Group #34174912 — subgroups

Subgroups — #34174912

Order 20224.

176 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 47 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	#23763553	C₂	1	1	yes	maximal, characteristic
2	#23761340	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1322	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₇₉⋊C₈)	C₄	1	1	yes	characteristic
5	C₂×Dic₇₉⋊C₈	—	2	1	no	—
6	C₂×Dic₇₉⋊C₈	—	2	1	no	—
7	#19098409	C₄	1	1	yes	characteristic
8	C₂²⋊(C₇₉⋊C₁₆)	—	2	1	no	—
9	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
10	C₂²⋊C₁₆×C₇₉	—	2	1	no	—
11	C₂²×C₇₉⋊C₈	—	2	1	no	—
12	C₂×C₄×Dic₇₉	—	2	1	no	—
13	C₂×C₇₉⋊C₁₆	—	4	1	no	—
14	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
15	Dic₇₉⋊C₈	—	4	2	no	—
16	Dic₇₉⋊C₈	—	4	2	no	—
17	C₇₉⋊#95	—	2	1	no	—
18	C₇₉⋊#95	C₈	1	1	yes	characteristic
19	C₇₉⋊#95	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₉	—	4	1	no	—
24	C₄⋊C₈×C₇₉	—	4	1	no	—
25	C₂²×Dic₇₉	—	2	1	no	—
26	C₂²×Dic₇₉	—	2	1	no	—
27	C₂×C₇₉⋊C₈	—	4	1	no	—
28	C₂×C₇₉⋊C₈	—	4	1	no	—
29	C₂×C₇₉⋊C₈	—	4	1	no	—
30	C₄×Dic₇₉	—	2	2	no	—
31	C₄×Dic₇₉	—	2	2	no	—
32	C₇₉⋊C₁₆	—	8	1	no	—
33	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₇₉	—	2	1	no	—
37	C₂×C₈×C₇₉	—	2	1	no	—
38	C₂×C₈×C₇₉	—	4	1	no	—
39	C₂×C₈×C₇₉	—	4	1	no	—
40	C₄²×C₇₉	—	2	1	no	—
41	C₄²×C₇₉	—	2	1	no	—
42	C₁₆×C₇₉	—	8	1	no	—
43	C₂³×C₇₉	C₂²⋊C₈	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	—	2	1	no	—
49	C₂×C₄×C₇₉	—	2	1	no	—
50	C₂×C₄×C₇₉	—	4	1	no	—
51	C₂×C₄×C₇₉	#99	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#101	1	1	yes	characteristic
53	C₂×Dic₇₉	—	2	2	no	—
54	C₂×Dic₇₉	—	2	2	no	—
55	C₂×Dic₇₉	—	4	1	no	—
56	C₂×Dic₇₉	—	4	1	no	—
57	C₂×Dic₇₉	—	4	1	no	—
58	C₂×Dic₇₉	—	4	1	no	—
59	C₈×C₇₉	—	4	1	no	—
60	C₈×C₇₉	—	4	1	no	—
61	C₇₉⋊C₈	—	8	1	no	—
62	C₂²×C₇₉	—	2	1	no	—
63	C₂²×C₇₉	—	2	1	no	—
64	C₂²×C₇₉	#324	1	1	yes	characteristic
65	C₂²×C₇₉	#350	1	1	yes	characteristic
66	C₂²×C₇₉	#350	1	1	yes	characteristic
67	C₄×C₇₉	—	2	1	no	—
68	C₄×C₇₉	—	4	1	no	—
69	C₄×C₇₉	—	4	1	no	—
70	C₄×C₇₉	#353	1	1	yes	characteristic
71	C₄×C₇₉	#356	1	1	yes	characteristic
72	Dic₇₉	—	4	2	no	—
73	Dic₇₉	—	4	2	no	—
74	#7066	—	79	1	no	maximal, Sylow
75	C₂×C₇₉	—	2	1	no	—
76	C₂×C₇₉	—	2	1	no	—
77	C₂×C₇₉	#1323	1	1	yes	characteristic
78	C₂×C₇₉	#1325	1	1	yes	characteristic
79	C₂×C₇₉	#1328	1	1	yes	characteristic
80	#1843	—	79	1	no	—
81	#1322	—	79	1	no	—
82	#1322	D₇₉	1	1	yes	characteristic
83	C₇₉	#7066	1	1	yes	characteristic, complemented, Sylow
84	C₂×C₄⋊C₈	—	79	1	no	—
85	C₂×C₄⋊C₈	—	158	1	no	—
86	C₂×C₄⋊C₈	—	158	1	no	—
87	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
88	#376	—	79	1	no	—
89	C₂²⋊C₁₆	—	2	1	no	—
90	C₂²⋊C₁₆	—	158	1	no	—
91	C₂²×C₈	—	158	1	no	—
92	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
93	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
94	C₂×C₄²	—	158	1	no	—
95	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
96	C₂×C₁₆	—	4	1	no	—
97	C₂×C₁₆	—	316	1	no	—
98	C₄⋊C₈	—	4	1	no	—
99	C₄⋊C₈	—	316	1	no	—
100	C₄⋊C₈	—	316	2	no	—
101	C₄⋊C₈	—	316	2	no	—
102	#95	—	79	1	no	—
103	#95	—	79	1	no	—
104	#95	—	158	1	no	—
105	C₂²×C₄	—	158	1	no	—
106	C₂²×C₄	—	158	1	no	—
107	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
108	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
109	C₂²×C₄	#11929607	1	1	yes	characteristic
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	4	1	no	—
114	C₂×C₈	—	316	1	no	—
115	C₂×C₈	—	316	1	no	—
116	C₂×C₈	—	316	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	158	2	no	—
120	C₄²	—	158	2	no	—
121	C₁₆	—	8	1	no	—
122	C₁₆	—	632	1	no	—
123	C₂³	(C₂×D₇₉)⋊C₈	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	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	316	1	no	—
132	C₂×C₄	—	316	1	no	—
133	C₂×C₄	—	316	1	no	—
134	C₂×C₄	—	316	1	no	—
135	C₂×C₄	—	158	2	no	—
136	C₂×C₄	—	158	2	no	—
137	C₂×C₄	#14579075	1	1	yes	characteristic
138	C₂×C₄	#14579079	1	1	yes	characteristic
139	C₈	—	4	1	no	—
140	C₈	—	4	1	no	—
141	C₈	—	632	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	#19098288	1	1	yes	characteristic
145	C₂²	#19098346	1	1	yes	characteristic
146	C₂²	#19098348	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	4	1	no	—
149	C₄	—	4	1	no	—
150	C₄	—	316	2	no	—
151	C₄	—	316	2	no	—
152	C₄	#19098355	1	1	yes	characteristic
153	C₄	#19098361	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#23761341	1	1	yes	characteristic
157	C₂	#23761345	1	1	yes	characteristic
158	C₂	#23761352	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763553

