Group #34208420 — subgroups

Subgroups — #34208420

Order 20224.

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

normal subgroups only

This group has 268 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	#23764148	C₂	1	1	yes	maximal, characteristic
2	#23763557	C₂	1	1	yes	maximal, characteristic
3	#23761506	C₂	1	2	yes	maximal
4	#23761499	C₂	1	2	yes	maximal
5	C₇₉×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579097	C₂²	1	1	yes	characteristic
7	#19098601	C₂²	1	1	yes	characteristic
8	C₂×#14579093	—	2	1	no	—
9	C₂×#14579093	C₂²	1	2	yes	—
10	#19098589	—	2	1	no	—
11	#19098433	—	2	1	no	—
12	#19098323	—	2	1	no	—
13	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#11929608	—	2	1	no	—
16	#14579120	—	2	1	no	—
17	C₂×C₄⋊Dic₇₉	—	2	2	no	—
18	C₂×C₄⋊Dic₇₉	—	4	2	no	—
19	#14579097	—	2	2	no	—
20	#14579095	Q₈	1	2	yes	—
21	#14579093	—	2	1	no	—
22	#14579093	—	2	4	no	—
23	#14579093	—	4	2	no	—
24	Dic₇₉⋊C₈	—	4	2	no	—
25	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
28	M₃₂×C₇₉	—	2	2	no	—
29	M₃₂×C₇₉	—	2	2	no	—
30	C₂×C₁₆×C₇₉	—	2	2	no	—
31	C₄×C₈×C₇₉	—	2	2	no	—
32	C₂²×Dic₇₉	—	4	2	no	—
33	#11929608	—	4	1	no	—
34	#11929608	—	4	1	no	—
35	C₂×C₇₉⋊C₈	—	4	1	no	—
36	C₄⋊Dic₇₉	—	2	2	no	—
37	C₄⋊Dic₇₉	—	4	2	no	—
38	C₄⋊Dic₇₉	—	4	2	no	—
39	C₄⋊Dic₇₉	—	8	4	no	—
40	C₂²⋊Dic₇₉	—	4	1	no	—
41	C₄×Dic₇₉	—	4	1	no	—
42	C₂²×C₄×C₇₉	—	2	1	no	—
43	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	—	2	1	no	—
45	C₂×C₈×C₇₉	—	2	1	no	—
46	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
47	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
49	C₂×C₈×C₇₉	#41	1	2	yes	—
50	C₂×C₈×C₇₉	#41	1	2	yes	—
51	C₄²×C₇₉	—	2	1	no	—
52	C₄²×C₇₉	Q₁₆	1	2	yes	—
53	C₁₆×C₇₉	—	4	2	no	—
54	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×C₄×C₇₉	—	2	4	no	—
56	C₂×C₄×C₇₉	—	2	2	no	—
57	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
59	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
62	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂×Dic₇₉	—	4	2	no	—
64	C₂×Dic₇₉	—	8	4	no	—
65	C₂×Dic₇₉	—	8	2	no	—
66	C₈×C₇₉	—	2	2	no	—
67	C₈×C₇₉	#125	1	2	yes	—
68	C₈×C₇₉	#126	1	2	yes	—
69	C₇₉⋊C₈	—	8	1	no	—
70	C₂²×C₇₉	—	2	2	no	—
71	C₂²×C₇₉	#476	1	1	yes	characteristic
72	C₂²×C₇₉	#484	1	1	yes	characteristic
73	C₂²×C₇₉	#490	1	1	yes	characteristic
74	C₄×C₇₉	—	2	4	no	—
75	C₄×C₇₉	#400	1	1	yes	characteristic
76	C₄×C₇₉	#477	1	1	yes	characteristic
77	C₄×C₇₉	#481	1	1	yes	characteristic
78	C₄×C₇₉	#487	1	1	yes	characteristic
79	Dic₇₉	—	8	2	no	—
80	Dic₇₉	—	16	2	no	—
81	#13543	—	79	1	no	maximal, Sylow
82	C₂×C₇₉	—	2	1	no	—
83	C₂×C₇₉	—	2	1	no	—
84	C₂×C₇₉	#2100	1	1	yes	characteristic
