Group #34175113 — subgroups

Subgroups — #34175113

Order 20224.

234 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 95 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	#23765203	C₂	1	2	yes	maximal
2	C₇₉×#1744	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₇₉⋊C₁₆)	—	2	2	no	—
4	C₂²⋊(C₇₉⋊C₁₆)	—	2	2	no	—
5	#19098339	C₄	1	4	yes	complemented
6	C₂×C₄×C₈×C₇₉	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
8	C₄³×C₇₉	C₄	1	1	yes	characteristic
9	#336×C₇₉	—	2	2	no	—
10	C₂×C₇₉⋊C₁₆	—	2	4	no	—
11	C₂²×C₈×C₇₉	—	2	1	no	—
12	C₂²×C₈×C₇₉	D₄	1	2	yes	—
13	C₂×C₄²×C₇₉	—	2	1	no	—
14	C₂×C₄²×C₇₉	—	2	1	no	—
15	C₂×C₄²×C₇₉	C₈	1	2	yes	—
16	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
17	C₄×C₈×C₇₉	—	2	1	no	—
18	C₄×C₈×C₇₉	C₈	1	2	yes	—
19	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
21	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
22	C₇₉⋊C₁₆	—	4	4	no	—
23	C₂²×C₄×C₇₉	—	2	2	no	—
24	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₇₉	—	2	2	no	—
28	C₂×C₈×C₇₉	—	4	2	no	—
29	C₂×C₈×C₇₉	—	4	2	no	—
30	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
31	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
32	C₄²×C₇₉	—	2	4	no	—
33	C₄²×C₇₉	—	4	2	no	—
34	C₄²×C₇₉	—	4	2	no	—
35	C₄²×C₇₉	C₄²	1	1	yes	characteristic
36	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
39	C₂³×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
40	C₂×C₄×C₇₉	—	2	2	no	—
41	C₂×C₄×C₇₉	—	2	2	no	—
42	C₂×C₄×C₇₉	—	2	2	no	—
43	C₂×C₄×C₇₉	—	2	4	no	—
44	C₂×C₄×C₇₉	—	4	2	no	—
45	C₂×C₄×C₇₉	—	4	2	no	—
46	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₉	#97	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
52	C₈×C₇₉	—	2	2	no	—
53	C₈×C₇₉	—	2	2	no	—
54	C₈×C₇₉	—	4	2	no	—
55	C₂²×C₇₉	—	2	1	no	—
56	C₂²×C₇₉	—	2	1	no	—
57	C₂²×C₇₉	#336	1	1	yes	characteristic
58	C₂²×C₇₉	#349	1	2	yes	—
59	C₄×C₇₉	—	2	1	no	—
60	C₄×C₇₉	—	2	1	no	—
61	C₄×C₇₉	—	2	1	no	—
62	C₄×C₇₉	—	2	1	no	—
63	C₄×C₇₉	—	4	4	no	—
64	C₄×C₇₉	H(3,ℤ/4)	1	2	yes	—
65	C₄×C₇₉	#364	1	2	yes	—
66	#7143	—	79	1	no	maximal, Sylow
67	C₂×C₇₉	—	2	2	no	—
68	C₂×C₇₉	#1274	1	1	yes	characteristic
69	C₂×C₇₉	#1376	1	1	yes	characteristic
70	C₂×C₇₉	#1377	1	1	yes	characteristic
71	#2163	—	79	2	no	—
72	#1744	D₇₉	1	1	yes	characteristic
73	C₇₉	#7143	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄×C₈	Dic₇₉	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
76	C₄³	Dic₇₉	1	1	yes	characteristic
77	C₄⋊C₁₆	—	158	2	no	—
78	C₂²⋊C₁₆	—	158	2	no	—
79	#346	—	79	4	no	—
80	#336	—	2	2	no	—
81	C₂²×C₈	—	2	1	no	—
82	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
83	C₂×C₄²	—	2	1	no	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	C₇₉⋊C₈	1	2	yes	—
86	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
87	C₂×C₁₆	—	158	4	no	—
88	C₄×C₈	—	2	1	no	—
89	C₄×C₈	C₇₉⋊C₈	1	2	yes	—
90	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
91	C₄×C₈	C₄×D₇₉	1	2	yes	—
92	C₄×C₈	D₃₁₆	1	1	yes	characteristic
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
95	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
96	C₂²×C₄	#11929608	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
101	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
102	C₄²	—	2	4	no	—
103	C₄²	—	4	2	no	—
104	C₄²	—	4	2	no	—
105	C₄²	C₄×Dic₇₉	1	1	yes	characteristic
106	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
107	C₄²	C₂×C₇₉⋊C₈	1	1	yes	characteristic
108	C₄²	#11929608	1	1	yes	characteristic
109	C₁₆	—	316	4	no	—
110	C₂³	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
118	C₂×C₄	C₄×C₇₉⋊C₈	1	1	yes	characteristic
119	C₂×C₄	#14579072	1	1	yes	characteristic
120	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
121	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
122	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	2	2	no	—
125	C₈	—	4	2	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	#19098318	1	1	yes	characteristic
129	C₂²	#19098345	1	2	yes	—
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	4	4	no	—
135	C₄	#19098319	1	2	yes	—
136	C₄	#19098384	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	#23761229	1	1	yes	characteristic
139	C₂	#23761482	1	1	yes	characteristic
140	C₂	#23761484	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765203

