Group #23762012 — subgroups

Subgroups — #23762012

Order 10112.

256 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₂×#14579072	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098302	C₂	1	4	yes	maximal, complemented
4	#432×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11929624	C₄	1	2	yes	—
6	C₂²×C₇₉⋊C₈	—	2	1	no	—
7	#14579161	C₄	1	1	yes	characteristic
8	#14579158	C₄	1	1	yes	characteristic
9	#14579149	C₂²	1	2	yes	—
10	#14579135	—	2	1	no	—
11	#14579135	—	2	1	no	—
12	C₄×D₃₁₆	—	2	1	no	—
13	C₄×D₃₁₆	—	2	1	no	—
14	#14579072	—	2	1	no	—
15	#14579072	C₂²	1	2	yes	—
16	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₇₉	—	2	1	no	—
18	C₂²⋊C₈×C₇₉	—	2	1	no	—
19	#97×C₇₉	C₂²	1	2	yes	—
20	#11929624	—	4	4	no	—
21	C₂×Dic₁₅₈	C₂×C₄	1	2	yes	—
22	C₂×D₃₁₆	—	2	1	no	—
23	C₂×D₃₁₆	—	2	1	no	—
24	C₂×D₃₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₂²⋊D₇₉	—	2	2	no	—
26	C₂×C₄×D₇₉	—	2	2	no	—
27	C₂×C₇₉⋊C₈	—	2	4	no	—
28	C₂×C₇₉⋊C₈	—	2	2	no	—
29	C₄⋊Dic₇₉	—	2	1	no	—
30	C₄⋊Dic₇₉	—	2	1	no	—
31	(C₂×D₇₉)⋊C₄	—	2	2	no	—
32	(C₂×D₇₉)⋊C₄	—	2	2	no	—
33	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
36	C₂×C₈×C₇₉	—	2	2	no	—
37	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₇₉	D₄	1	1	yes	characteristic
39	C₄²×C₇₉	D₄	1	1	yes	characteristic
40	C₄²×C₇₉	C₂³	1	1	yes	characteristic
41	C₂³×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×D₇₉	—	2	2	no	—
43	C₂×C₄×C₇₉	—	2	2	no	—
44	C₂×C₄×C₇₉	—	2	2	no	—
45	C₂×C₄×C₇₉	—	4	1	no	—
46	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂²⋊D₇₉	—	4	4	no	—
53	C₂×Dic₇₉	—	2	2	no	—
54	D₃₁₆	—	4	2	no	—
55	D₃₁₆	—	4	2	no	—
56	D₃₁₆	—	4	2	no	—
57	C₄×D₇₉	—	4	4	no	—
58	Dic₁₅₈	—	4	2	no	—
59	C₈×C₇₉	—	4	2	no	—
60	C₇₉⋊C₈	—	2	4	no	—
61	C₂²×C₇₉	—	2	1	no	—
62	C₂²×C₇₉	—	2	1	no	—
63	C₂²×C₇₉	#100	1	2	yes	—
64	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×D₇₉	—	4	2	no	—
66	C₂×D₇₉	—	4	2	no	—
67	C₂×D₇₉	—	4	2	no	—
68	C₄×C₇₉	—	2	1	no	—
69	C₄×C₇₉	—	2	1	no	—
70	C₄×C₇₉	—	2	1	no	—
71	C₄×C₇₉	—	2	1	no	—
72	C₄×C₇₉	—	2	1	no	—
73	C₄×C₇₉	#104	1	2	yes	—
74	Dic₇₉	—	4	2	no	—
75	C₂×C₇₉	—	2	2	no	—
76	C₂×C₇₉	C₂×#100	1	1	yes	characteristic
77	C₂×C₇₉	C₂×#104	1	1	yes	characteristic
78	C₂×C₇₉	#421	1	1	yes	characteristic
79	D₇₉	—	8	2	no	—
80	#1530	—	79	1	no	maximal, Sylow
81	C₇₉	#1530	1	1	yes	characteristic, complemented, Sylow
82	#532	—	79	1	no	—
83	#432	D₇₉	1	1	yes	characteristic, complemented
84	C₂×#97	—	79	1	no	—
85	#329	—	79	4	no	—
86	C₂×#41	—	79	2	no	—
87	C₂²×C₈	—	158	1	no	—
88	C₄⋊Q₈	—	79	1	no	—
89	#127	—	79	1	no	—
90	#124	—	79	2	no	—
91	#121	—	158	1	no	—
92	#121	—	158	1	no	—
93	C₄×D₄	—	158	1	no	—
94	C₄×D₄	—	158	1	no	—
95	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
96	C₄⋊C₈	—	2	1	no	—
97	C₂²⋊C₈	—	2	1	no	—
98	#97	—	79	2	no	—
99	#97	—	158	1	no	—
100	#97	C₂×D₇₉	1	2	yes	—
101	#41	—	316	4	no	—
102	C₂×Q₈	—	79	2	no	—
103	C₂×D₄	—	79	2	no	—
104	C₂×D₄	—	158	2	no	—
105	C₂×D₄	—	158	1	no	—
106	C₂×D₄	—	158	1	no	—
107	C₂²×C₄	—	158	2	no	—
108	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
109	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	158	4	no	—
113	C₂×C₈	—	158	2	no	—
114	C₄⋊C₄	—	158	1	no	—
115	C₄⋊C₄	—	158	1	no	—
116	C₂²⋊C₄	—	158	2	no	—
117	C₂²⋊C₄	—	158	2	no	—
118	C₄²	C₄×D₇₉	1	1	yes	characteristic
119	C₄²	D₃₁₆	1	1	yes	characteristic
120	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
121	C₄²	C₂²×D₇₉	1	1	yes	characteristic
122	C₂³	—	158	2	no	—
123	C₂³	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
124	Q₈	—	316	2	no	—
125	D₄	—	316	2	no	—
126	D₄	—	316	2	no	—
127	D₄	—	316	4	no	—
128	D₄	—	316	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	316	4	no	—
133	C₂×C₄	—	158	2	no	—
134	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
135	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
139	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
140	C₈	—	4	2	no	—
141	C₈	—	158	4	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	316	2	no	—
145	C₂²	—	316	2	no	—
146	C₂²	—	316	2	no	—
147	C₂²	#14579077	1	2	yes	—
148	C₂²	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	316	2	no	—
155	C₄	#14579087	1	2	yes	—
156	C₂	—	2	2	no	—
157	C₂	—	632	2	no	—
158	C₂	C₂×#14579077	1	1	yes	characteristic
159	C₂	C₂×#14579087	1	1	yes	characteristic
160	C₂	#19098581	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099113

