Group #34185791 — subgroups

Subgroups — #34185791

Order 20224.

384 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 99 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	#23773782	C₂	1	1	yes	maximal, characteristic
2	#23762118	C₂	1	2	yes	maximal
3	#23762038	C₂	1	1	yes	maximal, characteristic
4	#23761271	C₂	1	2	yes	maximal
5	C₇₉×#2987	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579161	C₂²	1	1	yes	characteristic
7	#19098933	—	2	2	no	—
8	#19098915	C₄	1	4	yes	—
9	#19098620	C₂²	1	2	yes	—
10	#19098571	—	2	2	no	—
11	C₄×#11929608	C₂²	1	1	yes	characteristic
12	#19098310	—	2	4	no	—
13	#19098305	—	2	2	no	—
14	#533×C₇₉	C₂²	1	1	yes	characteristic
15	#446×C₇₉	—	2	4	no	—
16	#433×C₇₉	—	2	2	no	—
17	#432×C₇₉	C₂²	1	2	yes	—
18	C₂²×Dic₁₅₈	—	2	1	no	—
19	C₂×#11929608	—	2	1	no	—
20	#14579161	—	2	2	no	—
21	#14579161	—	2	1	no	—
22	#14579161	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊Dic₇₉	—	2	2	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	#14579084	—	2	8	no	—
26	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
27	#14579072	—	2	1	no	—
28	#14579072	—	2	2	no	—
29	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
30	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
31	#125×C₇₉	D₄	1	2	yes	—
32	C₂²⋊Q₈×C₇₉	—	2	2	no	—
33	C₄×Q₈×C₇₉	—	2	2	no	—
34	#117×C₇₉	D₄	1	2	yes	—
35	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₇₉	—	2	4	no	—
37	C₄⋊C₈×C₇₉	D₄	1	4	yes	—
38	C₂²⋊C₈×C₇₉	—	2	2	no	—
39	#97×C₇₉	—	2	2	no	—
40	#97×C₇₉	—	2	2	no	—
41	C₄×C₈×C₇₉	—	2	2	no	—
42	C₂×Dic₁₅₈	—	2	2	no	—
43	C₂×Dic₁₅₈	—	4	4	no	—
44	C₂×Dic₁₅₈	—	4	1	no	—
45	C₂²×Dic₇₉	—	4	1	no	—
46	#11929608	—	4	2	no	—
47	C₂×C₇₉⋊C₈	—	2	2	no	—
48	C₂×C₇₉⋊C₈	—	4	2	no	—
49	C₄⋊Dic₇₉	—	2	4	no	—
50	C₄⋊Dic₇₉	—	4	2	no	—
51	C₂×Q₈×C₇₉	—	2	2	no	—
52	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₉	#41	1	2	yes	—
54	C₂×C₈×C₇₉	—	2	4	no	—
55	C₄⋊C₄×C₇₉	—	2	2	no	—
56	C₄⋊C₄×C₇₉	—	2	4	no	—
57	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
58	C₂²⋊C₄×C₇₉	—	2	2	no	—
59	C₄²×C₇₉	—	2	2	no	—
60	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₇₉	C₂×D₄	1	2	yes	—
63	C₂³×C₇₉	#123	1	1	yes	characteristic
64	Q₈×C₇₉	—	8	2	no	—
65	C₂×C₄×C₇₉	—	2	2	no	—
66	C₂×C₄×C₇₉	—	2	4	no	—
67	C₂×C₄×C₇₉	—	4	2	no	—
68	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
71	C₂×C₄×C₇₉	#121	1	2	yes	—
72	C₂×Dic₇₉	—	4	2	no	—
73	C₂×Dic₇₉	—	8	2	no	—
74	Dic₁₅₈	—	4	4	no	—
75	Dic₁₅₈	—	8	2	no	—
76	C₈×C₇₉	—	4	4	no	—
77	C₇₉⋊C₈	—	4	2	no	—
78	C₇₉⋊C₈	—	8	2	no	—
79	C₂²×C₇₉	—	2	1	no	—
80	C₂²×C₇₉	—	2	2	no	—
81	C₂²×C₇₉	#386	1	1	yes	characteristic
82	C₄×C₇₉	—	2	1	no	—
83	C₄×C₇₉	—	2	2	no	—
84	C₄×C₇₉	—	2	2	no	—
85	C₄×C₇₉	—	4	4	no	—
