Group #34175661 — subgroups

Subgroups — #34175661

Order 20224.

176 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 43 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	#23764364	C₂	1	1	yes	maximal, characteristic
2	#23761344	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1324	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579153	C₄	1	1	yes	characteristic
5	C₂×C₄×C₇₉⋊C₈	C₄	1	1	yes	characteristic
6	#19098325	—	2	1	no	—
7	#19098325	—	2	1	no	—
8	#19098289	—	2	1	no	—
9	#382×C₇₉	C₂²	1	1	yes	characteristic
10	#324×C₇₉	—	2	1	no	—
11	C₂²×C₇₉⋊C₈	—	2	1	no	—
12	#14579153	—	2	1	no	—
13	#14579153	—	4	1	no	—
14	#14579153	C₈	1	2	yes	complemented
15	C₂×C₄⋊Dic₇₉	—	2	1	no	—
16	C₂×Dic₇₉⋊C₄	—	4	1	no	—
17	C₂²⋊(C₇₉⋊C₈)	—	4	1	no	—
18	C₄×C₇₉⋊C₈	—	4	1	no	—
19	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂²⋊C₈×C₇₉	—	4	1	no	—
23	#95×C₇₉	—	4	1	no	—
24	C₂²×Dic₇₉	—	4	1	no	—
25	C₂×C₇₉⋊C₈	—	2	2	no	—
26	C₂×C₇₉⋊C₈	—	4	1	no	—
27	C₂×C₇₉⋊C₈	—	4	1	no	—
28	C₂×C₇₉⋊C₈	—	8	1	no	—
29	C₄⋊Dic₇₉	—	2	2	no	—
30	C₄⋊Dic₇₉	—	4	1	no	—
31	Dic₇₉⋊C₄	—	4	2	no	—
32	Dic₇₉⋊C₄	—	4	2	no	—
33	C₂²×C₄×C₇₉	—	2	1	no	—
34	C₂²×C₄×C₇₉	—	2	1	no	—
35	C₂²×C₄×C₇₉	—	2	1	no	—
36	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₉	—	8	1	no	—
38	C₄⋊C₄×C₇₉	—	4	1	no	—
39	C₄⋊C₄×C₇₉	—	4	1	no	—
40	C₄²×C₇₉	—	2	1	no	—
41	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
43	C₂³×C₇₉	#99	1	1	yes	characteristic
44	C₂×C₄×C₇₉	—	2	1	no	—
45	C₂×C₄×C₇₉	—	2	1	no	—
46	C₂×C₄×C₇₉	—	2	1	no	—
47	C₂×C₄×C₇₉	—	2	1	no	—
48	C₂×C₄×C₇₉	—	4	1	no	—
49	C₂×C₄×C₇₉	—	4	1	no	—
50	C₂×C₄×C₇₉	—	4	1	no	—
51	C₂×C₄×C₇₉	—	4	1	no	—
52	C₂×C₄×C₇₉	—	8	1	no	—
53	C₂×C₄×C₇₉	—	8	1	no	—
54	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
56	C₂×Dic₇₉	—	4	2	no	—
57	C₂×Dic₇₉	—	8	1	no	—
58	C₂×Dic₇₉	—	8	1	no	—
59	C₈×C₇₉	—	16	1	no	—
60	C₇₉⋊C₈	—	4	2	no	—
61	C₇₉⋊C₈	—	16	1	no	—
62	C₂²×C₇₉	—	2	1	no	—
63	C₂²×C₇₉	—	2	1	no	—
64	C₂²×C₇₉	#323	1	1	yes	characteristic
65	C₂²×C₇₉	#354	1	1	yes	characteristic
66	C₂²×C₇₉	#356	1	1	yes	characteristic
67	C₄×C₇₉	—	2	1	no	—
68	C₄×C₇₉	—	2	1	no	—
69	C₄×C₇₉	—	4	1	no	—
70	C₄×C₇₉	—	4	1	no	—
71	C₄×C₇₉	—	8	1	no	—
72	C₄×C₇₉	—	8	1	no	—
73	Dic₇₉	—	8	2	no	—
74	#7359	—	79	1	no	maximal, Sylow
75	C₂×C₇₉	—	2	1	no	—
76	C₂×C₇₉	—	2	1	no	—
77	C₂×C₇₉	#1319	1	1	yes	characteristic
78	C₂×C₇₉	#1414	1	1	yes	characteristic
79	C₂×C₇₉	#1414	1	1	yes	characteristic
80	#1991	—	79	1	no	—
81	#1324	—	79	1	no	—
82	#1324	D₇₉	1	1	yes	characteristic
83	C₇₉	#7359	1	1	yes	characteristic, complemented, Sylow
84	C₂×#125	—	79	1	no	—
85	C₂×C₄×C₈	—	79	1	no	—
86	#382	C₂×D₇₉	1	1	yes	characteristic
87	#340	—	158	1	no	—
88	#340	—	158	1	no	—
89	#324	—	2	1	no	—
90	#324	—	158	1	no	—
91	C₂²×C₈	—	158	1	no	—
92	#125	—	316	1	no	—
93	#125	—	79	2	no	—
94	#125	—	158	1	no	—
95	C₂×C₄⋊C₄	—	316	1	no	—
96	C₂×C₄⋊C₄	—	158	1	no	—
97	C₂×C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
98	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
99	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
100	C₂²⋊C₈	—	4	1	no	—
101	C₂²⋊C₈	—	316	1	no	—
102	C₄×C₈	—	316	1	no	—
103	#95	—	4	1	no	—
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	316	1	no	—
108	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
109	C₂×C₈	—	8	1	no	—
110	C₂×C₈	—	316	1	no	—
111	C₂×C₈	—	316	1	no	—
112	C₂×C₈	—	632	1	no	—
113	C₂×C₈	—	158	2	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	—	316	2	no	—
117	C₄⋊C₄	—	316	2	no	—
118	C₄⋊C₄	—	316	1	no	—
119	C₄⋊C₄	—	158	2	no	—
120	C₄²	—	2	1	no	—
121	C₄²	C₈×D₇₉	1	1	yes	characteristic
122	C₄²	#11929607	1	1	yes	characteristic
123	C₂³	#14579075	1	1	yes	characteristic
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	8	1	no	—
134	C₂×C₄	—	316	2	no	—
135	C₂×C₄	—	632	1	no	—
136	C₂×C₄	—	632	1	no	—
137	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
138	C₂×C₄	#14579077	1	1	yes	characteristic
139	C₈	—	16	1	no	—
140	C₈	—	316	2	no	—
141	C₈	—	1264	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	#19098286	1	1	yes	characteristic
145	C₂²	#19098357	1	1	yes	characteristic
146	C₂²	#19098361	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	4	1	no	—
150	C₄	—	4	1	no	—
151	C₄	—	8	1	no	—
152	C₄	—	8	1	no	—
153	C₄	—	632	2	no	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#23761333	1	1	yes	characteristic
157	C₂	#23761574	1	1	yes	characteristic
158	C₂	#23761576	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23764364

