Group #34208288 — subgroups

Subgroups — #34208288

Order 20224.

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

normal subgroups only

This group has 238 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	#23765160	C₂	1	1	yes	maximal, characteristic
2	#23764188	C₂	1	2	yes	maximal
3	#23761504	C₂	1	1	yes	maximal, characteristic
4	#23761493	C₂	1	2	yes	maximal
5	C₇₉×#2161	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	#19098611	—	2	2	no	—
8	C₂×#14579097	C₂²	1	2	yes	—
9	#19098518	C₂²	1	1	yes	characteristic
10	Dic₇₉⋊C₁₆	—	2	2	no	—
11	#19098334	—	2	2	no	—
12	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×#108×C₇₉	C₂²	1	2	yes	—
14	#368×C₇₉	—	2	2	no	—
15	C₂×#11929608	—	2	2	no	—
16	C₂×#11929608	D₄	1	1	yes	characteristic
17	#14579120	D₄	1	1	yes	characteristic
18	C₂×C₇₉⋊C₁₆	—	2	1	no	—
19	C₂×C₇₉⋊C₁₆	—	2	2	no	—
20	#14579097	—	2	2	no	—
21	#14579097	Q₈	1	4	yes	—
22	#14579096	—	4	4	no	—
23	#14579071	—	2	1	no	—
24	C₈×Dic₇₉	—	2	1	no	—
25	C₂×M₁₆×C₇₉	—	2	2	no	—
26	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
27	M₃₂×C₇₉	—	2	2	no	—
28	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
29	#108×C₇₉	—	2	2	no	—
30	#108×C₇₉	—	2	2	no	—
31	#11929608	—	2	4	no	—
32	#11929608	—	2	2	no	—
33	#11929608	—	4	2	no	—
34	C₂×C₇₉⋊C₈	—	2	1	no	—
35	C₂×C₇₉⋊C₈	—	4	2	no	—
36	C₄⋊Dic₇₉	—	2	1	no	—
37	C₂²⋊Dic₇₉	Q₁₆	1	2	yes	—
38	C₄×Dic₇₉	—	2	1	no	—
39	C₇₉⋊C₁₆	—	4	1	no	—
40	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
41	M₁₆×C₇₉	—	4	2	no	—
42	M₁₆×C₇₉	—	4	2	no	—
43	C₂×C₈×C₇₉	—	4	2	no	—
44	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
46	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
47	C₂×C₈×C₇₉	#41	1	2	yes	—
48	C₁₆×C₇₉	—	2	2	no	—
49	C₂³×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
52	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
53	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
54	C₂×Dic₇₉	—	2	2	no	—
55	C₈×C₇₉	—	8	2	no	—
56	C₈×C₇₉	#123	1	1	yes	characteristic
57	C₈×C₇₉	#125	1	2	yes	—
58	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
59	C₇₉⋊C₈	—	4	1	no	—
60	C₇₉⋊C₈	—	4	4	no	—
61	C₂²×C₇₉	—	4	1	no	—
62	C₂²×C₇₉	#467	1	1	yes	characteristic
63	C₂²×C₇₉	#477	1	2	yes	—
64	C₄×C₇₉	#398	1	1	yes	characteristic
65	C₄×C₇₉	#469	1	1	yes	characteristic
66	C₄×C₇₉	#478	1	2	yes	—
67	Dic₇₉	—	4	2	no	—
68	#13520	—	79	1	no	maximal, Sylow
69	C₂×C₇₉	—	2	1	no	—
70	C₂×C₇₉	—	2	2	no	—
71	C₂×C₇₉	#2087	1	1	yes	characteristic
72	#2161	D₇₉	1	1	yes	characteristic
73	#2151	—	79	1	no	—
74	#1953	—	79	2	no	—
75	#1385	—	79	1	no	—
76	#1381	—	79	2	no	—
77	C₇₉	#13520	1	1	yes	characteristic, complemented, Sylow
78	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
79	C₂²×C₁₆	—	79	1	no	—
80	#430	—	158	2	no	—
81	C₂×#108	—	79	2	no	—
82	C₂×#108	C₂×D₇₉	1	2	yes	—
83	#405	—	79	1	no	—
84	#368	—	2	2	no	—
85	C₄⋊C₁₆	—	158	2	no	—
86	#344	—	158	2	no	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	79	1	no	—
89	C₂×M₁₆	—	158	2	no	—
90	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
91	#117	—	79	1	no	—
92	M₃₂	—	2	2	no	—
93	C₂×C₁₆	—	158	1	no	—
94	C₂×C₁₆	—	158	2	no	—
95	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
96	#108	—	2	2	no	—
97	#108	—	2	2	no	—
98	#108	—	316	4	no	—
99	#108	—	79	4	no	—
100	#108	—	158	2	no	—
101	#97	—	158	1	no	—
102	C₄×C₈	—	158	1	no	—
103	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
104	M₁₆	—	4	2	no	—
105	M₁₆	—	4	2	no	—
106	M₁₆	—	316	2	no	—
107	M₁₆	—	158	4	no	—
108	M₁₆	—	158	2	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	316	2	no	—
111	C₂×C₈	—	158	1	no	—
112	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
113	C₂×C₈	Q₈×D₇₉	1	2	yes	—
114	C₂×C₈	#11929623	1	1	yes	characteristic
115	C₂×C₈	#11929624	1	2	yes	—
116	C₄⋊C₄	—	158	1	no	—
117	C₂²⋊C₄	—	79	2	no	—
118	C₄²	—	158	1	no	—
119	C₁₆	—	2	2	no	—
120	C₁₆	—	316	1	no	—
121	C₂³	Q₁₆×D₇₉	1	1	yes	characteristic
122	C₂×C₄	—	158	2	no	—
123	C₂×C₄	#14579132	1	1	yes	characteristic
124	C₂×C₄	#14579139	1	2	yes	—
125	C₂×C₄	#14579184	1	1	yes	characteristic
126	C₂×C₄	#14579185	1	2	yes	—
127	C₈	—	8	2	no	—
128	C₈	—	316	1	no	—
129	C₈	—	316	4	no	—
130	C₈	#14579144	1	1	yes	characteristic
131	C₈	#14579151	1	2	yes	—
132	C₈	#14579159	1	1	yes	characteristic
133	C₂²	—	4	1	no	—
134	C₂²	#19098801	1	1	yes	characteristic
135	C₂²	#19098857	1	2	yes	—
136	C₄	—	316	2	no	—
137	C₄	#19098494	1	1	yes	characteristic
138	C₄	#19098813	1	1	yes	characteristic
139	C₄	#19098866	1	2	yes	—
140	C₂	—	2	1	no	—
141	C₂	—	2	2	no	—
142	C₂	#23764894	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765160

