Group #23764343 — subgroups

Subgroups — #23764343

Order 10112.

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

normal subgroups only

This group has 260 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	#19099120	C₂	1	1	yes	maximal, characteristic
2	C₄×#11929608	C₂	1	1	yes	maximal, characteristic
3	#19098323	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#11929608	C₂²	1	2	yes	—
6	#14579161	C₄	1	1	yes	characteristic, complemented
7	#14579161	C₄	1	1	yes	characteristic, complemented
8	#14579153	C₄	1	2	yes	complemented
9	#14579140	—	2	1	no	—
10	#14579140	—	2	1	no	—
11	C₄×Dic₁₅₈	—	2	1	no	—
12	C₄×Dic₁₅₈	—	2	1	no	—
13	#14579120	C₂²	1	2	yes	—
14	#14579072	—	2	1	no	—
15	C₄×C₇₉⋊C₈	—	2	1	no	—
16	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
17	C₂×C₄²×C₇₉	C₂²	1	2	yes	—
18	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
19	#95×C₇₉	—	2	1	no	—
20	C₂×Dic₁₅₈	—	2	1	no	—
21	C₂×Dic₁₅₈	—	2	1	no	—
22	#11929608	—	2	2	no	—
23	#11929608	Q₈	1	4	yes	complemented
24	C₂×C₇₉⋊C₈	—	2	2	no	—
25	C₄⋊Dic₇₉	—	2	1	no	—
26	C₄⋊Dic₇₉	—	2	1	no	—
27	C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
28	C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
29	Dic₇₉⋊C₄	—	2	2	no	—
30	Dic₇₉⋊C₄	—	2	2	no	—
31	C₂²⋊Dic₇₉	—	2	2	no	—
32	C₄×Dic₇₉	—	2	2	no	—
33	C₂²×C₄×C₇₉	—	2	2	no	—
34	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₇₉	—	2	4	no	—
38	C₄²×C₇₉	—	2	4	no	—
39	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₇₉	D₄	1	1	yes	characteristic
42	C₄²×C₇₉	D₄	1	1	yes	characteristic
43	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇₉	—	2	2	no	—
45	C₂×C₄×C₇₉	—	2	4	no	—
46	C₂×C₄×C₇₉	—	4	1	no	—
47	C₂×C₄×C₇₉	—	4	2	no	—
48	C₂×C₄×C₇₉	—	4	2	no	—
49	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₇₉	#41	1	2	yes	—
57	C₂×Dic₇₉	—	2	2	no	—
58	C₂×Dic₇₉	—	2	2	no	—
59	Dic₁₅₈	—	4	2	no	—
60	Dic₁₅₈	—	4	2	no	—
61	C₇₉⋊C₈	—	2	4	no	—
62	C₂²×C₇₉	—	2	1	no	—
63	C₂²×C₇₉	—	2	1	no	—
64	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₇₉	#124	1	1	yes	characteristic
66	C₂²×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₄×C₇₉	—	2	1	no	—
68	C₄×C₇₉	—	2	1	no	—
69	C₄×C₇₉	—	2	1	no	—
70	C₄×C₇₉	—	4	4	no	—
71	C₄×C₇₉	#104	1	2	yes	—
72	C₄×C₇₉	C₄×Q₈	1	2	yes	—
73	C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
74	Dic₇₉	—	4	2	no	—
75	Dic₇₉	—	4	2	no	—
76	C₂×C₇₉	—	2	1	no	—
77	C₂×C₇₉	—	2	1	no	—
78	C₂×C₇₉	#391	1	1	yes	characteristic
79	C₂×C₇₉	C₂×#104	1	1	yes	characteristic
80	C₂×C₇₉	#421	1	1	yes	characteristic
81	#1986	—	79	1	no	maximal, Sylow
82	C₇₉	#1986	1	1	yes	characteristic, complemented, Sylow
83	#533	—	79	1	no	—
84	C₄×M₁₆	—	79	1	no	—
85	C₄×C₄⋊C₄	D₇₉	1	1	yes	characteristic
86	#339	—	79	4	no	—
87	C₂×M₁₆	—	79	2	no	—
88	C₄⋊Q₈	—	79	1	no	—
89	C₄⋊Q₈	—	79	1	no	—
90	#125	—	79	2	no	—
91	C₂²⋊Q₈	—	158	1	no	—
92	C₂²⋊Q₈	—	158	1	no	—
93	C₄×Q₈	—	158	1	no	—
94	C₄×Q₈	—	158	1	no	—
95	#117	—	79	2	no	—
96	C₂×C₄⋊C₄	—	2	1	no	—
97	C₂×C₄²	C₂×D₇₉	1	2	yes	—
98	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
99	#97	—	158	1	no	—
100	C₄×C₈	—	158	1	no	—
101	#95	—	2	1	no	—
102	C₂×Q₈	—	158	1	no	—
103	C₂×Q₈	—	158	1	no	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
106	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
107	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
108	M₁₆	—	79	4	no	—
109	M₁₆	—	158	2	no	—
110	C₂×C₈	—	158	2	no	—
111	C₄⋊C₄	—	2	4	no	—
112	C₄⋊C₄	—	79	2	no	—
113	C₄⋊C₄	—	79	2	no	—
114	C₄⋊C₄	—	158	2	no	—
115	C₄⋊C₄	—	158	2	no	—
116	C₄⋊C₄	—	158	1	no	—
117	C₄⋊C₄	—	158	1	no	—
118	C₂²⋊C₄	—	158	2	no	—
119	C₄²	—	2	4	no	—
120	C₄²	—	158	2	no	—
121	C₄²	C₄×D₇₉	1	1	yes	characteristic
122	C₄²	C₄×D₇₉	1	1	yes	characteristic
123	C₄²	D₃₁₆	1	1	yes	characteristic
124	C₄²	D₃₁₆	1	1	yes	characteristic
125	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
126	Q₈	—	316	2	no	—
127	Q₈	—	316	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	158	2	no	—
134	C₂×C₄	—	158	2	no	—
135	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
138	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
140	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
141	C₂×C₄	Q₈×D₇₉	1	2	yes	—
142	C₂×C₄	#11929623	1	2	yes	—
143	C₈	—	158	4	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
147	C₂²	#14579148	1	1	yes	characteristic
148	C₂²	#14579159	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	4	4	no	—
153	C₄	—	316	2	no	—
154	C₄	—	316	2	no	—
155	C₄	#14579089	1	2	yes	—
156	C₄	#14579126	1	2	yes	—
157	C₄	#14579141	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	#19098468	1	1	yes	characteristic
161	C₂	C₂×#14579089	1	1	yes	characteristic
162	C₂	#19098581	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099120

