Group #34188035 — subgroups

Subgroups — #34188035

Order 20224.

558 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 199 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×#19098943	C₂	1	1	yes	maximal, characteristic
2	#23764360	C₂	1	4	yes	maximal
3	C₂×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₁₆	—	2	2	no	—
6	C₂×#14579161	C₂²	1	2	yes	—
7	#19098943	—	2	2	no	—
8	#19098943	C₄	1	4	yes	complemented
9	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
10	#19098325	—	2	8	no	—
11	C₈×C₇₉⋊C₈	—	2	2	no	—
12	C₂×#107×C₇₉	—	2	2	no	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	#384×C₇₉	C₂²	1	2	yes	—
15	C₂²×Dic₁₅₈	—	2	2	no	—
16	C₂×Dic₃₁₆	—	2	8	no	—
17	C₂×Dic₃₁₆	—	4	4	no	—
18	C₂×Dic₃₁₆	—	4	4	no	—
19	C₂²×C₇₉⋊C₈	D₄	1	4	yes	—
20	#14579161	—	2	4	no	—
21	#14579161	—	2	2	no	—
22	#14579161	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₇₉	—	2	4	no	—
24	C₄×C₇₉⋊C₈	—	2	2	no	—
25	C₄×C₇₉⋊C₈	—	2	2	no	—
26	C₂²×C₈×C₇₉	D₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₇₉	—	2	4	no	—
29	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
30	#107×C₇₉	—	4	4	no	—
31	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₇₉	D₄	1	2	yes	—
33	C₂×Dic₁₅₈	—	2	4	no	—
34	C₂×Dic₁₅₈	—	4	2	no	—
35	C₂×Dic₁₅₈	—	4	4	no	—
36	C₂×Dic₁₅₈	—	4	4	no	—
37	C₂²×Dic₇₉	—	4	2	no	—
38	Dic₃₁₆	—	4	16	no	—
39	C₂×C₇₉⋊C₈	—	2	4	no	—
40	C₂×C₇₉⋊C₈	—	2	4	no	—
41	C₂×C₇₉⋊C₈	Q₁₆	1	8	yes	complemented
42	C₄⋊Dic₇₉	—	2	8	no	—
43	C₄⋊Dic₇₉	—	4	4	no	—
44	C₂²×C₄×C₇₉	—	4	2	no	—
45	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
48	C₂×C₈×C₇₉	D₈	1	4	yes	—
49	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
50	C₄⋊C₄×C₇₉	—	4	4	no	—
51	C₄⋊C₄×C₇₉	—	4	8	no	—
52	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₇₉	#41	1	2	yes	—
55	C₂³×C₇₉	#127	1	1	yes	characteristic
56	C₂×C₄×C₇₉	—	8	2	no	—
57	C₂×C₄×C₇₉	—	8	4	no	—
58	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
60	C₂×C₄×C₇₉	#121	1	2	yes	—
61	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
63	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
64	C₂×C₄×C₇₉	C₂×Q₁₆	1	4	yes	—
65	C₂×C₄×C₇₉	#135	1	4	yes	—
66	C₂×Dic₇₉	—	4	4	no	—
67	C₂×Dic₇₉	—	8	2	no	—
68	C₂×Dic₇₉	—	8	2	no	—
69	Dic₁₅₈	—	4	8	no	—
70	Dic₁₅₈	—	8	4	no	—
71	C₈×C₇₉	#102	1	8	yes	—
72	C₇₉⋊C₈	—	2	8	no	—
73	C₂²×C₇₉	#390	1	1	yes	characteristic
74	C₂²×C₇₉	#492	1	1	yes	characteristic
75	C₂²×C₇₉	#493	1	1	yes	characteristic
76	C₂²×C₇₉	#494	1	2	yes	—
77	C₂²×C₇₉	#495	1	2	yes	—
78	C₄×C₇₉	—	16	2	no	—
79	C₄×C₇₉	C₂×#102	1	2	yes	—
80	C₄×C₇₉	C₄×Q₁₆	1	4	yes	—
81	C₄×C₇₉	#467	1	4	yes	—
82	C₄×C₇₉	#486	1	2	yes	—
83	Dic₇₉	—	8	4	no	—
84	#9825	—	79	1	no	maximal, Sylow
85	C₂×C₇₉	#1707	1	2	yes	—
86	C₂×C₇₉	#1709	1	1	yes	characteristic
87	C₂×C₇₉	#1713	1	1	yes	characteristic
88	C₂×C₇₉	#1957	1	1	yes	characteristic
