Group #34175120 — subgroups

Subgroups — #34175120

Order 20224.

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

normal subgroups only

This group has 210 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	C₂×#322×C₇₉	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	—
6	C₂×C₄×C₈×C₇₉	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₇₉	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
9	#322×C₇₉	—	2	2	no	—
10	C₂×C₇₉⋊C₁₆	—	2	4	no	—
11	C₂²×C₈×C₇₉	—	2	1	no	—
12	C₂²×C₈×C₇₉	C₈	1	2	yes	—
13	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
14	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
15	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
16	C₄×C₈×C₇₉	—	2	1	no	—
17	C₄×C₈×C₇₉	—	2	1	no	—
18	C₄×C₈×C₇₉	—	2	1	no	—
19	C₄×C₈×C₇₉	C₈	1	2	yes	—
20	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₇₉	D₄	1	2	yes	—
22	C₇₉⋊C₁₆	—	4	4	no	—
23	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
25	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
26	C₂×C₈×C₇₉	—	2	4	no	—
27	C₂×C₈×C₇₉	—	2	2	no	—
28	C₂×C₈×C₇₉	—	2	2	no	—
29	C₂×C₈×C₇₉	—	4	1	no	—
30	C₂×C₈×C₇₉	—	4	1	no	—
31	C₂×C₈×C₇₉	—	4	2	no	—
32	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
33	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
34	C₄²×C₇₉	C₄²	1	1	yes	characteristic
35	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
38	C₂³×C₇₉	C₄⋊C₈	1	1	yes	characteristic
39	C₂×C₄×C₇₉	—	2	2	no	—
40	C₂×C₄×C₇₉	—	2	1	no	—
41	C₂×C₄×C₇₉	—	2	1	no	—
42	C₂×C₄×C₇₉	—	2	2	no	—
43	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
44	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄×C₇₉	#97	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
49	C₈×C₇₉	—	2	4	no	—
50	C₈×C₇₉	—	2	2	no	—
51	C₈×C₇₉	—	2	2	no	—
52	C₈×C₇₉	—	4	2	no	—
53	C₂²×C₇₉	—	2	1	no	—
54	C₂²×C₇₉	—	2	1	no	—
55	C₂²×C₇₉	#336	1	1	yes	characteristic
56	C₂²×C₇₉	#364	1	2	yes	—
57	C₄×C₇₉	—	2	1	no	—
58	C₄×C₇₉	—	2	1	no	—
59	C₄×C₇₉	—	2	1	no	—
60	C₄×C₇₉	—	2	1	no	—
61	C₄×C₇₉	H(3,ℤ/4)	1	1	yes	characteristic
62	C₄×C₇₉	#338	1	1	yes	characteristic
63	C₄×C₇₉	#349	1	2	yes	—
64	#7146	—	79	1	no	maximal, Sylow
65	C₂×C₇₉	—	2	2	no	—
66	C₂×C₇₉	#1275	1	1	yes	characteristic
67	C₂×C₇₉	#1376	1	1	yes	characteristic
68	C₂×C₇₉	#1377	1	1	yes	characteristic
69	#2163	—	79	2	no	—
70	C₂×#322	D₇₉	1	1	yes	characteristic
71	C₇₉	#7146	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₄×C₈	Dic₇₉	1	1	yes	characteristic
73	C₂×C₄×C₈	Dic₇₉	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
75	C₄⋊C₁₆	—	158	2	no	—
76	C₂²⋊C₁₆	—	158	2	no	—
77	#346	—	79	4	no	—
78	#322	—	2	2	no	—
79	C₂²×C₈	—	2	1	no	—
80	C₂²×C₈	C₇₉⋊C₈	1	2	yes	—
81	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
82	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
83	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
84	C₂×C₁₆	—	158	4	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	2	1	no	—
87	C₄×C₈	—	2	1	no	—
88	C₄×C₈	C₇₉⋊C₈	1	2	yes	—
89	C₄×C₈	C₄×D₇₉	1	2	yes	—
90	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
91	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
92	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
93	C₂²×C₄	#11929608	1	1	yes	characteristic
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	4	1	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	C₄×Dic₇₉	1	1	yes	characteristic
103	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
104	C₄²	C₂×C₇₉⋊C₈	1	1	yes	characteristic
105	C₄²	#11929608	1	1	yes	characteristic
106	C₁₆	—	316	4	no	—
107	C₂³	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
113	C₂×C₄	C₄×C₇₉⋊C₈	1	1	yes	characteristic
114	C₂×C₄	#14579072	1	1	yes	characteristic
115	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
116	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
117	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
118	C₈	—	2	4	no	—
119	C₈	—	2	2	no	—
120	C₈	—	2	2	no	—
121	C₈	—	4	2	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	#19098318	1	1	yes	characteristic
125	C₂²	#19098384	1	2	yes	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	#19098320	1	1	yes	characteristic
131	C₄	C₇₉⋊#338	1	1	yes	characteristic
132	C₄	#19098345	1	2	yes	—
133	C₂	—	2	2	no	—
134	C₂	#23761231	1	1	yes	characteristic
135	C₂	#23761482	1	1	yes	characteristic
136	C₂	#23761484	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765203

