Group #38738739 — subgroups

Subgroups — #1805×C₁₉₇ #38738739

Order 25216.

292 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 136 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×C₄×Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#103×C₁₉₇	C₂	1	2	yes	maximal
4	#382×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	#340×C₁₉₇	C₂	1	2	yes	maximal
6	C₂²×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₉₇	C₂²	1	2	yes	—
8	C₄×Q₈×C₁₉₇	—	2	2	no	—
9	C₄×Q₈×C₁₉₇	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₁₉₇	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₉₇	—	2	1	no	—
14	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₉₇	—	2	2	no	—
16	#103×C₁₉₇	—	2	4	no	—
17	#95×C₁₉₇	—	2	1	no	—
18	#95×C₁₉₇	—	2	1	no	—
19	C₂×Q₈×C₁₉₇	—	2	1	no	—
20	C₂×Q₈×C₁₉₇	—	2	1	no	—
21	C₂×Q₈×C₁₉₇	—	2	1	no	—
22	C₂×Q₈×C₁₉₇	C₂×C₄	1	2	yes	—
23	C₂×Q₈×C₁₉₇	D₄	1	2	yes	—
24	C₂×Q₈×C₁₉₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₉₇	—	2	1	no	—
26	C₂²×C₄×C₁₉₇	—	2	1	no	—
27	C₂²×C₄×C₁₉₇	—	2	2	no	—
28	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₉₇	—	2	2	no	—
32	C₂×C₈×C₁₉₇	—	2	2	no	—
33	C₂×C₈×C₁₉₇	—	2	2	no	—
34	C₄⋊C₄×C₁₉₇	—	2	4	no	—
35	C₄⋊C₄×C₁₉₇	—	2	1	no	—
36	C₄⋊C₄×C₁₉₇	—	2	2	no	—
37	C₄⋊C₄×C₁₉₇	—	2	2	no	—
38	C₄⋊C₄×C₁₉₇	C₂×C₄	1	2	yes	—
39	C₄²×C₁₉₇	—	2	4	no	—
40	C₄²×C₁₉₇	C₂×C₄	1	2	yes	—
41	C₄²×C₁₉₇	D₄	1	2	yes	—
42	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₁₉₇	—	2	2	no	—
44	Q₈×C₁₉₇	—	2	2	no	—
45	Q₈×C₁₉₇	—	2	2	no	—
46	Q₈×C₁₉₇	C₄⋊C₄	1	4	yes	—
47	C₂×C₄×C₁₉₇	—	2	2	no	—
48	C₂×C₄×C₁₉₇	—	2	2	no	—
49	C₂×C₄×C₁₉₇	—	2	2	no	—
50	C₂×C₄×C₁₉₇	—	2	2	no	—
51	C₂×C₄×C₁₉₇	—	2	2	no	—
52	C₂×C₄×C₁₉₇	—	2	2	no	—
53	C₂×C₄×C₁₉₇	—	4	2	no	—
54	C₂×C₄×C₁₉₇	—	4	2	no	—
55	C₂×C₄×C₁₉₇	—	4	2	no	—
56	C₂×C₄×C₁₉₇	—	4	2	no	—
57	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
64	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
65	C₈×C₁₉₇	—	4	2	no	—
66	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₁₉₇	#121	1	1	yes	characteristic
68	C₂²×C₁₉₇	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂²×C₁₉₇	#135	1	2	yes	—
70	C₂²×C₁₉₇	#137	1	2	yes	—
71	C₄×C₁₉₇	—	2	4	no	—
72	C₄×C₁₉₇	—	2	2	no	—
73	C₄×C₁₉₇	—	2	2	no	—
74	C₄×C₁₉₇	—	4	2	no	—
75	C₄×C₁₉₇	—	8	2	no	—
76	C₄×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₁₉₇	#117	1	1	yes	characteristic
78	C₄×C₁₉₇	#121	1	1	yes	characteristic
79	C₄×C₁₉₇	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₁₉₇	#380	1	1	yes	characteristic
