Group #23772470 — subgroups

Subgroups — #23772470

Order 10112.

394 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 157 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	#19099113	C₂	1	1	yes	maximal, characteristic
2	#19098636	C₂	1	4	yes	maximal, complemented
3	#19098633	C₂	1	4	yes	maximal, complemented
4	#19098620	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098528	C₂	1	4	yes	maximal, complemented
6	#432×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11929624	C₂²	1	2	yes	—
8	C₂×#11929607	C₂²	1	4	yes	—
9	C₂×C₈×D₇₉	C₂²	1	4	yes	—
10	#14579161	C₄	1	1	yes	characteristic
11	#14579158	C₄	1	1	yes	characteristic
12	#14579149	C₄	1	2	yes	—
13	#14579135	C₄	1	2	yes	—
14	#14579135	C₄	1	2	yes	—
15	C₄×D₃₁₆	C₂²	1	2	yes	—
16	C₄×D₃₁₆	C₂²	1	2	yes	—
17	C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
18	C₂²⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
19	(C₂×D₇₉)⋊C₈	C₂²	1	8	yes	—
20	#14579072	C₂²	1	2	yes	—
21	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
23	C₂²⋊C₈×C₇₉	C₂²	1	2	yes	—
24	#97×C₇₉	C₂²	1	2	yes	—
25	#11929624	—	4	2	no	—
26	#11929624	—	4	2	no	—
27	C₂×Dic₁₅₈	C₂×C₄	1	2	yes	—
28	C₂×D₃₁₆	C₂×C₄	1	2	yes	—
29	C₂×D₃₁₆	C₂×C₄	1	2	yes	—
30	C₂×D₃₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊D₇₉	C₂×C₄	1	4	yes	—
32	C₂×C₄×D₇₉	C₂³	1	4	yes	—
33	#11929607	—	4	4	no	—
34	C₂×C₇₉⋊C₈	C₂³	1	4	yes	—
35	C₈×D₇₉	—	4	4	no	—
36	C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
38	(C₂×D₇₉)⋊C₄	C₂×C₄	1	4	yes	—
39	(C₂×D₇₉)⋊C₄	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂³	1	4	yes	—
44	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₇₉	C₂³	1	1	yes	characteristic
47	C₄²×C₇₉	C₂³	1	1	yes	characteristic
48	C₂³×C₇₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×D₇₉	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₇₉	—	2	2	no	—
51	C₂×C₄×C₇₉	—	4	1	no	—
52	C₂×C₄×C₇₉	—	4	1	no	—
53	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
59	C₂²⋊D₇₉	—	4	4	no	—
60	C₂×Dic₇₉	C₂²×C₄	1	4	yes	—
61	D₃₁₆	—	4	2	no	—
62	D₃₁₆	—	4	2	no	—
63	D₃₁₆	—	4	2	no	—
64	C₄×D₇₉	—	2	4	no	—
65	C₄×D₇₉	—	2	4	no	—
66	Dic₁₅₈	—	4	2	no	—
67	C₈×C₇₉	—	2	4	no	—
68	C₇₉⋊C₈	—	2	4	no	—
69	C₂²×C₇₉	—	2	1	no	—
70	C₂²×C₇₉	—	2	1	no	—
71	C₂²×C₇₉	—	2	1	no	—
72	C₂²×C₇₉	C₂³×C₄	1	1	yes	characteristic
73	C₂×D₇₉	—	2	4	no	—
74	C₂×D₇₉	—	2	4	no	—
75	C₂×D₇₉	—	2	4	no	—
76	C₄×C₇₉	—	2	1	no	—
77	C₄×C₇₉	—	2	1	no	—
78	C₄×C₇₉	—	2	1	no	—
79	C₄×C₇₉	—	2	1	no	—
