Group #34209286 — subgroups

Subgroups — #34209286

Order 20224.

232 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 71 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	C₂×#19098403	C₂	1	2	yes	maximal, complemented
2	C₂×#19098401	C₂	1	2	yes	maximal
3	#23765653	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761534	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#2249	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₃₂	—	2	1	no	—
7	C₂×Dic₆₃₂	C₂²	1	2	yes	—
8	#19098943	—	2	1	no	—
9	#19098403	—	2	4	no	—
10	#19098401	—	2	4	no	—
11	C₂×C₇₉⋊C₃₂	C₂²	1	2	yes	—
12	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×SD₃₂×C₇₉	—	2	1	no	—
14	C₂×D₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	#495×C₇₉	—	2	1	no	—
16	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₁₆	—	2	2	no	—
18	C₂×Dic₃₁₆	—	4	1	no	—
19	#14579161	—	4	1	no	—
20	Dic₆₃₂	—	2	2	no	—
21	Dic₆₃₂	—	2	2	no	—
22	Dic₆₃₂	—	4	2	no	—
23	C₇₉⋊C₃₂	D₄	1	4	yes	—
24	C₂×Q₁₆×C₇₉	—	2	1	no	—
25	C₂×SD₁₆×C₇₉	—	4	1	no	—
26	C₂×D₈×C₇₉	—	2	1	no	—
27	#124×C₇₉	—	4	1	no	—
28	Q₃₂×C₇₉	—	2	2	no	—
29	SD₃₂×C₇₉	—	4	2	no	—
30	D₁₆×C₇₉	—	2	2	no	—
31	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
34	C₂×Dic₁₅₈	—	4	2	no	—
35	Dic₃₁₆	—	4	2	no	—
36	Dic₃₁₆	—	4	2	no	—
37	Dic₃₁₆	—	8	2	no	—
38	C₄⋊Dic₇₉	—	8	2	no	—
39	C₂×Q₈×C₇₉	—	4	1	no	—
40	C₂×D₄×C₇₉	—	4	1	no	—
41	Q₁₆×C₇₉	—	4	1	no	—
42	Q₁₆×C₇₉	—	4	1	no	—
43	SD₁₆×C₇₉	—	8	2	no	—
44	D₈×C₇₉	—	4	1	no	—
45	D₈×C₇₉	—	4	1	no	—
46	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
47	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊C₄×C₇₉	—	8	2	no	—
49	C₄²×C₇₉	D₈	1	1	yes	characteristic
50	C₁₆×C₇₉	D₈	1	2	yes	—
51	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
52	C₂³×C₇₉	—	8	1	no	—
53	Q₈×C₇₉	—	8	1	no	—
54	Q₈×C₇₉	—	8	1	no	—
55	D₄×C₇₉	—	8	1	no	—
56	D₄×C₇₉	—	8	1	no	—
57	C₂×C₄×C₇₉	—	8	1	no	—
58	C₂×C₄×C₇₉	D₁₆	1	2	yes	—
59	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
60	C₂×Dic₇₉	—	8	2	no	—
61	Dic₁₅₈	—	8	2	no	—
62	Dic₁₅₈	—	8	2	no	—
63	C₈×C₇₉	D₁₆	1	2	yes	—
64	C₈×C₇₉	#127	1	1	yes	characteristic
65	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
66	C₂²×C₇₉	—	16	1	no	—
67	C₂²×C₇₉	—	16	2	no	—
68	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
69	C₄×C₇₉	—	2	2	no	—
70	C₄×C₇₉	—	16	1	no	—
71	C₄×C₇₉	#493	1	1	yes	characteristic
72	C₄×C₇₉	C₂×D₁₆	1	1	yes	characteristic
