Group #23764734 — subgroups

Subgroups — #23764734

Order 10112.

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

normal subgroups only

This group has 282 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	C₂×#14579161	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579093	C₂	1	2	yes	maximal
3	#19098461	C₂	1	1	yes	maximal, characteristic
4	#19098326	C₂	1	2	yes	maximal
5	#336×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₅₈	—	2	1	no	—
7	#14579161	—	2	4	no	—
8	C₂×C₄⋊Dic₇₉	—	2	1	no	—
9	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₇₉⋊C₄	—	2	1	no	—
13	#14579093	—	2	4	no	—
14	C₇₉⋊#95	—	2	1	no	—
15	C₂²×C₈×C₇₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₅₈	—	2	4	no	—
18	C₂×Dic₁₅₈	—	4	2	no	—
19	C₂×Dic₁₅₈	—	4	2	no	—
20	C₂²×Dic₇₉	—	2	2	no	—
21	C₂²×Dic₇₉	—	2	1	no	—
22	C₂²×Dic₇₉	—	2	1	no	—
23	C₄⋊Dic₇₉	—	2	4	no	—
24	C₄⋊Dic₇₉	—	2	2	no	—
25	C₄⋊Dic₇₉	—	2	2	no	—
26	C₄⋊Dic₇₉	—	2	1	no	—
27	C₄⋊Dic₇₉	D₄	1	4	yes	—
28	C₄⋊Dic₇₉	Q₈	1	2	yes	complemented
29	Dic₇₉⋊C₄	—	4	2	no	—
30	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₇₉	—	2	2	no	—
34	C₂×C₈×C₇₉	—	2	2	no	—
35	C₂×C₈×C₇₉	Q₈	1	4	yes	complemented
36	C₄²×C₇₉	—	2	2	no	—
37	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₉	—	2	1	no	—
39	C₂×C₄×C₇₉	—	2	1	no	—
40	C₂×C₄×C₇₉	—	2	1	no	—
41	C₂×C₄×C₇₉	—	2	1	no	—
42	C₂×C₄×C₇₉	SD₁₆	1	2	yes	—
43	C₂×C₄×C₇₉	SD₁₆	1	2	yes	—
44	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
48	C₂×Dic₇₉	—	2	4	no	—
49	C₂×Dic₇₉	—	2	2	no	—
50	C₂×Dic₇₉	—	4	1	no	—
51	C₂×Dic₇₉	—	4	2	no	—
52	C₂×Dic₇₉	—	4	1	no	—
53	C₂×Dic₇₉	—	4	2	no	—
54	C₂×Dic₇₉	—	4	1	no	—
55	C₂×Dic₇₉	—	4	1	no	—
56	C₂×Dic₇₉	—	4	1	no	—
57	Dic₁₅₈	—	4	8	no	—
58	C₈×C₇₉	—	2	4	no	—
59	C₂²×C₇₉	#120	1	1	yes	characteristic
60	C₂²×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₂²×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₇₉	#137	1	1	yes	characteristic
66	C₄×C₇₉	—	2	2	no	—
67	C₄×C₇₉	—	2	2	no	—
68	C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
69	C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
70	Dic₇₉	—	4	4	no	—
71	Dic₇₉	—	4	2	no	—
72	Dic₇₉	—	8	1	no	—
73	C₂×C₇₉	#395	1	1	yes	characteristic
74	C₂×C₇₉	#448	1	1	yes	characteristic
75	C₂×C₇₉	#450	1	1	yes	characteristic
76	C₂×C₇₉	#475	1	1	yes	characteristic
77	C₂×C₇₉	#476	1	1	yes	characteristic
78	C₂×C₇₉	#498	1	1	yes	characteristic
79	C₂×C₇₉	#501	1	1	yes	characteristic
80	#2058	—	79	1	no	maximal, Sylow
81	C₇₉	#2058	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄⋊Q₈	—	79	1	no	—
83	C₂×#106	—	79	2	no	—
84	#389	—	79	1	no	—
85	#340	—	79	2	no	—
86	#336	D₇₉	1	1	yes	characteristic
87	C₂²×Q₈	—	158	1	no	—
88	C₂²×C₈	C₂×D₇₉	1	2	yes	—
89	C₄⋊Q₈	—	158	4	no	—
90	C₂×C₄⋊C₄	—	79	2	no	—
91	C₂×C₄⋊C₄	—	79	1	no	—
92	C₂×C₄⋊C₄	—	79	1	no	—
93	C₂×C₄⋊C₄	—	158	1	no	—
94	C₂×C₄⋊C₄	—	158	1	no	—
95	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
96	#106	—	158	4	no	—
97	#95	—	158	1	no	—
98	C₂×Q₈	—	316	2	no	—
99	C₂×Q₈	—	316	2	no	—
100	C₂×Q₈	—	158	4	no	—
101	C₂²×C₄	—	158	1	no	—
102	C₂²×C₄	—	158	1	no	—
103	C₂²×C₄	—	158	2	no	—
104	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
105	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	Dic₁₅₈	1	4	yes	complemented
110	C₄⋊C₄	—	316	2	no	—
111	C₄⋊C₄	—	79	4	no	—
112	C₄⋊C₄	—	79	2	no	—
113	C₄⋊C₄	—	158	4	no	—
114	C₄⋊C₄	—	158	2	no	—
115	C₄⋊C₄	—	158	2	no	—
116	C₄⋊C₄	—	158	1	no	—
117	C₄²	—	2	2	no	—
118	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
119	Q₈	—	316	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	316	1	no	—
125	C₂×C₄	—	316	2	no	—
126	C₂×C₄	—	316	2	no	—
127	C₂×C₄	—	316	1	no	—
128	C₂×C₄	—	316	1	no	—
129	C₂×C₄	—	316	1	no	—
130	C₂×C₄	—	316	1	no	—
131	C₂×C₄	—	158	4	no	—
132	C₂×C₄	—	158	2	no	—
133	C₂×C₄	#11929613	1	2	yes	—
134	C₂×C₄	#11929613	1	2	yes	—
135	C₂×C₄	D₄×D₇₉	1	2	yes	—
136	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
137	C₂×C₄	C₂×Dic₁₅₈	1	2	yes	—
138	C₂×C₄	#11929623	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₂²	#14579131	1	1	yes	characteristic
141	C₂²	#14579141	1	1	yes	characteristic
142	C₂²	#14579161	1	1	yes	characteristic
143	C₂²	C₂×#11929613	1	1	yes	characteristic
144	C₂²	C₂×#11929613	1	1	yes	characteristic
145	C₂²	#14579189	1	1	yes	characteristic
146	C₂²	#14579194	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	316	4	no	—
150	C₄	—	316	2	no	—
151	C₄	—	632	1	no	—
152	C₄	#14579138	1	2	yes	—
153	C₄	#14579160	1	2	yes	—
154	C₂	#19098484	1	1	yes	characteristic
155	C₂	#19098704	1	1	yes	characteristic
156	C₂	#19098716	1	1	yes	characteristic
157	C₂	#19098849	1	1	yes	characteristic
158	C₂	#19098854	1	1	yes	characteristic
159	C₂	#19098961	1	1	yes	characteristic
160	C₂	#19098972	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579161

