Group #34208415 — subgroups

Subgroups — #34208415

Order 20224.

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

normal subgroups only

This group has 252 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	#23763564	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×#14579095	—	2	1	no	—
9	C₂×#14579093	C₂²	1	2	yes	—
10	#19098589	—	2	1	no	—
11	#19098429	—	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	#14579097	—	2	2	no	—
19	#14579095	—	4	2	no	—
20	#14579095	Q₈	1	2	yes	—
21	#14579093	—	2	1	no	—
22	#14579093	—	2	4	no	—
23	Dic₇₉⋊C₈	—	4	2	no	—
24	C₇₉⋊#95	—	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	C₂²⋊Dic₇₉	—	4	1	no	—
40	C₄×Dic₇₉	—	4	1	no	—
41	C₂²×C₄×C₇₉	—	2	1	no	—
42	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	—	2	1	no	—
44	C₂×C₈×C₇₉	—	2	1	no	—
45	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
46	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
48	C₂×C₈×C₇₉	#41	1	2	yes	—
49	C₂×C₈×C₇₉	#41	1	2	yes	—
50	C₄²×C₇₉	—	2	1	no	—
51	C₄²×C₇₉	Q₁₆	1	2	yes	—
52	C₁₆×C₇₉	—	4	2	no	—
53	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₂×C₄×C₇₉	—	2	2	no	—
55	C₂×C₄×C₇₉	—	4	2	no	—
56	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
58	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
61	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₂×Dic₇₉	—	4	2	no	—
63	C₂×Dic₇₉	—	8	4	no	—
64	C₂×Dic₇₉	—	8	2	no	—
65	C₈×C₇₉	—	2	2	no	—
66	C₈×C₇₉	#125	1	2	yes	—
67	C₈×C₇₉	#126	1	2	yes	—
68	C₇₉⋊C₈	—	8	1	no	—
69	C₂²×C₇₉	—	2	2	no	—
70	C₂²×C₇₉	#476	1	1	yes	characteristic
71	C₂²×C₇₉	#484	1	1	yes	characteristic
72	C₂²×C₇₉	#490	1	1	yes	characteristic
73	C₄×C₇₉	—	4	2	no	—
74	C₄×C₇₉	#400	1	1	yes	characteristic
75	C₄×C₇₉	#477	1	1	yes	characteristic
76	C₄×C₇₉	#481	1	1	yes	characteristic
77	C₄×C₇₉	#487	1	1	yes	characteristic
78	Dic₇₉	—	8	2	no	—
79	Dic₇₉	—	16	2	no	—
80	#13542	—	79	1	no	maximal, Sylow
81	C₂×C₇₉	—	2	1	no	—
82	C₂×C₇₉	—	2	1	no	—
83	C₂×C₇₉	#2100	1	1	yes	characteristic
84	C₂×C₇₉	#2231	1	2	yes	—
85	#1945	—	79	1	no	—
86	#1846	—	79	1	no	—
87	#1386	—	79	2	no	—
88	#1383	—	79	2	no	—
89	#1376	D₇₉	1	1	yes	characteristic
90	C₇₉	#13542	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
92	C₂×#108	—	79	1	no	—
93	#427	—	79	1	no	—
94	C₂×#107	—	158	1	no	—
95	C₂×#106	—	79	2	no	—
96	#424	—	158	1	no	—
97	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
98	#382	—	158	1	no	—
99	#339	—	158	1	no	—
100	C₂×M₁₆	—	158	1	no	—
101	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
103	#117	—	158	1	no	—
104	C₂×C₄⋊C₄	—	158	2	no	—
105	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	2	2	no	—
108	C₂×C₁₆	—	2	2	no	—
109	#108	—	158	2	no	—
110	#107	—	316	2	no	—
111	#107	—	79	2	no	—
112	#106	—	158	4	no	—
113	#106	—	158	1	no	—
114	C₄⋊C₈	—	316	2	no	—
115	C₄×C₈	—	2	2	no	—
116	#95	—	316	2	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	316	2	no	—
119	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
120	M₁₆	—	316	1	no	—
121	M₁₆	—	316	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	316	1	no	—
125	C₂×C₈	#11929613	1	2	yes	—
126	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
127	C₂×C₈	#11929623	1	2	yes	—
128	C₂×C₈	#11929623	1	1	yes	characteristic
129	C₂×C₈	#11929624	1	2	yes	—
130	C₄⋊C₄	—	316	2	no	—
131	C₄⋊C₄	—	316	2	no	—
132	C₄⋊C₄	—	158	2	no	—
133	C₂²⋊C₄	—	316	1	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	316	1	no	—
136	C₄²	Dic₃₁₆	1	2	yes	—
137	C₁₆	—	4	2	no	—
138	C₂³	#14579189	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	316	2	no	—
142	C₂×C₄	—	632	4	no	—
143	C₂×C₄	—	632	2	no	—
144	C₂×C₄	#14579141	1	1	yes	characteristic
145	C₂×C₄	#14579145	1	1	yes	characteristic
146	C₂×C₄	#14579149	1	1	yes	characteristic
147	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
148	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
149	C₂×C₄	#14579189	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	632	1	no	—
152	C₈	#14579151	1	2	yes	—
153	C₈	#14579154	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	#19098854	1	1	yes	characteristic
156	C₂²	#19098899	1	1	yes	characteristic
157	C₂²	#19098928	1	1	yes	characteristic
158	C₄	—	4	2	no	—
159	C₄	—	632	2	no	—
160	C₄	—	1264	2	no	—
161	C₄	#19098503	1	1	yes	characteristic
162	C₄	#19098859	1	1	yes	characteristic
163	C₄	#19098882	1	1	yes	characteristic
164	C₄	#19098914	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#23764964	1	1	yes	characteristic
168	C₂	#23765577	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764148

