Group #23762098 — subgroups

Subgroups — #23762098

Order 10112.

208 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 77 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	#19098620	C₂	1	1	yes	maximal, characteristic
3	#19098310	C₂	1	2	yes	maximal, complemented
4	#19098308	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	#14579161	C₂²	1	1	yes	characteristic
7	#14579161	C₂²	1	1	yes	characteristic
8	#14579153	C₄	1	2	yes	—
9	#14579140	—	2	1	no	—
10	#14579140	—	2	1	no	—
11	C₄×Dic₁₅₈	—	2	1	no	—
12	C₄×Dic₁₅₈	—	2	1	no	—
13	#14579120	C₄	1	2	yes	—
14	C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
15	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
16	#14579072	—	2	1	no	—
17	C₂²×C₈×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	—	2	1	no	—
23	C₂×C₇₉⋊C₈	—	2	2	no	—
24	C₄⋊Dic₇₉	—	2	1	no	—
25	C₄⋊Dic₇₉	—	2	1	no	—
26	C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
27	C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
28	Dic₇₉⋊C₄	—	2	2	no	—
29	Dic₇₉⋊C₄	—	2	2	no	—
30	C₂²⋊Dic₇₉	—	2	2	no	—
31	C₄×Dic₇₉	—	2	2	no	—
32	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
35	C₂×C₈×C₇₉	—	2	2	no	—
36	C₂×C₈×C₇₉	—	4	2	no	—
37	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₇₉	D₄	1	1	yes	characteristic
39	C₄²×C₇₉	D₄	1	1	yes	characteristic
40	C₄²×C₇₉	C₂³	1	1	yes	characteristic
41	C₂³×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₉	—	2	2	no	—
43	C₂×C₄×C₇₉	—	2	2	no	—
44	C₂×C₄×C₇₉	—	2	1	no	—
45	C₂×C₄×C₇₉	—	2	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	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₇₉	—	2	2	no	—
53	C₂×Dic₇₉	—	2	2	no	—
54	Dic₁₅₈	—	4	2	no	—
55	Dic₁₅₈	—	4	1	no	—
56	Dic₁₅₈	—	4	1	no	—
57	C₈×C₇₉	—	4	2	no	—
58	C₇₉⋊C₈	—	4	2	no	—
59	C₂²×C₇₉	—	2	1	no	—
60	C₂²×C₇₉	—	2	1	no	—
61	C₂²×C₇₉	#101	1	2	yes	—
62	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₇₉	—	2	1	no	—
64	C₄×C₇₉	—	2	1	no	—
65	C₄×C₇₉	—	2	1	no	—
66	C₄×C₇₉	—	2	1	no	—
67	C₄×C₇₉	#135	1	2	yes	—
68	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₇₉	#137	1	1	yes	characteristic
70	Dic₇₉	—	4	2	no	—
71	Dic₇₉	—	4	2	no	—
72	C₂×C₇₉	—	2	2	no	—
73	C₂×C₇₉	C₂×#101	1	1	yes	characteristic
74	C₂×C₇₉	#416	1	1	yes	characteristic
75	C₂×C₇₉	#417	1	1	yes	characteristic
76	#1552	—	79	1	no	maximal, Sylow
77	C₇₉	#1552	1	1	yes	characteristic, complemented, Sylow
78	#533	—	79	1	no	—
79	#432	—	79	1	no	—
80	C₂×C₄⋊C₈	D₇₉	1	1	yes	characteristic
81	#332	—	79	2	no	—
82	#332	—	79	2	no	—
83	C₂²×C₈	—	2	1	no	—
84	C₄⋊Q₈	—	79	1	no	—
85	C₄⋊Q₈	—	79	1	no	—
86	#125	—	79	2	no	—
87	C₂²⋊Q₈	—	158	1	no	—
88	C₂²⋊Q₈	—	158	1	no	—
89	C₄×Q₈	—	158	1	no	—
90	C₄×Q₈	—	158	1	no	—
91	#117	—	79	2	no	—
92	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
93	C₄⋊C₈	—	2	1	no	—
94	C₄⋊C₈	—	79	2	no	—
95	C₄⋊C₈	C₂×D₇₉	1	2	yes	—
96	C₂²⋊C₈	—	158	1	no	—
97	#97	—	158	1	no	—
98	C₂×Q₈	—	158	1	no	—
99	C₂×Q₈	—	158	1	no	—
100	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
101	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	158	2	no	—
106	C₄⋊C₄	—	79	2	no	—
107	C₄⋊C₄	—	79	2	no	—
108	C₄⋊C₄	—	158	2	no	—
109	C₄⋊C₄	—	158	2	no	—
110	C₄⋊C₄	—	158	1	no	—
111	C₄⋊C₄	—	158	1	no	—
112	C₂²⋊C₄	—	158	2	no	—
113	C₄²	—	158	2	no	—
114	C₄²	C₄×D₇₉	1	1	yes	characteristic
115	C₄²	D₃₁₆	1	1	yes	characteristic
116	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
117	C₄²	C₂²×D₇₉	1	1	yes	characteristic
118	C₂³	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
119	Q₈	—	316	2	no	—
120	Q₈	—	316	1	no	—
121	Q₈	—	316	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	158	2	no	—
127	C₂×C₄	—	158	2	no	—
128	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
129	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
130	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
133	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
134	C₈	—	4	2	no	—
135	C₈	—	316	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	#14579079	1	2	yes	—
139	C₂²	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	316	2	no	—
145	C₄	—	316	2	no	—
146	C₄	#14579183	1	2	yes	—
147	C₄	#14579187	1	1	yes	characteristic
148	C₄	#14579192	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	C₂×#14579079	1	1	yes	characteristic
151	C₂	#19098560	1	1	yes	characteristic
152	C₂	#19098562	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099120

