Group #34174875 — subgroups

Subgroups — #34174875

Order 20224.

364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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	#23763162	C₂	1	2	yes	maximal
2	C₇₉×#1817	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
4	C₂×#14579072	—	2	2	no	—
5	#19098318	C₄	1	4	yes	—
6	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₈×C₇₉	C₄	1	2	yes	—
8	#336×C₇₉	—	2	2	no	—
9	C₂²×C₇₉⋊C₈	—	2	4	no	—
10	C₂²⋊(C₇₉⋊C₈)	—	2	8	no	—
11	#14579072	—	4	4	no	—
12	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
14	C₂³×C₄×C₇₉	Q₈	1	1	yes	characteristic
15	C₂²×C₈×C₇₉	—	2	2	no	—
16	C₂×C₄²×C₇₉	—	4	2	no	—
17	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₇₉	D₄	1	2	yes	—
19	C₂²⋊C₈×C₇₉	—	4	2	no	—
20	C₂×C₇₉⋊C₈	—	2	8	no	—
21	C₂×C₇₉⋊C₈	—	4	4	no	—
22	C₂×C₇₉⋊C₈	—	4	4	no	—
23	C₂⁴×C₇₉	C₄⋊C₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₉	—	2	2	no	—
25	C₂²×C₄×C₇₉	—	2	2	no	—
26	C₂²×C₄×C₇₉	—	2	4	no	—
27	C₂²×C₄×C₇₉	—	4	1	no	—
28	C₂²×C₄×C₇₉	—	4	1	no	—
29	C₂²×C₄×C₇₉	—	4	2	no	—
30	C₂²×C₄×C₇₉	—	4	1	no	—
31	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₇₉	—	4	4	no	—
36	C₂×C₈×C₇₉	—	4	2	no	—
37	C₄²×C₇₉	—	2	4	no	—
38	C₄²×C₇₉	—	4	2	no	—
39	C₂³×C₇₉	—	2	1	no	—
40	C₂³×C₇₉	—	2	2	no	—
41	C₂³×C₇₉	—	2	1	no	—
42	C₂³×C₇₉	—	2	2	no	—
43	C₂³×C₇₉	#95	1	1	yes	characteristic
44	C₂³×C₇₉	#108	1	2	yes	—
45	C₂×C₄×C₇₉	—	2	2	no	—
46	C₂×C₄×C₇₉	—	2	2	no	—
47	C₂×C₄×C₇₉	—	2	2	no	—
48	C₂×C₄×C₇₉	—	2	2	no	—
49	C₂×C₄×C₇₉	—	2	2	no	—
50	C₂×C₄×C₇₉	—	2	2	no	—
51	C₂×C₄×C₇₉	—	2	1	no	—
52	C₂×C₄×C₇₉	—	2	2	no	—
53	C₂×C₄×C₇₉	—	4	4	no	—
54	C₂×C₄×C₇₉	—	4	4	no	—
55	C₂×C₄×C₇₉	—	4	4	no	—
56	C₂×C₄×C₇₉	#99	1	2	yes	—
57	C₂×C₄×C₇₉	#104	1	4	yes	—
58	C₈×C₇₉	—	8	2	no	—
59	C₇₉⋊C₈	—	4	8	no	—
60	C₂²×C₇₉	—	2	2	no	—
61	C₂²×C₇₉	—	2	4	no	—
62	C₂²×C₇₉	—	4	2	no	—
63	C₂²×C₇₉	—	4	1	no	—
64	C₂²×C₇₉	—	4	1	no	—
65	C₂²×C₇₉	#338	1	1	yes	characteristic
66	C₂²×C₇₉	#339	1	2	yes	—
67	C₂²×C₇₉	#341	1	1	yes	characteristic
68	C₂²×C₇₉	#342	1	1	yes	characteristic
69	C₂²×C₇₉	#344	1	2	yes	—
70	C₄×C₇₉	—	2	4	no	—
71	C₄×C₇₉	—	2	2	no	—
72	C₄×C₇₉	—	4	2	no	—
73	C₄×C₇₉	—	4	1	no	—
74	#7046	—	79	1	no	maximal, Sylow
75	C₂×C₇₉	—	4	2	no	—
76	C₂×C₇₉	#1291	1	2	yes	—
77	C₂×C₇₉	#1293	1	1	yes	characteristic
78	C₂×C₇₉	#1297	1	1	yes	characteristic
79	C₂×C₇₉	#1302	1	2	yes	—
80	C₂×C₇₉	#1306	1	1	yes	characteristic
81	#1817	D₇₉	1	1	yes	characteristic
82	#1749	—	79	2	no	—
83	C₇₉	#7046	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
85	C₂×C₂²⋊C₈	—	158	2	no	—
86	C₂×C₂²⋊C₈	Dic₇₉	1	2	yes	—
87	C₂×#97	—	158	2	no	—
88	#336	—	2	2	no	—
89	#336	—	79	4	no	—
90	C₂³×C₄	Dic₁₅₈	1	1	yes	characteristic
91	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
92	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
93	C₂²×C₈	—	2	2	no	—
94	C₂²×C₈	—	158	4	no	—
95	C₂×C₄²	—	4	2	no	—
96	C₂×C₄²	C₄×D₇₉	1	2	yes	—
97	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
98	C₂²⋊C₈	—	4	2	no	—
99	C₂²⋊C₈	—	158	8	no	—
100	#97	—	316	4	no	—
101	C₂⁴	C₄⋊Dic₇₉	1	1	yes	characteristic
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	—	2	4	no	—
105	C₂²×C₄	—	4	1	no	—
106	C₂²×C₄	—	4	1	no	—
107	C₂²×C₄	—	4	1	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
110	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
111	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
112	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
113	C₂×C₈	—	4	4	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	316	4	no	—
116	C₂×C₈	—	316	4	no	—
117	C₂×C₈	—	158	8	no	—
118	C₄²	—	2	4	no	—
119	C₄²	—	4	2	no	—
120	C₂³	—	2	1	no	—
121	C₂³	—	2	2	no	—
122	C₂³	—	2	1	no	—
123	C₂³	—	2	2	no	—
124	C₂³	C₇₉⋊#95	1	1	yes	characteristic
125	C₂³	#14579097	1	2	yes	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	#14579076	1	2	yes	—
138	C₂×C₄	#14579087	1	4	yes	—
139	C₈	—	8	2	no	—
140	C₈	—	316	8	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	4	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	C₇₉⋊#338	1	1	yes	characteristic
147	C₂²	#19098324	1	2	yes	—
148	C₂²	#19098328	1	1	yes	characteristic
149	C₂²	#19098330	1	1	yes	characteristic
150	C₂²	#19098333	1	2	yes	—
151	C₄	—	2	4	no	—
152	C₄	—	2	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	4	1	no	—
155	C₂	—	4	2	no	—
156	C₂	#23761273	1	2	yes	—
157	C₂	#23761277	1	1	yes	characteristic
158	C₂	#23761285	1	1	yes	characteristic
159	C₂	#23761300	1	2	yes	—
160	C₂	#23761308	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763162

