Group #34443771 — subgroups

Subgroups — #34443771

Order 20224.

482 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 219 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	#23773914	C₂	1	2	yes	maximal
2	#23765686	C₂	1	8	yes	maximal, complemented
3	#23765184	C₂	1	4	yes	maximal
4	C₂×#346×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#19099120	—	2	2	no	—
6	#19098968	C₂²	1	4	yes	—
7	#19098965	—	2	2	no	—
8	#19098965	C₂²	1	4	yes	—
9	#19098961	—	2	2	no	—
10	#19098601	—	2	2	no	—
11	#19098601	C₂²	1	4	yes	—
12	#19098391	C₂²	1	8	yes	—
13	#19098389	C₂²	1	8	yes	—
14	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	#346×C₇₉	C₂²	1	4	yes	—
17	#14579161	—	2	4	no	—
18	#14579153	—	2	4	no	—
19	#14579140	—	4	4	no	—
20	C₄×Dic₁₅₈	—	4	4	no	—
21	#14579120	—	2	4	no	—
22	#14579095	—	2	4	no	—
23	#14579095	C₂³	1	8	yes	—
24	#14579093	—	2	4	no	—
25	#14579093	—	2	4	no	—
26	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₇₉	Q₈	1	8	yes	complemented
30	C₂×C₁₆×C₇₉	C₂³	1	4	yes	—
31	C₄×C₈×C₇₉	D₄	1	2	yes	—
32	C₄×C₈×C₇₉	C₂³	1	2	yes	—
33	C₂×Dic₁₅₈	—	4	4	no	—
34	C₄⋊Dic₇₉	—	2	8	no	—
35	C₄⋊Dic₇₉	—	4	4	no	—
36	Dic₇₉⋊C₄	—	4	8	no	—
37	C₂²⋊Dic₇₉	—	4	4	no	—
38	C₄×Dic₇₉	—	4	4	no	—
39	C₂²×C₄×C₇₉	D₈	1	2	yes	—
40	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	Q₁₆	1	4	yes	—
42	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
44	C₂×C₈×C₇₉	C₂×Q₈	1	4	yes	—
45	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
46	C₄²×C₇₉	D₈	1	2	yes	—
47	C₄²×C₇₉	C₂×D₄	1	2	yes	—
48	C₁₆×C₇₉	C₂×Q₈	1	8	yes	—
49	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₇₉	—	4	2	no	—
51	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₇₉	C₂×D₈	1	4	yes	—
53	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
55	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
56	C₂×Dic₇₉	—	4	8	no	—
57	Dic₁₅₈	—	8	8	no	—
58	C₈×C₇₉	C₄⋊Q₈	1	2	yes	—
59	C₈×C₇₉	C₂×Q₁₆	1	4	yes	—
60	C₈×C₇₉	C₂²×Q₈	1	2	yes	—
61	C₂²×C₇₉	—	2	2	no	—
62	C₂²×C₇₉	#500	1	2	yes	—
63	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
64	C₄×C₇₉	—	2	4	no	—
65	C₄×C₇₉	#500	1	2	yes	—
66	C₄×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
67	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
68	Dic₇₉	—	8	8	no	—
69	#33728	—	79	1	no	maximal, Sylow
70	C₂×C₇₉	—	2	2	no	—
71	C₂×C₇₉	C₂×#500	1	1	yes	characteristic
72	C₂×C₇₉	#3414	1	2	yes	—