C₂

yes

maximal, characteristic

#19098620

C₂

yes

maximal, characteristic

#19098310

C₂

yes

maximal, complemented

#19098308

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₇₉

C₂

yes

maximal, characteristic

#14579161

C₂²

yes

characteristic

#14579161

C₂²

yes

characteristic

#14579153

C₄

yes

—

#14579140

—

#14579140

—

C₄×Dic₁₅₈

—

C₄×Dic₁₅₈

—

#14579120

C₄

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

C₂²×C₈×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

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

C₂×C₈×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

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

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

#101

yes

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#135

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

—

#432

—

C₂×C₄⋊C₈

D₇₉

yes

characteristic

#332

—

#332

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

158

—

C₂²⋊Q₈

—

158

—

C₄×Q₈

—

158

—

C₄×Q₈

—

158

—

#117

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₉

yes

—

C₂²⋊C₈

—

158

—

#97

—

158

—

C₂×Q₈

—

158

—

C₂×Q₈

—

158

—

100

C₂²×C₄

C₄×D₇₉

yes

characteristic

101

C₂²×C₄

D₃₁₆

yes

characteristic

102

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

158

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

158

—

109

C₄⋊C₄

—

158

—

110

C₄⋊C₄

—

158

—

111

C₄⋊C₄

—

158

—

112

C₂²⋊C₄

—

158

—

113

C₄²

—

158

—

114

C₄²

C₄×D₇₉

yes

characteristic

115

C₄²

D₃₁₆

yes

characteristic

116

C₄²

C₂²⋊D₇₉

yes

characteristic

117

C₄²

C₂²×D₇₉

yes

characteristic

118

C₂³

(C₂×D₇₉)⋊C₄

yes

characteristic

119

Q₈

—

316

—

120

Q₈

—

316

—

121

Q₈

—

316

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

158

—

127

C₂×C₄

—

158

—

128

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

129

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

130

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

131

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

133

C₂×C₄

C₂×D₃₁₆

yes

characteristic

134

C₈

—

135

C₈

—

316

—

136

C₂²

—

137

C₂²

—

138

C₂²

#14579079

yes

—

139

C₂²

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

316

—

145

C₄

—

316

—

146

C₄

#14579183

yes

—

147

C₄

#14579187

yes

characteristic

148

C₄

#14579192

yes

characteristic

149

C₂

—

150

C₂

C₂×#14579079

yes

characteristic

151

C₂

#19098560

yes

characteristic

152

C₂

#19098562

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₄²	C₂²×D₇₉
Fitting	C₂×C₄⋊C₈×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₇₉	C₂²⋊C₄
cosocle	C₄²×C₇₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₇₉
hypocenter	C₇₉	#1552
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²

C₂×(C₂×D₇₉)⋊C₄

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₄²

C₂²×D₇₉

Fitting

C₂×C₄⋊C₈×C₇₉

C₂

socle

C₂²×C₇₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₇₉

C₂²⋊C₄

cosocle

C₄²×C₇₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₇₉

hypocenter

C₇₉

#1552

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #19098310	C₂	semidirect
4. #19098308	C₂	semidirect
77. C₇₉	#1552	semidirect

H ≅ G/N

type

3. #19098310

C₂

semidirect

4. #19098308

C₂

semidirect

77. C₇₉

#1552

semidirect

Subgroups — #23762098

Conjugacy classes of subgroups

Complements