73	Dic₇₉	—	16	2	no	—
74	#13730	—	79	1	no	maximal, Sylow
75	C₂×C₇₉	—	32	1	no	—
76	C₂×C₇₉	#2246	1	1	yes	characteristic
77	C₂×C₇₉	#2264	1	2	yes	—
78	C₂×Q₆₄	—	79	2	no	—
79	C₂×SD₆₄	—	79	2	no	—
80	#2249	D₇₉	1	1	yes	characteristic
81	#2247	—	79	1	no	—
82	#1397	—	79	1	no	—
83	C₇₉	#13730	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₃₂	—	79	2	no	—
85	C₂×Q₃₂	—	158	1	no	—
86	C₂×Q₃₂	C₂×D₇₉	1	1	yes	characteristic
87	C₂×SD₃₂	—	2	1	no	—
88	C₂×D₁₆	C₂×D₇₉	1	1	yes	characteristic
89	#495	—	2	1	no	—
90	#494	—	158	1	no	—
91	Q₆₄	—	158	4	no	—
92	SD₆₄	—	158	4	no	—
93	C₂×C₃₂	—	79	2	no	—
94	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
95	C₂×Q₁₆	—	2	1	no	—
96	C₂×Q₁₆	—	316	1	no	—
97	C₂×Q₁₆	—	158	2	no	—
98	C₂×SD₁₆	—	4	1	no	—
99	C₂×D₈	—	2	1	no	—
100	C₄⋊Q₈	—	316	1	no	—
101	#124	—	4	1	no	—
102	Q₃₂	—	2	2	no	—
103	Q₃₂	—	316	2	no	—
104	Q₃₂	—	158	2	no	—
105	Q₃₂	—	158	2	no	—
106	SD₃₂	—	4	2	no	—
107	D₁₆	—	2	2	no	—
108	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
109	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
110	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
111	C₃₂	—	79	4	no	—
112	C₂×Q₈	—	4	1	no	—
113	C₂×Q₈	—	316	2	no	—
114	C₂×D₄	—	4	1	no	—
115	Q₁₆	—	4	1	no	—
116	Q₁₆	—	4	1	no	—
117	Q₁₆	—	316	2	no	—
118	Q₁₆	—	316	2	no	—
119	Q₁₆	—	632	2	no	—
120	SD₁₆	—	8	2	no	—
121	D₈	—	4	1	no	—
122	D₈	—	4	1	no	—
123	C₂×C₈	#11929609	1	1	yes	characteristic
124	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
125	C₄⋊C₄	—	632	2	no	—
126	C₂²⋊C₄	—	8	2	no	—
127	C₄²	#11929609	1	1	yes	characteristic
128	C₁₆	#11929609	1	2	yes	—
129	C₁₆	D₄×D₇₉	1	2	yes	—
130	C₂³	—	8	1	no	—
131	Q₈	—	8	1	no	—
132	Q₈	—	8	1	no	—
133	Q₈	—	632	2	no	—
134	Q₈	—	632	2	no	—
135	D₄	—	8	1	no	—
136	D₄	—	8	1	no	—
137	C₂×C₄	—	8	1	no	—
138	C₂×C₄	—	632	2	no	—
139	C₂×C₄	#14579102	1	2	yes	—
140	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
141	C₈	#14579102	1	2	yes	—
142	C₈	#14579157	1	1	yes	characteristic
143	C₈	C₂×#11929609	1	1	yes	characteristic
144	C₂²	—	16	1	no	—
145	C₂²	—	16	2	no	—
146	C₂²	C₂×#14579102	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	16	1	no	—
149	C₄	—	1264	2	no	—
150	C₄	#19098939	1	1	yes	characteristic
151	C₄	C₂×#14579102	1	1	yes	characteristic
152	C₂	—	32	1	no	—
153	C₂	#23765649	1	1	yes	characteristic
154	C₂	#23765717	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098403

