Group #34175202 — subgroups

Subgroups — #34175202

Order 20224.

238 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₄×#14579097	C₂	1	1	yes	maximal, characteristic
2	#23762148	C₂	1	1	yes	maximal, characteristic
3	C₂×#346×C₇₉	C₂	1	1	yes	maximal, characteristic
4	#19098616	—	2	1	no	—
5	C₂×#14579097	C₄	1	2	yes	—
6	C₄×#11929608	—	2	1	no	—
7	#19098328	—	2	1	no	—
8	#19098316	—	2	1	no	—
9	#19098314	C₄	1	2	yes	—
10	C₂²×C₁₆×C₇₉	—	2	1	no	—
11	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	#346×C₇₉	—	2	1	no	—
13	#346×C₇₉	C₄	1	2	yes	—
14	C₂×#11929608	—	2	2	no	—
15	#14579097	—	2	2	no	—
16	#14579097	C₈	1	4	yes	complemented
17	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₇₉⋊C₈)	—	4	1	no	—
19	#14579072	—	4	1	no	—
20	C₄×C₇₉⋊C₈	—	4	1	no	—
21	C₄×C₇₉⋊C₈	—	4	1	no	—
22	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₉	—	2	4	no	—
26	C₂×C₁₆×C₇₉	—	2	2	no	—
27	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
31	#11929608	—	2	4	no	—
32	#11929608	—	4	2	no	—
33	C₂×C₇₉⋊C₈	—	4	2	no	—
34	C₂×C₇₉⋊C₈	—	4	2	no	—
35	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	Q₁₆	1	1	yes	characteristic
38	C₂×C₈×C₇₉	—	2	2	no	—
39	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
45	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₇₉	D₈	1	1	yes	characteristic
48	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
49	C₁₆×C₇₉	—	2	4	no	—
50	C₂³×C₇₉	#103	1	1	yes	characteristic
51	C₂×C₄×C₇₉	—	4	1	no	—
52	C₂×C₄×C₇₉	—	4	1	no	—
53	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
54	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
56	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
58	C₂×C₄×C₇₉	#104	1	2	yes	—
59	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
60	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
61	C₈×C₇₉	—	2	1	no	—
62	C₈×C₇₉	—	2	1	no	—
63	C₈×C₇₉	C₄×C₈	1	1	yes	characteristic
64	C₈×C₇₉	#97	1	1	yes	characteristic
65	C₈×C₇₉	C₄⋊C₈	1	2	yes	—
66	C₇₉⋊C₈	—	4	4	no	—
67	C₇₉⋊C₈	—	8	2	no	—
68	C₂²×C₇₉	—	2	1	no	—
69	C₂²×C₇₉	—	2	1	no	—
70	C₂²×C₇₉	Q₈⋊C₈	1	2	yes	—
71	C₂²×C₇₉	#340	1	1	yes	characteristic
72	C₄×C₇₉	—	2	1	no	—
73	C₄×C₇₉	—	2	1	no	—
74	C₄×C₇₉	—	2	1	no	—
75	C₄×C₇₉	—	2	1	no	—
76	C₄×C₇₉	D₄⋊C₈	1	2	yes	—
77	C₄×C₇₉	#336	1	1	yes	characteristic
78	C₄×C₇₉	#339	1	1	yes	characteristic
79	#7177	—	79	1	no	maximal, Sylow
80	C₂×C₇₉	—	2	2	no	—
81	C₂×C₇₉	#1279	1	1	yes	characteristic
82	C₂×C₇₉	#1388	1	1	yes	characteristic
83	C₂×C₇₉	#1389	1	1	yes	characteristic
84	C₂×#346	D₇₉	1	1	yes	characteristic
85	C₄×#108	—	79	1	no	—
86	#1564	—	79	1	no	—
87	C₇₉	#7177	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	2	1	no	—
89	#431	—	158	1	no	—
90	C₂×#108	—	79	2	no	—
91	C₄×M₁₆	—	158	1	no	—
92	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
93	#346	—	2	1	no	—
94	#346	Dic₇₉	1	2	yes	—
95	#341	—	158	1	no	—
96	#335	—	158	1	no	—
97	#334	—	79	2	no	—
98	C₂×M₁₆	—	158	2	no	—
99	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
101	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	—	2	2	no	—
104	#108	—	79	4	no	—
105	#108	—	158	2	no	—
106	C₄⋊C₈	—	158	2	no	—
107	C₂²⋊C₈	—	316	1	no	—
108	#97	—	316	1	no	—
109	C₄×C₈	—	316	1	no	—
110	C₄×C₈	—	316	1	no	—
111	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
112	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
113	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
114	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
115	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
116	C₂²×C₄	#11929613	1	1	yes	characteristic
117	C₂²×C₄	Dic₃₁₆	1	1	yes	characteristic
118	M₁₆	—	316	2	no	—
119	M₁₆	—	158	4	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	316	2	no	—
122	C₂×C₈	—	316	2	no	—
123	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
124	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
125	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
126	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
127	C₂×C₈	C₈×D₇₉	1	2	yes	—
128	C₂×C₈	#11929607	1	2	yes	—
129	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
130	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
131	C₄²	D₆₃₂	1	1	yes	characteristic
132	C₄²	#11929613	1	1	yes	characteristic
133	C₁₆	—	2	4	no	—
134	C₂³	#14579086	1	1	yes	characteristic
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
138	C₂×C₄	(C₂×D₇₉)⋊C₈	1	2	yes	—
139	C₂×C₄	#14579083	1	1	yes	characteristic
140	C₂×C₄	#14579083	1	1	yes	characteristic
141	C₂×C₄	#14579086	1	1	yes	characteristic
142	C₂×C₄	#14579089	1	2	yes	—
143	C₂×C₄	#14579093	1	1	yes	characteristic
144	C₂×C₄	#14579095	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	316	4	no	—
148	C₈	—	632	2	no	—
149	C₈	C₈×Dic₇₉	1	1	yes	characteristic
150	C₈	#14579071	1	1	yes	characteristic
151	C₈	Dic₇₉⋊C₈	1	2	yes	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	#19098295	1	2	yes	—
155	C₂²	#19098326	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	#19098292	1	2	yes	—
161	C₄	#19098317	1	1	yes	characteristic
162	C₄	#19098323	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	#23761240	1	1	yes	characteristic
165	C₂	#23761512	1	1	yes	characteristic
166	C₂	#23761515	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14579097

