Group #34175495 — subgroups

Subgroups — #34175495

Order 20224.

184 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 87 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	#23765199	C₂	1	1	yes	maximal, characteristic
2	#23762148	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1568	C₂	1	1	yes	maximal, characteristic
4	#19098616	—	2	1	no	—
5	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
6	C₂²⋊(C₇₉⋊C₁₆)	C₄	1	2	yes	—
7	C₄×C₇₉⋊C₁₆	—	2	1	no	—
8	#19098316	—	2	1	no	—
9	#19098314	C₄	1	2	yes	—
10	#432×C₇₉	—	2	1	no	—
11	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	#335×C₇₉	C₄	1	2	yes	—
13	#334×C₇₉	—	2	1	no	—
14	C₂×C₇₉⋊C₁₆	—	2	2	no	—
15	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
16	C₂²⋊(C₇₉⋊C₈)	—	4	1	no	—
17	C₄×C₇₉⋊C₈	—	4	1	no	—
18	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
20	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
21	C₄⋊C₈×C₇₉	—	2	2	no	—
22	C₂²⋊C₈×C₇₉	—	4	1	no	—
23	#97×C₇₉	—	4	1	no	—
24	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
27	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₂×C₇₉⋊C₈	—	4	2	no	—
29	C₇₉⋊C₁₆	—	4	2	no	—
30	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
32	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
33	C₂×C₈×C₇₉	—	2	2	no	—
34	C₂×C₈×C₇₉	—	4	1	no	—
35	C₂×C₈×C₇₉	—	4	2	no	—
36	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₇₉	D₈	1	1	yes	characteristic
43	C₄²×C₇₉	Q₁₆	1	1	yes	characteristic
44	C₂³×C₇₉	#106	1	1	yes	characteristic
45	C₂×C₄×C₇₉	—	2	1	no	—
46	C₂×C₄×C₇₉	—	2	1	no	—
47	C₂×C₄×C₇₉	—	2	2	no	—
48	C₂×C₄×C₇₉	—	4	1	no	—
49	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
50	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
51	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
55	C₈×C₇₉	—	2	1	no	—
56	C₈×C₇₉	—	2	1	no	—
57	C₈×C₇₉	—	2	1	no	—
58	C₈×C₇₉	—	8	2	no	—
59	C₈×C₇₉	#108	1	2	yes	—
60	C₇₉⋊C₈	—	8	2	no	—
61	C₂²×C₇₉	—	2	1	no	—
62	C₂²×C₇₉	—	2	1	no	—
63	C₂²×C₇₉	—	2	1	no	—
64	C₂²×C₇₉	#340	1	1	yes	characteristic
65	C₄×C₇₉	—	2	1	no	—
66	C₄×C₇₉	—	2	1	no	—
67	C₄×C₇₉	—	2	1	no	—
68	C₄×C₇₉	—	2	1	no	—
69	C₄×C₇₉	#341	1	1	yes	characteristic
70	C₄×C₇₉	#344	1	1	yes	characteristic
71	C₄×C₇₉	#359	1	2	yes	—
72	#7289	—	79	1	no	maximal, Sylow
73	C₂×C₇₉	—	4	1	no	—
74	C₂×C₇₉	#1301	1	1	yes	characteristic
75	C₂×C₇₉	#1382	1	1	yes	characteristic
76	C₂×C₇₉	#1389	1	1	yes	characteristic
77	#2162	—	79	1	no	—
78	#1568	D₇₉	1	1	yes	characteristic
79	#1564	—	79	1	no	—
80	C₇₉	#7289	1	1	yes	characteristic, complemented, Sylow
81	#432	—	2	1	no	—
82	#431	—	158	1	no	—
83	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
84	C₄⋊C₁₆	—	158	1	no	—
85	C₂²⋊C₁₆	—	79	2	no	—
86	C₄×C₁₆	—	158	1	no	—
87	#335	—	158	1	no	—
88	#335	Dic₇₉	1	2	yes	—
89	#334	—	2	1	no	—
90	#334	—	79	2	no	—
91	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
92	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
93	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
94	C₂×C₁₆	—	158	2	no	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	158	2	no	—
97	C₂²⋊C₈	—	4	1	no	—
98	C₂²⋊C₈	—	316	1	no	—
99	#97	—	4	1	no	—
100	C₄×C₈	—	316	1	no	—
101	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
102	C₄×C₈	D₃₁₆	1	1	yes	characteristic
103	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
104	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
105	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
106	C₂²×C₄	#11929611	1	1	yes	characteristic
107	C₂²×C₄	#11929612	1	1	yes	characteristic
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	4	1	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	316	2	no	—
112	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
113	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
114	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
115	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
116	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
117	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
118	C₄²	#11929609	1	1	yes	characteristic
119	C₄²	#11929614	1	1	yes	characteristic
120	C₁₆	—	316	2	no	—
121	C₂³	#14579092	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
127	C₂×C₄	#14579081	1	1	yes	characteristic
128	C₂×C₄	#14579082	1	1	yes	characteristic
129	C₂×C₄	#14579084	1	1	yes	characteristic
130	C₂×C₄	#14579085	1	1	yes	characteristic
131	C₂×C₄	#14579094	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	8	2	no	—
136	C₈	—	632	2	no	—
137	C₈	#14579096	1	2	yes	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#19098325	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	#19098327	1	1	yes	characteristic
147	C₄	#19098334	1	1	yes	characteristic
148	C₄	#19098370	1	2	yes	—
149	C₂	—	4	1	no	—
150	C₂	#23761295	1	1	yes	characteristic
151	C₂	#23761495	1	1	yes	characteristic
152	C₂	#23761514	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765199