81	C₂×C₁₉₇	#416	1	1	yes	characteristic
82	C₂×C₁₉₇	#419	1	1	yes	characteristic
83	C₂×C₁₉₇	#464	1	2	yes	—
84	C₂×C₁₉₇	#479	1	2	yes	—
85	C₁₉₇	#1805	1	1	yes	characteristic, direct product factor, Sylow
86	#1805	C₁₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂×C₄×Q₈	C₂×C₁₉₇	1	1	yes	characteristic
88	C₂×C₄⋊C₈	C₂×C₁₉₇	1	1	yes	characteristic
89	C₂×#103	C₂×C₁₉₇	1	2	yes	—
90	#382	C₂×C₁₉₇	1	1	yes	characteristic
91	#340	C₂×C₁₉₇	1	2	yes	—
92	C₂²×Q₈	C₂²×C₁₉₇	1	1	yes	characteristic
93	C₂²×C₈	C₂²×C₁₉₇	1	2	yes	—
94	C₄×Q₈	—	2	2	no	—
95	C₄×Q₈	C₄×C₁₉₇	1	4	yes	complemented
96	C₂×C₄⋊C₄	—	2	1	no	—
97	C₂×C₄⋊C₄	C₂²×C₁₉₇	1	2	yes	—
98	C₂×C₄⋊C₄	C₂²×C₁₉₇	1	1	yes	characteristic
99	C₂×C₄²	—	2	1	no	—
100	C₂×C₄²	C₂²×C₁₉₇	1	1	yes	characteristic
101	C₄⋊C₈	—	2	2	no	—
102	#103	—	2	4	no	—
103	#95	—	2	1	no	—
104	#95	—	2	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	C₂×C₄×C₁₉₇	1	2	yes	—
109	C₂×Q₈	D₄×C₁₉₇	1	2	yes	—
110	C₂×Q₈	Q₈×C₁₉₇	1	2	yes	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	D₄×C₁₉₇	1	1	yes	characteristic
115	C₂²×C₄	D₄×C₁₉₇	1	1	yes	characteristic
116	C₂²×C₄	C₂³×C₁₉₇	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	C₂×C₄×C₁₉₇	1	2	yes	—
125	C₄²	—	2	4	no	—
126	C₄²	C₂×C₄×C₁₉₇	1	2	yes	—
127	C₄²	D₄×C₁₉₇	1	2	yes	—
128	C₂³	C₂×D₄×C₁₉₇	1	1	yes	characteristic
129	Q₈	—	2	2	no	—
130	Q₈	—	2	2	no	—
131	Q₈	—	2	2	no	—
132	Q₈	C₄⋊C₄×C₁₉₇	1	4	yes	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	C₂²⋊C₄×C₁₉₇	1	2	yes	—
144	C₂×C₄	C₂²⋊C₄×C₁₉₇	1	2	yes	—
145	C₂×C₄	C₂²×C₄×C₁₉₇	1	1	yes	characteristic
146	C₂×C₄	C₂×D₄×C₁₉₇	1	1	yes	characteristic
147	C₂×C₄	C₂×D₄×C₁₉₇	1	1	yes	characteristic
148	C₂×C₄	C₂×Q₈×C₁₉₇	1	1	yes	characteristic
149	C₂×C₄	#41×C₁₉₇	1	1	yes	characteristic
150	C₂×C₄	#41×C₁₉₇	1	1	yes	characteristic
151	C₈	—	4	2	no	—
152	C₂²	C₂×C₂²⋊C₄×C₁₉₇	1	1	yes	characteristic
153	C₂²	#121×C₁₉₇	1	1	yes	characteristic
154	C₂²	C₂²⋊Q₈×C₁₉₇	1	1	yes	characteristic
155	C₂²	#135×C₁₉₇	1	2	yes	—
156	C₂²	#137×C₁₉₇	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	4	2	no	—
161	C₄	—	8	2	no	—
162	C₄	C₂×C₄⋊C₄×C₁₉₇	1	1	yes	characteristic
163	C₄	#117×C₁₉₇	1	1	yes	characteristic
164	C₄	#121×C₁₉₇	1	1	yes	characteristic
165	C₄	C₂²⋊Q₈×C₁₉₇	1	1	yes	characteristic
166	C₂	#380×C₁₉₇	1	1	yes	characteristic
167	C₂	#416×C₁₉₇	1	1	yes	characteristic
168	C₂	#419×C₁₉₇	1	1	yes	characteristic
169	C₂	#464×C₁₉₇	1	2	yes	—
170	C₂	#479×C₁₉₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Q₈×C₁₉₇

