Group #38742802 — subgroups

Subgroups — #38742802

Order 25216.

344 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₂²⋊#13096802	C₂	1	1	yes	maximal, characteristic
2	#27160939	C₂	1	1	yes	maximal, characteristic
3	#27160841	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄⋊C₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13096802	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
7	C₂×C₂²⋊Dic₁₉₇	—	2	1	no	—
8	C₂²⋊#13096802	—	2	2	no	—
9	#20263233	—	2	1	no	—
10	C₂³×C₄×C₁₉₇	—	2	1	no	—
11	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₉₇	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₉₇	—	2	2	no	—
14	C₂×C₄⋊C₄×C₁₉₇	C₄	1	4	yes	complemented
15	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
16	C₂²×Dic₁₉₇	—	2	2	no	—
17	C₂×#13096802	—	2	2	no	—
18	C₂×#13096802	—	2	4	no	—
19	C₄⋊Dic₁₉₇	—	2	4	no	—
20	C₂²⋊Dic₁₉₇	—	4	2	no	—
21	C₂⁴×C₁₉₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₉₇	—	2	4	no	—
23	C₂²×C₄×C₁₉₇	—	2	4	no	—
24	C₂²×C₄×C₁₉₇	—	2	2	no	—
25	C₂²×C₄×C₁₉₇	—	4	4	no	—
26	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
29	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₉₇	—	2	4	no	—
31	C₄⋊C₄×C₁₉₇	—	2	2	no	—
32	C₄⋊C₄×C₁₉₇	C₂×C₄	1	4	yes	—
33	C₂×Dic₁₉₇	—	4	2	no	—
34	C₂×Dic₁₉₇	—	4	2	no	—
35	C₂×Dic₁₉₇	—	4	2	no	—
36	#13096802	—	4	2	no	—
37	C₂³×C₁₉₇	—	2	2	no	—
38	C₂³×C₁₉₇	—	2	2	no	—
39	C₂³×C₁₉₇	C₂²⋊C₄	1	2	yes	—
40	C₂³×C₁₉₇	SD₁₆	1	2	yes	—
41	C₂³×C₁₉₇	Q₁₆	1	2	yes	—
42	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₉₇	—	2	2	no	—
44	C₂×C₄×C₁₉₇	—	2	4	no	—
45	C₂×C₄×C₁₉₇	—	2	4	no	—
46	C₂×C₄×C₁₉₇	—	2	4	no	—
47	C₂×C₄×C₁₉₇	—	4	8	no	—
48	C₂×C₄×C₁₉₇	—	4	4	no	—
49	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₇	#41	1	4	yes	—
53	C₂²×C₁₉₇	—	2	2	no	—
54	C₂²×C₁₉₇	—	2	2	no	—
55	C₂²×C₁₉₇	—	2	4	no	—
56	C₂²×C₁₉₇	—	2	4	no	—
57	C₂²×C₁₉₇	#103	1	4	yes	—
58	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₉₇	#123	1	2	yes	—
60	C₂²×C₁₉₇	C₂×SD₁₆	1	1	yes	characteristic
61	C₂²×C₁₉₇	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₁₉₇	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₁₉₇	—	2	2	no	—
64	C₄×C₁₉₇	—	4	4	no	—
65	C₄×C₁₉₇	#117	1	2	yes	—
66	C₄×C₁₉₇	#123	1	2	yes	—
67	Dic₁₉₇	—	8	2	no	—
68	C₂×C₁₉₇	—	2	4	no	—
69	C₂×C₁₉₇	#381	1	1	yes	characteristic