C₂

yes

maximal

C₇₉×#1817

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₇₉⋊C₈)

—

C₂×#14579072

—

#19098318

C₄

yes

—

C₂²×C₄²×C₇₉

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₇₉

C₄

yes

—

#336×C₇₉

—

C₂²×C₇₉⋊C₈

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

C₂³×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂³×C₄×C₇₉

D₄

yes

characteristic

C₂³×C₄×C₇₉

Q₈

yes

characteristic

C₂²×C₈×C₇₉

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

—

C₂²⋊C₈×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂⁴×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₄²

yes

characteristic

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

#95

yes

characteristic

C₂³×C₇₉

#108

yes

—

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

—

C₂×C₄×C₇₉

#99

yes

—

C₂×C₄×C₇₉

#104

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#338

yes

characteristic

C₂²×C₇₉

#339

yes

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

#7046

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1291

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#1306

yes

characteristic

#1817

D₇₉

yes

characteristic

#1749

—

C₇₉

#7046

yes

characteristic, complemented, Sylow

C₂²×C₄²

C₂×D₇₉

yes

characteristic

C₂×C₂²⋊C₈

—

158

—

C₂×C₂²⋊C₈

Dic₇₉

yes

—

C₂×#97

—

158

—

#336

—

#336

—

C₂³×C₄

Dic₁₅₈

yes

characteristic

C₂³×C₄

D₃₁₆

yes

characteristic

C₂³×C₄

C₂×Dic₇₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

158

—

C₂×C₄²

—

C₂×C₄²

C₄×D₇₉

yes

—

C₂×C₄²

C₂²⋊D₇₉

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

158

—

100

#97

—

316

—

101

C₂⁴

C₄⋊Dic₇₉

yes

characteristic

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₄×Dic₇₉

yes

characteristic

110

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

111

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

112

C₂²×C₄

Dic₇₉⋊C₄

yes

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

316

—

116

C₂×C₈

—

316

—

117

C₂×C₈

—

158

—

118

C₄²

—

119

C₄²

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

C₇₉⋊#95

yes

characteristic

125

C₂³

#14579097

yes

—

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

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#14579076

yes

—

138

C₂×C₄

#14579087

yes

—

139

C₈

—

140

C₈

—

316

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₇₉⋊#338

yes

characteristic

147

C₂²

#19098324

yes

—

148

C₂²

#19098328

yes

characteristic

149

C₂²

#19098330

yes

characteristic

150

C₂²

#19098333

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₂

—

156

C₂

#23761273

yes

—

157

C₂

#23761277

yes

characteristic

158

C₂

#23761285

yes

characteristic

159

C₂

#23761300

yes

—

160

C₂

#23761308

yes

characteristic

	subgroup	G/H
center	C₂³	C₇₉⋊#95
commutator	C₂²×C₄×C₇₉	C₄²
Frattini	C₂²×C₄²	C₂×D₇₉
Fitting	C₇₉×#1817	C₂
socle	C₂³×C₇₉	#95
prime radical	C₂⁴×C₇₉	C₄⋊C₄
cosocle	C₂²×C₄²×C₇₉	C₂²
hypercenter	#1817	D₇₉
hypocenter	C₇₉	#7046
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1817

subgroup

G/H

center

C₂³

C₇₉⋊#95

commutator

C₂²×C₄×C₇₉

C₄²

Frattini

C₂²×C₄²

C₂×D₇₉

Fitting

C₇₉×#1817

C₂

socle

C₂³×C₇₉

#95

prime radical

C₂⁴×C₇₉

C₄⋊C₄

cosocle

C₂²×C₄²×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1817

N	H ≅ G/N	type
83. C₇₉	#7046	semidirect

H ≅ G/N

type

83. C₇₉

#7046

semidirect

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

subgroup	index
#23763162	2
C₇₉×#1817	2
#7046	79

subgroup

index

#23763162

C₇₉×#1817

#7046

Subgroups — #34174875

Conjugacy classes of subgroups

Complements

Maximal subgroups