Group #23765104 — subgroups

Subgroups — #23765104

Order 10112.

228 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 69 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×#14579180	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098600	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098368	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098366	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098364	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929623	—	2	1	no	—
9	#14579180	—	2	2	no	—
10	#14579180	C₄	1	4	yes	complemented
11	C₂×Dic₃₁₆	C₂²	1	1	yes	characteristic
12	C₂×#11929612	—	2	1	no	—
13	C₂×C₈×D₇₉	C₂²	1	1	yes	characteristic
14	#14579119	—	2	1	no	—
15	C₂×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₁₆×D₇₉	—	2	2	no	—
17	#14579095	C₂²	1	1	yes	characteristic
18	#14579092	—	2	1	no	—
19	C₂×D₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
21	#107×C₇₉	C₂²	1	1	yes	characteristic
22	#11929623	—	2	4	no	—
23	#11929623	—	4	2	no	—
24	C₂×Dic₁₅₈	—	2	1	no	—
25	C₂×C₂²⋊D₇₉	—	4	1	no	—
26	C₂²×Dic₇₉	—	4	1	no	—
27	C₂×C₄×D₇₉	D₄	1	1	yes	characteristic
28	Dic₃₁₆	—	2	1	no	—
29	Dic₃₁₆	C₂×C₄	1	2	yes	—
30	#11929612	—	2	4	no	—
31	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
32	C₈×D₇₉	C₂×C₄	1	2	yes	—
33	C₈×D₇₉	D₄	1	2	yes	—
34	C₄⋊Dic₇₉	—	2	1	no	—
35	(C₂×D₇₉)⋊C₄	—	4	1	no	—
36	C₄×Dic₇₉	—	4	1	no	—
37	C₇₉⋊C₁₆	—	2	1	no	—
38	C₂×D₄×C₇₉	—	2	1	no	—
39	D₈×C₇₉	—	2	1	no	—
40	D₈×C₇₉	C₂×C₄	1	2	yes	—
41	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₇₉	—	2	1	no	—
43	C₁₆×C₇₉	—	2	1	no	—
44	C₂³×C₇₉	—	4	1	no	—
45	C₂²×D₇₉	D₈	1	1	yes	characteristic
46	D₄×C₇₉	—	2	2	no	—
47	D₄×C₇₉	—	4	1	no	—
48	C₂×C₄×C₇₉	—	4	1	no	—
49	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂²⋊D₇₉	—	4	4	no	—
51	C₂×Dic₇₉	—	4	1	no	—
52	C₂×Dic₇₉	—	4	4	no	—
53	C₂×Dic₇₉	—	4	1	no	—
54	C₂×Dic₇₉	D₈	1	1	yes	characteristic
55	C₄×D₇₉	C₂²⋊C₄	1	2	yes	—
56	C₄×D₇₉	SD₁₆	1	2	yes	—
57	Dic₁₅₈	—	2	2	no	—
58	Dic₁₅₈	—	4	1	no	—
59	C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
60	C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₇₉	—	4	2	no	—
63	C₂²×C₇₉	—	8	1	no	—
64	C₂²×C₇₉	—	8	1	no	—
65	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
66	C₂×D₇₉	—	2	1	no	—
67	C₂×D₇₉	—	2	1	no	—
68	C₂×D₇₉	#102	1	2	yes	—
69	C₄×C₇₉	—	8	1	no	—
70	C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
72	Dic₇₉	—	4	2	no	—
73	Dic₇₉	—	8	1	no	—
74	Dic₇₉	#102	1	2	yes	—
75	C₂×C₇₉	—	8	2	no	—
76	C₂×C₇₉	C₂×#102	1	1	yes	characteristic
77	C₂×C₇₉	#508	1	1	yes	characteristic
78	C₂×C₇₉	#508	1	1	yes	characteristic
79	D₇₉	—	2	2	no	—
80	#2138	—	79	1	no	maximal, Sylow
81	C₇₉	#2138	1	1	yes	characteristic, complemented, Sylow
82	C₂×#135	—	79	1	no	—
83	C₂²×C₁₆	—	79	1	no	—
84	#427	—	79	1	no	—
85	#358	—	79	1	no	—
86	#358	—	79	1	no	—
87	#357	—	79	1	no	—
88	#357	D₇₉	1	1	yes	characteristic, complemented
89	C₂×#41	—	158	1	no	—
90	#135	—	79	4	no	—
91	#135	—	158	2	no	—
92	C₂×Q₁₆	—	79	1	no	—
93	C₂×SD₁₆	—	158	1	no	—
94	C₂×D₈	C₂×D₇₉	1	1	yes	characteristic
95	C₂²×C₈	—	79	1	no	—
96	#117	—	158	1	no	—
97	C₂×C₁₆	—	79	1	no	—
98	C₂×C₁₆	—	158	2	no	—
99	C₂×C₁₆	C₂×D₇₉	1	1	yes	characteristic
100	#107	—	79	1	no	—
101	#107	C₂×D₇₉	1	1	yes	characteristic
102	#106	—	158	1	no	—
103	#41	—	316	2	no	—
104	#41	—	158	4	no	—
105	C₂×Q₈	—	158	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	316	1	no	—
108	C₂²×C₄	—	316	1	no	—
109	C₂²×C₄	—	79	1	no	—
110	Q₁₆	—	79	2	no	—
111	Q₁₆	—	158	1	no	—
112	SD₁₆	—	158	4	no	—
113	D₈	—	2	1	no	—
114	D₈	C₄×D₇₉	1	2	yes	—
115	C₂×C₈	—	79	1	no	—
116	C₂×C₈	—	79	2	no	—
117	C₂×C₈	—	79	2	no	—
118	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	158	1	no	—
121	C₂²⋊C₄	—	316	1	no	—
122	C₄²	—	316	1	no	—
123	C₁₆	—	2	1	no	—
124	C₁₆	—	158	1	no	—
125	C₂³	—	4	1	no	—
126	C₂³	—	79	1	no	—
127	Q₈	—	316	1	no	—
128	Q₈	—	158	2	no	—
129	D₄	—	2	2	no	—
130	D₄	—	4	1	no	—
131	D₄	—	316	4	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	316	1	no	—
134	C₂×C₄	—	316	4	no	—
135	C₂×C₄	—	316	1	no	—
136	C₂×C₄	—	79	1	no	—
137	C₂×C₄	—	79	2	no	—
138	C₂×C₄	—	79	2	no	—
139	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
140	C₈	—	79	2	no	—
141	C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
142	C₈	D₄×D₇₉	1	1	yes	characteristic
143	C₂²	—	4	2	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	no	—
146	C₂²	—	79	2	no	—
147	C₂²	—	158	1	no	—
148	C₂²	—	158	1	no	—
149	C₂²	D₈×D₇₉	1	1	yes	characteristic
150	C₄	—	8	1	no	—
151	C₄	—	316	2	no	—
152	C₄	—	79	2	no	—
153	C₄	—	632	1	no	—
154	C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
155	C₄	SD₁₆×D₇₉	1	1	yes	characteristic
156	C₂	—	8	2	no	—
157	C₂	—	158	2	no	—
158	C₂	#102×D₇₉	1	1	yes	characteristic
159	C₂	#19098992	1	1	yes	characteristic
160	C₂	#19098994	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579180

