Group #38742706 — subgroups

Subgroups — #38742706

Order 25216.

272 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 95 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×Q₈×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic
2	#27161102	C₂	1	2	yes	maximal
3	C₂×#20263245	C₂	1	1	yes	maximal, characteristic
4	#27160847	C₂	1	2	yes	maximal
5	C₂×#101×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691342	C₂²	1	2	yes	—
7	Q₈×Dic₁₉₇	—	2	2	no	—
8	Q₈×Dic₁₉₇	—	2	2	no	—
9	C₂×C₄⋊Dic₁₉₇	C₄	1	2	yes	—
10	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₉₇	C₄	1	2	yes	—
12	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
13	#20263255	—	2	2	no	—
14	#20263255	—	2	2	no	—
15	#20263245	—	2	2	no	—
16	C₂²×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
18	#101×C₁₉₇	—	2	2	no	—
19	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₁₉₇	C₂×C₄	1	2	yes	—
21	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
22	#15691342	—	2	2	no	—
23	#15691342	—	2	2	no	—
24	C₂×#13096802	—	2	2	no	—
25	C₄⋊Dic₁₉₇	—	2	4	no	—
26	C₄⋊Dic₁₉₇	—	2	2	no	—
27	C₄×Dic₁₉₇	—	2	2	no	—
28	C₄×Dic₁₉₇	—	2	2	no	—
29	C₄×Dic₁₉₇	D₄	1	2	yes	—
30	C₄×Dic₁₉₇	D₄	1	2	yes	—
31	C₂×Q₈×C₁₉₇	—	2	2	no	—
32	C₂×Q₈×C₁₉₇	—	2	2	no	—
33	C₂×Q₈×C₁₉₇	D₄	1	2	yes	—
34	C₂×Q₈×C₁₉₇	Q₈	1	2	yes	—
35	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
37	M₁₆×C₁₉₇	—	2	2	no	—
38	M₁₆×C₁₉₇	—	2	2	no	—
39	C₂×C₈×C₁₉₇	—	2	2	no	—
40	C₂×Dic₁₉₇	—	2	2	no	—
41	C₂×Dic₁₉₇	—	2	2	no	—
42	C₂×Dic₁₉₇	—	4	1	no	—
43	C₂×Dic₁₉₇	—	4	2	no	—
44	C₂×Dic₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×Dic₁₉₇	C₂²⋊C₄	1	2	yes	—
46	C₂×Dic₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
47	#13096802	—	4	2	no	—
48	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₁₉₇	—	2	4	no	—
50	Q₈×C₁₉₇	—	2	4	no	—
51	C₂×C₄×C₁₉₇	—	2	2	no	—
52	C₂×C₄×C₁₉₇	—	2	2	no	—
53	C₂×C₄×C₁₉₇	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₁₉₇	#41	1	2	yes	—
58	C₈×C₁₉₇	—	4	2	no	—
59	C₂²×C₁₉₇	—	2	1	no	—
60	C₂²×C₁₉₇	—	2	1	no	—
61	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₁₉₇	#117	1	1	yes	characteristic
64	C₄×C₁₉₇	—	2	4	no	—
65	C₄×C₁₉₇	#121	1	2	yes	—
66	C₄×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
67	Dic₁₉₇	—	2	1	no	—
68	Dic₁₉₇	—	2	2	no	—
69	Dic₁₉₇	—	4	2	no	—
70	Dic₁₉₇	#101	1	2	yes	complemented
