Group #34175352 — subgroups

Subgroups — #34175352

Order 20224.

264 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 83 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	#23763438	C₂	1	2	yes	maximal
2	C₇₉×#2112	C₂	1	1	yes	maximal, characteristic
3	C₂×#14579097	—	2	2	no	—
4	#19098527	—	2	2	no	—
5	#19098328	C₄	1	4	yes	—
6	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₇₉	C₄	1	2	yes	—
8	C₂²⋊C₁₆×C₇₉	—	2	2	no	—
9	C₂×#11929608	—	2	4	no	—
10	#14579097	—	4	4	no	—
11	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
12	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
13	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
14	C₂²×C₈×C₇₉	—	4	2	no	—
15	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₇₉	D₄	1	2	yes	—
18	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
19	M₃₂×C₇₉	—	4	2	no	—
20	C₂×C₁₆×C₇₉	—	2	2	no	—
21	#11929608	—	4	4	no	—
22	#11929608	—	4	4	no	—
23	C₂×C₇₉⋊C₈	—	4	4	no	—
24	C₂⁴×C₇₉	Q₁₆	1	1	yes	characteristic
25	C₂²×C₄×C₇₉	—	4	1	no	—
26	C₂²×C₄×C₇₉	—	4	1	no	—
27	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
28	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
31	C₂×C₈×C₇₉	—	2	2	no	—
32	C₂×C₈×C₇₉	—	2	4	no	—
33	C₂×C₈×C₇₉	—	4	2	no	—
34	C₂×C₈×C₇₉	—	4	1	no	—
35	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
38	C₁₆×C₇₉	—	4	2	no	—
39	C₂³×C₇₉	—	4	1	no	—
40	C₂³×C₇₉	—	4	2	no	—
41	C₂³×C₇₉	#103	1	2	yes	—
42	C₂³×C₇₉	#107	1	1	yes	characteristic
43	C₂×C₄×C₇₉	—	2	2	no	—
44	C₂×C₄×C₇₉	—	2	1	no	—
45	C₂×C₄×C₇₉	—	2	2	no	—
46	C₂×C₄×C₇₉	—	2	1	no	—
47	C₂×C₄×C₇₉	—	4	1	no	—
48	C₂×C₄×C₇₉	—	4	2	no	—
49	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
50	C₂×C₄×C₇₉	#102	1	2	yes	—
51	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
52	C₈×C₇₉	—	2	1	no	—
53	C₈×C₇₉	—	2	2	no	—
54	C₈×C₇₉	#99	1	2	yes	—
55	C₇₉⋊C₈	—	8	4	no	—
56	C₂²×C₇₉	—	2	2	no	—
57	C₂²×C₇₉	—	2	1	no	—
58	C₂²×C₇₉	—	2	4	no	—
59	C₂²×C₇₉	—	2	2	no	—
60	C₂²×C₇₉	—	4	4	no	—
61	C₂²×C₇₉	#340	1	1	yes	characteristic
62	C₄×C₇₉	—	2	2	no	—
63	C₄×C₇₉	—	2	1	no	—
64	C₄×C₇₉	H(3,ℤ/4)	1	1	yes	characteristic
65	C₄×C₇₉	#342	1	1	yes	characteristic
66	#7234	—	79	1	no	maximal, Sylow
67	C₂×C₇₉	—	4	2	no	—
68	C₂×C₇₉	—	4	1	no	—
69	C₂×C₇₉	#1289	1	1	yes	characteristic
70	C₂×C₇₉	#1388	1	2	yes	—
71	#2112	D₇₉	1	1	yes	characteristic
72	#1810	—	79	2	no	—
73	C₇₉	#7234	1	1	yes	characteristic, complemented, Sylow
74	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
75	C₂×M₃₂	Dic₇₉	1	2	yes	—
76	C₂×#108	—	158	2	no	—
77	#407	—	158	2	no	—
78	C₂²⋊C₁₆	—	2	2	no	—
79	#341	—	79	4	no	—
80	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
81	C₂×M₁₆	—	158	4	no	—
82	C₂²×C₈	—	4	2	no	—
83	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
84	C₂²×C₈	C₄×D₇₉	1	2	yes	—
85	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
86	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
87	M₃₂	—	4	2	no	—
88	C₂×C₁₆	—	2	2	no	—
89	#108	—	316	4	no	—
90	C₂²⋊C₈	—	316	2	no	—
91	C₂²⋊C₈	—	316	2	no	—
92	C₂⁴	Dic₃₁₆	1	1	yes	characteristic
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	—	4	1	no	—
95	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
96	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
97	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
98	C₂²×C₄	#11929613	1	2	yes	—
99	M₁₆	—	316	4	no	—
100	M₁₆	—	316	4	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	316	4	no	—
106	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
107	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
108	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
109	C₁₆	—	4	2	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	2	no	—
112	C₂³	#14579086	1	2	yes	—
113	C₂³	#14579095	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
121	C₂×C₄	#14579083	1	2	yes	—
122	C₂×C₄	#14579093	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	2	no	—
125	C₈	—	632	4	no	—
126	C₈	#14579076	1	2	yes	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	4	4	no	—
132	C₂²	#19098326	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	#19098320	1	1	yes	characteristic
136	C₄	#19098330	1	1	yes	characteristic
137	C₂	—	4	2	no	—
138	C₂	—	4	1	no	—
139	C₂	#23761268	1	1	yes	characteristic
140	C₂	#23761512	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763438