70	C₂×C₁₉₇	C₂×#103	1	1	yes	characteristic
71	C₂×C₁₉₇	#418	1	1	yes	characteristic
72	C₂×C₁₉₇	#481	1	2	yes	—
73	C₂×C₁₉₇	#484	1	2	yes	—
74	C₁₉₇	#1824	1	1	yes	characteristic, complemented, Sylow
75	#1824	—	197	1	no	maximal, Sylow
76	C₂²×C₄⋊C₄	D₁₉₇	1	1	yes	characteristic
77	C₂×C₂²⋊C₈	—	197	1	no	—
78	#380	—	197	1	no	—
79	#340	—	197	4	no	—
80	C₂³×C₄	—	2	1	no	—
81	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
82	C₂²×C₈	—	197	2	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	2	2	no	—
85	C₂×C₄⋊C₄	—	197	2	no	—
86	C₂×C₄⋊C₄	Dic₁₉₇	1	4	yes	complemented
87	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
88	C₂×C₂²⋊C₄	—	394	1	no	—
89	C₂²⋊C₈	—	394	2	no	—
90	#95	—	394	1	no	—
91	C₂⁴	C₂²⋊D₁₉₇	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	2	4	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	4	4	no	—
96	C₂²×C₄	—	394	2	no	—
97	C₂²×C₄	C₂×Dic₁₉₇	1	2	yes	—
98	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
100	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
101	C₂×C₈	—	394	2	no	—
102	C₂×C₈	—	394	4	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	394	4	no	—
106	C₄⋊C₄	C₂×Dic₁₉₇	1	4	yes	—
107	C₂²⋊C₄	—	788	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	C₂²⋊Dic₁₉₇	1	2	yes	—
111	C₂³	#15691345	1	2	yes	—
112	C₂³	#15691348	1	2	yes	—
113	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	8	no	—
119	C₂×C₄	—	4	4	no	—
120	C₂×C₄	—	788	2	no	—
121	C₂×C₄	—	788	2	no	—
122	C₂×C₄	—	788	2	no	—
123	C₂×C₄	C₂²⋊Dic₁₉₇	1	2	yes	—
124	C₂×C₄	C₂²×Dic₁₉₇	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
126	C₂×C₄	#15691358	1	4	yes	—
127	C₈	—	788	2	no	—
128	C₂²	—	2	4	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	#20263250	1	4	yes	—
133	C₂²	C₂×C₂²⋊Dic₁₉₇	1	1	yes	characteristic
134	C₂²	#20263309	1	2	yes	—
135	C₂²	C₂×#15691345	1	1	yes	characteristic
136	C₂²	C₂×#15691348	1	1	yes	characteristic
137	C₂²	#20263352	1	2	yes	—
138	C₄	—	2	2	no	—
139	C₄	—	4	4	no	—
140	C₄	—	1576	2	no	—
141	C₄	#20263285	1	2	yes	—
142	C₄	#20263309	1	2	yes	—
143	C₂	—	2	4	no	—
144	C₂	#27160942	1	1	yes	characteristic
145	C₂	C₂×#20263250	1	1	yes	characteristic
146	C₂	#27161084	1	1	yes	characteristic
147	C₂	#27161397	1	2	yes	—
148	C₂	#27161414	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊#13096802