C₂

yes

maximal, characteristic

C₂×#14579072

C₂

yes

maximal, characteristic, complemented

#19098302

C₂

yes

maximal, complemented

#432×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929624

C₄

yes

—

C₂²×C₇₉⋊C₈

—

#14579161

C₄

yes

characteristic

#14579158

C₄

yes

characteristic

#14579149

C₂²

yes

—

#14579135

—

#14579135

—

C₄×D₃₁₆

—

C₄×D₃₁₆

—

#14579072

—

#14579072

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

#97×C₇₉

C₂²

yes

—

#11929624

—

C₂×Dic₁₅₈

C₂×C₄

yes

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

C₂×C₄

yes

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×D₇₉

—

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

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#100

yes

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#104

yes

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

#1530

—

maximal, Sylow

C₇₉

#1530

yes

characteristic, complemented, Sylow

#532

—

#432

D₇₉

yes

characteristic, complemented

C₂×#97

—

#329

—

C₂×#41

—

C₂²×C₈

—

158

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

158

—

#121

—

158

—

C₄×D₄

—

158

—

C₄×D₄

—

158

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

158

—

100

#97

C₂×D₇₉

yes

—

101

#41

—

316

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

158

—

105

C₂×D₄

—

158

—

106

C₂×D₄

—

158

—

107

C₂²×C₄

—

158

—

108

C₂²×C₄

C₄×D₇₉

yes

characteristic

109

C₂²×C₄

D₃₁₆

yes

characteristic

110

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

158

—

113

C₂×C₈

—

158

—

114

C₄⋊C₄

—

158

—

115

C₄⋊C₄

—

158

—

116

C₂²⋊C₄

—

158

—

117

C₂²⋊C₄

—

158

—

118

C₄²

C₄×D₇₉

yes

characteristic

119

C₄²

D₃₁₆

yes

characteristic

120

C₄²

C₂²⋊D₇₉

yes

characteristic

121

C₄²

C₂²×D₇₉

yes

characteristic

122

C₂³

—

158

—

123

C₂³

(C₂×D₇₉)⋊C₄

yes

characteristic

124

Q₈

—

316

—

125

D₄

—

316

—

126

D₄

—

316

—

127

D₄

—

316

—

128

D₄

—

316

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

316

—

133

C₂×C₄

—

158

—

134

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

135

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

136

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

137

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

139

C₂×C₄

C₂×D₃₁₆

yes

characteristic

140

C₈

—

141

C₈

—

158

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

316

—

145

C₂²

—

316

—

146

C₂²

—

316

—

147

C₂²

#14579077

yes

—

148

C₂²

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

316

—

155

C₄

#14579087

yes

—

156

C₂

—

157

C₂

—

632

—

158

C₂

C₂×#14579077

yes

characteristic

159

C₂

C₂×#14579087

yes

characteristic

160

C₂

#19098581

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₄²	C₂²×D₇₉
Fitting	#432×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₄
prime radical	#19099113	C₂
cosocle	C₄²×C₇₉	C₂³
hypercenter	#432	D₇₉
hypocenter	C₇₉	#1530
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×(C₂×D₇₉)⋊C₄

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₄²

C₂²×D₇₉

Fitting

#432×C₇₉

C₂

socle

C₂²×C₇₉

C₂×C₂²⋊C₄

prime radical

#19099113

C₂

cosocle

C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
2. C₂×#14579072	C₂	semidirect
3. #19098302	C₂	semidirect
4. #432×C₇₉	C₂	semidirect
81. C₇₉	#1530	semidirect
83. #432	D₇₉	semidirect

H ≅ G/N

type

2. C₂×#14579072

C₂

semidirect

3. #19098302

C₂

semidirect

4. #432×C₇₉

C₂

semidirect

81. C₇₉

#1530

semidirect

83. #432

D₇₉

semidirect

Subgroups — #23762012

Conjugacy classes of subgroups

Complements