Group #34209285 — subgroups

Subgroups — #34209285

Order 20224.

232 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 75 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×Dic₁₂₆₄	C₂	1	2	yes	maximal, complemented
2	C₂×#19098402	C₂	1	2	yes	maximal
3	#23765661	C₂	1	1	yes	maximal, characteristic
4	#23765653	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₆₃₂	—	2	1	no	—
7	C₂×Dic₆₃₂	C₂²	1	2	yes	—
8	C₂×Dic₆₃₂	C₂²	1	1	yes	characteristic
9	C₂×#14579106	—	2	1	no	—
10	C₂×D₁₂₆₄	C₂²	1	1	yes	characteristic
11	#19098947	—	2	1	no	—
12	#19098943	—	2	1	no	—
13	Dic₁₂₆₄	—	2	4	no	—
14	#19098402	—	2	4	no	—
15	C₂×C₃₂×C₇₉	C₂²	1	2	yes	—
16	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₁₆	—	2	2	no	—
18	C₂×Dic₃₁₆	—	2	1	no	—
19	C₂×Dic₃₁₆	—	4	1	no	—
20	C₂×#11929613	—	4	1	no	—
21	C₂×D₆₃₂	—	2	1	no	—
22	#14579161	—	4	1	no	—
23	#14579149	—	4	1	no	—
24	Dic₆₃₂	—	2	2	no	—
25	Dic₆₃₂	—	2	2	no	—
26	Dic₆₃₂	—	2	2	no	—
27	Dic₆₃₂	—	4	2	no	—
28	#14579106	—	4	2	no	—
29	D₁₂₆₄	—	2	2	no	—
30	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
33	C₃₂×C₇₉	D₄	1	4	yes	—
34	C₂×Dic₁₅₈	—	4	1	no	—
35	C₂×Dic₁₅₈	—	4	2	no	—
36	C₂×D₃₁₆	—	4	1	no	—
37	Dic₃₁₆	—	4	2	no	—
38	Dic₃₁₆	—	4	1	no	—
39	Dic₃₁₆	—	4	1	no	—
40	Dic₃₁₆	—	4	2	no	—
41	Dic₃₁₆	—	8	2	no	—
42	#11929613	—	8	2	no	—
43	D₆₃₂	—	4	1	no	—
44	D₆₃₂	—	4	1	no	—
45	C₄⋊Dic₇₉	—	8	2	no	—
46	(C₂×D₇₉)⋊C₄	—	8	2	no	—
47	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
48	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₇₉	D₈	1	1	yes	characteristic
50	C₁₆×C₇₉	D₈	1	2	yes	—
51	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
52	C₂²×D₇₉	—	8	1	no	—
53	C₂×C₄×C₇₉	D₁₆	1	2	yes	—
54	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₇₉	—	8	1	no	—
56	C₂×Dic₇₉	—	8	2	no	—
57	D₃₁₆	—	8	1	no	—
58	D₃₁₆	—	8	1	no	—
59	Dic₁₅₈	—	8	1	no	—
60	Dic₁₅₈	—	8	2	no	—
61	Dic₁₅₈	—	8	1	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	C₂×D₁₆	1	1	yes	characteristic
67	C₂×D₇₉	—	16	1	no	—
68	C₂×D₇₉	—	16	2	no	—
69	C₄×C₇₉	—	2	2	no	—
70	C₄×C₇₉	#493	1	1	yes	characteristic
71	C₄×C₇₉	C₂×D₁₆	1	1	yes	characteristic
72	Dic₇₉	—	16	1	no	—
73	Dic₇₉	—	16	2	no	—
74	#13730	—	79	1	no	maximal, Sylow
75	C₂×C₇₉	#2246	1	1	yes	characteristic
76	C₂×C₇₉	#2264	1	2	yes	—
77	D₇₉	—	32	1	no	—
78	C₂×Q₆₄	—	79	2	no	—
79	C₂×SD₆₄	—	79	2	no	—
80	#2249	—	79	1	no	—
81	#2247	—	79	1	no	—
82	#1397	D₇₉	1	1	yes	characteristic, complemented
83	C₇₉	#13730	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₃₂	—	79	2	no	—
85	C₂×Q₃₂	—	79	1	no	—
86	C₂×Q₃₂	—	158	1	no	—
87	C₂×SD₃₂	—	158	1	no	—
88	C₂×D₁₆	—	79	1	no	—
89	#495	—	158	1	no	—
90	#494	—	158	1	no	—
91	Q₆₄	—	158	4	no	—
92	SD₆₄	—	158	4	no	—
93	C₂×C₃₂	C₂×D₇₉	1	2	yes	—
94	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
95	C₂×Q₁₆	—	316	1	no	—
96	C₂×Q₁₆	—	158	2	no	—
97	C₂×Q₁₆	—	158	1	no	—
98	C₂×SD₁₆	—	316	1	no	—
99	C₂×D₈	—	158	1	no	—
100	C₄⋊Q₈	—	316	1	no	—
101	#124	—	316	1	no	—
102	Q₃₂	—	316	2	no	—
103	Q₃₂	—	158	2	no	—
104	Q₃₂	—	158	2	no	—
105	Q₃₂	—	158	2	no	—
106	SD₃₂	—	316	2	no	—
107	D₁₆	—	158	2	no	—
108	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
109	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
110	C₄×C₈	D₃₁₆	1	1	yes	characteristic
111	C₃₂	D₃₁₆	1	4	yes	—
112	C₂×Q₈	—	316	1	no	—
113	C₂×Q₈	—	316	2	no	—
114	C₂×D₄	—	316	1	no	—
115	Q₁₆	—	316	2	no	—
116	Q₁₆	—	316	1	no	—
117	Q₁₆	—	316	1	no	—
118	Q₁₆	—	316	2	no	—
119	Q₁₆	—	632	2	no	—
120	SD₁₆	—	632	2	no	—
121	D₈	—	316	1	no	—
122	D₈	—	316	1	no	—
123	C₂×C₈	D₆₃₂	1	1	yes	characteristic
124	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
125	C₄⋊C₄	—	632	2	no	—
126	C₂²⋊C₄	—	632	2	no	—
127	C₄²	D₆₃₂	1	1	yes	characteristic
128	C₁₆	D₆₃₂	1	2	yes	—
129	C₁₆	C₂×D₃₁₆	1	2	yes	—
130	C₂³	—	632	1	no	—
131	Q₈	—	632	1	no	—
132	Q₈	—	632	2	no	—
133	Q₈	—	632	1	no	—
134	Q₈	—	632	2	no	—
135	D₄	—	632	1	no	—
136	D₄	—	632	1	no	—
137	C₂×C₄	—	632	1	no	—
138	C₂×C₄	—	632	2	no	—
139	C₂×C₄	D₁₂₆₄	1	2	yes	—
140	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
141	C₈	D₁₂₆₄	1	2	yes	—
142	C₈	#14579158	1	1	yes	characteristic
143	C₈	C₂×D₆₃₂	1	1	yes	characteristic
144	C₂²	—	1264	1	no	—
145	C₂²	—	1264	2	no	—
146	C₂²	C₂×D₁₂₆₄	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	1264	1	no	—
149	C₄	—	1264	2	no	—
150	C₄	#19098940	1	1	yes	characteristic
151	C₄	C₂×D₁₂₆₄	1	1	yes	characteristic
152	C₂	—	2528	1	no	—
153	C₂	#23765650	1	1	yes	characteristic
154	C₂	#23765720	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₂₆₄