C₂

yes

maximal, characteristic

C₂×C₁₆×D₇₉

C₂

yes

maximal, characteristic, complemented

#19098600

C₂

yes

maximal, characteristic, complemented

#19098368

C₂

yes

maximal, characteristic, complemented

#19098366

C₂

yes

maximal, characteristic, complemented

#19098364

C₂

yes

maximal, characteristic, complemented

#357×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929623

—

#14579180

—

#14579180

C₄

yes

complemented

C₂×Dic₃₁₆

C₂²

yes

characteristic

C₂×#11929612

—

C₂×C₈×D₇₉

C₂²

yes

characteristic

#14579119

—

C₂×C₇₉⋊C₁₆

C₂²

yes

characteristic

C₁₆×D₇₉

—

#14579095

C₂²

yes

characteristic

#14579092

—

C₂×D₈×C₇₉

C₂²

yes

characteristic

C₂×C₁₆×C₇₉

C₂²

yes

characteristic

#107×C₇₉

C₂²

yes

characteristic

#11929623

—

#11929623

—

C₂×Dic₁₅₈

—

C₂×C₂²⋊D₇₉

—

C₂²×Dic₇₉

—

C₂×C₄×D₇₉

D₄

yes

characteristic

Dic₃₁₆

—

Dic₃₁₆

C₂×C₄

yes

—

#11929612

—

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₈×D₇₉

C₂×C₄

yes

—

C₈×D₇₉

D₄

yes

—

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂×D₄×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

