Group #34174880 — subgroups

Subgroups — #34174880

Order 20224.

304 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 115 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	#23763253	C₂	1	2	yes	maximal
2	C₇₉×#1831	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
4	C₂×#14579072	—	2	2	no	—
5	#19098318	C₄	1	4	yes	—
6	C₂×C₄⋊C₈×C₇₉	C₄	1	2	yes	—
7	C₄³×C₇₉	C₂²	1	1	yes	characteristic
8	#336×C₇₉	—	2	2	no	—
9	C₂²×C₇₉⋊C₈	—	2	4	no	—
10	C₄⋊(C₇₉⋊C₈)	—	2	8	no	—
11	#14579072	—	4	2	no	—
12	#14579072	—	4	2	no	—
13	C₂²×C₈×C₇₉	—	2	2	no	—
14	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₇₉	D₄	1	2	yes	—
17	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₇₉	Q₈	1	1	yes	characteristic
19	C₄⋊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	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
24	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
25	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₇₉	—	4	2	no	—
29	C₂×C₈×C₇₉	—	4	4	no	—
30	C₄²×C₇₉	—	2	1	no	—
31	C₄²×C₇₉	—	2	1	no	—
32	C₄²×C₇₉	—	2	4	no	—
33	C₄²×C₇₉	—	4	2	no	—
34	C₄²×C₇₉	—	4	1	no	—
35	C₄²×C₇₉	D₈	1	1	yes	characteristic
36	C₄²×C₇₉	SD₁₆	1	2	yes	—
37	C₄²×C₇₉	Q₁₆	1	1	yes	characteristic
38	C₂³×C₇₉	#95	1	1	yes	characteristic
39	C₂×C₄×C₇₉	—	2	1	no	—
40	C₂×C₄×C₇₉	—	2	2	no	—
41	C₂×C₄×C₇₉	—	2	1	no	—
42	C₂×C₄×C₇₉	—	2	2	no	—
43	C₂×C₄×C₇₉	—	2	2	no	—
44	C₂×C₄×C₇₉	—	2	2	no	—
45	C₂×C₄×C₇₉	—	2	1	no	—
46	C₂×C₄×C₇₉	—	2	2	no	—
47	C₂×C₄×C₇₉	—	2	2	no	—
48	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
49	C₂×C₄×C₇₉	#101	1	1	yes	characteristic
50	C₂×C₄×C₇₉	#102	1	2	yes	—
51	C₂×C₄×C₇₉	#103	1	2	yes	—
52	C₂×C₄×C₇₉	#104	1	4	yes	—
53	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
55	C₈×C₇₉	—	8	2	no	—
56	C₇₉⋊C₈	—	4	8	no	—
57	C₂²×C₇₉	#338	1	1	yes	characteristic
58	C₂²×C₇₉	#339	1	2	yes	—
59	C₂²×C₇₉	#340	1	1	yes	characteristic
60	C₂²×C₇₉	#343	1	1	yes	characteristic
61	C₂²×C₇₉	#344	1	2	yes	—
62	C₄×C₇₉	—	2	1	no	—
63	C₄×C₇₉	—	2	1	no	—
64	C₄×C₇₉	—	2	2	no	—
65	C₄×C₇₉	—	2	2	no	—
66	C₄×C₇₉	—	4	2	no	—
67	C₄×C₇₉	—	4	1	no	—
68	C₄×C₇₉	#329	1	2	yes	—
69	C₄×C₇₉	#330	1	2	yes	—
70	#7049	—	79	1	no	maximal, Sylow
71	C₂×C₇₉	#1291	1	2	yes	—
72	C₂×C₇₉	#1292	1	1	yes	characteristic
73	C₂×C₇₉	#1295	1	1	yes	characteristic
74	C₂×C₇₉	#1303	1	2	yes	—
75	C₂×C₇₉	#1307	1	1	yes	characteristic
76	#1831	D₇₉	1	1	yes	characteristic
77	#1767	—	79	2	no	—
78	C₇₉	#7049	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₄⋊C₈	—	158	2	no	—
80	C₂×C₄⋊C₈	Dic₇₉	1	2	yes	—
81	C₂×#97	—	158	2	no	—
82	C₄³	C₂×D₇₉	1	1	yes	characteristic
83	#336	—	2	2	no	—
84	#336	—	79	4	no	—
85	C₂²×C₈	—	2	2	no	—
86	C₂²×C₈	—	158	4	no	—
87	C₂×C₄²	Dic₁₅₈	1	1	yes	characteristic
88	C₂×C₄²	C₄×D₇₉	1	2	yes	—
89	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
90	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
91	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
92	C₄⋊C₈	—	4	2	no	—
93	C₄⋊C₈	—	158	8	no	—
94	#97	—	316	2	no	—
95	#97	—	316	2	no	—
96	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
97	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
98	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
99	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
100	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	4	4	no	—
103	C₂×C₈	—	316	4	no	—
104	C₂×C₈	—	316	4	no	—
105	C₂×C₈	—	158	8	no	—
106	C₄²	—	2	1	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	2	4	no	—
109	C₄²	—	4	1	no	—
110	C₄²	—	4	2	no	—
111	C₄²	D₆₃₂	1	1	yes	characteristic
112	C₄²	#11929613	1	2	yes	—
113	C₄²	Dic₃₁₆	1	1	yes	characteristic
114	C₂³	C₇₉⋊#95	1	1	yes	characteristic
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	#14579078	1	1	yes	characteristic
125	C₂×C₄	#14579080	1	1	yes	characteristic
126	C₂×C₄	#14579083	1	2	yes	—
127	C₂×C₄	#14579086	1	2	yes	—
128	C₂×C₄	#14579088	1	4	yes	—
129	C₂×C₄	#14579093	1	1	yes	characteristic
130	C₂×C₄	#14579095	1	1	yes	characteristic
131	C₈	—	8	2	no	—
132	C₈	—	316	8	no	—
133	C₂²	C₇₉⋊#338	1	1	yes	characteristic
134	C₂²	#19098322	1	2	yes	—
135	C₂²	#19098326	1	1	yes	characteristic
136	C₂²	#19098332	1	1	yes	characteristic
137	C₂²	#19098335	1	2	yes	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	2	no	—
143	C₄	—	4	1	no	—
144	C₄	#19098303	1	2	yes	—
145	C₄	#19098305	1	2	yes	—
146	C₂	#23761271	1	2	yes	—
147	C₂	#23761275	1	1	yes	characteristic
148	C₂	#23761281	1	1	yes	characteristic
149	C₂	#23761302	1	2	yes	—
150	C₂	#23761311	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763253

