Group #34208227 — subgroups

Subgroups — #34208227

Order 20224.

268 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 97 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	#23765186	C₂	1	2	yes	maximal
2	#23764148	C₂	1	1	yes	maximal, characteristic
3	#23761508	C₂	1	2	yes	maximal
4	#23761482	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579101	C₂²	1	2	yes	—
7	C₂×#14579097	C₂²	1	1	yes	characteristic
8	#19098601	C₂²	1	1	yes	characteristic
9	#19098589	—	2	1	no	—
10	#19098390	—	2	2	no	—
11	#19098388	—	2	2	no	—
12	#19098323	—	2	1	no	—
13	C₂×#107×C₇₉	C₂²	1	2	yes	—
14	C₂×#106×C₇₉	—	2	1	no	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	#382×C₇₉	—	2	1	no	—
17	C₂×#11929608	—	2	1	no	—
18	#14579120	—	2	1	no	—
19	#14579101	—	2	2	no	—
20	#14579101	Q₈	1	4	yes	—
21	C₂×C₇₉⋊C₁₆	—	2	2	no	—
22	#14579097	—	2	2	no	—
23	#14579095	D₄	1	2	yes	—
24	#14579093	—	2	1	no	—
25	Dic₇₉⋊C₈	—	4	2	no	—
26	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
29	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
30	#107×C₇₉	—	2	2	no	—
31	#107×C₇₉	—	2	2	no	—
32	#106×C₇₉	—	4	2	no	—
33	C₄×C₈×C₇₉	—	2	2	no	—
34	#95×C₇₉	—	4	2	no	—
35	#11929608	—	4	1	no	—
36	#11929608	—	4	1	no	—
37	C₂×C₇₉⋊C₈	—	4	1	no	—
38	C₄⋊Dic₇₉	—	2	2	no	—
39	C₂²⋊Dic₇₉	—	4	1	no	—
40	C₄×Dic₇₉	—	4	1	no	—
41	C₇₉⋊C₁₆	—	2	4	no	—
42	C₂²×C₄×C₇₉	—	2	1	no	—
43	C₂²×C₄×C₇₉	—	4	2	no	—
44	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₇₉	—	2	1	no	—
46	C₂×C₈×C₇₉	—	2	1	no	—
47	C₂×C₈×C₇₉	D₈	1	2	yes	—
48	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
51	C₂×C₈×C₇₉	#41	1	2	yes	—
52	C₄⋊C₄×C₇₉	—	4	2	no	—
53	C₄⋊C₄×C₇₉	—	4	2	no	—
54	C₄²×C₇₉	—	2	1	no	—
55	C₄²×C₇₉	Q₁₆	1	2	yes	—
56	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
57	C₂×C₄×C₇₉	—	2	2	no	—
58	C₂×C₄×C₇₉	—	4	2	no	—
59	C₂×C₄×C₇₉	—	8	4	no	—
60	C₂×C₄×C₇₉	—	8	2	no	—
61	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
62	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
67	C₂×Dic₇₉	—	4	2	no	—
68	C₈×C₇₉	—	2	2	no	—
69	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
70	C₈×C₇₉	#125	1	2	yes	—
71	C₇₉⋊C₈	—	8	1	no	—
72	C₂²×C₇₉	—	2	2	no	—
73	C₂²×C₇₉	#459	1	1	yes	characteristic
74	C₂²×C₇₉	#484	1	1	yes	characteristic
75	C₂²×C₇₉	#501	1	1	yes	characteristic
76	C₄×C₇₉	—	4	2	no	—
77	C₄×C₇₉	—	16	2	no	—
78	C₄×C₇₉	#398	1	1	yes	characteristic
79	C₄×C₇₉	#462	1	1	yes	characteristic
80	C₄×C₇₉	#480	1	1	yes	characteristic
81	C₄×C₇₉	#500	1	1	yes	characteristic
82	Dic₇₉	—	8	2	no	—
83	#13509	—	79	1	no	maximal, Sylow
84	C₂×C₇₉	—	2	1	no	—
85	C₂×C₇₉	—	2	1	no	—
86	C₂×C₇₉	#2083	1	1	yes	characteristic
87	C₂×C₇₉	#2212	1	2	yes	—
88	#2159	—	79	2	no	—
89	#1945	—	79	1	no	—
90	#1846	D₇₉	1	1	yes	characteristic
91	#1387	—	79	2	no	—
92	#1376	—	79	1	no	—
93	C₇₉	#13509	1	1	yes	characteristic, complemented, Sylow
94	C₂×M₃₂	—	79	2	no	—
95	C₂×#108	—	79	1	no	—
96	#427	—	79	1	no	—
97	C₂×#107	C₂×D₇₉	1	2	yes	—
98	C₂×#106	—	2	1	no	—
99	#424	—	158	1	no	—
100	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
101	#382	—	2	1	no	—
102	#367	—	158	2	no	—
103	#366	—	158	2	no	—
104	#339	—	158	1	no	—
105	C₂×M₁₆	—	158	1	no	—
106	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
108	#117	—	158	1	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
111	M₃₂	—	79	4	no	—
112	M₃₂	—	158	2	no	—
113	C₂×C₁₆	—	158	2	no	—
114	#108	—	158	2	no	—
115	#107	—	2	2	no	—
116	#107	—	2	2	no	—
117	#107	—	79	2	no	—
118	#106	—	4	2	no	—
119	#106	—	158	1	no	—
120	C₄⋊C₈	—	316	2	no	—
121	C₄×C₈	—	2	2	no	—
122	#95	—	4	2	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	4	2	no	—
125	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
126	M₁₆	—	316	1	no	—
127	M₁₆	—	316	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	316	1	no	—
131	C₂×C₈	#11929609	1	2	yes	—
132	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
133	C₂×C₈	Q₈×D₇₉	1	2	yes	—
134	C₂×C₈	#11929623	1	1	yes	characteristic
135	C₂×C₈	#11929624	1	2	yes	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	158	2	no	—
139	C₂²⋊C₄	—	316	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	316	1	no	—
142	C₄²	#11929614	1	2	yes	—
143	C₁₆	—	158	4	no	—
144	C₂³	#14579187	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	8	4	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	316	2	no	—
150	C₂×C₄	#14579132	1	1	yes	characteristic
151	C₂×C₄	#14579144	1	1	yes	characteristic
152	C₂×C₄	#14579159	1	1	yes	characteristic
153	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
154	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
155	C₂×C₄	#14579192	1	1	yes	characteristic
156	C₈	—	2	2	no	—
157	C₈	—	632	1	no	—
158	C₈	#14579139	1	2	yes	—
159	C₈	#14579151	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	#19098755	1	1	yes	characteristic
162	C₂²	#19098898	1	1	yes	characteristic
163	C₂²	#19098969	1	1	yes	characteristic
164	C₄	—	4	2	no	—
165	C₄	—	16	2	no	—
166	C₄	—	632	2	no	—
167	C₄	#19098494	1	1	yes	characteristic
168	C₄	#19098770	1	1	yes	characteristic
169	C₄	#19098876	1	1	yes	characteristic
170	C₄	#19098966	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#23764872	1	1	yes	characteristic
174	C₂	#23765466	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765186