C₂

yes

maximal, characteristic

#23761340

C₂

yes

maximal, characteristic

C₇₉×#1322

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

C₄

yes

characteristic

C₂×Dic₇₉⋊C₈

—

C₂×Dic₇₉⋊C₈

—

#19098409

C₄

yes

characteristic

C₂²⋊(C₇₉⋊C₁₆)

—

C₂×C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₂²⋊C₁₆×C₇₉

—

C₂²×C₇₉⋊C₈

—

C₂×C₄×Dic₇₉

—

C₂×C₇₉⋊C₁₆

—

C₄⋊(C₇₉⋊C₈)

—

Dic₇₉⋊C₈

—

Dic₇₉⋊C₈

—

C₇₉⋊#95

—

C₇₉⋊#95

C₈

yes

characteristic

C₇₉⋊#95

C₈

yes

characteristic

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

—

C₄⋊C₈×C₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₉

M₁₆

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂²⋊C₈

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

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

#7066

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

D₇₉

yes

characteristic

C₇₉

#7066

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

158

—

C₂×C₄⋊C₈

—

158

—

C₂×C₄⋊C₈

C₂×D₇₉

yes

characteristic

#376

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

158

—

C₂²×C₈

—

158

—

C₂²×C₈

D₃₁₆

yes

characteristic

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

C₂×C₄²

—

158

—

C₂×C₄²

C₄×D₇₉

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

316

—

C₄⋊C₈

—

C₄⋊C₈

—

316

—

100

C₄⋊C₈

—

316

—

101

C₄⋊C₈

—

316

—

102

#95

—

103

#95

—

104

#95

—

158

—

105

C₂²×C₄

—

158

—

106

C₂²×C₄

—

158

—

107

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

108

C₂²×C₄

C₈×D₇₉

yes

characteristic

109

C₂²×C₄

#11929607

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

316

—

115

C₂×C₈

—

316

—

116

C₂×C₈

—

316

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

158

—

120

C₄²

—

158

—

121

C₁₆

—

122

C₁₆

—

632

—

123

C₂³

(C₂×D₇₉)⋊C₈

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

—

316

—

132

C₂×C₄

—

316

—

133

C₂×C₄

—

316

—

134

C₂×C₄

—

316

—

135

C₂×C₄

—

158

—

136

C₂×C₄

—

158

—

137

C₂×C₄

#14579075

yes

characteristic

138

C₂×C₄

#14579079

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

632

—

142

C₂²

—

143

C₂²

—

144

C₂²

#19098288

yes

characteristic

145

C₂²

#19098346

yes

characteristic

146

C₂²

#19098348

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

316

—

151

C₄

—

316

—

152

C₄

#19098355

yes

characteristic

153

C₄

#19098361

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

#23761341

yes

characteristic

157

C₂

#23761345

yes

characteristic

158

C₂

#23761352

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579075
commutator	C₂²×C₄×C₇₉	C₂×C₈
Frattini	C₂×C₄⋊C₈	C₂×D₇₉
Fitting	C₇₉×#1322	C₂
socle	C₂²×C₇₉	#324
prime radical	C₂³×C₇₉	C₂²⋊C₈
cosocle	C₂×C₄⋊C₈×C₇₉	C₂²
hypercenter	#1322	D₇₉
hypocenter	C₇₉	#7066
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1322

subgroup

G/H

center

C₂×C₄

#14579075

commutator

C₂²×C₄×C₇₉

C₂×C₈

Frattini

C₂×C₄⋊C₈

C₂×D₇₉

Fitting

C₇₉×#1322

C₂

socle

C₂²×C₇₉

#324

prime radical

C₂³×C₇₉

C₂²⋊C₈

cosocle

C₂×C₄⋊C₈×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1322

N	H ≅ G/N	type
83. C₇₉	#7066	semidirect

H ≅ G/N

type

83. C₇₉

#7066

semidirect

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

subgroup	index
#23763553	2
#23761340	2
C₇₉×#1322	2
#7066	79

subgroup

index

#23763553

#23761340

C₇₉×#1322

#7066

Subgroups — #34174912

Conjugacy classes of subgroups

Complements

Maximal subgroups