Group #34202646 — subgroups

Subgroups — #34202646

Order 20224.

200 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 67 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	#23765675	C₂	1	1	yes	maximal, characteristic
2	#23765308	C₂	1	1	yes	maximal, characteristic
3	#23762795	C₂	1	2	yes	maximal
4	#23761456	C₂	1	2	yes	maximal
5	C₇₉×#1372	C₂	1	1	yes	maximal, characteristic
6	#19098968	C₂²	1	2	yes	—
7	#19098774	—	2	2	no	—
8	C₈×Dic₁₅₈	C₂²	1	1	yes	characteristic
9	#19098389	—	2	1	no	—
10	#19098389	—	2	1	no	—
11	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
12	Dic₇₉⋊C₁₆	—	2	1	no	—
13	#19098338	—	2	1	no	—
14	#19098310	—	2	2	no	—
15	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	#335×C₇₉	C₂²	1	2	yes	—
17	C₂×Dic₃₁₆	—	4	2	no	—
18	#14579161	—	2	2	no	—
19	C₄×Dic₁₅₈	D₄	1	1	yes	characteristic
20	C₂×C₇₉⋊C₁₆	—	2	1	no	—
21	#14579095	—	2	2	no	—
22	#14579095	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
24	Dic₇₉⋊C₈	—	2	1	no	—
25	#14579086	—	4	2	no	—
26	C₈×Dic₇₉	—	2	1	no	—
27	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₇₉	—	2	2	no	—
30	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₅₈	—	4	2	no	—
32	C₂×Dic₁₅₈	D₈	1	1	yes	characteristic
33	Dic₃₁₆	—	8	4	no	—
34	C₂×C₇₉⋊C₈	—	2	1	no	—
35	C₄⋊Dic₇₉	—	4	2	no	—
36	C₄⋊Dic₇₉	—	4	2	no	—
37	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
38	Dic₇₉⋊C₄	—	2	1	no	—
39	C₄×Dic₇₉	—	2	1	no	—
40	C₇₉⋊C₁₆	—	4	1	no	—
41	C₂×C₈×C₇₉	—	4	2	no	—
42	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
44	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₁₆×C₇₉	—	2	1	no	—
46	C₁₆×C₇₉	Q₁₆	1	2	yes	complemented
47	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
50	C₂×Dic₇₉	—	2	1	no	—
51	C₂×Dic₇₉	—	2	1	no	—
52	C₂×Dic₇₉	—	8	2	no	—
53	Dic₁₅₈	—	2	2	no	—
54	Dic₁₅₈	—	8	2	no	—
55	Dic₁₅₈	—	8	2	no	—
56	C₈×C₇₉	—	2	1	no	—
57	C₈×C₇₉	—	8	2	no	—
58	C₈×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₇₉	#137	1	1	yes	characteristic
60	C₇₉⋊C₈	—	4	1	no	—
61	C₂²×C₇₉	#466	1	1	yes	characteristic
62	C₄×C₇₉	—	2	1	no	—
63	C₄×C₇₉	#462	1	1	yes	characteristic
64	C₄×C₇₉	#473	1	1	yes	characteristic
65	C₄×C₇₉	#509	1	2	yes	—
66	Dic₇₉	—	4	1	no	—
67	Dic₇₉	—	4	1	no	—
68	Dic₇₉	—	16	2	no	—
69	#12460	—	79	1	no	maximal, Sylow
70	C₂×C₇₉	#1674	1	1	yes	characteristic
71	C₂×C₇₉	#2218	1	1	yes	characteristic
72	C₂×C₇₉	#2222	1	1	yes	characteristic
73	#2252	—	79	1	no	—
74	#2183	—	79	1	no	—
75	#1674	—	79	2	no	—
76	#1372	D₇₉	1	1	yes	characteristic
77	#1364	—	79	2	no	—
78	C₇₉	#12460	1	1	yes	characteristic, complemented, Sylow
79	#500	—	79	2	no	—
80	#462	—	158	2	no	—
81	C₈×Q₈	—	79	1	no	—
82	#366	—	158	1	no	—
83	#366	—	158	1	no	—
84	C₄⋊C₁₆	—	79	1	no	—
85	C₄⋊C₁₆	—	158	1	no	—
86	#346	—	158	1	no	—
87	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
88	#335	C₂×D₇₉	1	2	yes	—
89	#332	—	158	2	no	—
90	C₂×Q₁₆	—	316	2	no	—
91	C₄⋊Q₈	—	158	2	no	—
92	C₄×Q₈	—	79	1	no	—
93	C₂×C₁₆	—	158	1	no	—
94	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
95	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
96	#107	—	158	2	no	—
97	#107	—	158	2	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	79	1	no	—
100	C₄⋊C₈	—	158	1	no	—
101	#103	—	316	2	no	—
102	C₄×C₈	—	158	1	no	—
103	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
104	C₂×Q₈	—	316	2	no	—
105	C₂×Q₈	—	79	1	no	—
106	Q₁₆	—	632	4	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	158	1	no	—
109	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
110	C₂×C₈	#11929625	1	1	yes	characteristic
111	C₄⋊C₄	—	316	2	no	—
112	C₄⋊C₄	—	316	2	no	—
113	C₄⋊C₄	—	79	1	no	—
114	C₄⋊C₄	—	158	1	no	—
115	C₄²	—	158	1	no	—
116	C₄²	D₄×D₇₉	1	1	yes	characteristic
117	C₁₆	—	2	1	no	—
118	C₁₆	—	316	1	no	—
119	C₁₆	Dic₃₁₆	1	2	yes	complemented
120	Q₈	—	632	2	no	—
121	Q₈	—	632	2	no	—
122	Q₈	—	158	2	no	—
123	C₂×C₄	—	632	2	no	—
124	C₂×C₄	—	158	1	no	—
125	C₂×C₄	—	158	1	no	—
126	C₂×C₄	#14579136	1	1	yes	characteristic
127	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
128	C₂×C₄	#14579184	1	1	yes	characteristic
129	C₈	—	2	1	no	—
130	C₈	—	8	2	no	—
131	C₈	—	316	1	no	—
132	C₈	C₂×Dic₃₁₆	1	1	yes	characteristic
133	C₈	#14579194	1	1	yes	characteristic
134	C₂²	#19098800	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	316	1	no	—
137	C₄	—	316	1	no	—
138	C₄	—	1264	2	no	—
139	C₄	#19098774	1	1	yes	characteristic
140	C₄	#19098839	1	1	yes	characteristic
141	C₄	#19099000	1	2	yes	—
142	C₂	#23762795	1	1	yes	characteristic
143	C₂	#23765504	1	1	yes	characteristic
144	C₂	#23765526	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765675

