Group #34208287 — subgroups

Subgroups — #34208287

Order 20224.

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

normal subgroups only

This group has 238 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	#23765195	C₂	1	1	yes	maximal, characteristic
2	#23764190	C₂	1	2	yes	maximal
3	#23761504	C₂	1	1	yes	maximal, characteristic
4	#23761494	C₂	1	2	yes	maximal
5	C₇₉×#2151	C₂	1	1	yes	maximal, characteristic
6	#19098611	—	2	2	no	—
7	C₂×#14579097	C₂²	1	2	yes	—
8	C₂×#14579097	C₂²	1	2	yes	—
9	#19098393	—	2	2	no	—
10	#19098333	—	2	2	no	—
11	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	#405×C₇₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₇₉	—	2	2	no	—
15	C₂×#11929608	—	2	2	no	—
16	C₂×#11929608	—	2	2	no	—
17	#14579097	—	2	2	no	—
18	#14579097	—	2	2	no	—
19	#14579097	—	2	2	no	—
20	#14579097	Q₈	1	4	yes	—
21	#14579096	—	4	4	no	—
22	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
24	#117×C₇₉	D₄	1	1	yes	characteristic
25	M₃₂×C₇₉	—	2	2	no	—
26	C₂×C₁₆×C₇₉	—	2	1	no	—
27	C₂×C₁₆×C₇₉	—	2	2	no	—
28	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
29	#97×C₇₉	—	2	1	no	—
30	C₄×C₈×C₇₉	—	2	1	no	—
31	#11929608	—	2	4	no	—
32	#11929608	—	4	2	no	—
33	#11929608	—	4	2	no	—
34	#11929608	—	4	2	no	—
35	C₂×C₇₉⋊C₈	—	4	2	no	—
36	C₂×C₇₉⋊C₈	—	4	2	no	—
37	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₇₉	—	2	2	no	—
39	C₂×C₈×C₇₉	—	2	1	no	—
40	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
42	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
43	C₂×C₈×C₇₉	#41	1	2	yes	—
44	C₄⋊C₄×C₇₉	—	2	1	no	—
45	C₂²⋊C₄×C₇₉	Q₁₆	1	2	yes	—
46	C₄²×C₇₉	—	2	1	no	—
47	C₁₆×C₇₉	—	2	2	no	—
48	C₁₆×C₇₉	—	4	1	no	—
49	C₂³×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₄×C₇₉	—	2	2	no	—
51	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
53	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
54	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
55	C₈×C₇₉	—	4	1	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	4	no	—
60	C₇₉⋊C₈	—	8	2	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	—	4	2	no	—
65	C₄×C₇₉	#398	1	1	yes	characteristic
66	C₄×C₇₉	#469	1	1	yes	characteristic
67	C₄×C₇₉	#478	1	2	yes	—
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	—	79	1	no	—
73	#2151	D₇₉	1	1	yes	characteristic
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	C₂×D₇₉	1	1	yes	characteristic
80	#430	—	158	2	no	—
81	C₂×#108	—	79	2	no	—
82	C₂×#108	—	79	2	no	—
83	#405	C₂×D₇₉	1	1	yes	characteristic
84	#368	—	158	2	no	—
85	C₄⋊C₁₆	—	2	2	no	—
86	#344	—	158	2	no	—
87	C₂×M₁₆	—	158	2	no	—
88	C₂×M₁₆	—	158	2	no	—
89	C₂×M₁₆	D₃₁₆	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
91	#117	D₃₁₆	1	1	yes	characteristic
92	M₃₂	—	2	2	no	—
93	C₂×C₁₆	—	2	1	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
96	#108	—	316	4	no	—
97	#108	—	79	4	no	—
98	#108	—	158	2	no	—
99	#108	—	158	2	no	—
100	#108	—	158	2	no	—
101	#97	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
104	M₁₆	—	2	2	no	—
105	M₁₆	—	316	2	no	—
106	M₁₆	—	316	2	no	—
107	M₁₆	—	316	2	no	—
108	M₁₆	—	158	4	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	316	2	no	—
111	C₂×C₈	—	316	2	no	—
112	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
113	C₂×C₈	Q₈×D₇₉	1	2	yes	—
114	C₂×C₈	#11929623	1	2	yes	—
115	C₂×C₈	#11929624	1	1	yes	characteristic
116	C₄⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	Dic₃₁₆	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₁₆	—	2	2	no	—
120	C₁₆	—	4	1	no	—
121	C₂³	C₂×Dic₃₁₆	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	#14579135	1	1	yes	characteristic
124	C₂×C₄	#14579141	1	2	yes	—
125	C₂×C₄	#14579183	1	1	yes	characteristic
126	C₂×C₄	#14579189	1	2	yes	—
127	C₈	—	4	1	no	—
128	C₈	—	316	4	no	—
129	C₈	—	632	2	no	—
130	C₈	#14579142	1	1	yes	characteristic
131	C₈	#14579151	1	2	yes	—
132	C₈	#14579159	1	1	yes	characteristic
133	C₂²	—	4	1	no	—
134	C₂²	#19098804	1	1	yes	characteristic
135	C₂²	#19098859	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	#19098493	1	1	yes	characteristic
138	C₄	#19098816	1	1	yes	characteristic
139	C₄	#19098864	1	2	yes	—
140	C₂	—	2	1	no	—
141	C₂	—	2	2	no	—
142	C₂	#23764893	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765195