C₂

yes

maximal

C₂×#322×C₇₉

C₂

yes

maximal, characteristic

C₄⋊(C₇₉⋊C₁₆)

—

C₂²⋊(C₇₉⋊C₁₆)

—

#19098339

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₄

yes

characteristic

C₂×C₄×C₈×C₇₉

C₄

yes

characteristic

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#322×C₇₉

—

C₂×C₇₉⋊C₁₆

—

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

C₈

yes

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

Q₈

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₈

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₄⋊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

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₄²×C₇₉

C₄²

yes

characteristic

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₄²×C₇₉

C₂×C₈

yes

characteristic

C₄²×C₇₉

M₁₆

yes

characteristic

C₂³×C₇₉

C₄⋊C₈

yes

characteristic

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

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₈

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#336

yes

characteristic

C₂²×C₇₉

#364

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

#7146

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

yes

characteristic

C₇₉

#7146

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

Dic₇₉

yes

characteristic

C₂×C₄×C₈

Dic₇₉

yes

characteristic

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₄⋊C₁₆

—

158

—

C₂²⋊C₁₆

—

158

—

#346

—

#322

—

C₂²×C₈

—

C₂²×C₈

C₇₉⋊C₈

yes

—

C₂²×C₈

Dic₁₅₈

yes

characteristic

C₂²×C₈

D₃₁₆

yes

characteristic

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

C₂×C₁₆

—

158

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₇₉⋊C₈

yes

—

C₄×C₈

C₄×D₇₉

yes

—

C₄×C₈

C₂²⋊D₇₉

yes

—

C₂²×C₄

C₄⋊Dic₇₉

yes

characteristic

C₂²×C₄

C₂×C₇₉⋊C₈

yes

characteristic

C₂²×C₄

#11929608

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

101

C₂×C₈

Dic₇₉⋊C₄

yes

—

102

C₄²

C₄×Dic₇₉

yes

characteristic

103

C₄²

C₂²⋊Dic₇₉

yes

characteristic

104

C₄²

C₂×C₇₉⋊C₈

yes

characteristic

105

C₄²

#11929608

yes

characteristic

106

C₁₆

—

316

—

107

C₂³

C₄⋊(C₇₉⋊C₈)

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₇₉⋊#95

yes

characteristic

113

C₂×C₄

C₄×C₇₉⋊C₈

yes

characteristic

114

C₂×C₄

#14579072

yes

characteristic

115

C₂×C₄

C₂²⋊(C₇₉⋊C₈)

yes

characteristic

116

C₂×C₄

C₂²⋊(C₇₉⋊C₈)

yes

characteristic

117

C₂×C₄

C₄⋊(C₇₉⋊C₈)

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

#19098318

yes

characteristic

125

C₂²

#19098384

yes

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#19098320

yes

characteristic

131

C₄

C₇₉⋊#338

yes

characteristic

132

C₄

#19098345

yes

—

133

C₂

—

134

C₂

#23761231

yes

characteristic

135

C₂

#23761482

yes

characteristic

136

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	C₂×#322×C₇₉	C₂
socle	C₂²×C₇₉	#336
prime radical	C₂³×C₇₉	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	C₂×#322	D₇₉
hypocenter	C₇₉	#7146
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

C₇₉⋊#95

commutator

C₂×C₄×C₇₉

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₉

Fitting

C₂×#322×C₇₉

C₂

socle

C₂²×C₇₉

#336

prime radical

C₂³×C₇₉

C₄⋊C₈

cosocle

C₂×C₄×C₈×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

N	H ≅ G/N	type
71. C₇₉	#7146	semidirect

H ≅ G/N

type

71. C₇₉

#7146

semidirect

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

subgroup	index
#23765203	2
C₂×#322×C₇₉	2
#7146	79

subgroup

index

#23765203

C₂×#322×C₇₉

#7146

Subgroups — #34175120

Conjugacy classes of subgroups

Complements

Maximal subgroups