89	C₂×C₇₉	#1969	1	2	yes	—
90	C₂×#494	—	79	1	no	—
91	#1990	—	79	4	no	—
92	#1845	D₇₉	1	1	yes	characteristic
93	C₂×C₈²	—	79	1	no	—
94	C₇₉	#9825	1	1	yes	characteristic, complemented, Sylow
95	C₂²×Q₁₆	—	158	2	no	—
96	C₂×C₄⋊Q₈	—	79	2	no	—
97	#494	—	79	4	no	—
98	#494	—	158	2	no	—
99	C₂×#107	—	2	2	no	—
100	C₂×C₄×C₈	—	79	2	no	—
101	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
102	#384	C₂×D₇₉	1	2	yes	—
103	#340	—	158	8	no	—
104	C₈²	—	158	2	no	—
105	C₂²×Q₈	—	158	2	no	—
106	C₂×Q₁₆	—	316	4	no	—
107	C₂×Q₁₆	—	316	4	no	—
108	C₂×Q₁₆	—	158	8	no	—
109	C₂²×C₈	—	79	4	no	—
110	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
111	C₄⋊Q₈	—	79	4	no	—
112	C₄⋊Q₈	—	158	2	no	—
113	C₄⋊Q₈	—	158	4	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	158	4	no	—
117	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
118	#107	—	4	4	no	—
119	C₄×C₈	—	158	2	no	—
120	C₄×C₈	—	158	2	no	—
121	C₄×C₈	C₄×D₇₉	1	2	yes	—
122	C₄×C₈	D₃₁₆	1	2	yes	—
123	C₂×Q₈	—	316	2	no	—
124	C₂×Q₈	—	316	4	no	—
125	C₂×Q₈	—	316	4	no	—
126	C₂×Q₈	—	158	4	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	316	2	no	—
129	C₂²×C₄	D₄×D₇₉	1	2	yes	—
130	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
131	Q₁₆	—	316	16	no	—
132	C₂×C₈	—	79	8	no	—
133	C₂×C₈	—	158	4	no	—
134	C₂×C₈	—	158	4	no	—
135	C₂×C₈	(C₂×D₇₉)⋊C₄	1	4	yes	—
136	C₂×C₈	#11929609	1	4	yes	—
137	C₂×C₈	#11929611	1	4	yes	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄⋊C₄	—	4	8	no	—
140	C₄⋊C₄	—	316	4	no	—
141	C₄⋊C₄	—	158	8	no	—
142	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
143	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
144	C₄²	#11929625	1	2	yes	—
145	C₂³	#14579157	1	1	yes	characteristic
146	Q₈	—	316	8	no	—
147	Q₈	—	632	4	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	8	4	no	—
150	C₂×C₄	—	316	4	no	—
151	C₂×C₄	—	632	2	no	—
152	C₂×C₄	—	632	2	no	—
153	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
154	C₂×C₄	#14579123	1	2	yes	—
155	C₂×C₄	#14579136	1	2	yes	—
156	C₂×C₄	#14579150	1	1	yes	characteristic
157	C₂×C₄	C₂×#11929609	1	2	yes	—
158	C₂×C₄	C₂×#11929611	1	2	yes	—
159	C₂×C₄	Q₁₆×D₇₉	1	4	yes	—
160	C₂×C₄	#14579180	1	4	yes	—
161	C₈	—	158	8	no	—
162	C₈	#14579081	1	8	yes	—
163	C₂²	#19098464	1	1	yes	characteristic
164	C₂²	#19098937	1	1	yes	characteristic
165	C₂²	#19098939	1	1	yes	characteristic
166	C₂²	#19098941	1	2	yes	—
167	C₂²	#19098946	1	2	yes	—
168	C₄	—	16	2	no	—
169	C₄	—	632	4	no	—
170	C₄	C₂×#14579081	1	2	yes	—
171	C₄	#19098657	1	4	yes	—
172	C₄	#19098805	1	4	yes	—
173	C₄	#19098911	1	2	yes	—
174	C₂	#23762998	1	2	yes	—
175	C₂	#23763005	1	1	yes	characteristic
176	C₂	#23763016	1	1	yes	characteristic
177	C₂	#23764205	1	1	yes	characteristic
178	C₂	#23764259	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098943