C₂

yes

maximal, characteristic

#23764190

C₂

yes

maximal

#23761504

C₂

yes

maximal, characteristic

#23761494

C₂

yes

maximal

C₇₉×#2151

C₂

yes

maximal, characteristic

#19098611

—

C₂×#14579097

C₂²

yes

—

C₂×#14579097

C₂²

yes

—

#19098393

—

#19098333

—

C₂×M₃₂×C₇₉

C₂²

yes

characteristic

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

#405×C₇₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₉

—

—

—

—

—

—

Q₈

yes

—

#14579096

—

C₂×M₁₆×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

#117×C₇₉

D₄

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

—

#97×C₇₉

—

C₄×C₈×C₇₉

—

—

—

—

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

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

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

Q₁₆

yes

—

C₄²×C₇₉

—

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₉

—

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

—

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

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

#13520

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

D₇₉

yes

—

—

—

C₇₉

#13520

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₇₉

yes

characteristic

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

#430

—

158

—

C₂×#108

—

C₂×#108

—

#405

C₂×D₇₉

yes

characteristic

#368

—

158

—

C₄⋊C₁₆

—

#344

—

158

—

C₂×M₁₆

—

158

—

C₂×M₁₆

—

158

—

C₂×M₁₆

D₃₁₆

yes

characteristic

C₂²×C₈

C₂²×D₇₉

yes

characteristic

#117

D₃₁₆

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₉

yes

—

#108

—

316

—

#108

—

#108

—

158

—

#108

—

158

—

100

#108

—

158

—

101

#97

—

102

C₄×C₈

—

103

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

104

M₁₆

—

105

M₁₆

—

316

—

106

M₁₆

—

316

—

107

M₁₆

—

316

—

108

M₁₆

—

158

—

109

C₂×C₈

—

110

C₂×C₈

—

316

—

111

C₂×C₈

—

316

—

112

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

113

C₂×C₈

Q₈×D₇₉

yes

—

114

C₂×C₈

#11929623

yes

—

115

C₂×C₈

#11929624

yes

characteristic

116

C₄⋊C₄

—

117

C₂²⋊C₄

Dic₃₁₆

yes

—

118

C₄²

—

119

C₁₆

—

120

C₁₆

—

121

C₂³

C₂×Dic₃₁₆

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

#14579135

yes

characteristic

124

C₂×C₄

#14579141

yes

—

125

C₂×C₄

#14579183

yes

characteristic

126

C₂×C₄

#14579189

yes

—

127

C₈

—

128

C₈

—

316

—

129

C₈

—

632

—

130

C₈

#14579142

yes

characteristic

131

C₈

#14579151

yes

—

132

C₈

#14579159

yes

characteristic

133

C₂²

—

134

C₂²

#19098804

yes

characteristic

135

C₂²

#19098859

yes

—

136

C₄

—

137

C₄

#19098493

yes

characteristic

138

C₄

#19098816

yes

characteristic

139

C₄

#19098864

yes

—

140

C₂

—

141

C₂

—

142

C₂

#23764893

yes

characteristic

	subgroup	G/H
center	C₄	#19098816
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2151	C₂
socle	C₂×C₇₉	#2087
prime radical	C₂³×C₇₉	C₂×Q₁₆
hypercenter	#2151	D₇₉
hypocenter	C₇₉	#13520
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405 ◁ #2151

subgroup

G/H

center

C₄

#19098816

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#2151

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 ◁ #405 ◁ #2151

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
#23765195	2
#23764190	2
#23761504	2
#23761494	2
C₇₉×#2151	2
#13520	79

subgroup

index

C₇₉×#2151

Subgroups — #34208287

Conjugacy classes of subgroups

Complements

Maximal subgroups