C₂

yes

maximal, complemented

C₂×#19098401

C₂

yes

maximal

#23765653

C₂

yes

maximal, characteristic, complemented

#23761534

C₂

yes

maximal, characteristic, complemented

C₇₉×#2249

C₂

yes

maximal, characteristic

C₂×Dic₆₃₂

—

C₂×Dic₆₃₂

C₂²

yes

—

#19098943

—

#19098403

—

#19098401

—

C₂×C₇₉⋊C₃₂

C₂²

yes

—

C₂×Q₃₂×C₇₉

C₂²

yes

characteristic

C₂×SD₃₂×C₇₉

—

C₂×D₁₆×C₇₉

C₂²

yes

characteristic

#495×C₇₉

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

#14579161

—

Dic₆₃₂

—

Dic₆₃₂

—

Dic₆₃₂

—

C₇₉⋊C₃₂

D₄

yes

—

C₂×Q₁₆×C₇₉

—

C₂×SD₁₆×C₇₉

—

C₂×D₈×C₇₉

—

#124×C₇₉

—

Q₃₂×C₇₉

—

SD₃₂×C₇₉

—

D₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

SD₁₆×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

D₈

yes

characteristic

C₁₆×C₇₉

D₈

yes

—

C₁₆×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

D₁₆

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

D₁₆

yes

—

C₈×C₇₉

#127

yes

characteristic

C₈×C₇₉

C₂×D₈

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#493

yes

characteristic

C₄×C₇₉

C₂×D₁₆

yes

characteristic

Dic₇₉

—

#13730

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#2246

yes

characteristic

C₂×C₇₉

#2264

yes

—

C₂×Q₆₄

—

C₂×SD₆₄

—

#2249

D₇₉

yes

characteristic

#2247

—

#1397

—

C₇₉

#13730

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

158

—

C₂×Q₃₂

C₂×D₇₉

yes

characteristic

C₂×SD₃₂

—

C₂×D₁₆

C₂×D₇₉

yes

characteristic

#495

—

#494

—

158

—

Q₆₄

—

158

—

SD₆₄

—

158

—

C₂×C₃₂

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

316

—

C₂×Q₁₆

—

158

—

C₂×SD₁₆

—

C₂×D₈

—

100

C₄⋊Q₈

—

316

—

101

#124

—

102

Q₃₂

—

103

Q₃₂

—

316

—

104

Q₃₂

—

158

—

105

Q₃₂

—

158

—

106

SD₃₂

—

107

D₁₆

—

108

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

109

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

110

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

111

C₃₂

—

112

C₂×Q₈

—

113

C₂×Q₈

—

316

—

114

C₂×D₄

—

115

Q₁₆

—

116

Q₁₆

—

117

Q₁₆

—

316

—

118

Q₁₆

—

316

—

119

Q₁₆

—

632

—

120

SD₁₆

—

121

D₈

—

122

D₈

—

123

C₂×C₈

#11929609

yes

characteristic

124

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

125

C₄⋊C₄

—

632

—

126

C₂²⋊C₄

—

127

C₄²

#11929609

yes

characteristic

128

C₁₆

#11929609

yes

—

129

C₁₆

D₄×D₇₉

yes

—

130

C₂³

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

632

—

134

Q₈

—

632

—

135

D₄

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

632

—

139

C₂×C₄

#14579102

yes

—

140

C₂×C₄

C₂×#11929609

yes

characteristic

141

C₈

#14579102

yes

—

142

C₈

#14579157

yes

characteristic

143

C₈

C₂×#11929609

yes

characteristic

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×#14579102

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

1264

—

150

C₄

#19098939

yes

characteristic

151

C₄

C₂×#14579102

yes

characteristic

152

C₂

—

153

C₂

#23765649

yes

characteristic

154

C₂

#23765717

yes

—

	subgroup	G/H
center	C₂²	C₂×#14579102
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₇₉×#2249	C₂
socle	C₂²×C₇₉	C₂×D₁₆
prime radical	C₂×D₁₆×C₇₉	C₂²
hypercenter	#2249	D₇₉
hypocenter	C₇₉	#13730
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2249

subgroup

G/H

center

C₂²

C₂×#14579102

commutator, cosocle

C₂×C₁₆×C₇₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₉

Fitting

C₇₉×#2249

C₂

socle

C₂²×C₇₉

C₂×D₁₆

prime radical

C₂×D₁₆×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2249

N	H ≅ G/N	type
1. C₂×#19098403	C₂	semidirect
3. #23765653	C₂	semidirect
4. #23761534	C₂	semidirect
83. C₇₉	#13730	semidirect

H ≅ G/N

type

1. C₂×#19098403

C₂

semidirect

3. #23765653

C₂

semidirect

4. #23761534

C₂

semidirect

83. C₇₉

#13730

semidirect

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

subgroup	index
C₂×#19098403	2
C₂×#19098401	2
#23765653	2
#23761534	2
C₇₉×#2249	2
#13730	79

subgroup

index

C₇₉×#2249

Subgroups — #34209286

Conjugacy classes of subgroups

Complements

Maximal subgroups