73	#3160	—	79	2	no	—
74	#2255	—	79	8	no	—
75	#2158	—	79	4	no	—
76	C₂×#346	D₇₉	1	1	yes	characteristic
77	C₇₉	#33728	1	1	yes	characteristic, complemented, Sylow
78	#533	—	158	2	no	—
79	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
80	#500	—	79	4	no	—
81	#499	—	79	4	no	—
82	#499	—	158	2	no	—
83	#498	—	158	2	no	—
84	#427	—	79	4	no	—
85	#427	—	158	2	no	—
86	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
87	#367	—	79	8	no	—
88	#366	—	79	8	no	—
89	#346	C₂×D₇₉	1	4	yes	—
90	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
91	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
92	C₄⋊Q₈	—	158	4	no	—
93	#125	—	158	4	no	—
94	C₂²⋊Q₈	—	316	4	no	—
95	C₄×Q₈	—	316	4	no	—
96	#117	—	158	4	no	—
97	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
98	C₂×C₁₆	Dic₁₅₈	1	8	yes	complemented
99	C₂×C₁₆	C₂²×D₇₉	1	4	yes	—
100	#107	—	79	8	no	—
101	#107	—	158	4	no	—
102	#106	—	158	4	no	—
103	#106	—	158	4	no	—
104	C₄×C₈	D₃₁₆	1	2	yes	—
105	C₄×C₈	C₂²×D₇₉	1	2	yes	—
106	C₂×Q₈	—	316	4	no	—
107	C₂²×C₄	D₆₃₂	1	2	yes	—
108	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
109	C₂×C₈	Dic₃₁₆	1	4	yes	—
110	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
111	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
112	C₂×C₈	C₂×Dic₁₅₈	1	4	yes	—
113	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
114	C₄⋊C₄	—	316	8	no	—
115	C₄⋊C₄	—	316	4	no	—
116	C₄⋊C₄	—	158	8	no	—
117	C₂²⋊C₄	—	316	4	no	—
118	C₄²	—	316	4	no	—
119	C₄²	D₆₃₂	1	2	yes	—
120	C₄²	C₂×D₃₁₆	1	2	yes	—
121	C₁₆	C₂×Dic₁₅₈	1	8	yes	—
122	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
123	Q₈	—	632	8	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	316	8	no	—
126	C₂×C₄	#14579161	1	2	yes	—
127	C₂×C₄	C₂×D₆₃₂	1	4	yes	—
128	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₃₁₆	1	2	yes	—
130	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
131	C₈	#14579161	1	2	yes	—
132	C₈	C₂×Dic₃₁₆	1	4	yes	—
133	C₈	C₂²×Dic₁₅₈	1	2	yes	—
134	C₂²	—	2	2	no	—
135	C₂²	#19098968	1	2	yes	—
136	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
137	C₄	—	2	4	no	—
138	C₄	—	632	8	no	—
139	C₄	#19098968	1	2	yes	—
140	C₄	C₂×#14579161	1	1	yes	characteristic
141	C₄	C₂²×Dic₃₁₆	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#19098968	1	1	yes	characteristic
144	C₂	#23776536	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23773914