C₂

yes

maximal, characteristic

#23762148

C₂

yes

maximal, characteristic

C₂×#346×C₇₉

C₂

yes

maximal, characteristic

#19098616

—

C₂×#14579097

C₄

yes

—

—

—

—

C₄

yes

—

C₂²×C₁₆×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#346×C₇₉

—

#346×C₇₉

C₄

yes

—

C₂×#11929608

—

#14579097

—

#14579097

C₈

yes

complemented

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

Q₈

yes

characteristic

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

characteristic

C₂²×C₄×C₇₉

Q₁₆

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄²

yes

characteristic

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×C₈

yes

—

C₂×C₈×C₇₉

M₁₆

yes

—

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₄²×C₇₉

C₄⋊C₄

yes

characteristic

C₄²×C₇₉

D₈

yes

characteristic

C₄²×C₇₉

SD₁₆

yes

characteristic

C₁₆×C₇₉

—

C₂³×C₇₉

#103

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#95

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₈

yes

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₄×C₈

yes

characteristic

C₈×C₇₉

#97

yes

characteristic

C₈×C₇₉

C₄⋊C₈

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

Q₈⋊C₈

yes

—

C₂²×C₇₉

#340

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

D₄⋊C₈

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

yes

characteristic

C₄×#108

—

#1564

—

C₇₉

#7177

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

#431

—

158

—

C₂×#108

—

C₄×M₁₆

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#346

—

#346

Dic₇₉

yes

—

#341

—

158

—

#335

—

158

—

#334

—

C₂×M₁₆

—

158

—

C₂²×C₈

C₄×D₇₉

yes

characteristic

100

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

101

C₂×C₄²

D₃₁₆

yes

characteristic

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

#108

—

105

#108

—

158

—

106

C₄⋊C₈

—

158

—

107

C₂²⋊C₈

—

316

—

108

#97

—

316

—

109

C₄×C₈

—

316

—

110

C₄×C₈

—

316

—

111

C₄×C₈

Dic₁₅₈

yes

characteristic

112

C₄×C₈

C₄×D₇₉

yes

characteristic

113

C₄×C₈

C₂×Dic₇₉

yes

characteristic

114

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

115

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

116

C₂²×C₄

#11929613

yes

characteristic

117

C₂²×C₄

Dic₃₁₆

yes

characteristic

118

M₁₆

—

316

—

119

M₁₆

—

158

—

120

C₂×C₈

—

121

C₂×C₈

—

316

—

122

C₂×C₈

—

316

—

123

C₂×C₈

C₄×Dic₇₉

yes

characteristic

124

C₂×C₈

C₂²⋊Dic₇₉

yes

characteristic

125

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

126

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

127

C₂×C₈

C₈×D₇₉

yes

—

128

C₂×C₈

#11929607

yes

—

129

C₄²

(C₂×D₇₉)⋊C₄

yes

characteristic

130

C₄²

C₄⋊Dic₇₉

yes

characteristic

131

C₄²

D₆₃₂

yes

characteristic

132

C₄²

#11929613

yes

characteristic

133

C₁₆

—

134

C₂³

#14579086

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₇₉⋊#95

yes

characteristic

138

C₂×C₄

(C₂×D₇₉)⋊C₈

yes

—

139

C₂×C₄

#14579083

yes

characteristic

140

C₂×C₄

#14579083

yes

characteristic

141

C₂×C₄

#14579086

yes

characteristic

142

C₂×C₄

#14579089

yes

—

143

C₂×C₄

#14579093

yes

characteristic

144

C₂×C₄

#14579095

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

316

—

148

C₈

—

632

—

149

C₈

C₈×Dic₇₉

yes

characteristic

150

C₈

#14579071

yes

characteristic

151

C₈

Dic₇₉⋊C₈

yes

—

152

C₂²

—

153

C₂²

—

154

C₂²

#19098295

yes

—

155

C₂²

#19098326

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#19098292

yes

—

161

C₄

#19098317

yes

characteristic

162

C₄

#19098323

yes

characteristic

163

C₂

—

164

C₂

#23761240

yes

characteristic

165

C₂

#23761512

yes

characteristic

166

C₂

#23761515

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579083
commutator	C₈×C₇₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₂×#346×C₇₉	C₂
socle	C₂²×C₇₉	#340
prime radical	C₂³×C₇₉	#103
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	C₂×#346	D₇₉
hypocenter	C₇₉	#7177
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

#14579083

commutator

C₈×C₇₉

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₉

Fitting

C₂×#346×C₇₉

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄×C₈×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
16. #14579097	C₈	semidirect
87. C₇₉	#7177	semidirect

H ≅ G/N

type

16. #14579097

C₈

semidirect

87. C₇₉

#7177

semidirect

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

subgroup	index
C₄×#14579097	2
#23762148	2
C₂×#346×C₇₉	2
#7177	79

subgroup

index

C₄×#14579097

#23762148

C₂×#346×C₇₉

#7177

Subgroups — #34175202

Conjugacy classes of subgroups

Complements

Maximal subgroups