C₂

yes

maximal, characteristic

#23763564

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×#14579095

—

C₂×#14579093

C₂²

yes

—

#19098589

—

#19098429

—

#19098323

—

C₂×M₃₂×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

#14579120

—

C₂×C₄⋊Dic₇₉

—

#14579097

—

#14579095

—

#14579095

Q₈

yes

—

#14579093

—

#14579093

—

Dic₇₉⋊C₈

—

C₇₉⋊#95

—

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

—

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

—

#13542

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#2100

yes

characteristic

C₂×C₇₉

#2231

yes

—

—

—

—

—

D₇₉

yes

characteristic

C₇₉

#13542

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₇₉

yes

—

C₂×#108

—

#427

—

C₂×#107

—

158

—

C₂×#106

—

#424

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#382

—

158

—

#339

—

158

—

100

C₂×M₁₆

—

158

—

101

C₂²×C₈

D₃₁₆

yes

characteristic

102

C₂²×C₈

C₂²×D₇₉

yes

characteristic

103

#117

—

158

—

104

C₂×C₄⋊C₄

—

158

—

105

C₂×C₄²

D₃₁₆

yes

characteristic

106

M₃₂

—

107

M₃₂

—

108

C₂×C₁₆

—

109

#108

—

158

—

110

#107

—

316

—

111

#107

—

112

#106

—

158

—

113

#106

—

158

—

114

C₄⋊C₈

—

316

—

115

C₄×C₈

—

116

#95

—

316

—

117

C₂²×C₄

—

118

C₂²×C₄

—

316

—

119

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

120

M₁₆

—

316

—

121

M₁₆

—

316

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

316

—

125

C₂×C₈

#11929613

yes

—

126

C₂×C₈

Q₈×D₇₉

yes

characteristic

127

C₂×C₈

#11929623

yes

—

128

C₂×C₈

#11929623

yes

characteristic

129

C₂×C₈

#11929624

yes

—

130

C₄⋊C₄

—

316

—

131

C₄⋊C₄

—

316

—

132

C₄⋊C₄

—

158

—

133

C₂²⋊C₄

—

316

—

134

C₄²

—

135

C₄²

—

316

—

136

C₄²

Dic₃₁₆

yes

—

137

C₁₆

—

138

C₂³

#14579189

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

316

—

142

C₂×C₄

—

632

—

143

C₂×C₄

—

632

—

144

C₂×C₄

#14579141

yes

characteristic

145

C₂×C₄

#14579145

yes

characteristic

146

C₂×C₄

#14579149

yes

characteristic

147

C₂×C₄

C₂×#11929613

yes

characteristic

148

C₂×C₄

C₂×Dic₃₁₆

yes

characteristic

149

C₂×C₄

#14579189

yes

characteristic

150

C₈

—

151

C₈

—

632

—

152

C₈

#14579151

yes

—

153

C₈

#14579154

yes

—

154

C₂²

—

155

C₂²

#19098854

yes

characteristic

156

C₂²

#19098899

yes

characteristic

157

C₂²

#19098928

yes

characteristic

158

C₄

—

159

C₄

—

632

—

160

C₄

—

1264

—

161

C₄

#19098503

yes

characteristic

162

C₄

#19098859

yes

characteristic

163

C₄

#19098882

yes

characteristic

164

C₄

#19098914

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

#23764964

yes

characteristic

168

C₂

#23765577

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	#13542
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
90. C₇₉	#13542	semidirect

H ≅ G/N

type

90. C₇₉

#13542

semidirect

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

subgroup	index
#23764148	2
#23763564	2
#23761506	2
#23761499	2
C₇₉×#1376	2
#13542	79

subgroup

index

C₇₉×#1376

Subgroups — #34208415

Conjugacy classes of subgroups

Complements

Maximal subgroups