Group #23762091 — subgroups

Subgroups — #23762091

Order 10112.

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

normal subgroups only

This group has 216 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	#19099120	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#19098311	C₂	1	2	yes	maximal, complemented
4	#19098308	C₂	1	2	yes	maximal, complemented
5	#423×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₉⋊C₈	—	2	1	no	—
7	#14579161	C₂²	1	1	yes	characteristic
8	#14579161	C₂²	1	1	yes	characteristic
9	#14579153	—	2	1	no	—
10	#14579140	—	2	1	no	—
11	#14579140	C₄	1	2	yes	—
12	C₄×Dic₁₅₈	—	2	1	no	—
13	C₄×Dic₁₅₈	C₄	1	2	yes	—
14	#14579120	—	2	1	no	—
15	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
16	C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
17	C₂×M₁₆×C₇₉	—	2	1	no	—
18	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₇₉	—	2	1	no	—
20	C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
21	C₂×Dic₁₅₈	—	2	1	no	—
22	C₂×Dic₁₅₈	C₂×C₄	1	2	yes	—
23	C₂×C₇₉⋊C₈	—	2	2	no	—
24	C₂×C₇₉⋊C₈	—	4	2	no	—
25	C₄⋊Dic₇₉	—	2	1	no	—
26	C₄⋊Dic₇₉	—	2	1	no	—
27	C₄⋊Dic₇₉	—	2	1	no	—
28	C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
29	Dic₇₉⋊C₄	—	2	2	no	—
30	Dic₇₉⋊C₄	—	2	2	no	—
31	C₂²⋊Dic₇₉	—	2	2	no	—
32	C₄×Dic₇₉	—	2	2	no	—
33	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
36	M₁₆×C₇₉	—	4	2	no	—
37	C₂×C₈×C₇₉	—	2	2	no	—
38	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₇₉	D₄	1	1	yes	characteristic
40	C₄²×C₇₉	D₄	1	1	yes	characteristic
41	C₄²×C₇₉	C₂³	1	1	yes	characteristic
42	C₂³×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₉	—	2	1	no	—
44	C₂×C₄×C₇₉	—	2	1	no	—
45	C₂×C₄×C₇₉	—	4	1	no	—
46	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	SD₁₆	1	2	yes	—
50	C₂×C₄×C₇₉	Q₁₆	1	2	yes	—
51	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₇₉	—	2	2	no	—
55	C₂×Dic₇₉	—	2	2	no	—
56	Dic₁₅₈	—	4	1	no	—
57	Dic₁₅₈	—	4	1	no	—
58	Dic₁₅₈	—	4	2	no	—
59	C₈×C₇₉	—	4	2	no	—
60	C₇₉⋊C₈	—	4	2	no	—
61	C₂²×C₇₉	—	2	1	no	—
62	C₂²×C₇₉	—	2	1	no	—
63	C₂²×C₇₉	#103	1	2	yes	—
64	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₇₉	—	2	1	no	—
66	C₄×C₇₉	—	2	1	no	—
67	C₄×C₇₉	—	2	1	no	—
68	C₄×C₇₉	#103	1	2	yes	—
69	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
70	C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
71	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₄×C₇₉	#137	1	1	yes	characteristic
73	Dic₇₉	—	4	2	no	—
74	Dic₇₉	—	4	2	no	—
75	C₂×C₇₉	—	2	2	no	—
76	C₂×C₇₉	#413	1	1	yes	characteristic
77	C₂×C₇₉	C₂×#103	1	1	yes	characteristic
78	C₂×C₇₉	#417	1	1	yes	characteristic
79	#1550	—	79	1	no	maximal, Sylow
80	C₇₉	#1550	1	1	yes	characteristic, complemented, Sylow
81	#533	—	79	1	no	—
82	#423	D₇₉	1	1	yes	characteristic
83	C₂×C₄⋊C₈	—	79	1	no	—
84	#333	—	79	2	no	—
85	#332	—	79	2	no	—
86	C₂×M₁₆	—	2	1	no	—
87	C₂²×C₈	—	158	1	no	—
88	C₄⋊Q₈	—	79	1	no	—
89	C₄⋊Q₈	—	79	1	no	—
90	#125	—	158	1	no	—
91	C₂²⋊Q₈	—	79	2	no	—
92	C₂²⋊Q₈	—	158	1	no	—
93	C₄×Q₈	—	79	2	no	—
94	C₄×Q₈	—	158	1	no	—
95	#117	—	158	1	no	—
96	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
97	C₄⋊C₈	—	2	1	no	—
98	C₄⋊C₈	—	79	2	no	—
99	C₄⋊C₈	—	158	1	no	—
100	C₄⋊C₈	C₂×D₇₉	1	2	yes	—
101	C₂×Q₈	—	79	2	no	—
102	C₂×Q₈	—	158	1	no	—
103	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
104	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
106	M₁₆	—	4	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	316	2	no	—
109	C₂×C₈	—	158	2	no	—
110	C₄⋊C₄	—	79	2	no	—
111	C₄⋊C₄	—	158	2	no	—
112	C₄⋊C₄	—	158	2	no	—
113	C₄⋊C₄	—	158	1	no	—
114	C₄⋊C₄	—	158	1	no	—
115	C₄⋊C₄	—	158	1	no	—
116	C₂²⋊C₄	—	158	2	no	—
117	C₄²	—	158	2	no	—
118	C₄²	C₄×D₇₉	1	1	yes	characteristic
119	C₄²	D₃₁₆	1	1	yes	characteristic
120	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
121	C₄²	C₂²×D₇₉	1	1	yes	characteristic
122	C₂³	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
123	Q₈	—	316	2	no	—
124	Q₈	—	316	1	no	—
125	Q₈	—	316	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	158	2	no	—
130	C₂×C₄	—	158	2	no	—
131	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
132	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
133	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
134	C₂×C₄	#11929612	1	2	yes	—
135	C₂×C₄	#11929614	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
138	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
139	C₈	—	4	2	no	—
140	C₈	—	316	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#14579084	1	2	yes	—
144	C₂²	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	316	2	no	—
149	C₄	—	316	2	no	—
150	C₄	#14579084	1	2	yes	—
151	C₄	C₂×#11929612	1	1	yes	characteristic
152	C₄	C₂×#11929614	1	1	yes	characteristic
153	C₄	#14579189	1	1	yes	characteristic
154	C₄	#14579194	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	#19098545	1	1	yes	characteristic
157	C₂	C₂×#14579084	1	1	yes	characteristic
158	C₂	#19098565	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099120