85	C₂×C₇₉	#2230	1	2	yes	—
86	#1945	—	79	1	no	—
87	#1844	—	79	1	no	—
88	#1386	—	79	2	no	—
89	#1383	—	79	2	no	—
90	#1376	D₇₉	1	1	yes	characteristic
91	C₇₉	#13543	1	1	yes	characteristic, complemented, Sylow
92	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
93	C₂×#108	—	79	1	no	—
94	#427	—	79	1	no	—
95	C₂×#106	—	79	2	no	—
96	C₂×#106	—	158	1	no	—
97	#424	—	158	1	no	—
98	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
99	#384	—	158	1	no	—
100	#339	—	158	1	no	—
101	C₂×M₁₆	—	158	1	no	—
102	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
104	#117	—	158	1	no	—
105	C₂×C₄⋊C₄	—	316	2	no	—
106	C₂×C₄⋊C₄	—	158	2	no	—
107	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
108	M₃₂	—	2	2	no	—
109	M₃₂	—	2	2	no	—
110	C₂×C₁₆	—	2	2	no	—
111	#108	—	158	2	no	—
112	#107	—	79	2	no	—
113	#106	—	316	2	no	—
114	#106	—	158	4	no	—
115	#106	—	158	1	no	—
116	C₄⋊C₈	—	316	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	316	2	no	—
120	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
121	M₁₆	—	316	1	no	—
122	M₁₆	—	316	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	316	1	no	—
126	C₂×C₈	#11929613	1	2	yes	—
127	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
128	C₂×C₈	#11929623	1	2	yes	—
129	C₂×C₈	#11929623	1	1	yes	characteristic
130	C₂×C₈	#11929624	1	2	yes	—
131	C₄⋊C₄	—	316	2	no	—
132	C₄⋊C₄	—	316	2	no	—
133	C₄⋊C₄	—	632	4	no	—
134	C₄⋊C₄	—	158	2	no	—
135	C₂²⋊C₄	—	316	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	316	1	no	—
138	C₄²	Dic₃₁₆	1	2	yes	—
139	C₁₆	—	4	2	no	—
140	C₂³	#14579189	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	316	2	no	—
144	C₂×C₄	—	632	4	no	—
145	C₂×C₄	—	632	2	no	—
146	C₂×C₄	#14579141	1	1	yes	characteristic
147	C₂×C₄	#14579145	1	1	yes	characteristic
148	C₂×C₄	#14579149	1	1	yes	characteristic
149	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
150	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
151	C₂×C₄	#14579189	1	1	yes	characteristic
152	C₈	—	2	2	no	—
153	C₈	—	632	1	no	—
154	C₈	#14579151	1	2	yes	—
155	C₈	#14579154	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	#19098854	1	1	yes	characteristic
158	C₂²	#19098899	1	1	yes	characteristic
159	C₂²	#19098928	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	632	2	no	—
162	C₄	—	1264	2	no	—
163	C₄	#19098503	1	1	yes	characteristic
164	C₄	#19098859	1	1	yes	characteristic
165	C₄	#19098882	1	1	yes	characteristic
166	C₄	#19098914	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#23764964	1	1	yes	characteristic
170	C₂	#23765572	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764148