C₂

yes

maximal, characteristic

#23761344

C₂

yes

maximal, characteristic

C₇₉×#1324

C₂

yes

maximal, characteristic

C₂×#14579153

C₄

yes

characteristic

C₂×C₄×C₇₉⋊C₈

C₄

yes

—

—

—

#382×C₇₉

C₂²

yes

characteristic

#324×C₇₉

—

C₂²×C₇₉⋊C₈

—

#14579153

—

#14579153

—

#14579153

C₈

yes

complemented

C₂×C₄⋊Dic₇₉

—

C₂×Dic₇₉⋊C₄

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₂×C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₂×C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂²⋊C₈×C₇₉

—

#95×C₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×C₈

yes

characteristic

C₄²×C₇₉

M₁₆

yes

characteristic

C₂³×C₇₉

#99

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₉

#100

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

#7359

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

D₇₉

yes

characteristic

C₇₉

#7359

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×C₄×C₈

—

#382

C₂×D₇₉

yes

characteristic

#340

—

158

—

#340

—

158

—

#324

—

#324

—

158

—

C₂²×C₈

—

158

—

#125

—

316

—

#125

—

#125

—

158

—

C₂×C₄⋊C₄

—

316

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

D₃₁₆

yes

characteristic

C₂×C₄⋊C₄

C₂²⋊D₇₉

yes

characteristic

C₂×C₄²

C₄×D₇₉

yes

characteristic

100

C₂²⋊C₈

—

101

C₂²⋊C₈

—

316

—

102

C₄×C₈

—

316

—

103

#95

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

316

—

108

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

316

—

111

C₂×C₈

—

316

—

112

C₂×C₈

—

632

—

113

C₂×C₈

—

158

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

316

—

117

C₄⋊C₄

—

316

—

118

C₄⋊C₄

—

316

—

119

C₄⋊C₄

—

158

—

120

C₄²

—

121

C₄²

C₈×D₇₉

yes

characteristic

122

C₄²

#11929607

yes

characteristic

123

C₂³

#14579075

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

316

—

135

C₂×C₄

—

632

—

136

C₂×C₄

—

632

—

137

C₂×C₄

(C₂×D₇₉)⋊C₈

yes

characteristic

138

C₂×C₄

#14579077

yes

characteristic

139

C₈

—

140

C₈

—

316

—

141

C₈

—

1264

—

142

C₂²

—

143

C₂²

—

144

C₂²

#19098286

yes

characteristic

145

C₂²

#19098357

yes

characteristic

146

C₂²

#19098361

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

632

—

154

C₂

—

155

C₂

—

156

C₂

#23761333

yes

characteristic

157

C₂

#23761574

yes

characteristic

158

C₂

#23761576

yes

characteristic

	subgroup	G/H
center	C₂²	#19098357
commutator	C₄²×C₇₉	C₂×C₈
Frattini	#382	C₂×D₇₉
Fitting	C₇₉×#1324	C₂
socle	C₂²×C₇₉	#354
prime radical	C₂³×C₇₉	#99
cosocle	#382×C₇₉	C₂²
hypercenter	#1324	D₇₉
hypocenter	C₇₉	#7359
LCS	G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #382 ◁ #1324

subgroup

G/H

center

C₂²

#19098357

commutator

C₄²×C₇₉

C₂×C₈

Frattini

#382

C₂×D₇₉

Fitting

C₇₉×#1324

C₂

C₂²×C₇₉

C₂³×C₇₉

#382×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #382 ◁ #1324

N	H ≅ G/N	type
14. #14579153	C₈	semidirect
83. C₇₉	#7359	semidirect

H ≅ G/N

type

14. #14579153

C₈

semidirect

83. C₇₉

#7359

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23764364	2
#23761344	2
C₇₉×#1324	2
#7359	79

subgroup

index

#23764364

#23761344

C₇₉×#1324

#7359

Subgroups — #34175661

Conjugacy classes of subgroups

Complements

Maximal subgroups