Group #34208632 — subgroups

Subgroups — #34208632

Order 20224.

312 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 79 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×#19098362	C₂	1	2	yes	maximal
2	#23761573	C₂	1	4	yes	maximal, complemented
3	C₇₉×#3139	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579080	—	2	2	no	—
5	#19098362	—	2	2	no	—
6	#19098362	C₂²	1	4	yes	—
7	#533×C₇₉	C₄	1	1	yes	characteristic
8	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
9	#491×C₇₉	C₄	1	2	yes	—
10	#491×C₇₉	C₂²	1	2	yes	—
11	#487×C₇₉	—	2	2	no	—
12	#419×C₇₉	—	2	2	no	—
13	C₄×M₁₆×C₇₉	C₄	1	1	yes	characteristic
14	C₂×#11929608	—	4	2	no	—
15	#14579080	—	2	4	no	—
16	#14579080	—	4	2	no	—
17	C₂²×Q₈×C₇₉	D₄	1	2	yes	—
18	C₂×M₁₆×C₇₉	—	2	1	no	—
19	C₄⋊Q₈×C₇₉	—	2	1	no	—
20	C₄⋊Q₈×C₇₉	—	4	1	no	—
21	C₄⋊Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Q₈×C₇₉	D₄	1	2	yes	—
23	C₄⋊Q₈×C₇₉	C₂³	1	1	yes	characteristic
24	#125×C₇₉	C₂×C₄	1	2	yes	—
25	C₂²⋊Q₈×C₇₉	—	4	1	no	—
26	C₄×Q₈×C₇₉	—	4	1	no	—
27	#117×C₇₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
29	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
30	#103×C₇₉	—	2	4	no	—
31	#103×C₇₉	—	2	4	no	—
32	#97×C₇₉	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₇₉	—	2	1	no	—
34	#11929608	—	4	4	no	—
35	#11929608	—	8	2	no	—
36	C₂×C₇₉⋊C₈	—	8	2	no	—
37	C₂×Q₈×C₇₉	—	2	2	no	—
38	C₂×Q₈×C₇₉	—	4	1	no	—
39	C₂×Q₈×C₇₉	—	4	2	no	—
40	C₂×Q₈×C₇₉	—	4	2	no	—
41	C₂×Q₈×C₇₉	C₂²⋊C₄	1	2	yes	—
42	C₂×Q₈×C₇₉	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₇₉	—	2	1	no	—
44	C₂²×C₄×C₇₉	—	2	2	no	—
45	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
46	M₁₆×C₇₉	—	4	2	no	—
47	C₂×C₈×C₇₉	—	2	2	no	—
48	C₄⋊C₄×C₇₉	—	2	2	no	—
49	C₄⋊C₄×C₇₉	—	4	1	no	—
50	C₄⋊C₄×C₇₉	—	4	2	no	—
51	C₄⋊C₄×C₇₉	—	4	1	no	—
52	C₄⋊C₄×C₇₉	—	4	1	no	—
53	C₄⋊C₄×C₇₉	—	4	2	no	—
54	C₂²⋊C₄×C₇₉	—	4	1	no	—
55	C₄²×C₇₉	—	2	1	no	—
56	C₄²×C₇₉	—	4	1	no	—
57	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
59	C₂³×C₇₉	#99	1	1	yes	characteristic
60	Q₈×C₇₉	—	4	2	no	—
61	Q₈×C₇₉	—	4	2	no	—
62	Q₈×C₇₉	—	8	2	no	—
63	Q₈×C₇₉	—	8	2	no	—
64	C₂×C₄×C₇₉	—	2	2	no	—
65	C₂×C₄×C₇₉	—	2	1	no	—
66	C₂×C₄×C₇₉	—	2	2	no	—
67	C₂×C₄×C₇₉	—	2	1	no	—
68	C₂×C₄×C₇₉	—	2	2	no	—
69	C₂×C₄×C₇₉	—	4	2	no	—
70	C₂×C₄×C₇₉	—	4	2	no	—
71	C₂×C₄×C₇₉	—	4	2	no	—
72	C₂×C₄×C₇₉	—	4	2	no	—
73	C₂×C₄×C₇₉	#99	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₈×C₇₉	—	4	2	no	—
76	C₇₉⋊C₈	—	8	4	no	—
77	C₂²×C₇₉	—	4	1	no	—
78	C₂²×C₇₉	#354	1	2	yes	—
79	C₂²×C₇₉	C₂×#99	1	1	yes	characteristic
80	C₄×C₇₉	—	2	2	no	—