C₂

yes

maximal, characteristic

#23763557

C₂

yes

maximal, characteristic

#23761506

C₂

yes

maximal

#23761499

C₂

yes

maximal

C₇₉×#1376

C₂

yes

maximal, characteristic

C₂×#14579097

C₂²

yes

characteristic

#19098601

C₂²

yes

characteristic

C₂×#14579093

—

C₂×#14579093

C₂²

yes

—

#19098589

—

#19098433

—

#19098323

—

C₂×M₃₂×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

#14579120

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

#14579097

—

#14579095

Q₈

yes

—

#14579093

—

#14579093

—

#14579093

—

Dic₇₉⋊C₈

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

M₃₂×C₇₉

—

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

C₄×C₈×C₇₉

—

C₂²×Dic₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

C₂×Q₈

yes

C₂×C₈×C₇₉

yes

C₂×C₈×C₇₉

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

Q₁₆

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#125

yes

—

C₈×C₇₉

#126

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

—

Dic₇₉

—

#13543

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#2100

yes

characteristic

C₂×C₇₉

#2230

yes

—

—

—

—

—

D₇₉

yes

characteristic

C₇₉

#13543

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₇₉

yes

—

—

—

—

—

158

—

#424

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#384

—

158

—

100

#339

—

158

—

101

C₂×M₁₆

—

158

—

102

C₂²×C₈

D₃₁₆

yes

characteristic

103

C₂²×C₈

C₂²×D₇₉

yes

characteristic

104

#117

—

158

—

105

C₂×C₄⋊C₄

—

316

—

106

C₂×C₄⋊C₄

—

158

—

107

C₂×C₄²

D₃₁₆

yes

characteristic

108

M₃₂

—

109

M₃₂

—

110

C₂×C₁₆

—

111

#108

—

158

—

112

#107

—

113

#106

—

316

—

114

#106

—

158

—

115

#106

—

158

—

116

C₄⋊C₈

—

316

—

117

C₄×C₈

—

118

C₂²×C₄

—

119

C₂²×C₄

—

316

—

120

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

121

M₁₆

—

316

—

122

M₁₆

—

316

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

316

—

126

C₂×C₈

#11929613

yes

—

127

C₂×C₈

Q₈×D₇₉

yes

characteristic

128

C₂×C₈

#11929623

yes

—

129

C₂×C₈

#11929623

yes

characteristic

130

C₂×C₈

#11929624

yes

—

131

C₄⋊C₄

—

316

—

132

C₄⋊C₄

—

316

—

133

C₄⋊C₄

—

632

—

134

C₄⋊C₄

—

158

—

135

C₂²⋊C₄

—

316

—

136

C₄²

—

137

C₄²

—

316

—

138

C₄²

Dic₃₁₆

yes

—

139

C₁₆

—

140

C₂³

#14579189

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

316

—

144

C₂×C₄

—

632

—

145

C₂×C₄

—

632

—

146

C₂×C₄

#14579141

yes

characteristic

147

C₂×C₄

#14579145

yes

characteristic

148

C₂×C₄

#14579149

yes

characteristic

149

C₂×C₄

C₂×#11929613

yes

characteristic

150

C₂×C₄

C₂×Dic₃₁₆

yes

characteristic

151

C₂×C₄

#14579189

yes

characteristic

152

C₈

—

153

C₈

—

632

—

154

C₈

#14579151

yes

—

155

C₈

#14579154

yes

—

156

C₂²

—

157

C₂²

#19098854

yes

characteristic

158

C₂²

#19098899

yes

characteristic

159

C₂²

#19098928

yes

characteristic

160

C₄

—

161

C₄

—

632

—

162

C₄

—

1264

—

163

C₄

#19098503

yes

characteristic

164

C₄

#19098859

yes

characteristic

165

C₄

#19098882

yes

characteristic

166

C₄

#19098914

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#23764964

yes

characteristic

170

C₂

#23765572

yes

—

	subgroup	G/H
center	C₂²	#19098854
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1376	C₂
socle	C₂²×C₇₉	#476
prime radical	C₂³×C₇₉	AGL(1,ℤ/8)
hypercenter	#1376	D₇₉
hypocenter	C₇₉	#13543
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#19098854

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1376

C₂

C₂²×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
91. C₇₉	#13543	semidirect

H ≅ G/N

type

91. C₇₉

#13543

semidirect

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

subgroup	index
#23764148	2
#23763557	2
#23761506	2
#23761499	2
C₇₉×#1376	2
#13543	79

subgroup

index

C₇₉×#1376

Subgroups — #34208420

Conjugacy classes of subgroups

Complements

Maximal subgroups