Group #23768385 — subgroups

Subgroups — #23768385

Order 10112.

540 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 197 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×#14579159	C₂	1	2	yes	maximal
3	#19098493	C₂	1	4	yes	maximal
4	#19098468	C₂	1	2	yes	maximal
5	#19098461	C₂	1	4	yes	maximal
6	C₄×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₅₈	C₂²	1	2	yes	—
9	#14579161	—	2	4	no	—
10	#14579159	—	2	8	no	—
11	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
14	C₂×Dic₇₉⋊C₄	C₂²	1	8	yes	—
15	C₂×C₄×Dic₇₉	C₂²	1	2	yes	—
16	C₇₉⋊#95	C₂²	1	2	yes	—
17	C₇₉⋊#95	C₂²	1	8	yes	—
18	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₇₉	—	2	4	no	—
20	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₇₉	C₂²	1	2	yes	—
23	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₅₈	—	2	4	no	—
25	C₂×Dic₁₅₈	—	2	4	no	—
26	C₂×Dic₁₅₈	—	2	4	no	—
27	C₂²×Dic₇₉	C₂³	1	4	yes	—
28	C₂²×Dic₇₉	C₂³	1	4	yes	—
29	C₄⋊Dic₇₉	—	2	4	no	—
30	C₄⋊Dic₇₉	—	2	4	no	—
31	C₄⋊Dic₇₉	Q₈	1	8	yes	complemented
32	Dic₇₉⋊C₄	—	2	16	no	—
33	C₄×Dic₇₉	—	2	4	no	—
34	C₂×Q₈×C₇₉	—	2	2	no	—
35	C₂×Q₈×C₇₉	—	2	2	no	—
36	C₂×Q₈×C₇₉	D₄	1	4	yes	—
37	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₇₉	—	2	4	no	—
43	C₄⋊C₄×C₇₉	—	2	2	no	—
44	C₄²×C₇₉	—	2	4	no	—
45	C₄²×C₇₉	D₄	1	4	yes	—
46	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
47	Q₈×C₇₉	—	2	8	no	—
48	C₂×C₄×C₇₉	—	2	2	no	—
49	C₂×C₄×C₇₉	—	2	2	no	—
50	C₂×C₄×C₇₉	—	2	2	no	—
51	C₂×C₄×C₇₉	—	2	2	no	—
52	C₂×C₄×C₇₉	—	2	4	no	—
53	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₇₉	C₂×D₄	1	4	yes	—
57	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₇₉	C₂×Q₈	1	4	yes	—
59	C₂×C₄×C₇₉	#41	1	2	yes	—
60	C₂×Dic₇₉	—	2	4	no	—
61	C₂×Dic₇₉	—	2	4	no	—
62	C₂×Dic₇₉	—	2	4	no	—
63	C₂×Dic₇₉	—	2	4	no	—
64	C₂×Dic₇₉	—	2	4	no	—
65	C₂×Dic₇₉	C₂×Q₈	1	8	yes	—
66	Dic₁₅₈	—	4	8	no	—
67	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
68	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
69	C₂²×C₇₉	C₂²×Q₈	1	2	yes	—
70	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
71	C₂²×C₇₉	2₋¹⁺⁴	1	2	yes	—
72	C₄×C₇₉	—	2	4	no	—
73	C₄×C₇₉	—	2	2	no	—
74	C₄×C₇₉	—	4	2	no	—
75	C₄×C₇₉	#121	1	4	yes	—
76	C₄×C₇₉	C₄⋊Q₈	1	4	yes	complemented
77	Dic₇₉	—	2	8	no	—
78	Dic₇₉	—	4	4	no	—
79	C₂×C₇₉	C₂×#121	1	1	yes	characteristic
80	C₂×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₇₉	#536	1	1	yes	characteristic
82	C₂×C₇₉	D₄×Q₈	1	2	yes	—
83	C₂×C₇₉	#557	1	2	yes	—
84	#2546	—	79	1	no	maximal, Sylow
