Group #34175153 — subgroups

Subgroups — #34175153

Order 20224.

210 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 119 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×#19098384	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#14579101	—	2	2	no	—
4	#19098384	—	2	2	no	—
5	#19098384	C₄	1	4	yes	—
6	C₂²×C₁₆×C₇₉	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₇₉	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₇₉	—	2	2	no	—
10	#14579101	—	2	4	no	—
11	#14579101	—	4	2	no	—
12	C₂×C₇₉⋊C₁₆	—	4	2	no	—
13	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
15	C₂×C₄²×C₇₉	Q₈	1	1	yes	characteristic
16	C₂×C₁₆×C₇₉	—	2	1	no	—
17	C₂×C₁₆×C₇₉	—	2	1	no	—
18	C₂×C₁₆×C₇₉	—	2	1	no	—
19	C₂×C₁₆×C₇₉	—	2	1	no	—
20	C₂×C₁₆×C₇₉	C₈	1	2	yes	—
21	C₂×C₁₆×C₇₉	C₈	1	2	yes	—
22	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₇₉	D₄	1	2	yes	—
24	C₇₉⋊C₁₆	—	4	4	no	—
25	C₂²×C₄×C₇₉	—	2	1	no	—
26	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
28	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
29	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
34	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
35	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
36	C₂×C₈×C₇₉	Q₁₆	1	1	yes	characteristic
37	C₄²×C₇₉	—	2	1	no	—
38	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
39	C₁₆×C₇₉	—	2	2	no	—
40	C₁₆×C₇₉	—	2	2	no	—
41	C₂³×C₇₉	#108	1	1	yes	characteristic
42	C₂×C₄×C₇₉	—	2	4	no	—
43	C₂×C₄×C₇₉	—	2	2	no	—
44	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
45	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
48	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
49	C₂×C₄×C₇₉	#108	1	1	yes	characteristic
50	C₈×C₇₉	C₄×C₈	1	1	yes	characteristic
51	C₈×C₇₉	#97	1	1	yes	characteristic
52	C₈×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
53	C₈×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
54	C₈×C₇₉	#102	1	2	yes	—
55	C₈×C₇₉	#103	1	2	yes	—
56	C₂²×C₇₉	—	2	2	no	—
57	C₂²×C₇₉	#334	1	1	yes	characteristic
58	C₂²×C₇₉	#335	1	1	yes	characteristic
59	C₂²×C₇₉	#341	1	1	yes	characteristic
60	C₄×C₇₉	—	2	4	no	—
61	C₄×C₇₉	#334	1	1	yes	characteristic
62	C₄×C₇₉	#335	1	1	yes	characteristic
63	C₄×C₇₉	#336	1	1	yes	characteristic
64	C₄×C₇₉	#340	1	1	yes	characteristic
65	#7160	—	79	1	no	maximal, Sylow
66	C₂×C₇₉	—	2	1	no	—
67	C₂×C₇₉	—	2	1	no	—
68	C₂×C₇₉	#1277	1	1	yes	characteristic
69	C₂×C₇₉	#1373	1	2	yes	—
70	C₂×#364	—	79	2	no	—
71	C₂×C₄×C₁₆	D₇₉	1	1	yes	characteristic
72	C₇₉	#7160	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₃₂	—	158	2	no	—
74	C₂²×C₁₆	Dic₇₉	1	1	yes	characteristic
75	C₂²×C₁₆	Dic₇₉	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
77	#364	—	79	4	no	—
78	#364	—	158	2	no	—
79	C₄×C₁₆	—	2	2	no	—
80	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
81	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
82	C₂×C₄²	Dic₁₅₈	1	1	yes	characteristic
83	M₃₂	—	316	2	no	—
84	M₃₂	—	158	4	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	2	1	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	—	316	2	no	—
90	C₂×C₁₆	C₇₉⋊C₈	1	2	yes	—
91	C₂×C₁₆	C₇₉⋊C₈	1	2	yes	—
92	C₄×C₈	C₄×D₇₉	1	2	yes	—
93	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
96	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
97	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
98	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
99	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
100	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
101	C₂×C₈	#11929608	1	1	yes	characteristic
102	C₂×C₈	#11929608	1	1	yes	characteristic
103	C₂×C₈	D₆₃₂	1	1	yes	characteristic
104	C₂×C₈	#11929613	1	2	yes	—
105	C₂×C₈	Dic₃₁₆	1	1	yes	characteristic
106	C₄²	—	2	1	no	—
107	C₄²	Dic₇₉⋊C₄	1	2	yes	—
108	C₁₆	—	2	2	no	—
109	C₁₆	—	2	2	no	—
110	C₁₆	—	316	4	no	—
111	C₂³	#14579097	1	1	yes	characteristic
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
115	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
116	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
117	C₂×C₄	#14579093	1	1	yes	characteristic
118	C₂×C₄	#14579095	1	1	yes	characteristic
119	C₂×C₄	#14579097	1	1	yes	characteristic
120	C₈	C₄×C₇₉⋊C₈	1	1	yes	characteristic
121	C₈	#14579072	1	1	yes	characteristic
122	C₈	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
123	C₈	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
124	C₈	#14579083	1	2	yes	—
125	C₈	#14579086	1	2	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	#19098314	1	1	yes	characteristic
128	C₂²	#19098316	1	1	yes	characteristic
129	C₂²	#19098328	1	1	yes	characteristic
130	C₄	—	2	4	no	—
131	C₄	#19098314	1	1	yes	characteristic
132	C₄	#19098316	1	1	yes	characteristic
133	C₄	#19098318	1	1	yes	characteristic
134	C₄	#19098326	1	1	yes	characteristic
135	C₂	—	2	1	no	—
136	C₂	—	2	1	no	—
137	C₂	#23761235	1	1	yes	characteristic
138	C₂	#23761476	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098384

