Group #23764723 — subgroups

Subgroups — #23764723

Order 10112.

288 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂²×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579086	C₂	1	2	yes	maximal
3	#19098469	C₂	1	2	yes	maximal
4	#19098326	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₅₈	C₂²	1	2	yes	—
7	C₂×Dic₃₁₆	—	2	2	no	—
8	C₂×Dic₃₁₆	—	2	4	no	—
9	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
10	#14579086	—	2	4	no	—
11	C₇₉⋊#95	—	2	2	no	—
12	C₇₉⋊#95	—	2	2	no	—
13	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₇₉	—	2	2	no	—
17	C₂×Dic₁₅₈	—	2	2	no	—
18	C₂×Dic₁₅₈	—	2	2	no	—
19	C₂×Dic₁₅₈	—	2	2	no	—
20	C₂×Dic₁₅₈	—	2	2	no	—
21	C₂×Dic₁₅₈	—	2	2	no	—
22	C₂×Dic₁₅₈	D₄	1	4	yes	—
23	C₂²×Dic₇₉	—	2	2	no	—
24	C₂²×Dic₇₉	—	2	2	no	—
25	Dic₃₁₆	—	2	8	no	—
26	C₄⋊Dic₇₉	—	2	2	no	—
27	C₄⋊Dic₇₉	—	2	2	no	—
28	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₇₉	—	2	2	no	—
32	C₂×C₈×C₇₉	—	2	1	no	—
33	C₂×C₈×C₇₉	—	2	2	no	—
34	C₂×C₈×C₇₉	D₄	1	2	yes	—
35	C₄²×C₇₉	—	2	1	no	—
36	C₄²×C₇₉	—	2	1	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	2	no	—
42	C₂×C₄×C₇₉	Q₁₆	1	2	yes	complemented
43	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
46	C₂×C₄×C₇₉	#41	1	2	yes	—
47	C₂×Dic₇₉	—	2	4	no	—
48	C₂×Dic₇₉	—	4	2	no	—
49	C₂×Dic₇₉	—	4	2	no	—
50	C₂×Dic₇₉	—	4	2	no	—
51	C₂×Dic₇₉	—	4	2	no	—
52	C₂×Dic₇₉	—	4	2	no	—
53	Dic₁₅₈	—	2	4	no	—
54	Dic₁₅₈	—	2	4	no	—
55	Dic₁₅₈	—	4	2	no	—
56	Dic₁₅₈	—	4	2	no	—
57	C₈×C₇₉	—	2	2	no	—
58	C₈×C₇₉	—	4	1	no	—
59	C₂²×C₇₉	#121	1	1	yes	characteristic
60	C₂²×C₇₉	#121	1	2	yes	—
61	C₂²×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₇₉	#135	1	1	yes	characteristic
63	C₂²×C₇₉	#137	1	2	yes	—
64	C₄×C₇₉	—	2	2	no	—
65	C₄×C₇₉	—	4	1	no	—
66	C₄×C₇₉	#120	1	1	yes	characteristic
67	C₄×C₇₉	#123	1	2	yes	—
68	C₄×C₇₉	#124	1	1	yes	characteristic
69	Dic₇₉	—	4	4	no	—
70	Dic₇₉	—	8	2	no	—
71	C₂×C₇₉	#394	1	1	yes	characteristic
72	C₂×C₇₉	#462	1	2	yes	—
73	C₂×C₇₉	#464	1	2	yes	—
74	C₂×C₇₉	#467	1	1	yes	characteristic
75	C₂×C₇₉	#470	1	1	yes	characteristic
76	#2056	—	79	1	no	maximal, Sylow
77	C₇₉	#2056	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₁₆	—	79	1	no	—
79	C₂×C₄⋊C₈	D₇₉	1	1	yes	characteristic
80	C₂×#103	—	79	2	no	—
81	#391	—	79	2	no	—
82	#340	—	79	1	no	—
83	C₂²×Q₈	—	79	2	no	—
84	C₂×Q₁₆	—	158	4	no	—
85	C₂×Q₁₆	—	158	2	no	—
86	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	79	2	no	—
89	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	#103	—	158	4	no	—
92	#95	—	158	2	no	—
93	#95	—	158	2	no	—
94	C₂×Q₈	—	79	4	no	—
95	C₂×Q₈	—	158	2	no	—
96	C₂×Q₈	—	158	2	no	—
97	C₂×Q₈	—	158	2	no	—
98	C₂×Q₈	—	158	2	no	—
99	C₂×Q₈	—	158	2	no	—
100	C₂²×C₄	—	158	2	no	—
101	C₂²×C₄	—	158	2	no	—
102	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
103	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
105	Q₁₆	—	158	8	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₂²⋊D₇₉	1	2	yes	—
110	C₄⋊C₄	—	158	2	no	—
111	C₄⋊C₄	—	158	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
115	Q₈	—	316	2	no	—
116	Q₈	—	316	2	no	—
117	Q₈	—	158	4	no	—
118	Q₈	—	158	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	316	2	no	—
124	C₂×C₄	—	316	2	no	—
125	C₂×C₄	—	316	2	no	—
126	C₂×C₄	—	316	2	no	—
127	C₂×C₄	—	316	2	no	—
128	C₂×C₄	—	158	4	no	—
129	C₂×C₄	Dic₃₁₆	1	2	yes	complemented
130	C₂×C₄	D₄×D₇₉	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
132	C₂×C₄	#11929624	1	1	yes	characteristic
133	C₂×C₄	#11929625	1	2	yes	—
134	C₈	—	2	2	no	—
135	C₈	—	4	1	no	—
136	C₂²	#14579135	1	1	yes	characteristic
137	C₂²	#14579136	1	2	yes	—
138	C₂²	C₂×Dic₃₁₆	1	1	yes	characteristic
139	C₂²	#14579183	1	1	yes	characteristic
140	C₂²	#14579194	1	2	yes	—
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	—	316	4	no	—
144	C₄	—	632	2	no	—
145	C₄	#14579129	1	1	yes	characteristic
146	C₄	#14579146	1	2	yes	—
147	C₄	#14579150	1	1	yes	characteristic
148	C₂	#19098480	1	1	yes	characteristic
149	C₂	#19098774	1	2	yes	—
150	C₂	#19098786	1	2	yes	—
151	C₂	#19098804	1	1	yes	characteristic
152	C₂	#19098821	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₁₆