86	C₄×C₇₉	#454	1	2	yes	—
87	Dic₇₉	—	8	2	no	—
88	#9422	—	79	1	no	maximal, Sylow
89	C₂×C₇₉	—	4	1	no	—
90	C₂×C₇₉	#1888	1	1	yes	characteristic
91	C₂×C₇₉	#1902	1	2	yes	—
92	#3139	—	79	1	no	—
93	#2987	D₇₉	1	1	yes	characteristic
94	#1556	—	79	2	no	—
95	#1537	—	79	1	no	—
96	#1291	—	79	2	no	—
97	C₇₉	#9422	1	1	yes	characteristic, complemented, Sylow
98	#533	C₂×D₇₉	1	1	yes	characteristic
99	C₂×C₄⋊Q₈	—	79	1	no	—
100	#491	—	158	2	no	—
101	#487	—	79	4	no	—
102	#446	—	2	4	no	—
103	#433	—	2	2	no	—
104	#432	—	79	2	no	—
105	#432	C₂×D₇₉	1	2	yes	—
106	#419	—	158	2	no	—
107	C₄×M₁₆	—	79	1	no	—
108	#332	—	158	4	no	—
109	#330	—	158	2	no	—
110	C₂²×Q₈	—	158	1	no	—
111	C₂×M₁₆	—	158	1	no	—
112	C₄⋊Q₈	—	79	2	no	—
113	C₄⋊Q₈	—	158	2	no	—
114	C₄⋊Q₈	—	158	1	no	—
115	C₄⋊Q₈	C₄×D₇₉	1	2	yes	—
116	#125	D₃₁₆	1	2	yes	—
117	C₂²⋊Q₈	—	2	2	no	—
118	C₄×Q₈	—	2	2	no	—
119	#117	C₂²⋊D₇₉	1	2	yes	—
120	C₂×C₄⋊C₄	—	158	2	no	—
121	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
122	C₄⋊C₈	—	2	4	no	—
123	C₄⋊C₈	—	158	2	no	—
124	C₄⋊C₈	C₂²⋊D₇₉	1	4	yes	—
125	#103	—	158	8	no	—
126	C₂²⋊C₈	—	2	2	no	—
127	C₂²⋊C₈	—	158	2	no	—
128	#97	—	2	2	no	—
129	#97	—	2	2	no	—
130	#97	—	158	2	no	—
131	#97	—	158	1	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	79	2	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	316	4	no	—
136	C₂×Q₈	—	316	1	no	—
137	C₂×Q₈	—	158	2	no	—
138	C₂²×C₄	—	316	1	no	—
139	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
140	C₂²×C₄	#11929624	1	2	yes	—
141	M₁₆	—	316	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	316	2	no	—
144	C₂×C₈	—	158	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	316	2	no	—
148	C₄⋊C₄	—	158	4	no	—
149	C₄⋊C₄	(C₂×D₇₉)⋊C₄	1	4	yes	—
150	C₂²⋊C₄	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
153	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
154	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
155	C₂³	#14579144	1	1	yes	characteristic
156	Q₈	—	8	2	no	—
157	Q₈	—	316	4	no	—
158	Q₈	—	632	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	316	2	no	—
163	C₂×C₄	—	632	2	no	—
164	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
165	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
166	C₂×C₄	#14579130	1	1	yes	characteristic
167	C₂×C₄	#14579135	1	2	yes	—
168	C₈	—	4	4	no	—
169	C₈	—	316	2	no	—
170	C₈	—	632	2	no	—
171	C₂²	—	2	1	no	—
172	C₂²	—	2	2	no	—
173	C₂²	#19098442	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	4	4	no	—
178	C₄	—	632	2	no	—
179	C₄	#19098736	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	#23763803	1	1	yes	characteristic
182	C₂	#23763873	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23773782