C₂×C₄

yes

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

—

C₂²×D₇₉

D₈

yes

characteristic

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

D₈

yes

characteristic

C₄×D₇₉

C₂²⋊C₄

yes

—

C₄×D₇₉

SD₁₆

yes

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

C₂²×C₄

yes

characteristic

C₈×C₇₉

C₂×D₄

yes

characteristic

C₇₉⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

#102

yes

—

C₄×C₇₉

—

C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

C₂×SD₁₆

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

#102

yes

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

#2138

—

maximal, Sylow

C₇₉

#2138

yes

characteristic, complemented, Sylow

C₂×#135

—

C₂²×C₁₆

—

—

—

—

—

D₇₉

yes

characteristic, complemented

C₂×#41

—

158

—

#135

—

#135

—

158

—

C₂×Q₁₆

—

C₂×SD₁₆

—

158

—

C₂×D₈

C₂×D₇₉

yes

characteristic

C₂²×C₈

—

#117

—

158

—

C₂×C₁₆

—

C₂×C₁₆

—

158

—

C₂×C₁₆

C₂×D₇₉

yes

characteristic

100

#107

—

101

#107

C₂×D₇₉

yes

characteristic

102

#106

—

158

—

103

#41

—

316

—

104

#41

—

158

—

105

C₂×Q₈

—

158

—

106

C₂×D₄

—

107

C₂×D₄

—

316

—

108

C₂²×C₄

—

316

—

109

C₂²×C₄

—

110

Q₁₆

—

111

Q₁₆

—

158

—

112

SD₁₆

—

158

—

113

D₈

—

114

D₈

C₄×D₇₉

yes

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₂²×D₇₉

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

158

—

121

C₂²⋊C₄

—

316

—

122

C₄²

—

316

—

123

C₁₆

—

124

C₁₆

—

158

—

125

C₂³

—

126

C₂³

—

127

Q₈

—

316

—

128

Q₈

—

158

—

129

D₄

—

130

D₄

—

131

D₄

—

316

—

132

C₂×C₄

—

133

C₂×C₄

—

316

—

134

C₂×C₄

—

316

—

135

C₂×C₄

—

316

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

D₄×D₇₉

yes

characteristic

140

C₈

—

141

C₈

C₂×C₄×D₇₉

yes

characteristic

142

C₈

D₄×D₇₉

yes

characteristic

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

158

—

148

C₂²

—

158

—

149

C₂²

D₈×D₇₉

yes

characteristic

150

C₄

—

151

C₄

—

316

—

152

C₄

—

153

C₄

—

632

—

154

C₄

C₂²⋊C₄×D₇₉

yes

characteristic

155

C₄

SD₁₆×D₇₉

yes

characteristic

156

C₂

—

157

C₂

—

158

—

158

C₂

#102×D₇₉

yes

characteristic

159

C₂

#19098992

yes

characteristic

160

C₂

#19098994

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₉
commutator	C₈×C₇₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	#357×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
prime radical	C₂×#14579180	C₂
cosocle	C₂×C₈×C₇₉	C₂³
hypercenter	#357	D₇₉
hypocenter	C₇₉	#2138
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₇₉

commutator

C₈×C₇₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

#357×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

prime radical

C₂×#14579180

C₂

cosocle

C₂×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
2. C₂×C₁₆×D₇₉	C₂	semidirect
3. #19098600	C₂	semidirect
4. #19098368	C₂	semidirect
5. #19098366	C₂	semidirect
6. #19098364	C₂	semidirect
7. #357×C₇₉	C₂	semidirect
10. #14579180	C₄	semidirect
81. C₇₉	#2138	semidirect
88. #357	D₇₉	semidirect

H ≅ G/N

type

2. C₂×C₁₆×D₇₉

C₂

semidirect

3. #19098600

C₂

semidirect

4. #19098368

C₂

semidirect

5. #19098366

C₂

semidirect

6. #19098364

C₂

semidirect

7. #357×C₇₉

C₂

semidirect

10. #14579180

C₄

semidirect

81. C₇₉

#2138

semidirect

88. #357

D₇₉

semidirect

Subgroups — #23765104

Conjugacy classes of subgroups

Complements