C₂

yes

maximal, characteristic

C₄×#11929608

C₂

yes

maximal, characteristic

#19098323

C₂

yes

maximal

C₄×C₄⋊C₄×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929608

C₂²

yes

—

#14579161

C₄

yes

characteristic, complemented

#14579161

C₄

yes

characteristic, complemented

#14579153

C₄

yes

complemented

#14579140

—

#14579140

—

C₄×Dic₁₅₈

—

C₄×Dic₁₅₈

—

#14579120

C₂²

yes

—

#14579072

—

C₄×C₇₉⋊C₈

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

#95×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

#11929608

—

#11929608

Q₈

yes

complemented

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

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

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₉

#124

yes

characteristic

C₂²×C₇₉

C₄⋊Q₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#104

yes

—

C₄×C₇₉

C₄×Q₈

yes

—

C₄×C₇₉

C₂²⋊Q₈

yes

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#533

—

C₄×M₁₆

—

C₄×C₄⋊C₄

D₇₉

yes

characteristic

#339

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

158

—

C₂²⋊Q₈

—

158

—

C₄×Q₈

—

158

—

C₄×Q₈

—

158

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇₉

yes

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

#97

—

158

—

100

C₄×C₈

—

158

—

101

#95

—

102

C₂×Q₈

—

158

—

103

C₂×Q₈

—

158

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

106

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

107

C₂²×C₄

C₂²×D₇₉

yes

characteristic

108

M₁₆

—

109

M₁₆

—

158

—

110

C₂×C₈

—

158

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

158

—

115

C₄⋊C₄

—

158

—

116

C₄⋊C₄

—

158

—

117

C₄⋊C₄

—

158

—

118

C₂²⋊C₄

—

158

—

119

C₄²

—

120

C₄²

—

158

—

121

C₄²

C₄×D₇₉

yes

characteristic

122

C₄²

C₄×D₇₉

yes

characteristic

123

C₄²

D₃₁₆

yes

characteristic

124

C₄²

D₃₁₆

yes

characteristic

125

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

126

Q₈

—

316

—

127

Q₈

—

316

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

158

—

134

C₂×C₄

—

158

—

135

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

136

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

137

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

138

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

139

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

140

C₂×C₄

C₂×D₃₁₆

yes

characteristic

141

C₂×C₄

Q₈×D₇₉

yes

—

142

C₂×C₄

#11929623

yes

—

143

C₈

—

158

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

147

C₂²

#14579148

yes

characteristic

148

C₂²

#14579159

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

316

—

154

C₄

—

316

—

155

C₄

#14579089

yes

—

156

C₄

#14579126

yes

—

157

C₄

#14579141

yes

—

158

C₂

—

159

C₂

—

160

C₂

#19098468

yes

characteristic

161

C₂

C₂×#14579089

yes

characteristic

162

C₂

#19098581

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₄×C₄⋊C₄×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₇₉
hypocenter	C₇₉	#1986
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂×C₄

C₂×D₃₁₆

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₄×C₄⋊C₄×C₇₉

C₂

socle

C₂²×C₇₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₇₉

C₂×D₄

cosocle

C₂²×C₄×C₇₉

C₂³

hypercenter

C₄×C₄⋊C₄

D₇₉

hypocenter

C₇₉

#1986

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
6. #14579161	C₄	semidirect
7. #14579161	C₄	semidirect
8. #14579153	C₄	semidirect
23. #11929608	Q₈	semidirect
82. C₇₉	#1986	semidirect

H ≅ G/N

type

6. #14579161

C₄

semidirect

7. #14579161

C₄

semidirect

8. #14579153

C₄

semidirect

23. #11929608

Q₈

semidirect

82. C₇₉

#1986

semidirect

Subgroups — #23764343

Conjugacy classes of subgroups

Complements