C₂

yes

maximal

C₂×C₄×C₁₆×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579101

—

#19098384

—

#19098384

C₄

yes

—

C₂²×C₁₆×C₇₉

C₄

yes

characteristic

C₂²×C₁₆×C₇₉

C₄

yes

characteristic

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

—

#14579101

—

#14579101

—

C₂×C₇₉⋊C₁₆

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

Q₈

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

C₈

yes

—

C₂×C₁₆×C₇₉

C₈

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄²

yes

characteristic

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×C₈

yes

characteristic

C₂×C₈×C₇₉

C₂×C₈

yes

characteristic

C₂×C₈×C₇₉

M₁₆

yes

characteristic

C₂×C₈×C₇₉

M₁₆

yes

characteristic

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

Q₁₆

yes

characteristic

C₄²×C₇₉

—

C₄²×C₇₉

C₄⋊C₄

yes

—

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

#108

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#95

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₈×C₇₉

C₄×C₈

yes

characteristic

C₈×C₇₉

#97

yes

characteristic

C₈×C₇₉

C₂²⋊C₈

yes

characteristic

C₈×C₇₉

C₂²⋊C₈

yes

characteristic

C₈×C₇₉

#102

yes

—

C₈×C₇₉

#103

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1277

yes

characteristic

C₂×C₇₉

#1373

yes

—

C₂×#364

—

C₂×C₄×C₁₆

D₇₉

yes

characteristic

C₇₉

#7160

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

158

—

C₂²×C₁₆

Dic₇₉

yes

characteristic

C₂²×C₁₆

Dic₇₉

yes

characteristic

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#364

—

#364

—

158

—

C₄×C₁₆

—

C₂²×C₈

D₃₁₆

yes

characteristic

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

C₂×C₄²

Dic₁₅₈

yes

characteristic

M₃₂

—

316

—

M₃₂

—

158

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

316

—

C₂×C₁₆

C₇₉⋊C₈

yes

—

C₂×C₁₆

C₇₉⋊C₈

yes

—

C₄×C₈

C₄×D₇₉

yes

—

C₄×C₈

C₂²⋊D₇₉

yes

—

C₂²×C₄

—

C₂²×C₄

C₄⋊Dic₇₉

yes

characteristic

C₂×C₈

C₄×Dic₇₉

yes

characteristic

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

C₂×C₈

C₂²⋊Dic₇₉

yes

characteristic

C₂×C₈

C₂×C₇₉⋊C₈

yes

characteristic

100

C₂×C₈

C₂×C₇₉⋊C₈

yes

characteristic

101

C₂×C₈

#11929608

yes

characteristic

102

C₂×C₈

#11929608

yes

characteristic

103

C₂×C₈

D₆₃₂

yes

characteristic

104

C₂×C₈

#11929613

yes

—

105

C₂×C₈

Dic₃₁₆

yes

characteristic

106

C₄²

—

107

C₄²

Dic₇₉⋊C₄

yes

—

108

C₁₆

—

109

C₁₆

—

110

C₁₆

—

316

—

111

C₂³

#14579097

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₇₉⋊#95

yes

characteristic

115

C₂×C₄

C₄⋊(C₇₉⋊C₈)

yes

characteristic

116

C₂×C₄

C₄⋊(C₇₉⋊C₈)

yes

characteristic

117

C₂×C₄

#14579093

yes

characteristic

118

C₂×C₄

#14579095

yes

characteristic

119

C₂×C₄

#14579097

yes

characteristic

120

C₈

C₄×C₇₉⋊C₈

yes

characteristic

121

C₈

#14579072

yes

characteristic

122

C₈

C₂²⋊(C₇₉⋊C₈)

yes

characteristic

123

C₈

C₂²⋊(C₇₉⋊C₈)

yes

characteristic

124

C₈

#14579083

yes

—

125

C₈

#14579086

yes

—

126

C₂²

—

127

C₂²

#19098314

yes

characteristic

128

C₂²

#19098316

yes

characteristic

129

C₂²

#19098328

yes

characteristic

130

C₄

—

131

C₄

#19098314

yes

characteristic

132

C₄

#19098316

yes

characteristic

133

C₄

#19098318

yes

characteristic

134

C₄

#19098326

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

#23761235

yes

characteristic

138

C₂

#23761476

yes

—

	subgroup	G/H
center	C₂×C₈	D₆₃₂
commutator	C₈×C₇₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₂×C₄×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	#335
prime radical	C₂³×C₇₉	#108
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	C₂×C₄×C₁₆	D₇₉
hypocenter	C₇₉	#7160
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂×C₈

D₆₃₂

commutator

C₈×C₇₉

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₉

Fitting

C₂×C₄×C₁₆×C₇₉

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄×C₈×C₇₉

C₂²

hypercenter

C₂×C₄×C₁₆

D₇₉

hypocenter

C₇₉

#7160

LCS

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

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

N	H ≅ G/N	type
72. C₇₉	#7160	semidirect

H ≅ G/N

type

72. C₇₉

#7160

semidirect

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

subgroup	index
C₂×#19098384	2
C₂×C₄×C₁₆×C₇₉	2
#7160	79

subgroup

index

C₂×#19098384

C₂×C₄×C₁₆×C₇₉

#7160

Subgroups — #34175153

Conjugacy classes of subgroups

Complements

Maximal subgroups