C₂

yes

maximal

C₇₉×#2112

C₂

yes

maximal, characteristic

C₂×#14579097

—

#19098527

—

#19098328

C₄

yes

—

C₂³×C₈×C₇₉

C₂²

yes

characteristic

C₂×M₃₂×C₇₉

C₄

yes

—

C₂²⋊C₁₆×C₇₉

—

C₂×#11929608

—

#14579097

—

C₂²⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₂³×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

C₂×C₄

yes

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

D₄

yes

—

C₂²×C₈×C₇₉

Q₈

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂⁴×C₇₉

Q₁₆

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

D₈

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

—

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

—

C₁₆×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

#103

yes

—

C₂³×C₇₉

#107

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#95

yes

characteristic

C₂×C₄×C₇₉

#102

yes

—

C₂×C₄×C₇₉

#106

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#99

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#340

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1289

yes

characteristic

C₂×C₇₉

#1388

yes

—

#2112

D₇₉

yes

characteristic

#1810

—

C₇₉

#7234

yes

characteristic, complemented, Sylow

C₂³×C₈

C₂×D₇₉

yes

characteristic

C₂×M₃₂

Dic₇₉

yes

—

C₂×#108

—

158

—

#407

—

158

—

C₂²⋊C₁₆

—

#341

—

C₂³×C₄

D₃₁₆

yes

characteristic

C₂×M₁₆

—

158

—

C₂²×C₈

—

C₂²×C₈

Dic₁₅₈

yes

characteristic

C₂²×C₈

C₄×D₇₉

yes

—

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

C₂²×C₈

C₂²⋊D₇₉

yes

—

M₃₂

—

C₂×C₁₆

—

#108

—

316

—

C₂²⋊C₈

—

316

—

C₂²⋊C₈

—

316

—

C₂⁴

Dic₃₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

C₂²×C₄

C₄⋊Dic₇₉

yes

characteristic

C₂²×C₄

D₆₃₂

yes

characteristic

C₂²×C₄

#11929613

yes

—

M₁₆

—

316

—

100

M₁₆

—

316

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

316

—

106

C₂×C₈

C₄×Dic₇₉

yes

characteristic

107

C₂×C₈

C₂²⋊Dic₇₉

yes

characteristic

108

C₂×C₈

Dic₇₉⋊C₄

yes

—

109

C₁₆

—

110

C₂³

—

111

C₂³

—

112

C₂³

#14579086

yes

—

113

C₂³

#14579095

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₇₉⋊#95

yes

characteristic

121

C₂×C₄

#14579083

yes

—

122

C₂×C₄

#14579093

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

632

—

126

C₈

#14579076

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#19098326

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

#19098320

yes

characteristic

136

C₄

#19098330

yes

characteristic

137

C₂

—

138

C₂

—

139

C₂

#23761268

yes

characteristic

140

C₂

#23761512

yes

—

	subgroup	G/H
center	C₂²	#19098326
commutator	C₂×C₈×C₇₉	C₄²
Frattini	C₂³×C₈	C₂×D₇₉
Fitting	C₇₉×#2112	C₂
socle	C₂²×C₇₉	#340
prime radical	C₂⁴×C₇₉	Q₁₆
cosocle	C₂³×C₈×C₇₉	C₂²
hypercenter	#2112	D₇₉
hypocenter	C₇₉	#7234
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #2112

subgroup

G/H

center

C₂²

#19098326

commutator

C₂×C₈×C₇₉

C₄²

Frattini

C₂³×C₈

C₂×D₇₉

Fitting

C₇₉×#2112

C₂

socle

C₂²×C₇₉

#340

prime radical

C₂⁴×C₇₉

Q₁₆

cosocle

C₂³×C₈×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #2112

N	H ≅ G/N	type
73. C₇₉	#7234	semidirect

H ≅ G/N

type

73. C₇₉

#7234

semidirect

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

subgroup	index
#23763438	2
C₇₉×#2112	2
#7234	79

subgroup

index

#23763438

C₇₉×#2112

#7234

Subgroups — #34175352

Conjugacy classes of subgroups

Complements

Maximal subgroups