Group #23775830 — subgroups

Subgroups — #23775830

Order 10112.

340 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 117 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	#19099287	C₂	1	2	yes	maximal
2	#19098906	C₂	1	4	yes	maximal, complemented
3	#19098845	C₂	1	2	yes	maximal, complemented
4	#19098805	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×D₃₁₆	C₂	1	1	yes	maximal, characteristic
7	#19098600	C₂	1	2	yes	maximal, complemented
8	#500×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₃₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₇₉	C₂²	1	2	yes	—
11	#14579156	—	2	4	no	—
12	#14579150	—	2	2	no	—
13	#14579141	C₂²	1	4	yes	—
14	Q₈×Dic₇₉	—	2	2	no	—
15	C₄×Dic₁₅₈	C₂²	1	2	yes	—
16	C₄×D₃₁₆	C₂²	1	1	yes	characteristic
17	#14579119	C₂²	1	4	yes	—
18	#14579095	C₂²	1	1	yes	characteristic
19	#14579092	C₂²	1	4	yes	—
20	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
21	#14579086	C₂²	1	2	yes	—
22	#14579084	C₂²	1	4	yes	—
23	(C₂×D₇₉)⋊C₈	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
25	#107×C₇₉	C₂²	1	2	yes	—
26	#107×C₇₉	C₂²	1	2	yes	—
27	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₅₈	C₂³	1	2	yes	—
29	C₂×D₃₁₆	D₄	1	1	yes	characteristic
30	C₂×C₄×D₇₉	C₂³	1	2	yes	—
31	Dic₃₁₆	—	2	2	no	—
32	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
33	C₈×D₇₉	—	2	4	no	—
34	C₄⋊Dic₇₉	—	2	4	no	—
35	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
36	C₄⋊Dic₇₉	C₂³	1	2	yes	—
37	Dic₇₉⋊C₄	—	2	2	no	—
38	(C₂×D₇₉)⋊C₄	—	2	4	no	—
39	(C₂×D₇₉)⋊C₄	—	2	4	no	—
40	(C₂×D₇₉)⋊C₄	D₄	1	2	yes	—
41	C₄×Dic₇₉	—	2	4	no	—
42	C₄×Dic₇₉	—	2	2	no	—
43	C₂×Q₈×C₇₉	—	2	2	no	—
44	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₇₉	—	2	2	no	—
47	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
48	C₄²×C₇₉	C₂³	1	1	yes	characteristic
49	C₂²×D₇₉	C₂×D₄	1	2	yes	—
50	Q₈×C₇₉	—	4	4	no	—
51	C₂×C₄×C₇₉	—	2	4	no	—
52	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₇₉	—	2	2	no	—
56	C₂×Dic₇₉	—	2	2	no	—
57	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
58	D₃₁₆	—	2	2	no	—
59	C₄×D₇₉	—	2	2	no	—
60	C₄×D₇₉	—	2	2	no	—
61	Dic₁₅₈	—	2	2	no	—
62	Dic₁₅₈	—	2	2	no	—
63	C₈×C₇₉	—	2	1	no	—
64	C₈×C₇₉	#41	1	2	yes	—
65	C₇₉⋊C₈	—	2	2	no	—
66	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₇₉	—	2	2	no	—
68	C₂×D₇₉	—	2	2	no	—
69	C₂×D₇₉	—	2	2	no	—
70	C₄×C₇₉	—	2	1	no	—
71	C₄×C₇₉	—	4	4	no	—
72	C₄×C₇₉	#135	1	2	yes	—
73	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
74	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
75	Dic₇₉	—	2	2	no	—