C₂

yes

maximal

#23764148

C₂

yes

maximal, characteristic

#23761508

C₂

yes

maximal

#23761482

C₂

yes

maximal, characteristic

C₇₉×#1846

C₂

yes

maximal, characteristic

C₂×#14579101

C₂²

yes

—

C₂×#14579097

C₂²

yes

characteristic

#19098601

C₂²

yes

—

—

—

—

C₂×#107×C₇₉

C₂²

yes

—

C₂×#106×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#382×C₇₉

—

—

—

—

Q₈

yes

—

C₂×C₇₉⋊C₁₆

—

#14579097

—

#14579095

D₄

yes

—

#14579093

—

Dic₇₉⋊C₈

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

#107×C₇₉

—

#107×C₇₉

—

#106×C₇₉

—

C₄×C₈×C₇₉

—

#95×C₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

#41

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

Q₁₆

yes

—

C₂³×C₇₉

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₉

#137

yes

characteristic

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂²⋊Q₈

yes

—

C₈×C₇₉

#125

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

#13509

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#2083

yes

characteristic

C₂×C₇₉

#2212

yes

—

#2159

—

#1945

—

#1846

D₇₉

yes

characteristic

#1387

—

#1376

—

C₇₉

#13509

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×#108

—

#427

—

C₂×#107

C₂×D₇₉

yes

—

C₂×#106

—

#424

—

158

—

100

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

101

#382

—

102

#367

—

158

—

103

#366

—

158

—

104

#339

—

158

—

105

C₂×M₁₆

—

158

—

106

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

107

C₂²×C₈

C₂²×D₇₉

yes

characteristic

108

#117

—

158

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

111

M₃₂

—

112

M₃₂

—

158

—

113

C₂×C₁₆

—

158

—

114

#108

—

158

—

115

#107

—

116

#107

—

117

#107

—

118

#106

—

119

#106

—

158

—

120

C₄⋊C₈

—

316

—

121

C₄×C₈

—

122

#95

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

126

M₁₆

—

316

—

127

M₁₆

—

316

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

316

—

131

C₂×C₈

#11929609

yes

—

132

C₂×C₈

D₄×D₇₉

yes

characteristic

133

C₂×C₈

Q₈×D₇₉

yes

—

134

C₂×C₈

#11929623

yes

characteristic

135

C₂×C₈

#11929624

yes

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

158

—

139

C₂²⋊C₄

—

316

—

140

C₄²

—

141

C₄²

—

316

—

142

C₄²

#11929614

yes

—

143

C₁₆

—

158

—

144

C₂³

#14579187

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

316

—

150

C₂×C₄

#14579132

yes

characteristic

151

C₂×C₄

#14579144

yes

characteristic

152

C₂×C₄

#14579159

yes

characteristic

153

C₂×C₄

C₂×#11929609

yes

characteristic

154

C₂×C₄

C₂×#11929614

yes

characteristic

155

C₂×C₄

#14579192

yes

characteristic

156

C₈

—

157

C₈

—

632

—

158

C₈

#14579139

yes

—

159

C₈

#14579151

yes

—

160

C₂²

—

161

C₂²

#19098755

yes

characteristic

162

C₂²

#19098898

yes

characteristic

163

C₂²

#19098969

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

632

—

167

C₄

#19098494

yes

characteristic

168

C₄

#19098770

yes

characteristic

169

C₄

#19098876

yes

characteristic

170

C₄

#19098966

yes

characteristic

171

C₂

—

172

C₂

—

173

C₂

#23764872

yes

characteristic

174

C₂

#23765466

yes

—

	subgroup	G/H
center	C₂²	#19098755
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1846	C₂
socle	C₂²×C₇₉	#459
prime radical	C₂³×C₇₉	AGL(1,ℤ/8)
hypercenter	#1846	D₇₉
hypocenter	C₇₉	#13509
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂²

#19098755

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1846

C₂

C₂²×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
93. C₇₉	#13509	semidirect

H ≅ G/N

type

93. C₇₉

#13509

semidirect

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

subgroup	index
#23765186	2
#23764148	2
#23761508	2
#23761482	2
C₇₉×#1846	2
#13509	79

subgroup

index

C₇₉×#1846

Subgroups — #34208227

Conjugacy classes of subgroups

Complements

Maximal subgroups