C₂

yes

maximal, characteristic

#23765308

C₂

yes

maximal, characteristic

#23762795

C₂

yes

maximal

#23761456

C₂

yes

maximal

C₇₉×#1372

C₂

yes

maximal, characteristic

#19098968

C₂²

yes

—

#19098774

—

C₈×Dic₁₅₈

C₂²

yes

characteristic

#19098389

—

#19098389

—

C₄⋊(C₇₉⋊C₁₆)

C₂²

yes

characteristic

Dic₇₉⋊C₁₆

—

#19098338

—

#19098310

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

#335×C₇₉

C₂²

yes

—

C₂×Dic₃₁₆

—

#14579161

—

C₄×Dic₁₅₈

D₄

yes

characteristic

C₂×C₇₉⋊C₁₆

—

#14579095

—

#14579095

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

characteristic

Dic₇₉⋊C₈

—

#14579086

—

C₈×Dic₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

D₈

yes

characteristic

Dic₃₁₆

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₈

yes

characteristic

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

Q₁₆

yes

complemented

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

#135

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂×Q₁₆

yes

characteristic

C₈×C₇₉

#137

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

#466

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

—

D₇₉

yes

characteristic

#1364

—

C₇₉

#12460

yes

characteristic, complemented, Sylow

#500

—

#462

—

158

—

C₈×Q₈

—

#366

—

158

—

#366

—

158

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

158

—

#346

—

158

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

#335

C₂×D₇₉

yes

—

#332

—

158

—

C₂×Q₁₆

—

316

—

C₄⋊Q₈

—

158

—

C₄×Q₈

—

C₂×C₁₆

—

158

—

C₂×C₁₆

D₃₁₆

yes

characteristic

C₂×C₁₆

D₃₁₆

yes

characteristic

#107

—

158

—

#107

—

158

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

158

—

101

#103

—

316

—

102

C₄×C₈

—

158

—

103

C₄×C₈

C₂²×D₇₉

yes

characteristic

104

C₂×Q₈

—

316

—

105

C₂×Q₈

—

106

Q₁₆

—

632

—

107

C₂×C₈

—

108

C₂×C₈

—

158

—

109

C₂×C₈

C₂×D₃₁₆

yes

characteristic

110

C₂×C₈

#11929625

yes

characteristic

111

C₄⋊C₄

—

316

—

112

C₄⋊C₄

—

316

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

158

—

115

C₄²

—

158

—

116

C₄²

D₄×D₇₉

yes

characteristic

117

C₁₆

—

118

C₁₆

—

316

—

119

C₁₆

Dic₃₁₆

yes

complemented

120

Q₈

—

632

—

121

Q₈

—

632

—

122

Q₈

—

158

—

123

C₂×C₄

—

632

—

124

C₂×C₄

—

158

—

125

C₂×C₄

—

158

—

126

C₂×C₄

#14579136

yes

characteristic

127

C₂×C₄

D₈×D₇₉

yes

characteristic

128

C₂×C₄

#14579184

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

316

—

132

C₈

C₂×Dic₃₁₆

yes

characteristic

133

C₈

#14579194

yes

characteristic

134

C₂²

#19098800

yes

characteristic

135

C₄

—

136

C₄

—

316

—

137

C₄

—

316

—

138

C₄

—

1264

—

139

C₄

#19098774

yes

characteristic

140

C₄

#19098839

yes

characteristic

141

C₄

#19099000

yes

—

142

C₂

#23762795

yes

characteristic

143

C₂

#23765504

yes

characteristic

144

C₂

#23765526

yes

characteristic

	subgroup	G/H
center	C₂²	#19098800
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1372	C₂
socle, prime radical	C₂²×C₇₉	#466
hypercenter	#1372	D₇₉
hypocenter	C₇₉	#12460
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

subgroup

G/H

center

C₂²

#19098800

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1372

C₂

C₂²×C₇₉

D₇₉

C₇₉

G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

N	H ≅ G/N	type
46. C₁₆×C₇₉	Q₁₆	semidirect
78. C₇₉	#12460	semidirect
119. C₁₆	Dic₃₁₆	semidirect

H ≅ G/N

type

46. C₁₆×C₇₉

Q₁₆

semidirect

78. C₇₉

#12460

semidirect

119. C₁₆

Dic₃₁₆

semidirect

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

subgroup	index
#23765675	2
#23765308	2
#23762795	2
#23761456	2
C₇₉×#1372	2
#12460	79

subgroup

index

C₇₉×#1372

Subgroups — #34202646

Conjugacy classes of subgroups

Complements

Maximal subgroups