76	Dic₇₉	—	4	2	no	—
77	Dic₇₉	—	4	2	no	—
78	C₂×C₇₉	#547	1	1	yes	characteristic
79	C₂×C₇₉	C₂×#135	1	1	yes	characteristic
80	C₂×C₇₉	#578	1	1	yes	characteristic
81	D₇₉	—	4	2	no	—
82	#3342	—	79	1	no	maximal, Sylow
83	C₇₉	#3342	1	1	yes	characteristic, complemented, Sylow
84	#556	—	79	2	no	—
85	#500	D₇₉	1	1	yes	characteristic, complemented
86	#485	—	79	4	no	—
87	#475	—	79	2	no	—
88	#467	—	79	2	no	—
89	C₄×Q₁₆	—	79	1	no	—
90	C₈×D₄	—	79	1	no	—
91	#427	—	79	2	no	—
92	C₂×Q₁₆	—	79	1	no	—
93	C₂²×C₈	—	79	2	no	—
94	C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
95	#126	—	158	4	no	—
96	#124	—	158	2	no	—
97	C₂²⋊Q₈	—	79	4	no	—
98	C₄×Q₈	—	79	2	no	—
99	C₄×Q₈	—	158	2	no	—
100	C₄×D₄	—	79	1	no	—
101	#117	—	79	4	no	—
102	#107	—	79	1	no	—
103	#107	C₂×D₇₉	1	2	yes	—
104	#107	C₂×D₇₉	1	2	yes	—
105	#106	—	79	4	no	—
106	C₄⋊C₈	—	79	1	no	—
107	#103	—	79	2	no	—
108	#103	—	79	4	no	—
109	C₂²⋊C₈	—	79	2	no	—
110	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
111	C₂×Q₈	—	2	2	no	—
112	C₂×Q₈	—	79	2	no	—
113	C₂×D₄	—	79	1	no	—
114	C₂²×C₄	—	79	2	no	—
115	Q₁₆	—	158	2	no	—
116	C₂×C₈	—	79	2	no	—
117	C₂×C₈	—	158	4	no	—
118	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
119	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	79	1	no	—
122	C₄⋊C₄	—	79	2	no	—
123	C₄⋊C₄	—	158	4	no	—
124	C₄⋊C₄	—	158	2	no	—
125	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
126	C₂²⋊C₄	—	79	2	no	—
127	C₂²⋊C₄	—	158	4	no	—
128	C₂²⋊C₄	—	158	4	no	—
129	C₄²	—	158	4	no	—
130	C₄²	—	158	2	no	—
131	C₄²	C₂²×D₇₉	1	1	yes	characteristic
132	C₂³	—	79	2	no	—
133	Q₈	—	4	4	no	—
134	Q₈	—	158	2	no	—
135	Q₈	—	158	2	no	—
136	D₄	—	158	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	79	2	no	—
139	C₂×C₄	—	158	2	no	—
140	C₂×C₄	—	158	2	no	—
141	C₂×C₄	—	158	2	no	—
142	C₂×C₄	—	158	2	no	—
143	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
144	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
145	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	158	2	no	—
148	C₈	#11929625	1	2	yes	—
149	C₂²	—	158	2	no	—
150	C₂²	—	158	2	no	—
151	C₂²	—	158	2	no	—
152	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	4	no	—
155	C₄	—	316	2	no	—
156	C₄	—	316	2	no	—
157	C₄	—	158	2	no	—
158	C₄	#14579180	1	2	yes	—
159	C₄	C₂×#11929625	1	1	yes	characteristic
160	C₄	#14579209	1	1	yes	characteristic
161	C₂	—	316	2	no	—
162	C₂	#19099216	1	1	yes	characteristic
163	C₂	C₂×#14579180	1	1	yes	characteristic
164	C₂	#19099376	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099287