C₂

yes

maximal, characteristic

#27160939

C₂

yes

maximal, characteristic

#27160841

C₂

yes

maximal, complemented

C₂²×C₄⋊C₄×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×#13096802

C₂²

yes

—

C₂×C₄⋊Dic₁₉₇

C₂²

yes

—

C₂×C₂²⋊Dic₁₉₇

—

C₂²⋊#13096802

—

#20263233

—

C₂³×C₄×C₁₉₇

—

C₂³×C₄×C₁₉₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₉₇

C₂²

yes

—

C₂²×Dic₁₉₇

—

C₂×#13096802

—

C₂×#13096802

—

C₄⋊Dic₁₉₇

—

C₂²⋊Dic₁₉₇

—

C₂⁴×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

C₂×C₄

yes

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

C₂²⋊C₄

yes

—

C₂³×C₁₉₇

SD₁₆

yes

—

C₂³×C₁₉₇

Q₁₆

yes

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

#103

yes

—

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₇

#123

yes

—

C₂²×C₁₉₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉₇

C₂×Q₁₆

yes

characteristic

C₂²×C₁₉₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#117

yes

—

C₄×C₁₉₇

#123

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

—

C₂×C₁₉₇

#484

yes

—

C₁₉₇

#1824

yes

characteristic, complemented, Sylow

#1824

—

197

maximal, Sylow

C₂²×C₄⋊C₄

D₁₉₇

yes

characteristic

C₂×C₂²⋊C₈

—

197

—

#380

—

197

—

#340

—

197

—

C₂³×C₄

—

C₂³×C₄

C₂×D₁₉₇

yes

characteristic

C₂²×C₈

—

197

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄⋊C₄

Dic₁₉₇

yes

complemented

C₂×C₄⋊C₄

C₂×D₁₉₇

yes

—

C₂×C₂²⋊C₄

—

394

—

C₂²⋊C₈

—

394

—

#95

—

394

—

C₂⁴

C₂²⋊D₁₉₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

394

—

C₂²×C₄

C₂×Dic₁₉₇

yes

—

C₂²×C₄

C₂²⋊D₁₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₇

yes

—

100

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

101

C₂×C₈

—

394

—

102

C₂×C₈

—

394

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

394

—

106

C₄⋊C₄

C₂×Dic₁₉₇

yes

—

107

C₂²⋊C₄

—

788

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₂²⋊Dic₁₉₇

yes

—

111

C₂³

#15691345

yes

—

112

C₂³

#15691348

yes

—

113

C₂³

C₂×C₂²⋊D₁₉₇

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

788

—

121

C₂×C₄

—

788

—

122

C₂×C₄

—

788

—

123

C₂×C₄

C₂²⋊Dic₁₉₇

yes

—

124

C₂×C₄

C₂²×Dic₁₉₇

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

126

C₂×C₄

#15691358

yes

—

127

C₈

—

788

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#20263250

yes

—

133

C₂²

C₂×C₂²⋊Dic₁₉₇

yes

characteristic

134

C₂²

#20263309

yes

—

135

C₂²

C₂×#15691345

yes

characteristic

136

C₂²

C₂×#15691348

yes

characteristic

137

C₂²

#20263352

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

1576

—

141

C₄

#20263285

yes

—

142

C₄

#20263309

yes

—

143

C₂

—

144

C₂

#27160942

yes

characteristic

145

C₂

C₂×#20263250

yes

characteristic

146

C₂

#27161084

yes

characteristic

147

C₂

#27161397

yes

—

148

C₂

#27161414

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₇
commutator	C₂×C₄×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	C₂²×C₄⋊C₄×C₁₉₇	C₂
socle	C₂³×C₁₉₇	C₂×D₄
prime radical	C₂⁴×C₁₉₇	D₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₂²×C₄⋊C₄	D₁₉₇
hypocenter	C₁₉₇	#1824
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₇

commutator

C₂×C₄×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

C₂²×C₄⋊C₄×C₁₉₇

C₂

socle

C₂³×C₁₉₇

C₂×D₄

prime radical

C₂⁴×C₁₉₇

D₄

cosocle

C₂²×C₄×C₁₉₇

C₂³

hypercenter

C₂²×C₄⋊C₄

D₁₉₇

hypocenter

C₁₉₇

#1824

LCS

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
3. #27160841	C₂	semidirect
14. C₂×C₄⋊C₄×C₁₉₇	C₄	semidirect
74. C₁₉₇	#1824	semidirect
86. C₂×C₄⋊C₄	Dic₁₉₇	semidirect

H ≅ G/N

type

3. #27160841

C₂

semidirect

14. C₂×C₄⋊C₄×C₁₉₇

C₄

semidirect

74. C₁₉₇

#1824

semidirect

86. C₂×C₄⋊C₄

Dic₁₉₇

semidirect

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

subgroup	index
C₂×C₂²⋊#13096802	2
#27160939	2
#27160841	2
C₂²×C₄⋊C₄×C₁₉₇	2
#1824	197

subgroup

index

C₂×C₂²⋊#13096802

#27160939

#27160841

C₂²×C₄⋊C₄×C₁₉₇

#1824

197

Subgroups — #38742802

Conjugacy classes of subgroups

Complements

Maximal subgroups