71	C₂×C₁₉₇	—	2	1	no	—
72	C₂×C₁₉₇	—	2	1	no	—
73	C₂×C₁₉₇	#380	1	1	yes	characteristic
74	C₂×C₁₉₇	C₂×#101	1	1	yes	characteristic
75	C₂×C₁₉₇	#413	1	1	yes	characteristic
76	C₁₉₇	#1801	1	1	yes	characteristic, complemented, Sylow
77	#1801	—	197	1	no	maximal, Sylow
78	C₂×C₄×Q₈	—	197	1	no	—
79	#423	—	197	2	no	—
80	C₂×#101	—	197	1	no	—
81	C₂×#101	D₁₉₇	1	1	yes	characteristic
82	#343	—	197	2	no	—
83	C₂²×Q₈	C₂×D₁₉₇	1	1	yes	characteristic
84	C₂×M₁₆	—	197	2	no	—
85	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
86	C₄×Q₈	—	394	2	no	—
87	C₄×Q₈	—	394	2	no	—
88	C₂×C₄⋊C₄	—	197	2	no	—
89	C₂×C₄⋊C₄	—	197	1	no	—
90	C₂×C₄²	—	197	2	no	—
91	C₂×C₄²	—	197	1	no	—
92	C₄⋊C₈	—	394	2	no	—
93	C₄⋊C₈	—	394	2	no	—
94	#101	—	2	2	no	—
95	#101	—	394	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	C₂²⋊D₁₉₇	1	2	yes	—
99	C₂×Q₈	Dic₃₉₄	1	2	yes	—
100	C₂²×C₄	—	197	2	no	—
101	C₂²×C₄	—	197	1	no	—
102	C₂²×C₄	—	197	1	no	—
103	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
104	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
105	M₁₆	—	2	2	no	—
106	M₁₆	—	2	2	no	—
107	M₁₆	—	394	2	no	—
108	M₁₆	—	394	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	394	2	no	—
111	C₄⋊C₄	—	394	4	no	—
112	C₄⋊C₄	—	394	2	no	—
113	C₄²	—	394	2	no	—
114	C₄²	—	394	2	no	—
115	C₄²	—	197	2	no	—
116	C₄²	—	197	2	no	—
117	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	788	2	no	—
123	C₂×C₄	—	788	1	no	—
124	C₂×C₄	—	394	2	no	—
125	C₂×C₄	—	394	2	no	—
126	C₂×C₄	—	197	1	no	—
127	C₂×C₄	—	197	2	no	—
128	C₂×C₄	—	197	1	no	—
129	C₂×C₄	#15691337	1	4	yes	—
130	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
132	C₂×C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
133	C₂×C₄	#15691357	1	2	yes	—
134	C₈	—	4	2	no	—
135	C₈	—	788	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	C₂²⋊C₄×D₁₉₇	1	1	yes	characteristic
139	C₂²	C₂×#15691337	1	1	yes	characteristic
140	C₂²	#20263282	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	788	2	no	—
143	C₄	—	394	1	no	—
144	C₄	—	394	2	no	—
145	C₄	—	197	2	no	—
146	C₄	#20263298	1	2	yes	—
147	C₄	#20263303	1	2	yes	—
148	C₂	—	2	1	no	—
149	C₂	—	2	1	no	—
150	C₂	#27160937	1	1	yes	characteristic
151	C₂	#101×D₁₉₇	1	1	yes	characteristic
152	C₂	#27161062	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₁₉₇