C₂

yes

maximal

#19098906

C₂

yes

maximal, complemented

#19098845

C₂

yes

maximal, complemented

#19098805

C₂

yes

maximal, complemented

C₄×Dic₃₁₆

C₂

yes

maximal, characteristic, complemented

C₈×D₃₁₆

C₂

yes

maximal, characteristic

#19098600

C₂

yes

maximal, complemented

#500×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₁₆

C₂²

yes

characteristic

C₂×C₈×D₇₉

C₂²

yes

—

#14579156

—

#14579150

—

#14579141

C₂²

yes

—

Q₈×Dic₇₉

—

C₄×Dic₁₅₈

C₂²

yes

—

C₄×D₃₁₆

C₂²

yes

characteristic

#14579119

C₂²

yes

—

#14579095

C₂²

yes

characteristic

#14579092

C₂²

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

#14579086

C₂²

yes

—

#14579084

C₂²

yes

—

(C₂×D₇₉)⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₇₉

C₂²

yes

—

#107×C₇₉

C₂²

yes

—

#107×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

C₂³

yes

—

C₂×D₃₁₆

D₄

yes

characteristic

C₂×C₄×D₇₉

C₂³

yes

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₈×D₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

characteristic

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

D₄

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂²×D₇₉

C₂×D₄

yes

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

#41

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#135

yes

—

C₄×C₇₉

C₂×#41

yes

characteristic

C₄×C₇₉

2₋¹⁺⁴

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

#3342

—

maximal, Sylow

C₇₉

#3342

yes

characteristic, complemented, Sylow

#556

—

#500

D₇₉

yes

characteristic, complemented

#485

—

#475

—

#467

—

C₄×Q₁₆

—

C₈×D₄

—

#427

—

C₂×Q₁₆

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₇₉

yes

—

#126

—

158

—

#124

—

158

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

158

—

100

C₄×D₄

—

101

#117

—

102

#107

—

103

#107

C₂×D₇₉

yes

—

104

#107

C₂×D₇₉

yes

—

105

#106

—

106

C₄⋊C₈

—

107

#103

—

108

#103

—

109

C₂²⋊C₈

—

110

C₄×C₈

C₂×D₇₉

yes

characteristic

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

Q₁₆

—

158

—

116

C₂×C₈

—

117

C₂×C₈

—

158

—

118

C₂×C₈

C₂²×D₇₉

yes

characteristic

119

C₂×C₈

C₂²×D₇₉

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

158

—

124

C₄⋊C₄

—

158

—

125

C₄⋊C₄

C₂²×D₇₉

yes

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

158

—

128

C₂²⋊C₄

—

158

—

129

C₄²

—

158

—

130

C₄²

—

158

—

131

C₄²

C₂²×D₇₉

yes

characteristic

132

C₂³

—

133

Q₈

—

134

Q₈

—

158

—

135

Q₈

—

158

—

136

D₄

—

158

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

158

—

140

C₂×C₄

—

158

—

141

C₂×C₄

—

158

—

142

C₂×C₄

—

158

—

143

C₂×C₄

D₄×D₇₉

yes

characteristic

144

C₂×C₄

D₄×D₇₉

yes

characteristic

145

C₂×C₄

C₂³×D₇₉

yes

characteristic

146

C₈

—

147

C₈

—

158

—

148

C₈

#11929625

yes

—

149

C₂²

—

158

—

150

C₂²

—

158

—

151

C₂²

—

158

—

152

C₂²

C₂×D₄×D₇₉

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

316

—

156

C₄

—

316

—

157

C₄

—

158

—

158

C₄

#14579180

yes

—

159

C₄

C₂×#11929625

yes

characteristic

160

C₄

#14579209

yes

characteristic

161

C₂

—

316

—

162

C₂

#19099216

yes

characteristic

163

C₂

C₂×#14579180

yes

characteristic

164

C₂

#19099376

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#500×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂×D₃₁₆	D₄
hypercenter	#500	D₇₉
hypocenter	C₇₉	#3342
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#500×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

prime radical

C₂×D₃₁₆

D₄

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
2. #19098906	C₂	semidirect
3. #19098845	C₂	semidirect
4. #19098805	C₂	semidirect
5. C₄×Dic₃₁₆	C₂	semidirect
7. #19098600	C₂	semidirect
8. #500×C₇₉	C₂	semidirect
83. C₇₉	#3342	semidirect
85. #500	D₇₉	semidirect

H ≅ G/N

type

2. #19098906

C₂

semidirect

3. #19098845

C₂

semidirect

4. #19098805

C₂

semidirect

5. C₄×Dic₃₁₆

C₂

semidirect

7. #19098600

C₂

semidirect

8. #500×C₇₉

C₂

semidirect

83. C₇₉

#3342

semidirect

85. #500

D₇₉

semidirect

Subgroups — #23775830

Conjugacy classes of subgroups

Complements