Group #34208237 — subgroups

Subgroups — #34208237

Order 20224.

250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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	#23765174	C₂	1	2	yes	maximal
2	#23764153	C₂	1	1	yes	maximal, characteristic
3	#23761503	C₂	1	2	yes	maximal
4	#23761480	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1849	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	#19098387	—	2	4	no	—
11	#19098323	—	2	1	no	—
12	C₂×#108×C₇₉	—	2	1	no	—
13	C₂×#108×C₇₉	C₂²	1	2	yes	—
14	#423×C₇₉	—	2	1	no	—
15	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×#11929608	—	2	1	no	—
17	#14579120	—	2	1	no	—
18	#14579101	—	2	2	no	—
19	#14579101	—	2	2	no	—
20	C₂×C₇₉⋊C₁₆	Q₈	1	4	yes	—
21	#14579097	—	2	2	no	—
22	#14579095	—	2	1	no	—
23	#14579093	D₄	1	2	yes	—
24	Dic₇₉⋊C₈	—	4	2	no	—
25	C₂×M₁₆×C₇₉	—	2	2	no	—
26	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
29	#108×C₇₉	—	2	2	no	—
30	#108×C₇₉	—	2	2	no	—
31	#108×C₇₉	—	4	2	no	—
32	C₄⋊C₈×C₇₉	—	4	2	no	—
33	#97×C₇₉	—	2	2	no	—
34	#11929608	—	4	1	no	—
35	#11929608	—	4	1	no	—
36	C₂×C₇₉⋊C₈	—	4	1	no	—
37	C₄⋊Dic₇₉	—	2	2	no	—
38	C₂²⋊Dic₇₉	—	4	1	no	—
39	C₄×Dic₇₉	—	4	1	no	—
40	C₇₉⋊C₁₆	—	2	4	no	—
41	C₂²×C₄×C₇₉	—	2	1	no	—
42	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
43	M₁₆×C₇₉	—	4	2	no	—
44	M₁₆×C₇₉	—	4	2	no	—
45	C₂×C₈×C₇₉	—	2	1	no	—
46	C₂×C₈×C₇₉	—	2	1	no	—
47	C₂×C₈×C₇₉	—	4	2	no	—
48	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
49	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
52	C₂×C₈×C₇₉	#41	1	2	yes	—
53	C₄²×C₇₉	—	2	1	no	—
54	C₄²×C₇₉	SD₁₆	1	2	yes	—
55	C₂³×C₇₉	#137	1	1	yes	characteristic
56	C₂×C₄×C₇₉	—	2	2	no	—
57	C₂×C₄×C₇₉	—	8	1	no	—
58	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
59	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×Dic₇₉	—	4	2	no	—
65	C₈×C₇₉	—	2	2	no	—
66	C₈×C₇₉	—	8	2	no	—
67	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
68	C₈×C₇₉	#125	1	2	yes	—
69	C₇₉⋊C₈	—	8	1	no	—
70	C₂²×C₇₉	—	4	1	no	—
71	C₂²×C₇₉	#461	1	1	yes	characteristic
72	C₂²×C₇₉	#483	1	1	yes	characteristic
73	C₂²×C₇₉	#501	1	1	yes	characteristic
74	C₄×C₇₉	—	4	2	no	—
75	C₄×C₇₉	#398	1	1	yes	characteristic
76	C₄×C₇₉	#460	1	1	yes	characteristic
77	C₄×C₇₉	#481	1	1	yes	characteristic
78	C₄×C₇₉	#498	1	1	yes	characteristic
79	Dic₇₉	—	8	2	no	—
80	#13511	—	79	1	no	maximal, Sylow
81	C₂×C₇₉	—	2	1	no	—
82	C₂×C₇₉	—	2	1	no	—
83	C₂×C₇₉	—	2	1	no	—
84	C₂×C₇₉	#2084	1	1	yes	characteristic
85	#2155	—	79	2	no	—
86	#1946	—	79	1	no	—
87	#1849	D₇₉	1	1	yes	characteristic
88	#1385	—	79	2	no	—
89	#1375	—	79	1	no	—
90	C₇₉	#13511	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	—	79	2	no	—
92	C₂×#108	—	2	1	no	—
93	C₂×#108	—	79	1	no	—
94	C₂×#108	C₂×D₇₉	1	2	yes	—
95	#427	—	79	1	no	—
96	#424	—	158	1	no	—
97	#423	—	2	1	no	—
98	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
99	#365	—	158	4	no	—
100	#339	—	158	1	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂×M₁₆	—	158	1	no	—
103	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
104	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
105	#117	—	158	1	no	—
106	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
107	M₃₂	—	158	2	no	—
108	M₃₂	—	158	2	no	—
109	C₂×C₁₆	—	79	4	no	—
110	#108	—	2	2	no	—
111	#108	—	2	2	no	—
112	#108	—	4	2	no	—
113	#108	—	158	2	no	—
114	#107	—	158	1	no	—
115	#106	—	79	2	no	—
116	C₄⋊C₈	—	4	2	no	—
117	C₄⋊C₈	—	316	2	no	—
118	#97	—	2	2	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
121	M₁₆	—	4	2	no	—
122	M₁₆	—	4	2	no	—
123	M₁₆	—	316	1	no	—
124	M₁₆	—	316	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	—	316	1	no	—
129	C₂×C₈	#11929611	1	2	yes	—
130	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
131	C₂×C₈	Q₈×D₇₉	1	2	yes	—
132	C₂×C₈	#11929623	1	1	yes	characteristic
133	C₂×C₈	#11929624	1	2	yes	—
134	C₄⋊C₄	—	158	2	no	—
135	C₂²⋊C₄	—	316	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	316	1	no	—
138	C₄²	#11929612	1	2	yes	—
139	C₁₆	—	158	4	no	—
140	C₂³	#14579192	1	1	yes	characteristic
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	316	2	no	—
144	C₂×C₄	#14579132	1	1	yes	characteristic
145	C₂×C₄	#14579144	1	1	yes	characteristic
146	C₂×C₄	#14579159	1	1	yes	characteristic
147	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
148	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
149	C₂×C₄	#14579187	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	8	2	no	—
152	C₈	—	632	1	no	—
153	C₈	#14579139	1	2	yes	—
154	C₈	#14579151	1	2	yes	—
155	C₂²	—	4	1	no	—
156	C₂²	#19098765	1	1	yes	characteristic
157	C₂²	#19098893	1	1	yes	characteristic
158	C₂²	#19098969	1	1	yes	characteristic
159	C₄	—	4	2	no	—
160	C₄	—	632	2	no	—
161	C₄	#19098494	1	1	yes	characteristic
162	C₄	#19098760	1	1	yes	characteristic
163	C₄	#19098881	1	1	yes	characteristic
164	C₄	#19098959	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	#23764877	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765174