C₂

yes

maximal, characteristic

#23764188

C₂

yes

maximal

#23761504

C₂

yes

maximal, characteristic

#23761493

C₂

yes

maximal

C₇₉×#2161

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₁₆

C₂²

yes

characteristic

#19098611

—

C₂×#14579097

C₂²

yes

—

#19098518

C₂²

yes

characteristic

Dic₇₉⋊C₁₆

—

#19098334

—

C₂×M₃₂×C₇₉

C₂²

yes

characteristic

C₂×#108×C₇₉

C₂²

yes

—

#368×C₇₉

—

C₂×#11929608

—

C₂×#11929608

D₄

yes

characteristic

#14579120

D₄

yes

characteristic

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

#14579097

—

#14579097

Q₈

yes

—

#14579096

—

#14579071

—

C₈×Dic₇₉

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

—

#108×C₇₉

—

#108×C₇₉

—

#11929608

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

Q₁₆

yes

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

#41

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

#135

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#123

yes

characteristic

C₈×C₇₉

#125

yes

—

C₈×C₇₉

C₄⋊Q₈

yes

characteristic

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#467

yes

characteristic

C₂²×C₇₉

#477

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

#13520

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#2087

yes

characteristic

#2161

D₇₉

yes

—

—

—

—

C₇₉

#13520

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₇₉

yes

characteristic

C₂²×C₁₆

—

#430

—

158

—

C₂×#108

—

C₂×#108

C₂×D₇₉

yes

—

#405

—

#368

—

C₄⋊C₁₆

—

158

—

#344

—

158

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

158

—

C₂²×C₈

C₂²×D₇₉

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

158

—

C₂×C₁₆

—

158

—

C₂×C₁₆

C₂²⋊D₇₉

yes

—

#108

—

#108

—

#108

—

316

—

#108

—

100

#108

—

158

—

101

#97

—

158

—

102

C₄×C₈

—

158

—

103

C₂²×C₄

D₄×D₇₉

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

316

—

107

M₁₆

—

158

—

108

M₁₆

—

158

—

109

C₂×C₈

—

110

C₂×C₈

—

316

—

111

C₂×C₈

—

158

—

112

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

113

C₂×C₈

Q₈×D₇₉

yes

—

114

C₂×C₈

#11929623

yes

characteristic

115

C₂×C₈

#11929624

yes

—

116

C₄⋊C₄

—

158

—

117

C₂²⋊C₄

—

118

C₄²

—

158

—

119

C₁₆

—

120

C₁₆

—

316

—

121

C₂³

Q₁₆×D₇₉

yes

characteristic

122

C₂×C₄

—

158

—

123

C₂×C₄

#14579132

yes

characteristic

124

C₂×C₄

#14579139

yes

—

125

C₂×C₄

#14579184

yes

characteristic

126

C₂×C₄

#14579185

yes

—

127

C₈

—

128

C₈

—

316

—

129

C₈

—

316

—

130

C₈

#14579144

yes

characteristic

131

C₈

#14579151

yes

—

132

C₈

#14579159

yes

characteristic

133

C₂²

—

134

C₂²

#19098801

yes

characteristic

135

C₂²

#19098857

yes

—

136

C₄

—

316

—

137

C₄

#19098494

yes

characteristic

138

C₄

#19098813

yes

characteristic

139

C₄

#19098866

yes

—

140

C₂

—

141

C₂

—

142

C₂

#23764894

yes

characteristic

	subgroup	G/H
center	C₄	#19098813
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2161	C₂
socle	C₂×C₇₉	#2087
prime radical	C₂³×C₇₉	C₂×Q₁₆
hypercenter	#2161	D₇₉
hypocenter	C₇₉	#13520
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

subgroup

G/H

center

C₄

#19098813

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#2161

C₂

socle

C₂×C₇₉

#2087

prime radical

C₂³×C₇₉

C₂×Q₁₆

D₇₉

C₇₉

G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

N	H ≅ G/N	type
77. C₇₉	#13520	semidirect

H ≅ G/N

type

77. C₇₉

#13520

semidirect

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

subgroup	index
#23765160	2
#23764188	2
#23761504	2
#23761493	2
C₇₉×#2161	2
#13520	79

subgroup

index

C₇₉×#2161

Subgroups — #34208288

Conjugacy classes of subgroups

Complements

Maximal subgroups