C₂

yes

maximal, characteristic

#23762118

C₂

yes

maximal

#23762038

C₂

yes

maximal, characteristic

#23761271

C₂

yes

maximal

C₇₉×#2987

C₂

yes

maximal, characteristic

C₂×#14579161

C₂²

yes

characteristic

#19098933

—

#19098915

C₄

yes

—

#19098620

C₂²

yes

—

#19098571

—

C₄×#11929608

C₂²

yes

characteristic

#19098310

—

#19098305

—

#533×C₇₉

C₂²

yes

characteristic

#446×C₇₉

—

#433×C₇₉

—

#432×C₇₉

C₂²

yes

—

C₂²×Dic₁₅₈

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₇₉

—

C₄⋊(C₇₉⋊C₈)

—

#14579084

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

#14579072

—

C₄×C₇₉⋊C₈

C₂×C₄

yes

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

#125×C₇₉

D₄

yes

—

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

#117×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

D₄

yes

—

C₂²⋊C₈×C₇₉

—

#97×C₇₉

—

#97×C₇₉

—

C₄×C₈×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

#123

yes

characteristic

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#120

yes

characteristic

C₂×C₄×C₇₉

#121

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#386

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#454

yes

—

Dic₇₉

—

#9422

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1888

yes

characteristic

C₂×C₇₉

#1902

yes

—

#3139

—

#2987

D₇₉

yes

—

—

—

C₇₉

#9422

yes

characteristic, complemented, Sylow

#533

C₂×D₇₉

yes

characteristic

C₂×C₄⋊Q₈

—

100

#491

—

158

—

101

#487

—

102

#446

—

103

#433

—

104

#432

—

105

#432

C₂×D₇₉

yes

—

106

#419

—

158

—

107

C₄×M₁₆

—

108

#332

—

158

—

109

#330

—

158

—

110

C₂²×Q₈

—

158

—

111

C₂×M₁₆

—

158

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

158

—

114

C₄⋊Q₈

—

158

—

115

C₄⋊Q₈

C₄×D₇₉

yes

—

116

#125

D₃₁₆

yes

—

117

C₂²⋊Q₈

—

118

C₄×Q₈

—

119

#117

C₂²⋊D₇₉

yes

—

120

C₂×C₄⋊C₄

—

158

—

121

C₂×C₄²

C₂²×D₇₉

yes

characteristic

122

C₄⋊C₈

—

123

C₄⋊C₈

—

158

—

124

C₄⋊C₈

C₂²⋊D₇₉

yes

—

125

#103

—

158

—

126

C₂²⋊C₈

—

127

C₂²⋊C₈

—

158

—

128

#97

—

129

#97

—

130

#97

—

158

—

131

#97

—

158

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

316

—

136

C₂×Q₈

—

316

—

137

C₂×Q₈

—

158

—

138

C₂²×C₄

—

316

—

139

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

140

C₂²×C₄

#11929624

yes

—

141

M₁₆

—

316

—

142

C₂×C₈

—

143

C₂×C₈

—

316

—

144

C₂×C₈

—

158

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

316

—

148

C₄⋊C₄

—

158

—

149

C₄⋊C₄

(C₂×D₇₉)⋊C₄

yes

—

150

C₂²⋊C₄

—

151

C₄²

—

152

C₄²

C₂×C₄×D₇₉

yes

characteristic

153

C₄²

C₂×C₂²⋊D₇₉

yes

—

154

C₄²

C₂×D₃₁₆

yes

characteristic

155

C₂³

#14579144

yes

characteristic

156

Q₈

—

157

Q₈

—

316

—

158

Q₈

—

632

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

316

—

163

C₂×C₄

—

632

—

164

C₂×C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

165

C₂×C₄

C₄×C₂²⋊D₇₉

yes

—

166

C₂×C₄

#14579130

yes

characteristic

167

C₂×C₄

#14579135

yes

—

168

C₈

—

169

C₈

—

316

—

170

C₈

—

632

—

171

C₂²

—

172

C₂²

—

173

C₂²

#19098442

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

632

—

179

C₄

#19098736

yes

—

180

C₂

—

181

C₂

#23763803

yes

characteristic

182

C₂

#23763873

yes

—

	subgroup	G/H
center	C₂²	#19098442
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2987	C₂
socle	C₂²×C₇₉	#386
prime radical	C₂³×C₇₉	#123
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2987	D₇₉
hypocenter	C₇₉	#9422
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2987

subgroup

G/H

center

C₂²

#19098442

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#2987

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2987

N	H ≅ G/N	type
97. C₇₉	#9422	semidirect

H ≅ G/N

type

97. C₇₉

#9422

semidirect

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

subgroup	index
#23773782	2
#23762118	2
#23762038	2
#23761271	2
C₇₉×#2987	2
#9422	79

subgroup

index

C₇₉×#2987

Subgroups — #34185791

Conjugacy classes of subgroups

Complements

Maximal subgroups