Group #34175139 — subgroups

Subgroups — #34175139

Order 20224.

242 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 151 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	C₂×#19098316	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
4	#19098316	—	2	2	no	—
5	#19098316	C₄	1	4	yes	—
6	C₂²×C₁₆×C₇₉	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
8	C₄×C₁₆×C₇₉	—	2	2	no	—
9	C₂²×C₇₉⋊C₈	—	4	2	no	—
10	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
11	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
12	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₇₉	—	2	2	no	—
16	C₂×C₁₆×C₇₉	—	2	2	no	—
17	C₂×C₁₆×C₇₉	C₈	1	4	yes	complemented
18	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₇₉	D₄	1	2	yes	—
20	C₂×C₇₉⋊C₈	—	4	4	no	—
21	C₂×C₇₉⋊C₈	—	8	2	no	—
22	C₂×C₇₉⋊C₈	—	8	2	no	—
23	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
25	C₂²×C₄×C₇₉	Q₁₆	1	1	yes	characteristic
26	C₂×C₈×C₇₉	—	2	2	no	—
27	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
28	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
32	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
33	C₄²×C₇₉	SD₁₆	1	2	yes	—
34	C₁₆×C₇₉	—	2	4	no	—
35	C₂³×C₇₉	#107	1	1	yes	characteristic
36	C₂×C₄×C₇₉	—	2	2	no	—
37	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
38	C₂×C₄×C₇₉	#102	1	2	yes	—
39	C₂×C₄×C₇₉	#103	1	2	yes	—
40	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
41	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
42	C₂×C₄×C₇₉	#108	1	2	yes	—
43	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
44	C₂×C₄×C₇₉	SD₃₂	1	2	yes	—
45	C₂×C₄×C₇₉	Q₃₂	1	1	yes	characteristic
46	C₈×C₇₉	—	2	2	no	—
47	C₈×C₇₉	C₄×C₈	1	1	yes	characteristic
48	C₈×C₇₉	#97	1	1	yes	characteristic
49	C₈×C₇₉	C₂²⋊C₈	1	2	yes	—
50	C₇₉⋊C₈	—	8	4	no	—
51	C₂²×C₇₉	#335	1	2	yes	—
52	C₂²×C₇₉	#340	1	1	yes	characteristic
53	C₂²×C₇₉	#366	1	1	yes	characteristic
54	C₂²×C₇₉	#367	1	1	yes	characteristic
55	C₂²×C₇₉	#368	1	2	yes	—
56	C₄×C₇₉	—	2	2	no	—
57	C₄×C₇₉	#334	1	2	yes	—
58	C₄×C₇₉	#336	1	1	yes	characteristic
59	C₄×C₇₉	#341	1	1	yes	characteristic
60	C₄×C₇₉	#357	1	2	yes	—
61	C₄×C₇₉	#358	1	2	yes	—
62	#7154	—	79	1	no	maximal, Sylow
63	C₂×C₇₉	#1277	1	1	yes	characteristic
64	C₂×C₇₉	#1371	1	2	yes	—
65	C₂×C₇₉	#1372	1	2	yes	—
66	C₂×C₇₉	#1380	1	1	yes	characteristic
67	C₂×C₇₉	#1381	1	1	yes	characteristic
68	C₂×C₄×C₁₆	D₇₉	1	1	yes	characteristic
69	C₂×#335	—	79	2	no	—
70	C₇₉	#7154	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₁₆	Dic₇₉	1	2	yes	—
72	C₂×C₄⋊C₈	—	158	2	no	—
73	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
74	C₄×C₁₆	—	2	2	no	—
75	#335	—	79	4	no	—
76	#335	—	158	2	no	—
77	C₂²×C₈	—	316	2	no	—
78	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
79	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
80	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
81	C₂×C₁₆	—	2	2	no	—
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	C₇₉⋊C₈	1	4	yes	complemented
84	C₄⋊C₈	—	316	2	no	—
85	C₄⋊C₈	—	158	4	no	—
86	C₄×C₈	C₄×D₇₉	1	2	yes	—
87	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
88	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
89	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
90	C₂²×C₄	Dic₃₁₆	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	316	4	no	—
93	C₂×C₈	—	632	2	no	—
94	C₂×C₈	—	632	2	no	—
95	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
96	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
97	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
98	C₂×C₈	C₂×C₇₉⋊C₈	1	2	yes	—
99	C₂×C₈	#11929608	1	2	yes	—
100	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
101	C₄²	#11929613	1	2	yes	—
102	C₁₆	—	2	4	no	—
103	C₂³	#14579095	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
106	C₂×C₄	#14579083	1	2	yes	—
107	C₂×C₄	#14579086	1	2	yes	—
108	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
109	C₂×C₄	#14579093	1	1	yes	characteristic
110	C₂×C₄	#14579097	1	2	yes	—
111	C₂×C₄	D₁₂₆₄	1	1	yes	characteristic
112	C₂×C₄	#14579106	1	2	yes	—
113	C₂×C₄	Dic₆₃₂	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	632	4	no	—
116	C₈	C₄×C₇₉⋊C₈	1	1	yes	characteristic
117	C₈	#14579072	1	1	yes	characteristic
118	C₈	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
119	C₂²	#19098316	1	2	yes	—
120	C₂²	#19098326	1	1	yes	characteristic
121	C₂²	#19098389	1	1	yes	characteristic
122	C₂²	#19098391	1	1	yes	characteristic
123	C₂²	#19098393	1	2	yes	—
124	C₄	—	2	2	no	—
125	C₄	#19098314	1	2	yes	—
126	C₄	#19098318	1	1	yes	characteristic
127	C₄	#19098328	1	1	yes	characteristic
128	C₄	#19098365	1	2	yes	—
129	C₄	#19098368	1	2	yes	—
130	C₂	#23761235	1	1	yes	characteristic
131	C₂	#23761472	1	2	yes	—
132	C₂	#23761474	1	2	yes	—
133	C₂	#23761492	1	1	yes	characteristic
134	C₂	#23761494	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098316