C₂

yes

maximal, characteristic

#23764360

C₂

yes

maximal

C₂×C₈×C₇₉⋊C₈

C₂

yes

maximal, characteristic

C₇₉×#1845

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₆

—

C₂×#14579161

C₂²

yes

—

#19098943

—

#19098943

C₄

yes

complemented

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

#19098325

—

C₈×C₇₉⋊C₈

—

C₂×#107×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#384×C₇₉

C₂²

yes

—

C₂²×Dic₁₅₈

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂²×C₇₉⋊C₈

D₄

yes

—

#14579161

—

#14579161

—

#14579161

C₂×C₄

yes

—

C₂×C₄⋊Dic₇₉

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

Q₁₆

yes

complemented

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₂×C₄×C₇₉

C₂×Q₁₆

yes

—

C₂×C₄×C₇₉

#135

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

#102

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

#495

yes

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#102

yes

—

C₄×C₇₉

C₄×Q₁₆

yes

—

C₄×C₇₉

#467

yes

—

C₄×C₇₉

#486

yes

—

Dic₇₉

—

#9825

—

maximal, Sylow

C₂×C₇₉

#1707

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×#494

—

#1990

—

#1845

D₇₉

yes

characteristic

C₂×C₈²

—

C₇₉

#9825

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

158

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

158

—

C₂×#107

—

100

C₂×C₄×C₈

—

101

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

102

#384

C₂×D₇₉

yes

—

103

#340

—

158

—

104

C₈²

—

158

—

105

C₂²×Q₈

—

158

—

106

C₂×Q₁₆

—

316

—

107

C₂×Q₁₆

—

316

—

108

C₂×Q₁₆

—

158

—

109

C₂²×C₈

—

110

C₂²×C₈

C₂²⋊D₇₉

yes

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

158

—

113

C₄⋊Q₈

—

158

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

158

—

117

C₂×C₄²

C₂²×D₇₉

yes

characteristic

118

#107

—

119

C₄×C₈

—

158

—

120

C₄×C₈

—

158

—

121

C₄×C₈

C₄×D₇₉

yes

—

122

C₄×C₈

D₃₁₆

yes

—

123

C₂×Q₈

—

316

—

124

C₂×Q₈

—

316

—

125

C₂×Q₈

—

316

—

126

C₂×Q₈

—

158

—

127

C₂²×C₄

—

128

C₂²×C₄

—

316

—

129

C₂²×C₄

D₄×D₇₉

yes

—

130

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

131

Q₁₆

—

316

—

132

C₂×C₈

—

133

C₂×C₈

—

158

—

134

C₂×C₈

—

158

—

135

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

136

C₂×C₈

#11929609

yes

—

137

C₂×C₈

#11929611

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

316

—

141

C₄⋊C₄

—

158

—

142

C₄²

C₂×C₄×D₇₉

yes

characteristic

143

C₄²

C₂×D₃₁₆

yes

characteristic

144

C₄²

#11929625

yes

—

145

C₂³

#14579157

yes

characteristic

146

Q₈

—

316

—

147

Q₈

—

632

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

316

—

151

C₂×C₄

—

632

—

152

C₂×C₄

—

632

—

153

C₂×C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

154

C₂×C₄

#14579123

yes

—

155

C₂×C₄

#14579136

yes

—

156

C₂×C₄

#14579150

yes

characteristic

157

C₂×C₄

C₂×#11929609

yes

—

158

C₂×C₄

C₂×#11929611

yes

—

159

C₂×C₄

Q₁₆×D₇₉

yes

—

160

C₂×C₄

#14579180

yes

—

161

C₈

—

158

—

162

C₈

#14579081

yes

—

163

C₂²

#19098464

yes

characteristic

164

C₂²

#19098937

yes

characteristic

165

C₂²

#19098939

yes

characteristic

166

C₂²

#19098941

yes

—

167

C₂²

#19098946

yes

—

168

C₄

—

169

C₄

—

632

—

170

C₄

C₂×#14579081

yes

—

171

C₄

#19098657

yes

—

172

C₄

#19098805

yes

—

173

C₄

#19098911

yes

—

174

C₂

#23762998

yes

—

175

C₂

#23763005

yes

characteristic

176

C₂

#23763016

yes

characteristic

177

C₂

#23764205

yes

characteristic

178

C₂

#23764259

yes

—

	subgroup	G/H
center	C₂³	#14579157
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1845	C₂
socle, prime radical	C₂³×C₇₉	#127
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1845	D₇₉
hypocenter	C₇₉	#9825
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#14579157

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1845

C₂

socle, prime radical

C₂³×C₇₉

#127

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
8. #19098943	C₄	semidirect
41. C₂×C₇₉⋊C₈	Q₁₆	semidirect
94. C₇₉	#9825	semidirect

H ≅ G/N

type

8. #19098943

C₄

semidirect

41. C₂×C₇₉⋊C₈

Q₁₆

semidirect

94. C₇₉

#9825

semidirect

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

subgroup	index
C₂×#19098943	2
#23764360	2
C₂×C₈×C₇₉⋊C₈	2
C₇₉×#1845	2
#9825	79

subgroup

index

C₂×#19098943

#23764360

C₂×C₈×C₇₉⋊C₈

C₇₉×#1845

#9825

Subgroups — #34188035

Conjugacy classes of subgroups

Complements

Maximal subgroups