80	C₄×C₇₉	—	2	1	no	—
81	C₄×C₇₉	H(5,𝔽₂)	1	2	yes	—
82	Dic₇₉	—	2	4	no	—
83	C₂×C₇₉	—	4	1	no	—
84	C₂×C₇₉	#518	1	1	yes	characteristic
85	C₂×C₇₉	#568	1	1	yes	characteristic
86	C₂×C₇₉	#568	1	1	yes	characteristic
87	D₇₉	—	4	4	no	—
88	#2980	—	79	1	no	maximal, Sylow
89	C₇₉	#2980	1	1	yes	characteristic, complemented, Sylow
90	#532	—	79	1	no	—
91	#435	—	79	4	no	—
92	#435	—	79	4	no	—
93	#432	—	79	1	no	—
94	#432	D₇₉	1	1	yes	characteristic, complemented
95	#408	—	79	4	no	—
96	C₂×#41	—	79	2	no	—
97	C₂×M₁₆	—	79	4	no	—
98	C₂²×C₈	—	79	4	no	—
99	C₄⋊Q₈	—	79	1	no	—
100	#127	—	79	1	no	—
101	#124	—	79	2	no	—
102	#121	—	79	2	no	—
103	#121	—	79	2	no	—
104	C₄×D₄	—	79	2	no	—
105	C₄×D₄	—	79	2	no	—
106	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
107	C₄⋊C₈	—	79	2	no	—
108	C₄⋊C₈	C₂×D₇₉	1	2	yes	—
109	C₂²⋊C₈	—	79	2	no	—
110	C₂²⋊C₈	—	79	8	no	—
111	C₂²⋊C₈	C₂×D₇₉	1	2	yes	—
112	#97	—	79	2	no	—
113	#97	C₂×D₇₉	1	2	yes	—
114	#41	—	316	2	no	—
115	#41	—	316	2	no	—
116	C₂×Q₈	—	79	2	no	—
117	C₂×D₄	—	79	4	no	—
118	C₂×D₄	—	79	2	no	—
119	C₂×D₄	—	79	2	no	—
120	C₂×D₄	—	79	2	no	—
121	C₂²×C₄	—	79	4	no	—
122	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
123	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
125	M₁₆	—	316	4	no	—
126	C₂×C₈	—	316	4	no	—
127	C₂×C₈	—	79	4	no	—
128	C₂×C₈	C₂²×D₇₉	1	4	yes	—
129	C₄⋊C₄	—	79	2	no	—
130	C₄⋊C₄	—	79	2	no	—
131	C₂²⋊C₄	—	79	4	no	—
132	C₂²⋊C₄	—	79	4	no	—
133	C₄²	C₄×D₇₉	1	1	yes	characteristic
134	C₄²	C₄×D₇₉	1	1	yes	characteristic
135	C₄²	C₂²×D₇₉	1	1	yes	characteristic
136	C₄²	C₂²×D₇₉	1	1	yes	characteristic
137	C₂³	—	79	4	no	—
138	C₂³	C₂×C₄×D₇₉	1	1	yes	characteristic
139	Q₈	—	316	2	no	—
140	D₄	—	316	2	no	—
141	D₄	—	316	2	no	—
142	D₄	—	316	2	no	—
143	D₄	—	316	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	79	4	no	—
148	C₂×C₄	—	158	4	no	—
149	C₂×C₄	—	158	4	no	—
150	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
155	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
156	C₈	—	2	4	no	—
157	C₈	—	158	4	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	158	4	no	—
162	C₂²	—	158	4	no	—
163	C₂²	—	158	4	no	—
164	C₂²	C₂²×C₄×D₇₉	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	158	4	no	—
171	C₄	#14579207	1	2	yes	—
172	C₂	—	4	1	no	—
173	C₂	—	316	4	no	—
174	C₂	#19099037	1	1	yes	characteristic
175	C₂	#19099329	1	1	yes	characteristic
176	C₂	#19099329	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099113