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

C₂

yes

maximal, characteristic

#19098311

C₂

yes

maximal, complemented

#19098308

C₂

yes

maximal, complemented

#423×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

—

#14579161

C₂²

yes

characteristic

#14579161

C₂²

yes

—

—

C₄

yes

—

C₄×Dic₁₅₈

—

C₄×Dic₁₅₈

C₄

yes

—

#14579120

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

C₂×M₁₆×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

Q₁₆

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#103

yes

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#103

yes

—

C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₄×C₇₉

C₂×Q₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#533

—

#423

D₇₉

yes

characteristic

C₂×C₄⋊C₈

—

#333

—

#332

—

C₂×M₁₆

—

C₂²×C₈

—

158

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

158

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

158

—

C₄×Q₈

—

C₄×Q₈

—

158

—

#117

—

158

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

158

—

100

C₄⋊C₈

C₂×D₇₉

yes

—

101

C₂×Q₈

—

102

C₂×Q₈

—

158

—

103

C₂²×C₄

C₄×D₇₉

yes

characteristic

104

C₂²×C₄

D₃₁₆

yes

characteristic

105

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

106

M₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

316

—

109

C₂×C₈

—

158

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

158

—

112

C₄⋊C₄

—

158

—

113

C₄⋊C₄

—

158

—

114

C₄⋊C₄

—

158

—

115

C₄⋊C₄

—

158

—

116

C₂²⋊C₄

—

158

—

117

C₄²

—

158

—

118

C₄²

C₄×D₇₉

yes

characteristic

119

C₄²

D₃₁₆

yes

characteristic

120

C₄²

C₂²⋊D₇₉

yes

characteristic

121

C₄²

C₂²×D₇₉

yes

characteristic

122

C₂³

(C₂×D₇₉)⋊C₄

yes

characteristic

123

Q₈

—

316

—

124

Q₈

—

316

—

125

Q₈

—

316

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

158

—

130

C₂×C₄

—

158

—

131

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

132

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

133

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

134

C₂×C₄

#11929612

yes

—

135

C₂×C₄

#11929614

yes

—

136

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

138

C₂×C₄

C₂×D₃₁₆

yes

characteristic

139

C₈

—

140

C₈

—

316

—

141

C₂²

—

142

C₂²

—

143

C₂²

#14579084

yes

—

144

C₂²

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

316

—

149

C₄

—

316

—

150

C₄

#14579084

yes

—

151

C₄

C₂×#11929612

yes

characteristic

152

C₄

C₂×#11929614

yes

characteristic

153

C₄

#14579189

yes

characteristic

154

C₄

#14579194

yes

characteristic

155

C₂

—

156

C₂

#19098545

yes

characteristic

157

C₂

C₂×#14579084

yes

characteristic

158

C₂

#19098565

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₄²	C₂²×D₇₉
Fitting	#423×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₇₉	C₂²⋊C₄
cosocle	C₄²×C₇₉	C₂³
hypercenter	#423	D₇₉
hypocenter	C₇₉	#1550
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₇₉)⋊C₄

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₄²

C₂²×D₇₉

Fitting

#423×C₇₉

C₂

socle

C₂²×C₇₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₇₉

C₂²⋊C₄

cosocle

C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #19098311	C₂	semidirect
4. #19098308	C₂	semidirect
80. C₇₉	#1550	semidirect

H ≅ G/N

type

3. #19098311

C₂

semidirect

4. #19098308

C₂

semidirect

80. C₇₉

#1550

semidirect

Subgroups — #23762091

Conjugacy classes of subgroups

Complements