C₂

yes

maximal, characteristic

#23762148

C₂

yes

maximal, characteristic

C₇₉×#1568

C₂

yes

maximal, characteristic

#19098616

—

C₄⋊(C₇₉⋊C₁₆)

—

C₂²⋊(C₇₉⋊C₁₆)

C₄

yes

—

C₄×C₇₉⋊C₁₆

—

#19098316

—

#19098314

C₄

yes

—

#432×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#335×C₇₉

C₄

yes

—

#334×C₇₉

—

C₂×C₇₉⋊C₁₆

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

Q₈

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

#97×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₂×C₇₉⋊C₈

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄²

yes

characteristic

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₄²×C₇₉

D₈

yes

characteristic

C₄²×C₇₉

Q₁₆

yes

characteristic

C₂³×C₇₉

#106

yes

characteristic

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

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#108

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#340

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

#7289

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

yes

characteristic

#1564

—

C₇₉

#7289

yes

characteristic, complemented, Sylow

#432

—

#431

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₄⋊C₁₆

—

158

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

158

—

#335

—

158

—

#335

Dic₇₉

yes

—

#334

—

#334

—

C₂²×C₈

Dic₁₅₈

yes

characteristic

C₂²×C₈

C₄×D₇₉

yes

characteristic

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

C₂×C₁₆

—

158

—

C₄⋊C₈

—

C₄⋊C₈

—

158

—

C₂²⋊C₈

—

C₂²⋊C₈

—

316

—

#97

—

100

C₄×C₈

—

316

—

101

C₄×C₈

C₄×D₇₉

yes

characteristic

102

C₄×C₈

D₃₁₆

yes

characteristic

103

C₄×C₈

C₂×Dic₇₉

yes

characteristic

104

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

105

C₂²×C₄

Dic₇₉⋊C₄

yes

characteristic

106

C₂²×C₄

#11929611

yes

characteristic

107

C₂²×C₄

#11929612

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

316

—

112

C₂×C₈

C₄×Dic₇₉

yes

characteristic

113

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

characteristic

114

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

115

C₂×C₈

C₄⋊Dic₇₉

yes

characteristic

116

C₄²

(C₂×D₇₉)⋊C₄

yes

characteristic

117

C₄²

C₂²⋊Dic₇₉

yes

characteristic

118

C₄²

#11929609

yes

characteristic

119

C₄²

#11929614

yes

characteristic

120

C₁₆

—

316

—

121

C₂³

#14579092

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₇₉⋊#95

yes

characteristic

127

C₂×C₄

#14579081

yes

characteristic

128

C₂×C₄

#14579082

yes

characteristic

129

C₂×C₄

#14579084

yes

characteristic

130

C₂×C₄

#14579085

yes

characteristic

131

C₂×C₄

#14579094

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

632

—

137

C₈

#14579096

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

#19098325

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#19098327

yes

characteristic

147

C₄

#19098334

yes

characteristic

148

C₄

#19098370

yes

—

149

C₂

—

150

C₂

#23761295

yes

characteristic

151

C₂

#23761495

yes

characteristic

152

C₂

#23761514

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579082
commutator	C₂×C₈×C₇₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₇₉×#1568	C₂
socle	C₂²×C₇₉	#340
prime radical	C₂³×C₇₉	#106
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	#1568	D₇₉
hypocenter	C₇₉	#7289
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1568

subgroup

G/H

center

C₂×C₄

#14579082

commutator

C₂×C₈×C₇₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₇₉

Fitting

C₇₉×#1568

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄×C₈×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1568

N	H ≅ G/N	type
80. C₇₉	#7289	semidirect

H ≅ G/N

type

80. C₇₉

#7289

semidirect

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

subgroup	index
#23765199	2
#23762148	2
C₇₉×#1568	2
#7289	79

subgroup

index

#23765199

#23762148

C₇₉×#1568

#7289

Subgroups — #34175495

Conjugacy classes of subgroups

Complements

Maximal subgroups