C₂

yes

maximal, characteristic

C₂×#14579093

C₂

yes

maximal

#19098461

C₂

yes

maximal, characteristic

#19098326

C₂

yes

maximal

#336×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

—

#14579161

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×Dic₇₉⋊C₄

—

#14579093

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

—

C₄⋊Dic₇₉

Q₈

yes

complemented

Dic₇₉⋊C₄

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

Q₈

yes

complemented

C₄²×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₂²×C₇₉

#120

yes

characteristic

C₂²×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₉

C₄⋊Q₈

yes

characteristic

C₂²×C₇₉

C₂×SD₁₆

yes

characteristic

C₂²×C₇₉

C₂×SD₁₆

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊Q₈

yes

—

C₄×C₇₉

C₄⋊Q₈

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

D₇₉

yes

characteristic

C₂²×Q₈

—

158

—

C₂²×C₈

C₂×D₇₉

yes

—

C₄⋊Q₈

—

158

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

#106

—

158

—

#95

—

158

—

C₂×Q₈

—

316

—

C₂×Q₈

—

316

—

100

C₂×Q₈

—

158

—

101

C₂²×C₄

—

158

—

102

C₂²×C₄

—

158

—

103

C₂²×C₄

—

158

—

104

C₂²×C₄

D₃₁₆

yes

characteristic

105

C₂²×C₄

D₃₁₆

yes

characteristic

106

C₂²×C₄

C₂²×D₇₉

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

Dic₁₅₈

yes

complemented

110

C₄⋊C₄

—

316

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

158

—

114

C₄⋊C₄

—

158

—

115

C₄⋊C₄

—

158

—

116

C₄⋊C₄

—

158

—

117

C₄²

—

118

C₂³

C₂×D₃₁₆

yes

characteristic

119

Q₈

—

316

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

316

—

125

C₂×C₄

—

316

—

126

C₂×C₄

—

316

—

127

C₂×C₄

—

316

—

128

C₂×C₄

—

316

—

129

C₂×C₄

—

316

—

130

C₂×C₄

—

316

—

131

C₂×C₄

—

158

—

132

C₂×C₄

—

158

—

133

C₂×C₄

#11929613

yes

—

134

C₂×C₄

#11929613

yes

—

135

C₂×C₄

D₄×D₇₉

yes

—

136

C₂×C₄

Q₈×D₇₉

yes

characteristic

137

C₂×C₄

C₂×Dic₁₅₈

yes

—

138

C₂×C₄

#11929623

yes

characteristic

139

C₈

—

140

C₂²

#14579131

yes

characteristic

141

C₂²

#14579141

yes

characteristic

142

C₂²

#14579161

yes

characteristic

143

C₂²

C₂×#11929613

yes

characteristic

144

C₂²

C₂×#11929613

yes

characteristic

145

C₂²

#14579189

yes

characteristic

146

C₂²

#14579194

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

316

—

150

C₄

—

316

—

151

C₄

—

632

—

152

C₄

#14579138

yes

—

153

C₄

#14579160

yes

—

154

C₂

#19098484

yes

characteristic

155

C₂

#19098704

yes

characteristic

156

C₂

#19098716

yes

characteristic

157

C₂

#19098849

yes

characteristic

158

C₂

#19098854

yes

characteristic

159

C₂

#19098961

yes

characteristic

160

C₂

#19098972

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₁₆
commutator, cosocle	C₂²×C₄×C₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#336×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	#336	D₇₉
hypocenter	C₇₉	#2058
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₃₁₆

commutator, cosocle

C₂²×C₄×C₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#336×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
28. C₄⋊Dic₇₉	Q₈	semidirect
35. C₂×C₈×C₇₉	Q₈	semidirect
81. C₇₉	#2058	semidirect
109. C₂×C₈	Dic₁₅₈	semidirect

H ≅ G/N

type

28. C₄⋊Dic₇₉

Q₈

semidirect

35. C₂×C₈×C₇₉

Q₈

semidirect

81. C₇₉

#2058

semidirect

109. C₂×C₈

Dic₁₅₈

semidirect

Subgroups — #23764734

Conjugacy classes of subgroups

Complements