81	C₄×C₇₉	—	2	1	no	—
82	C₄×C₇₉	—	2	1	no	—
83	C₄×C₇₉	—	4	1	no	—
84	C₄×C₇₉	—	4	2	no	—
85	C₄×C₇₉	—	4	2	no	—
86	C₄×C₇₉	—	8	2	no	—
87	#13599	—	79	1	no	maximal, Sylow
88	C₂×C₇₉	—	2	1	no	—
89	C₂×C₇₉	—	2	2	no	—
90	C₂×C₇₉	C₂×#354	1	1	yes	characteristic
91	#3139	D₇₉	1	1	yes	characteristic
92	C₂×#356	—	79	2	no	—
93	#1413	—	79	4	no	—
94	C₇₉	#13599	1	1	yes	characteristic, complemented, Sylow
95	#533	Dic₇₉	1	1	yes	characteristic
96	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
97	#491	Dic₇₉	1	2	yes	—
98	#491	C₂×D₇₉	1	2	yes	—
99	#487	—	2	2	no	—
100	#419	—	2	2	no	—
101	C₂×#101	—	158	2	no	—
102	C₄×M₁₆	Dic₇₉	1	1	yes	characteristic
103	#356	—	79	4	no	—
104	#356	—	158	2	no	—
105	C₂²×Q₈	C₂²⋊D₇₉	1	2	yes	—
106	C₂×M₁₆	—	2	1	no	—
107	C₂×M₁₆	—	316	2	no	—
108	C₄⋊Q₈	—	2	1	no	—
109	C₄⋊Q₈	—	4	1	no	—
110	C₄⋊Q₈	C₂×Dic₇₉	1	1	yes	characteristic
111	C₄⋊Q₈	C₂²⋊D₇₉	1	2	yes	—
112	C₄⋊Q₈	C₂²×D₇₉	1	1	yes	characteristic
113	#125	C₂×Dic₇₉	1	2	yes	—
114	C₂²⋊Q₈	—	4	1	no	—
115	C₄×Q₈	—	4	1	no	—
116	#117	—	2	1	no	—
117	C₂×C₄⋊C₄	—	4	1	no	—
118	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
119	#103	—	2	4	no	—
120	#103	—	2	4	no	—
121	#101	—	316	2	no	—
122	#101	—	158	4	no	—
123	#97	C₂×Dic₇₉	1	2	yes	—
124	C₄×C₈	—	2	1	no	—
125	C₂×Q₈	—	2	2	no	—
126	C₂×Q₈	—	4	1	no	—
127	C₂×Q₈	—	4	2	no	—
128	C₂×Q₈	—	4	2	no	—
129	C₂×Q₈	C₂²⋊Dic₇₉	1	2	yes	—
130	C₂×Q₈	C₂×C₂²⋊D₇₉	1	2	yes	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
134	M₁₆	—	4	2	no	—
135	M₁₆	—	316	4	no	—
136	M₁₆	—	632	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	632	2	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	4	1	no	—
142	C₄⋊C₄	—	4	1	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	1	no	—
145	C₂²⋊C₄	—	4	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	4	1	no	—
148	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
149	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
150	C₂³	#14579076	1	1	yes	characteristic
151	Q₈	—	4	2	no	—
152	Q₈	—	4	2	no	—
153	Q₈	—	8	2	no	—
154	Q₈	—	8	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	#14579076	1	1	yes	characteristic
165	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
166	C₈	—	4	2	no	—
167	C₈	—	632	4	no	—
168	C₂²	—	4	1	no	—
169	C₂²	#19098358	1	2	yes	—
170	C₂²	C₂×#14579076	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	4	1	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	—	8	2	no	—
178	C₂	—	2	1	no	—
179	C₂	—	2	2	no	—
180	C₂	C₂×#19098358	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098362