C₂

yes

maximal, characteristic

#19098636

C₂

yes

maximal, complemented

#19098633

C₂

yes

maximal, complemented

#19098620

C₂

yes

maximal, characteristic, complemented

#19098528

C₂

yes

maximal, complemented

#432×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929624

C₂²

yes

—

C₂×#11929607

C₂²

yes

—

C₂×C₈×D₇₉

C₂²

yes

—

#14579161

C₄

yes

characteristic

#14579158

C₄

yes

characteristic

#14579149

C₄

yes

—

#14579135

C₄

yes

—

#14579135

C₄

yes

—

C₄×D₃₁₆

C₂²

yes

—

C₄×D₃₁₆

C₂²

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₇₉⋊C₈)

C₂²

yes

—

(C₂×D₇₉)⋊C₈

C₂²

yes

—

#14579072

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

—

C₂²⋊C₈×C₇₉

C₂²

yes

—

#97×C₇₉

C₂²

yes

—

#11929624

—

#11929624

—

C₂×Dic₁₅₈

C₂×C₄

yes

—

C₂×D₃₁₆

C₂×C₄

yes

—

C₂×D₃₁₆

C₂×C₄

yes

—

C₂×D₃₁₆

C₂×C₄

yes

—

C₂×C₂²⋊D₇₉

C₂×C₄

yes

—

C₂×C₄×D₇₉

C₂³

yes

—

#11929607

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₈×D₇₉

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

(C₂×D₇₉)⋊C₄

C₂×C₄

yes

—

(C₂×D₇₉)⋊C₄

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

—

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

C₂²×C₄

yes

characteristic

C₂²×D₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

C₂²×C₄

yes

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂³×C₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

H(5,𝔽₂)

yes

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

#2980

—

maximal, Sylow

C₇₉

#2980

yes

characteristic, complemented, Sylow

—

—

—

—

D₇₉

yes

characteristic, complemented

#408

—

C₂×#41

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

100

#127

—

101

#124

—

102

#121

—

103

#121

—

104

C₄×D₄

—

105

C₄×D₄

—

106

C₂×C₄²

C₂×D₇₉

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

C₂×D₇₉

yes

—

109

C₂²⋊C₈

—

110

C₂²⋊C₈

—

111

C₂²⋊C₈

C₂×D₇₉

yes

—

112

#97

—

113

#97

C₂×D₇₉

yes

—

114

#41

—

316

—

115

#41

—

316

—

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₄×D₇₉

yes

characteristic

123

C₂²×C₄

C₄×D₇₉

yes

characteristic

124

C₂²×C₄

C₂²×D₇₉

yes

characteristic

125

M₁₆

—

316

—

126

C₂×C₈

—

316

—

127

C₂×C₈

—

128

C₂×C₈

C₂²×D₇₉

yes

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

C₄×D₇₉

yes

characteristic

134

C₄²

C₄×D₇₉

yes

characteristic

135

C₄²

C₂²×D₇₉

yes

characteristic

136

C₄²

C₂²×D₇₉

yes

characteristic

137

C₂³

—

138

C₂³

C₂×C₄×D₇₉

yes

characteristic

139

Q₈

—

316

—

140

D₄

—

316

—

141

D₄

—

316

—

142

D₄

—

316

—

143

D₄

—

316

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

158

—

149

C₂×C₄

—

158

—

150

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

151

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

152

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

153

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

154

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

155

C₂×C₄

C₂³×D₇₉

yes

characteristic

156

C₈

—

157

C₈

—

158

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

158

—

162

C₂²

—

158

—

163

C₂²

—

158

—

164

C₂²

C₂²×C₄×D₇₉

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

158

—

171

C₄

#14579207

yes

—

172

C₂

—

173

C₂

—

316

—

174

C₂

#19099037

yes

characteristic

175

C₂

#19099329

yes

characteristic

176

C₂

#19099329

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₉
commutator, socle	C₂²×C₇₉	C₂³×C₄
Fitting	#432×C₇₉	C₂
prime radical	#19099113	C₂
cosocle	C₂×C₄×C₇₉	C₂⁴
hypercenter	#432	D₇₉
hypocenter	C₇₉	#2980
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ #432

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₇₉

commutator, socle

C₂²×C₇₉

C₂³×C₄

Fitting

#432×C₇₉

C₂

prime radical

#19099113

C₂

cosocle

C₂×C₄×C₇₉

C₂⁴

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂×C₄ ◁ #432

N	H ≅ G/N	type
2. #19098636	C₂	semidirect
3. #19098633	C₂	semidirect
4. #19098620	C₂	semidirect
5. #19098528	C₂	semidirect
6. #432×C₇₉	C₂	semidirect
89. C₇₉	#2980	semidirect
94. #432	D₇₉	semidirect

H ≅ G/N

type

2. #19098636

C₂

semidirect

3. #19098633

C₂

semidirect

4. #19098620

C₂

semidirect

5. #19098528

C₂

semidirect

6. #432×C₇₉

C₂

semidirect

89. C₇₉

#2980

semidirect

94. #432

D₇₉

semidirect

Subgroups — #23772470

Conjugacy classes of subgroups

Complements