85	C₇₉	#2546	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₄⋊Q₈	—	79	2	no	—
87	C₂×C₄⋊Q₈	—	79	1	no	—
88	C₂×C₄×Q₈	D₇₉	1	1	yes	characteristic
89	#398	—	79	4	no	—
90	#391	—	79	2	no	—
91	#389	—	79	4	no	—
92	C₄×C₄⋊C₄	—	79	1	no	—
93	C₂²×Q₈	—	79	2	no	—
94	C₂²×Q₈	C₂×D₇₉	1	1	yes	characteristic
95	C₄⋊Q₈	—	158	8	no	—
96	C₄⋊Q₈	—	158	4	no	—
97	C₄×Q₈	—	2	4	no	—
98	C₂×C₄⋊C₄	—	79	2	no	—
99	C₂×C₄⋊C₄	—	79	2	no	—
100	C₂×C₄⋊C₄	—	79	8	no	—
101	C₂×C₄⋊C₄	—	79	2	no	—
102	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
103	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
104	C₂×C₄²	—	79	2	no	—
105	C₂×C₄²	C₂×D₇₉	1	2	yes	—
106	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
107	#95	—	79	2	no	—
108	#95	—	79	8	no	—
109	C₂×Q₈	—	2	2	no	—
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	158	4	no	—
112	C₂×Q₈	—	158	4	no	—
113	C₂×Q₈	—	158	4	no	—
114	C₂×Q₈	D₃₁₆	1	4	yes	—
115	C₂²×C₄	—	79	4	no	—
116	C₂²×C₄	—	79	4	no	—
117	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
118	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
119	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
120	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
121	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	79	8	no	—
125	C₄⋊C₄	—	158	4	no	—
126	C₄⋊C₄	—	158	4	no	—
127	C₄⋊C₄	—	158	16	no	—
128	C₄²	—	2	4	no	—
129	C₄²	—	158	4	no	—
130	C₄²	C₂²⋊D₇₉	1	4	yes	—
131	C₂³	C₂³×D₇₉	1	1	yes	characteristic
132	Q₈	—	2	8	no	—
133	Q₈	—	316	8	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	79	8	no	—
140	C₂×C₄	—	158	4	no	—
141	C₂×C₄	—	158	4	no	—
142	C₂×C₄	—	158	4	no	—
143	C₂×C₄	—	158	4	no	—
144	C₂×C₄	—	158	4	no	—
145	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
148	C₂×C₄	C₂×D₃₁₆	1	4	yes	—
149	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
150	C₂×C₄	Q₈×D₇₉	1	4	yes	—
151	C₂×C₄	#11929624	1	2	yes	—
152	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
153	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
154	C₂²	C₂×Q₈×D₇₉	1	2	yes	—
155	C₂²	C₂×#11929624	1	1	yes	characteristic
156	C₂²	#14579208	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	2	no	—
160	C₄	—	316	4	no	—
161	C₄	—	158	8	no	—
162	C₄	#14579135	1	4	yes	—
163	C₄	#14579159	1	4	yes	complemented
164	C₂	C₂×#14579135	1	1	yes	characteristic
165	C₂	C₂×#14579159	1	1	yes	characteristic
166	C₂	#19099134	1	1	yes	characteristic
167	C₂	Q₈×D₃₁₆	1	2	yes	—
168	C₂	#19099291	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579161