C₂

yes

maximal

#23761573

C₂

yes

maximal, complemented

C₇₉×#3139

C₂

yes

maximal, characteristic

C₂×#14579080

—

#19098362

—

#19098362

C₂²

yes

—

#533×C₇₉

C₄

yes

characteristic

C₂×C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

#491×C₇₉

C₄

yes

—

#491×C₇₉

C₂²

yes

—

#487×C₇₉

—

#419×C₇₉

—

C₄×M₁₆×C₇₉

C₄

yes

—

—

—

C₂²×Q₈×C₇₉

D₄

yes

—

C₂×M₁₆×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₇₉

D₄

yes

—

C₄⋊Q₈×C₇₉

C₂³

yes

characteristic

#125×C₇₉

C₂×C₄

yes

—

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

#117×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

#103×C₇₉

—

#103×C₇₉

—

#97×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

C₂²⋊C₄

yes

—

C₂×Q₈×C₇₉

C₂×D₄

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₂³×C₇₉

#99

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#99

yes

characteristic

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#354

yes

—

C₂²×C₇₉

C₂×#99

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

#13599

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂×#354

yes

characteristic

#3139

D₇₉

yes

characteristic

C₂×#356

—

#1413

—

C₇₉

#13599

yes

characteristic, complemented, Sylow

#533

Dic₇₉

yes

characteristic

C₂×C₄⋊Q₈

C₂×D₇₉

yes

characteristic

#491

Dic₇₉

yes

—

#491

C₂×D₇₉

yes

—

#487

—

100

#419

—

101

C₂×#101

—

158

—

102

C₄×M₁₆

Dic₇₉

yes

characteristic

103

#356

—

104

#356

—

158

—

105

C₂²×Q₈

C₂²⋊D₇₉

yes

—

106

C₂×M₁₆

—

107

C₂×M₁₆

—

316

—

108

C₄⋊Q₈

—

109

C₄⋊Q₈

—

110

C₄⋊Q₈

C₂×Dic₇₉

yes

characteristic

111

C₄⋊Q₈

C₂²⋊D₇₉

yes

—

112

C₄⋊Q₈

C₂²×D₇₉

yes

characteristic

113

#125

C₂×Dic₇₉

yes

—

114

C₂²⋊Q₈

—

115

C₄×Q₈

—

116

#117

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

119

#103

—

120

#103

—

121

#101

—

316

—

122

#101

—

158

—

123

#97

C₂×Dic₇₉

yes

—

124

C₄×C₈

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

C₂²⋊Dic₇₉

yes

—

130

C₂×Q₈

C₂×C₂²⋊D₇₉

yes

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

134

M₁₆

—

135

M₁₆

—

316

—

136

M₁₆

—

632

—

137

C₂×C₈

—

138

C₂×C₈

—

632

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₂²⋊C₄

—

146

C₄²

—

147

C₄²

—

148

C₄²

C₂²⋊Dic₇₉

yes

characteristic

149

C₄²

C₂²×Dic₇₉

yes

characteristic

150

C₂³

#14579076

yes

characteristic

151

Q₈

—

152

Q₈

—

153

Q₈

—

154

Q₈

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

#14579076

yes

characteristic

165

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

166

C₈

—

167

C₈

—

632

—

168

C₂²

—

169

C₂²

#19098358

yes

—

170

C₂²

C₂×#14579076

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₂

—

179

C₂

—

180

C₂

C₂×#19098358

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#19098358
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₄⋊Q₈	C₂²×D₇₉
Fitting	C₇₉×#3139	C₂
socle	C₂×C₇₉	C₂×#354
prime radical	C₂³×C₇₉	#99
cosocle	C₄⋊Q₈×C₇₉	C₂³
hypercenter	#3139	D₇₉
hypocenter	C₇₉	#13599
LCS	G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈ ◁ #3139

subgroup

G/H

center

C₂

C₂×#19098358

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₄⋊Q₈

C₂²×D₇₉

Fitting

C₇₉×#3139

C₂

C₂×C₇₉

C₂³×C₇₉

C₄⋊Q₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈ ◁ #3139

N	H ≅ G/N	type
2. #23761573	C₂	semidirect
94. C₇₉	#13599	semidirect

H ≅ G/N

type

2. #23761573

C₂

semidirect

94. C₇₉

#13599

semidirect

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

subgroup	index
C₂×#19098362	2
#23761573	2
C₇₉×#3139	2
#13599	79

subgroup

index

C₂×#19098362

#23761573

C₇₉×#3139

#13599

Subgroups — #34208632

Conjugacy classes of subgroups

Complements

Maximal subgroups