C₂

yes

maximal

C₇₉×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

—

C₂×#14579072

—

#19098318

C₄

yes

—

C₂×C₄⋊C₈×C₇₉

C₄

yes

—

C₄³×C₇₉

C₂²

yes

characteristic

#336×C₇₉

—

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

—

#14579072

—

#14579072

—

C₂²×C₈×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

Q₈

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₄²

yes

characteristic

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

D₈

yes

characteristic

C₄²×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

Q₁₆

yes

characteristic

C₂³×C₇₉

#95

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

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

—

C₂×C₄×C₇₉

#103

yes

—

C₂×C₄×C₇₉

#104

yes

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₈×C₇₉

—

C₇₉⋊C₈

—

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

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#329

yes

—

C₄×C₇₉

#330

yes

—

#7049

—

maximal, Sylow

C₂×C₇₉

#1291

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#1307

yes

characteristic

#1831

D₇₉

yes

characteristic

#1767

—

C₇₉

#7049

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

158

—

C₂×C₄⋊C₈

Dic₇₉

yes

—

C₂×#97

—

158

—

C₄³

C₂×D₇₉

yes

characteristic

#336

—

#336

—

C₂²×C₈

—

C₂²×C₈

—

158

—

C₂×C₄²

Dic₁₅₈

yes

characteristic

C₂×C₄²

C₄×D₇₉

yes

—

C₂×C₄²

D₃₁₆

yes

characteristic

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

C₂×C₄²

C₂²⋊D₇₉

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

158

—

#97

—

316

—

#97

—

316

—

C₂²×C₄

C₄×Dic₇₉

yes

characteristic

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

C₂²×C₄

Dic₇₉⋊C₄

yes

—

100

C₂²×C₄

C₄⋊Dic₇₉

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

316

—

104

C₂×C₈

—

316

—

105

C₂×C₈

—

158

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

D₆₃₂

yes

characteristic

112

C₄²

#11929613

yes

—

113

C₄²

Dic₃₁₆

yes

characteristic

114

C₂³

C₇₉⋊#95

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#14579078

yes

characteristic

125

C₂×C₄

#14579080

yes

characteristic

126

C₂×C₄

#14579083

yes

—

127

C₂×C₄

#14579086

yes

—

128

C₂×C₄

#14579088

yes

—

129

C₂×C₄

#14579093

yes

characteristic

130

C₂×C₄

#14579095

yes

characteristic

131

C₈

—

132

C₈

—

316

—

133

C₂²

C₇₉⋊#338

yes

characteristic

134

C₂²

#19098322

yes

—

135

C₂²

#19098326

yes

characteristic

136

C₂²

#19098332

yes

characteristic

137

C₂²

#19098335

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#19098303

yes

—

145

C₄

#19098305

yes

—

146

C₂

#23761271

yes

—

147

C₂

#23761275

yes

characteristic

148

C₂

#23761281

yes

characteristic

149

C₂

#23761302

yes

—

150

C₂

#23761311

yes

characteristic

	subgroup	G/H
center	C₂³	C₇₉⋊#95
commutator	C₂²×C₄×C₇₉	C₄²
Frattini	C₄³	C₂×D₇₉
Fitting	C₇₉×#1831	C₂
socle, prime radical	C₂³×C₇₉	#95
cosocle	C₄³×C₇₉	C₂²
hypercenter	#1831	D₇₉
hypocenter	C₇₉	#7049
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₇₉⋊#95

commutator

C₂²×C₄×C₇₉

C₄²

Frattini

C₄³

C₂×D₇₉

Fitting

C₇₉×#1831

C₂

socle, prime radical

C₂³×C₇₉

#95

cosocle

C₄³×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
78. C₇₉	#7049	semidirect

H ≅ G/N

type

78. C₇₉

#7049

semidirect

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

subgroup	index
#23763253	2
C₇₉×#1831	2
#7049	79

subgroup

index

#23763253

C₇₉×#1831

#7049

Subgroups — #34174880

Conjugacy classes of subgroups

Complements

Maximal subgroups