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#103×C₁₉₇

C₂

yes

maximal

#382×C₁₉₇

C₂

yes

maximal, characteristic

#340×C₁₉₇

C₂

yes

maximal

C₂²×Q₈×C₁₉₇

C₂²

yes

characteristic

C₂²×C₈×C₁₉₇

C₂²

yes

—

C₄×Q₈×C₁₉₇

—

C₄×Q₈×C₁₉₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉₇

C₂²

yes

characteristic

C₂×C₄²×C₁₉₇

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₇

—

#103×C₁₉₇

—

#95×C₁₉₇

—

#95×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

C₂×C₄

yes

—

C₂×Q₈×C₁₉₇

D₄

yes

—

C₂×Q₈×C₁₉₇

Q₈

yes

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

C₂×C₄

yes

—

C₄²×C₁₉₇

—

C₄²×C₁₉₇

C₂×C₄

yes

—

C₄²×C₁₉₇

D₄

yes

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

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

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

C₂×C₄×C₁₉₇

yes

C₂×C₄×C₁₉₇

yes

C₈×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₇

#121

yes

characteristic

C₂²×C₁₉₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₉₇

#135

yes

—

C₂²×C₁₉₇

#137

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂×C₄⋊C₄

yes

C₄×C₁₉₇

yes

C₄×C₁₉₇

yes

C₄×C₁₉₇

C₂²⋊Q₈

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

—

C₂×C₁₉₇

#479

yes

—

C₁₉₇

#1805

yes

characteristic, direct product factor, Sylow

#1805

C₁₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄×Q₈

C₂×C₁₉₇

yes

characteristic

C₂×C₄⋊C₈

C₂×C₁₉₇

yes

characteristic

C₂×#103

C₂×C₁₉₇

yes

—

#382

C₂×C₁₉₇

yes

characteristic

#340

C₂×C₁₉₇

yes

—

C₂²×Q₈

C₂²×C₁₉₇

yes

characteristic

C₂²×C₈

C₂²×C₁₉₇

yes

—

C₄×Q₈

—

C₄×Q₈

C₄×C₁₉₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₉₇

yes

—

C₂×C₄⋊C₄

C₂²×C₁₉₇

yes

characteristic

C₂×C₄²

—

100

C₂×C₄²

C₂²×C₁₉₇

yes

characteristic

101

C₄⋊C₈

—

102

#103

—

103

#95

—

104

#95

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

C₂×C₄×C₁₉₇

yes

—

109

C₂×Q₈

D₄×C₁₉₇

yes

—

110

C₂×Q₈

Q₈×C₁₉₇

yes

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

D₄×C₁₉₇

yes

characteristic

115

C₂²×C₄

D₄×C₁₉₇

yes

characteristic

116

C₂²×C₄

C₂³×C₁₉₇

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

C₂×C₄×C₁₉₇

yes

—

125

C₄²

—

126

C₄²

C₂×C₄×C₁₉₇

yes

—

127

C₄²

D₄×C₁₉₇

yes

—

128

C₂³

C₂×D₄×C₁₉₇

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

Q₈

C₄⋊C₄×C₁₉₇

yes

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

C₂²⋊C₄×C₁₉₇

yes

—

144

C₂×C₄

C₂²⋊C₄×C₁₉₇

yes

—

145

C₂×C₄

C₂²×C₄×C₁₉₇

yes

characteristic

146

C₂×C₄

C₂×D₄×C₁₉₇

yes

characteristic

147

C₂×C₄

C₂×D₄×C₁₉₇

yes

characteristic

148

C₂×C₄

C₂×Q₈×C₁₉₇

yes

characteristic

149

C₂×C₄

#41×C₁₉₇

yes

characteristic

150

C₂×C₄

#41×C₁₉₇

yes

characteristic

151

C₈

—

152

C₂²

C₂×C₂²⋊C₄×C₁₉₇

yes

characteristic

153

C₂²

#121×C₁₉₇

yes

characteristic

154

C₂²

C₂²⋊Q₈×C₁₉₇

yes

characteristic

155

C₂²

#135×C₁₉₇

yes

—

156

C₂²

#137×C₁₉₇

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

C₂×C₄⋊C₄×C₁₉₇

yes

characteristic

163

C₄

#117×C₁₉₇

yes

characteristic

164

C₄

#121×C₁₉₇

yes

characteristic

165

C₄

C₂²⋊Q₈×C₁₉₇

yes

characteristic

166

C₂

#380×C₁₉₇

yes

characteristic

167

C₂

#416×C₁₉₇

yes

characteristic

168

C₂

#419×C₁₉₇

yes

characteristic

169

C₂

#464×C₁₉₇

yes

—

170

C₂

#479×C₁₉₇

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₉₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₉₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉₇ ◁ C₂×C₄²×C₁₉₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₉₇

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₁₉₇

Frattini, cosocle

C₂²×C₄

C₂³×C₁₉₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₉₇ ◁ C₂×C₄²×C₁₉₇ ◁ G

N	H ≅ G/N	type
9. C₄×Q₈×C₁₉₇	C₄	semidirect
85. C₁₉₇	#1805	direct (N × H)
86. #1805	C₁₉₇	direct (N × H)
95. C₄×Q₈	C₄×C₁₉₇	semidirect

H ≅ G/N

type

9. C₄×Q₈×C₁₉₇

C₄

semidirect

85. C₁₉₇

#1805

direct (N × H)

86. #1805

C₁₉₇

direct (N × H)

95. C₄×Q₈

C₄×C₁₉₇

semidirect

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

subgroup	index
C₂×C₄×Q₈×C₁₉₇	2
C₂×C₄⋊C₈×C₁₉₇	2
C₂×#103×C₁₉₇	2
#382×C₁₉₇	2
#340×C₁₉₇	2
#1805	197

subgroup

index

C₂×C₄×Q₈×C₁₉₇

C₂×C₄⋊C₈×C₁₉₇

C₂×#103×C₁₉₇

#382×C₁₉₇

#340×C₁₉₇

#1805

197

Subgroups — #1805×C₁₉₇ #38738739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1805×C197 #38738739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1805×C₁₉₇ #38738739