C₂

yes

maximal

#23765686

C₂

yes

maximal, complemented

#23765184

C₂

yes

maximal

C₂×#346×C₇₉

C₂

yes

maximal, characteristic

#19099120

—

#19098968

C₂²

yes

—

#19098965

—

#19098965

C₂²

yes

—

#19098961

—

#19098601

—

#19098601

C₂²

yes

—

#19098391

C₂²

yes

—

#19098389

C₂²

yes

—

C₂²×C₁₆×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#346×C₇₉

C₂²

yes

—

#14579161

—

#14579153

—

#14579140

—

C₄×Dic₁₅₈

—

#14579120

—

#14579095

—

#14579095

C₂³

yes

—

#14579093

—

#14579093

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

Q₈

yes

complemented

C₂×C₁₆×C₇₉

C₂³

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

C₂³

yes

—

C₂×Dic₁₅₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

D₈

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

Q₁₆

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

C₂⁴

yes

characteristic

C₄²×C₇₉

D₈

yes

—

C₄²×C₇₉

C₂×D₄

yes

—

C₁₆×C₇₉

C₂×Q₈

yes

—

C₂³×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄⋊Q₈

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₁₆

yes

—

C₂×C₄×C₇₉

C₂²×D₄

yes

characteristic

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

C₄⋊Q₈

yes

—

C₈×C₇₉

C₂×Q₁₆

yes

—

C₈×C₇₉

C₂²×Q₈

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

#500

yes

—

C₂²×C₇₉

C₂²×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#500

yes

—

C₄×C₇₉

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₇₉

C₂²×Q₁₆

yes

characteristic

Dic₇₉

—

#33728

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

C₂×#500

yes

characteristic

C₂×C₇₉

#3414

yes

—

—

—

—

D₇₉

yes

characteristic

C₇₉

#33728

yes

characteristic, complemented, Sylow

#533

—

158

—

C₂²×C₁₆

C₂×D₇₉

yes

—

#500

—

#499

—

#499

—

158

—

#498

—

158

—

#427

—

#427

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

—

—

C₂×D₇₉

yes

—

C₂²×C₈

D₃₁₆

yes

characteristic

C₂²×C₈

C₂²×D₇₉

yes

characteristic

C₄⋊Q₈

—

158

—

#125

—

158

—

C₂²⋊Q₈

—

316

—

C₄×Q₈

—

316

—

#117

—

158

—

C₂×C₄²

D₃₁₆

yes

characteristic

C₂×C₁₆

Dic₁₅₈

yes

complemented

C₂×C₁₆

C₂²×D₇₉

yes

—

100

#107

—

101

#107

—

158

—

102

#106

—

158

—

103

#106

—

158

—

104

C₄×C₈

D₃₁₆

yes

—

105

C₄×C₈

C₂²×D₇₉

yes

—

106

C₂×Q₈

—

316

—

107

C₂²×C₄

D₆₃₂

yes

—

108

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

109

C₂×C₈

Dic₃₁₆

yes

—

110

C₂×C₈

C₂×D₃₁₆

yes

characteristic

111

C₂×C₈

C₂×D₃₁₆

yes

—

112

C₂×C₈

C₂×Dic₁₅₈

yes

—

113

C₂×C₈

C₂³×D₇₉

yes

characteristic

114

C₄⋊C₄

—

316

—

115

C₄⋊C₄

—

316

—

116

C₄⋊C₄

—

158

—

117

C₂²⋊C₄

—

316

—

118

C₄²

—

316

—

119

C₄²

D₆₃₂

yes

—

120

C₄²

C₂×D₃₁₆

yes

—

121

C₁₆

C₂×Dic₁₅₈

yes

—

122

C₂³

C₂×D₆₃₂

yes

characteristic

123

Q₈

—

632

—

124

C₂×C₄

—

125

C₂×C₄

—

316

—

126

C₂×C₄

#14579161

yes

—

127

C₂×C₄

C₂×D₆₃₂

yes

—

128

C₂×C₄

C₂×D₆₃₂

yes

characteristic

129

C₂×C₄

C₂×Dic₃₁₆

yes

—

130

C₂×C₄

C₂²×D₃₁₆

yes

characteristic

131

C₈

#14579161

yes

—

132

C₈

C₂×Dic₃₁₆

yes

—

133

C₈

C₂²×Dic₁₅₈

yes

—

134

C₂²

—

135

C₂²

#19098968

yes

—

136

C₂²

C₂²×D₆₃₂

yes

characteristic

137

C₄

—

138

C₄

—

632

—

139

C₄

#19098968

yes

—

140

C₄

C₂×#14579161

yes

characteristic

141

C₄

C₂²×Dic₃₁₆

yes

characteristic

142

C₂

—

143

C₂

C₂×#19098968

yes

characteristic

144

C₂

#23776536

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₃₂
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#346×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂³×C₇₉	C₂×D₈
hypercenter	C₂×#346	D₇₉
hypocenter	C₇₉	#33728
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₆₃₂

commutator, cosocle

C₂×C₈×C₇₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₇₉

Fitting

C₂×#346×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₈

prime radical

C₂³×C₇₉

C₂×D₈

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
2. #23765686	C₂	semidirect
29. C₂×C₁₆×C₇₉	Q₈	semidirect
77. C₇₉	#33728	semidirect
98. C₂×C₁₆	Dic₁₅₈	semidirect

H ≅ G/N

type

2. #23765686

C₂

semidirect

29. C₂×C₁₆×C₇₉

Q₈

semidirect

77. C₇₉

#33728

semidirect

98. C₂×C₁₆

Dic₁₅₈

semidirect

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

subgroup	index
#23773914	2
#23765686	2
#23765184	2
C₂×#346×C₇₉	2
#33728	79

subgroup

index

C₂×#346×C₇₉

Subgroups — #34443771

Conjugacy classes of subgroups

Complements

Maximal subgroups