C₂

yes

maximal

C₂×C₄×C₁₆×C₇₉

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

—

#19098316

—

#19098316

C₄

yes

—

C₂²×C₁₆×C₇₉

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

—

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

Q₈

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

C₈

yes

complemented

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

D₈

yes

characteristic

C₂²×C₄×C₇₉

Q₁₆

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄²

yes

characteristic

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₂×C₈

yes

—

C₂×C₈×C₇₉

M₁₆

yes

—

C₄²×C₇₉

C₂²⋊C₄

yes

—

C₄²×C₇₉

SD₁₆

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

#107

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#95

yes

characteristic

C₂×C₄×C₇₉

#102

yes

—

C₂×C₄×C₇₉

#103

yes

—

C₂×C₄×C₇₉

C₄⋊C₈

yes

—

C₂×C₄×C₇₉

#106

yes

characteristic

C₂×C₄×C₇₉

#108

yes

—

C₂×C₄×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

SD₃₂

yes

—

C₂×C₄×C₇₉

Q₃₂

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

C₄×C₈

yes

characteristic

C₈×C₇₉

#97

yes

characteristic

C₈×C₇₉

C₂²⋊C₈

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

#335

yes

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₄×C₇₉

—

C₄×C₇₉

#334

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₄×C₇₉

#358

yes

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#1372

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₄×C₁₆

D₇₉

yes

characteristic

C₂×#335

—

C₇₉

#7154

yes

characteristic, complemented, Sylow

C₂²×C₁₆

Dic₇₉

yes

—

C₂×C₄⋊C₈

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₄×C₁₆

—

#335

—

#335

—

158

—

C₂²×C₈

—

316

—

C₂²×C₈

Dic₁₅₈

yes

characteristic

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

C₂×C₄²

D₃₁₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₇₉⋊C₈

yes

complemented

C₄⋊C₈

—

316

—

C₄⋊C₈

—

158

—

C₄×C₈

C₄×D₇₉

yes

—

C₄×C₈

C₂²⋊D₇₉

yes

—

C₂²×C₄

C₄⋊Dic₇₉

yes

characteristic

C₂²×C₄

D₆₃₂

yes

characteristic

C₂²×C₄

Dic₃₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

316

—

C₂×C₈

—

632

—

C₂×C₈

—

632

—

C₂×C₈

C₄×Dic₇₉

yes

characteristic

C₂×C₈

C₂²⋊Dic₇₉

yes

characteristic

C₂×C₈

Dic₇₉⋊C₄

yes

—

C₂×C₈

C₂×C₇₉⋊C₈

yes

—

C₂×C₈

#11929608

yes

—

100

C₄²

(C₂×D₇₉)⋊C₄

yes

—

101

C₄²

#11929613

yes

—

102

C₁₆

—

103

C₂³

#14579095

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

C₇₉⋊#95

yes

characteristic

106

C₂×C₄

#14579083

yes

—

107

C₂×C₄

#14579086

yes

—

108

C₂×C₄

C₄⋊(C₇₉⋊C₈)

yes

—

109

C₂×C₄

#14579093

yes

characteristic

110

C₂×C₄

#14579097

yes

—

111

C₂×C₄

D₁₂₆₄

yes

characteristic

112

C₂×C₄

#14579106

yes

—

113

C₂×C₄

Dic₆₃₂

yes

characteristic

114

C₈

—

115

C₈

—

632

—

116

C₈

C₄×C₇₉⋊C₈

yes

characteristic

117

C₈

#14579072

yes

characteristic

118

C₈

C₂²⋊(C₇₉⋊C₈)

yes

—

119

C₂²

#19098316

yes

—

120

C₂²

#19098326

yes

characteristic

121

C₂²

#19098389

yes

characteristic

122

C₂²

#19098391

yes

characteristic

123

C₂²

#19098393

yes

—

124

C₄

—

125

C₄

#19098314

yes

—

126

C₄

#19098318

yes

characteristic

127

C₄

#19098328

yes

characteristic

128

C₄

#19098365

yes

—

129

C₄

#19098368

yes

—

130

C₂

#23761235

yes

characteristic

131

C₂

#23761472

yes

—

132

C₂

#23761474

yes

—

133

C₂

#23761492

yes

characteristic

134

C₂

#23761494

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₆₃₂
commutator	C₈×C₇₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₂×C₄×C₁₆×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#107
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	C₂×C₄×C₁₆	D₇₉
hypocenter	C₇₉	#7154
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂²×C₄

D₆₃₂

commutator

C₈×C₇₉

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₉

Fitting

C₂×C₄×C₁₆×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

#107

cosocle

C₂×C₄×C₈×C₇₉

C₂²

hypercenter

C₂×C₄×C₁₆

D₇₉

hypocenter

C₇₉

#7154

LCS

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
17. C₂×C₁₆×C₇₉	C₈	semidirect
70. C₇₉	#7154	semidirect
83. C₂×C₁₆	C₇₉⋊C₈	semidirect

H ≅ G/N

type

17. C₂×C₁₆×C₇₉

C₈

semidirect

70. C₇₉

#7154

semidirect

83. C₂×C₁₆

C₇₉⋊C₈

semidirect

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

subgroup	index
C₂×#19098316	2
C₂×C₄×C₁₆×C₇₉	2
#7154	79

subgroup

index

C₂×#19098316

C₂×C₄×C₁₆×C₇₉

#7154

Subgroups — #34175139

Conjugacy classes of subgroups

Complements

Maximal subgroups