4	no	—
37	C₂×C₄×C₁₉₇	—	2	4	no	—
38	C₂×C₄×C₁₉₇	—	4	4	no	—
39	C₂×C₄×C₁₉₇	—	4	4	no	—
40	C₂×C₄×C₁₉₇	—	4	4	no	—
41	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₉₇	#41	1	4	yes	—
44	C₂²×C₁₉₇	—	2	8	no	—
45	C₂²×C₁₉₇	—	2	1	no	—
46	C₂²×C₁₉₇	—	2	1	no	—
47	C₂²×C₁₉₇	—	2	1	no	—
48	C₂²×C₁₉₇	—	2	1	no	—
49	C₂²×C₁₉₇	—	2	1	no	—
50	C₂²×C₁₉₇	—	2	1	no	—
51	C₂²×C₁₉₇	#100	1	2	yes	—
52	C₂²×C₁₉₇	#101	1	2	yes	—
53	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₁₉₇	#117	1	2	yes	—
55	C₄×C₁₉₇	—	2	2	no	—
56	C₄×C₁₉₇	—	4	4	no	—
57	C₄×C₁₉₇	#123	1	4	yes	—
58	C₂×C₁₉₇	—	2	2	no	—
59	C₂×C₁₉₇	—	2	2	no	—
60	C₂×C₁₉₇	—	2	2	no	—
61	C₂×C₁₉₇	#381	1	1	yes	characteristic
62	C₂×C₁₉₇	C₂×#100	1	1	yes	characteristic
63	C₂×C₁₉₇	C₂×#101	1	1	yes	characteristic
64	C₁₉₇	#1821	1	1	yes	characteristic, complemented, Sylow
65	#1821	—	197	1	no	maximal, Sylow
66	C₂²×C₄⋊C₄	D₁₉₇	1	1	yes	characteristic
67	#407	—	197	2	no	—
68	#343	—	197	4	no	—
69	C₂³×C₄	Dic₁₉₇	1	2	yes	—
70	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
71	C₂×M₁₆	—	197	4	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	2	2	no	—
74	C₂×C₄⋊C₄	Dic₁₉₇	1	2	yes	—
75	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
76	C₂²⋊C₈	—	394	2	no	—
77	C₂²⋊C₈	—	394	2	no	—
78	C₂⁴	C₂×Dic₁₉₇	1	1	yes	characteristic
79	C₂²×C₄	—	2	4	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	4	4	no	—
83	C₂²×C₄	C₂×Dic₁₉₇	1	1	yes	characteristic
84	C₂²×C₄	C₂×Dic₁₉₇	1	4	yes	—
85	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
86	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
88	M₁₆	—	394	8	no	—
89	C₂×C₈	—	394	4	no	—
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₂³	—	2	2	no	—
93	C₂³	—	2	2	no	—
94	C₂³	—	2	2	no	—
95	C₂³	C₂²⋊Dic₁₉₇	1	1	yes	characteristic
96	C₂³	C₂²⋊Dic₁₉₇	1	1	yes	characteristic
97	C₂³	C₂²×Dic₁₉₇	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₂²⋊Dic₁₉₇	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
107	C₂×C₄	#15691358	1	4	yes	—
108	C₈	—	788	4	no	—
109	C₂²	—	2	8	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	#20263243	1	2	yes	—
117	C₂²	#20263245	1	2	yes	—
118	C₂²	C₂×C₂²⋊Dic₁₉₇	1	1	yes	characteristic
119	C₂²	#20263285	1	2	yes	—
120	C₄	—	2	2	no	—
121	C₄	—	4	4	no	—
122	C₄	#20263309	1	4	yes	—
123	C₂	—	2	2	no	—
124	C₂	—	2	2	no	—
125	C₂	—	2	2	no	—
126	C₂	#27160942	1	1	yes	characteristic
127	C₂	C₂×#20263243	1	1	yes	characteristic
128	C₂	C₂×#20263245	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161043

C₂

yes

maximal

#27160848

C₂

yes

maximal, complemented

C₂²×C₄⋊C₄×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

C₂²⋊#13096802

—

C₂²⋊#13096802

—

C₂³×C₄×C₁₉₇

C₄

yes

—

C₂³×C₄×C₁₉₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

C₄

yes

—

C₂×C₄⋊C₄×C₁₉₇

C₂²

yes

—

#15691342

—

C₂×#13096802

—

C₂⁴×C₁₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

#13096802

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

C₂²⋊C₄

yes

characteristic

C₂³×C₁₉₇

C₂²⋊C₄

yes

characteristic

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

#41

yes

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

#100

yes

—

C₂²×C₁₉₇

#101

yes

—

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₇

#117

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#123

yes

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, complemented, Sylow

#1821

—

197

maximal, Sylow

C₂²×C₄⋊C₄

D₁₉₇

yes

characteristic

#407

—

197

—

#343

—

197

—

C₂³×C₄

Dic₁₉₇

yes

—

C₂³×C₄

C₂×D₁₉₇

yes

characteristic

C₂×M₁₆

—

197

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₁₉₇

yes

—

C₂×C₄⋊C₄

C₂×D₁₉₇

yes

—

C₂²⋊C₈

—

394

—

C₂²⋊C₈

—

394

—

C₂⁴

C₂×Dic₁₉₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₉₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₉₇

yes

—

C₂²×C₄

C₂²⋊D₁₉₇

yes

—

C₂²×C₄

C₂²⋊D₁₉₇

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

394

—

C₂×C₈

—

394

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₁₉₇

yes

characteristic

C₂³

C₂²⋊Dic₁₉₇

yes

characteristic

C₂³

C₂²×Dic₁₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊Dic₁₉₇

yes

—

106

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

107

C₂×C₄

#15691358

yes

—

108

C₈

—

788

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#20263243

yes

—

117

C₂²

#20263245

yes

—

118

C₂²

C₂×C₂²⋊Dic₁₉₇

yes

characteristic

119

C₂²

#20263285

yes

—

120

C₄

—

121

C₄

—

122

C₄

#20263309

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

#27160942

yes

characteristic

127

C₂

C₂×#20263243

yes

characteristic

128

C₂

C₂×#20263245

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₉₇
commutator	C₂³×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	C₂²×C₄⋊C₄×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉₇	C₂×C₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₂²×C₄⋊C₄	D₁₉₇
hypocenter	C₁₉₇	#1821
LCS	G ▷ C₂³×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₉₇

commutator

C₂³×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

C₂²×C₄⋊C₄×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₁₉₇

C₂×C₄

cosocle

C₂²×C₄×C₁₉₇

C₂³

hypercenter

C₂²×C₄⋊C₄

D₁₉₇

hypocenter

C₁₉₇

#1821

LCS

G ▷ C₂³×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
2. #27160848	C₂	semidirect
64. C₁₉₇	#1821	semidirect

H ≅ G/N

type

2. #27160848

C₂

semidirect

64. C₁₉₇

#1821

semidirect

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

subgroup	index
#27161043	2
#27160848	2
C₂²×C₄⋊C₄×C₁₉₇	2
#1821	197

subgroup

index

#27161043

#27160848

C₂²×C₄⋊C₄×C₁₉₇

#1821

197

Subgroups — #38742792

Conjugacy classes of subgroups

Complements

Maximal subgroups