C₂

yes

maximal, characteristic

C₂×#14579086

C₂

yes

maximal

#19098469

C₂

yes

maximal

#19098326

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

#14579086

—

C₇₉⋊#95

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

D₄

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

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

—

C₂×C₈×C₇₉

D₄

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

Q₁₆

yes

complemented

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₂²×C₇₉

#121

yes

characteristic

C₂²×C₇₉

#121

yes

—

C₂²×C₇₉

C₂×Q₁₆

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#120

yes

characteristic

C₄×C₇₉

#123

yes

—

C₄×C₇₉

#124

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

#394

yes

characteristic

C₂×C₇₉

#462

yes

—

C₂×C₇₉

#464

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊C₈

D₇₉

yes

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

158

—

C₂×Q₁₆

—

158

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

#103

—

158

—

#95

—

158

—

#95

—

158

—

C₂×Q₈

—

C₂×Q₈

—

158

—

C₂×Q₈

—

158

—

C₂×Q₈

—

158

—

C₂×Q₈

—

158

—

C₂×Q₈

—

158

—

100

C₂²×C₄

—

158

—

101

C₂²×C₄

—

158

—

102

C₂²×C₄

D₃₁₆

yes

characteristic

103

C₂²×C₄

D₃₁₆

yes

characteristic

104

C₂²×C₄

C₂²×D₇₉

yes

characteristic

105

Q₁₆

—

158

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂²⋊D₇₉

yes

—

110

C₄⋊C₄

—

158

—

111

C₄⋊C₄

—

158

—

112

C₄²

—

113

C₄²

—

114

C₂³

C₂×D₃₁₆

yes

characteristic

115

Q₈

—

316

—

116

Q₈

—

316

—

117

Q₈

—

158

—

118

Q₈

—

158

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

316

—

124

C₂×C₄

—

316

—

125

C₂×C₄

—

316

—

126

C₂×C₄

—

316

—

127

C₂×C₄

—

316

—

128

C₂×C₄

—

158

—

129

C₂×C₄

Dic₃₁₆

yes

complemented

130

C₂×C₄

D₄×D₇₉

yes

—

131

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

132

C₂×C₄

#11929624

yes

characteristic

133

C₂×C₄

#11929625

yes

—

134

C₈

—

135

C₈

—

136

C₂²

#14579135

yes

characteristic

137

C₂²

#14579136

yes

—

138

C₂²

C₂×Dic₃₁₆

yes

characteristic

139

C₂²

#14579183

yes

characteristic

140

C₂²

#14579194

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

316

—

144

C₄

—

632

—

145

C₄

#14579129

yes

characteristic

146

C₄

#14579146

yes

—

147

C₄

#14579150

yes

characteristic

148

C₂

#19098480

yes

characteristic

149

C₂

#19098774

yes

—

150

C₂

#19098786

yes

—

151

C₂

#19098804

yes

characteristic

152

C₂

#19098821

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₁₆
commutator, cosocle	C₂²×C₄×C₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂×C₄⋊C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₇₉
hypocenter	C₇₉	#2056
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₃₁₆

commutator, cosocle

C₂²×C₄×C₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂×C₄⋊C₈×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₇₉

hypocenter

C₇₉

#2056

LCS

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
42. C₂×C₄×C₇₉	Q₁₆	semidirect
77. C₇₉	#2056	semidirect
129. C₂×C₄	Dic₃₁₆	semidirect

H ≅ G/N

type

42. C₂×C₄×C₇₉

Q₁₆

semidirect

77. C₇₉

#2056

semidirect

129. C₂×C₄

Dic₃₁₆

semidirect

Subgroups — #23764723

Conjugacy classes of subgroups

Complements