C₂

yes

maximal

C₇₉×#1744

C₂

yes

maximal, characteristic

C₄⋊(C₇₉⋊C₁₆)

—

C₂²⋊(C₇₉⋊C₁₆)

—

#19098339

C₄

yes

complemented

C₂×C₄×C₈×C₇₉

C₄

yes

characteristic

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄³×C₇₉

C₄

yes

characteristic

#336×C₇₉

—

C₂×C₇₉⋊C₁₆

—

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

C₈

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₈

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

Q₈

yes

characteristic

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₉

M₁₆

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₄²

yes

characteristic

C₄²×C₇₉

C₄⋊C₄

yes

characteristic

C₄²×C₇₉

C₂×C₈

yes

characteristic

C₄²×C₇₉

M₁₆

yes

characteristic

C₂³×C₇₉

C₂²⋊C₈

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

C₄×C₈

yes

characteristic

C₂×C₄×C₇₉

#97

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₈

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#336

yes

characteristic

C₂²×C₇₉

#349

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

H(3,ℤ/4)

yes

—

C₄×C₇₉

#364

yes

—

#7143

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

yes

characteristic

C₇₉

#7143

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

Dic₇₉

yes

characteristic

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₄³

Dic₇₉

yes

characteristic

C₄⋊C₁₆

—

158

—

C₂²⋊C₁₆

—

158

—

#346

—

#336

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₇₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₇₉⋊C₈

yes

—

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

C₂×C₁₆

—

158

—

C₄×C₈

—

C₄×C₈

C₇₉⋊C₈

yes

—

C₄×C₈

Dic₁₅₈

yes

characteristic

C₄×C₈

C₄×D₇₉

yes

—

C₄×C₈

D₃₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

C₂²×C₄

C₂×C₇₉⋊C₈

yes

characteristic

C₂²×C₄

#11929608

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

101

C₂×C₈

Dic₇₉⋊C₄

yes

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₄×Dic₇₉

yes

characteristic

106

C₄²

C₄⋊Dic₇₉

yes

characteristic

107

C₄²

C₂×C₇₉⋊C₈

yes

characteristic

108

C₄²

#11929608

yes

characteristic

109

C₁₆

—

316

—

110

C₂³

C₂²⋊(C₇₉⋊C₈)

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₇₉⋊#95

yes

characteristic

118

C₂×C₄

C₄×C₇₉⋊C₈

yes

characteristic

119

C₂×C₄

#14579072

yes

characteristic

120

C₂×C₄

C₂²⋊(C₇₉⋊C₈)

yes

characteristic

121

C₂×C₄

C₄⋊(C₇₉⋊C₈)

yes

characteristic

122

C₂×C₄

C₄⋊(C₇₉⋊C₈)

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₂²

—

127

C₂²

—

128

C₂²

#19098318

yes

characteristic

129

C₂²

#19098345

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#19098319

yes

—

136

C₄

#19098384

yes

—

137

C₂

—

138

C₂

#23761229

yes

characteristic

139

C₂

#23761482

yes

characteristic

140

C₂

#23761484

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₇₉⋊#95
commutator	C₂×C₄×C₇₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₇₉×#1744	C₂
socle	C₂²×C₇₉	#336
prime radical	C₂³×C₇₉	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	#1744	D₇₉
hypocenter	C₇₉	#7143
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1744

subgroup

G/H

center

C₂×C₄

C₇₉⋊#95

commutator

C₂×C₄×C₇₉

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₉

Fitting

C₇₉×#1744

C₂

socle

C₂²×C₇₉

#336

prime radical

C₂³×C₇₉

C₂²⋊C₈

cosocle

C₂×C₄×C₈×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1744

N	H ≅ G/N	type
5. #19098339	C₄	semidirect
73. C₇₉	#7143	semidirect

H ≅ G/N

type

5. #19098339

C₄

semidirect

73. C₇₉

#7143

semidirect

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

subgroup	index
#23765203	2
C₇₉×#1744	2
#7143	79

subgroup

index

#23765203

C₇₉×#1744

#7143

Subgroups — #34175113

Conjugacy classes of subgroups

Complements

Maximal subgroups