C₂

yes

maximal, characteristic

C₂×#14579159

C₂

yes

maximal

#19098493

C₂

yes

maximal

#19098468

C₂

yes

maximal

#19098461

C₂

yes

maximal

C₄×C₄⋊Dic₇₉

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

—

#14579161

—

#14579159

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×Dic₇₉⋊C₄

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₂²×Q₈×C₇₉

C₂²

yes

characteristic

C₄×Q₈×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

C₂³

yes

—

C₂²×Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

Q₈

yes

complemented

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

D₄

yes

—

C₂³×C₇₉

C₂⁴

yes

characteristic

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×Q₈

yes

—

Dic₁₅₈

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂²×C₇₉

C₂²×Q₈

yes

—

C₂²×C₇₉

C₂×#41

yes

characteristic

C₂²×C₇₉

2₋¹⁺⁴

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#121

yes

—

C₄×C₇₉

C₄⋊Q₈

yes

complemented

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

C₂×#121

yes

characteristic

C₂×C₇₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₇₉

#536

yes

characteristic

C₂×C₇₉

D₄×Q₈

yes

—

C₂×C₇₉

#557

yes

—

#2546

—

maximal, Sylow

C₇₉

#2546

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

D₇₉

yes

—

—

—

C₄×C₄⋊C₄

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₇₉

yes

characteristic

C₄⋊Q₈

—

158

—

C₄⋊Q₈

—

158

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

103

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

104

C₂×C₄²

—

105

C₂×C₄²

C₂×D₇₉

yes

—

106

C₂×C₄²

C₂×D₇₉

yes

characteristic

107

#95

—

108

#95

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

158

—

112

C₂×Q₈

—

158

—

113

C₂×Q₈

—

158

—

114

C₂×Q₈

D₃₁₆

yes

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂²×D₇₉

yes

—

118

C₂²×C₄

C₂²×D₇₉

yes

characteristic

119

C₂²×C₄

C₂²×D₇₉

yes

—

120

C₂²×C₄

C₂²×D₇₉

yes

characteristic

121

C₂²×C₄

C₂²×D₇₉

yes

characteristic

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

158

—

126

C₄⋊C₄

—

158

—

127

C₄⋊C₄

—

158

—

128

C₄²

—

129

C₄²

—

158

—

130

C₄²

C₂²⋊D₇₉

yes

—

131

C₂³

C₂³×D₇₉

yes

characteristic

132

Q₈

—

133

Q₈

—

316

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

158

—

141

C₂×C₄

—

158

—

142

C₂×C₄

—

158

—

143

C₂×C₄

—

158

—

144

C₂×C₄

—

158

—

145

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

146

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

147

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

148

C₂×C₄

C₂×D₃₁₆

yes

—

149

C₂×C₄

C₂×D₃₁₆

yes

—

150

C₂×C₄

Q₈×D₇₉

yes

—

151

C₂×C₄

#11929624

yes

—

152

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

153

C₂²

C₂²×D₃₁₆

yes

characteristic

154

C₂²

C₂×Q₈×D₇₉

yes

—

155

C₂²

C₂×#11929624

yes

characteristic

156

C₂²

#14579208

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

316

—

161

C₄

—

158

—

162

C₄

#14579135

yes

—

163

C₄

#14579159

yes

complemented

164

C₂

C₂×#14579135

yes

characteristic

165

C₂

C₂×#14579159

yes

characteristic

166

C₂

#19099134

yes

characteristic

167

C₂

Q₈×D₃₁₆

yes

—

168

C₂

#19099291

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator, socle, prime radical, cosocle	C₂³×C₇₉	C₂⁴
Fitting	C₂×C₄×Q₈×C₇₉	C₂
hypercenter	C₂×C₄×Q₈	D₇₉
hypocenter	C₇₉	#2546
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₉

commutator, socle, prime radical, cosocle

C₂³×C₇₉

C₂⁴

Fitting

C₂×C₄×Q₈×C₇₉

C₂

hypercenter

C₂×C₄×Q₈

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
31. C₄⋊Dic₇₉	Q₈	semidirect
76. C₄×C₇₉	C₄⋊Q₈	semidirect
85. C₇₉	#2546	semidirect
163. C₄	#14579159	semidirect

H ≅ G/N

type

31. C₄⋊Dic₇₉

Q₈

semidirect

76. C₄×C₇₉

C₄⋊Q₈

semidirect

85. C₇₉

#2546

semidirect

163. C₄

#14579159

semidirect

Subgroups — #23768385

Conjugacy classes of subgroups

Complements