C₂

yes

maximal

#23764153

C₂

yes

maximal, characteristic

#23761503

C₂

yes

maximal

#23761480

C₂

yes

maximal, characteristic

C₇₉×#1849

C₂

yes

maximal, characteristic

C₂×#14579101

C₂²

yes

—

C₂×#14579097

C₂²

yes

characteristic

#19098601

C₂²

yes

—

—

—

C₂×#108×C₇₉

—

C₂×#108×C₇₉

C₂²

yes

—

#423×C₇₉

—

C₂×#97×C₇₉

C₂²

yes

—

—

—

—

C₂×C₇₉⋊C₁₆

Q₈

yes

—

#14579097

—

#14579095

—

#14579093

D₄

yes

—

Dic₇₉⋊C₈

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

#108×C₇₉

—

#108×C₇₉

—

#108×C₇₉

—

C₄⋊C₈×C₇₉

—

#97×C₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

SD₁₆

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

—

C₄²×C₇₉

—

C₄²×C₇₉

SD₁₆

yes

—

C₂³×C₇₉

#137

yes

characteristic

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×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₇₉

—

C₈×C₇₉

—

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

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

#13511

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

—

D₇₉

yes

characteristic

#1385

—

#1375

—

C₇₉

#13511

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×#108

—

C₂×#108

—

C₂×#108

C₂×D₇₉

yes

—

#427

—

#424

—

158

—

#423

—

C₂×#97

C₂×D₇₉

yes

characteristic

#365

—

158

—

100

#339

—

158

—

101

C₂×M₁₆

—

102

C₂×M₁₆

—

158

—

103

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

104

C₂²×C₈

C₂²×D₇₉

yes

characteristic

105

#117

—

158

—

106

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

107

M₃₂

—

158

—

108

M₃₂

—

158

—

109

C₂×C₁₆

—

110

#108

—

111

#108

—

112

#108

—

113

#108

—

158

—

114

#107

—

158

—

115

#106

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

316

—

118

#97

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

121

M₁₆

—

122

M₁₆

—

123

M₁₆

—

316

—

124

M₁₆

—

316

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

316

—

129

C₂×C₈

#11929611

yes

—

130

C₂×C₈

D₄×D₇₉

yes

characteristic

131

C₂×C₈

Q₈×D₇₉

yes

—

132

C₂×C₈

#11929623

yes

characteristic

133

C₂×C₈

#11929624

yes

—

134

C₄⋊C₄

—

158

—

135

C₂²⋊C₄

—

316

—

136

C₄²

—

137

C₄²

—

316

—

138

C₄²

#11929612

yes

—

139

C₁₆

—

158

—

140

C₂³

#14579192

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

316

—

144

C₂×C₄

#14579132

yes

characteristic

145

C₂×C₄

#14579144

yes

characteristic

146

C₂×C₄

#14579159

yes

characteristic

147

C₂×C₄

C₂×#11929611

yes

characteristic

148

C₂×C₄

C₂×#11929612

yes

characteristic

149

C₂×C₄

#14579187

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

632

—

153

C₈

#14579139

yes

—

154

C₈

#14579151

yes

—

155

C₂²

—

156

C₂²

#19098765

yes

characteristic

157

C₂²

#19098893

yes

characteristic

158

C₂²

#19098969

yes

characteristic

159

C₄

—

160

C₄

—

632

—

161

C₄

#19098494

yes

characteristic

162

C₄

#19098760

yes

characteristic

163

C₄

#19098881

yes

characteristic

164

C₄

#19098959

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

—

168

C₂

#23764877

yes

characteristic

	subgroup	G/H
center	C₄	#19098959
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1849	C₂
socle	C₂×C₇₉	#2084
prime radical	C₂³×C₇₉	#137
hypercenter	#1849	D₇₉
hypocenter	C₇₉	#13511
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

subgroup

G/H

center

C₄

#19098959

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1849

C₂

C₂×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

N	H ≅ G/N	type
90. C₇₉	#13511	semidirect

H ≅ G/N

type

90. C₇₉

#13511

semidirect

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

subgroup	index
#23765174	2
#23764153	2
#23761503	2
#23761480	2
C₇₉×#1849	2
#13511	79

subgroup

index

C₇₉×#1849

Subgroups — #34208237

Conjugacy classes of subgroups

Complements

Maximal subgroups