C₂

yes

maximal, characteristic

#27161102

C₂

yes

maximal

C₂×#20263245

C₂

yes

maximal, characteristic

#27160847

C₂

yes

maximal

C₂×#101×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

Q₈×Dic₁₉₇

—

Q₈×Dic₁₉₇

—

C₂×C₄⋊Dic₁₉₇

C₄

yes

—

C₂×C₄⋊Dic₁₉₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₉₇

C₄

yes

—

C₂×C₄×Dic₁₉₇

C₂²

yes

—

—

—

C₂²×Q₈×C₁₉₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₉₇

C₂²

yes

—

#101×C₁₉₇

—

C₂²×Dic₁₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₉₇

C₂×C₄

yes

—

C₂²×Dic₁₉₇

C₂×C₄

yes

—

—

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

D₄

yes

—

C₄×Dic₁₉₇

D₄

yes

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

D₄

yes

—

C₂×Q₈×C₁₉₇

Q₈

yes

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

M₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₉₇

C₂²⋊C₄

yes

—

C₂×Dic₁₉₇

C₂²⋊C₄

yes

characteristic

#13096802

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₇

#117

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

#121

yes

—

C₄×C₁₉₇

C₂²⋊Q₈

yes

—

Dic₁₉₇

—

Dic₁₉₇

—

Dic₁₉₇

—

Dic₁₉₇

#101

yes

complemented

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, complemented, Sylow

#1801

—

197

maximal, Sylow

C₂×C₄×Q₈

—

197

—

#423

—

197

—

C₂×#101

—

197

—

C₂×#101

D₁₉₇

yes

characteristic

#343

—

197

—

C₂²×Q₈

C₂×D₁₉₇

yes

characteristic

C₂×M₁₆

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

—

C₄×Q₈

—

394

—

C₄×Q₈

—

394

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄²

—

197

—

C₂×C₄²

—

197

—

C₄⋊C₈

—

394

—

C₄⋊C₈

—

394

—

#101

—

#101

—

394

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₉₇

yes

—

C₂×Q₈

Dic₃₉₄

yes

—

100

C₂²×C₄

—

197

—

101

C₂²×C₄

—

197

—

102

C₂²×C₄

—

197

—

103

C₂²×C₄

C₄×D₁₉₇

yes

—

104

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

394

—

108

M₁₆

—

394

—

109

C₂×C₈

—

110

C₂×C₈

—

394

—

111

C₄⋊C₄

—

394

—

112

C₄⋊C₄

—

394

—

113

C₄²

—

394

—

114

C₄²

—

394

—

115

C₄²

—

197

—

116

C₄²

—

197

—

117

C₂³

C₂×C₄×D₁₉₇

yes

characteristic

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

788

—

123

C₂×C₄

—

788

—

124

C₂×C₄

—

394

—

125

C₂×C₄

—

394

—

126

C₂×C₄

—

197

—

127

C₂×C₄

—

197

—

128

C₂×C₄

—

197

—

129

C₂×C₄

#15691337

yes

—

130

C₂×C₄

D₄×D₁₉₇

yes

—

131

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

132

C₂×C₄

C₂×Dic₃₉₄

yes

characteristic

133

C₂×C₄

#15691357

yes

—

134

C₈

—

135

C₈

—

788

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂²⋊C₄×D₁₉₇

yes

characteristic

139

C₂²

C₂×#15691337

yes

characteristic

140

C₂²

#20263282

yes

characteristic

141

C₄

—

142

C₄

—

788

—

143

C₄

—

394

—

144

C₄

—

394

—

145

C₄

—

197

—

146

C₄

#20263298

yes

—

147

C₄

#20263303

yes

—

148

C₂

—

149

C₂

—

150

C₂

#27160937

yes

characteristic

151

C₂

#101×D₁₉₇

yes

characteristic

152

C₂

#27161062

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₉₇
commutator, prime radical	C₂³×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	C₂×#101×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₂×#101	D₁₉₇
hypocenter	C₁₉₇	#1801
LCS	G ▷ C₂³×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₉₇

commutator, prime radical

C₂³×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

C₂×#101×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂³×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
70. Dic₁₉₇	#101	semidirect
76. C₁₉₇	#1801	semidirect

H ≅ G/N

type

70. Dic₁₉₇

#101

semidirect

76. C₁₉₇

#1801

semidirect

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

subgroup	index
C₂×Q₈×Dic₁₉₇	2
#27161102	2
C₂×#20263245	2
#27160847	2
C₂×#101×C₁₉₇	2
#1801	197

subgroup

index

C₂×Q₈×Dic₁₉₇

C₂×#101×C₁₉₇

197

Subgroups — #38742706

Conjugacy classes of subgroups

Complements

Maximal subgroups