C₂

yes

maximal, complemented

C₂×#19098402

C₂

yes

maximal

#23765661

C₂

yes

maximal, characteristic

#23765653

C₂

yes

maximal, characteristic, complemented

C₇₉×#1397

C₂

yes

maximal, characteristic, complemented

C₂×Dic₆₃₂

—

C₂×Dic₆₃₂

C₂²

yes

—

C₂×Dic₆₃₂

C₂²

yes

characteristic

C₂×#14579106

—

C₂×D₁₂₆₄

C₂²

yes

characteristic

#19098947

—

#19098943

—

Dic₁₂₆₄

—

#19098402

—

C₂×C₃₂×C₇₉

C₂²

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×#11929613

—

C₂×D₆₃₂

—

#14579161

—

#14579149

—

Dic₆₃₂

—

Dic₆₃₂

—

Dic₆₃₂

—

Dic₆₃₂

—

#14579106

—

D₁₂₆₄

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₃₂×C₇₉

D₄

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

—

#11929613

—

D₆₃₂

—

D₆₃₂

—

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

D₈

yes

characteristic

C₁₆×C₇₉

D₈

yes

—

C₁₆×C₇₉

C₂×D₄

yes

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

D₁₆

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

D₁₆

yes

—

C₈×C₇₉

#127

yes

characteristic

C₈×C₇₉

C₂×D₈

yes

characteristic

C₂²×C₇₉

C₂×D₁₆

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#493

yes

characteristic

C₄×C₇₉

C₂×D₁₆

yes

characteristic

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

—

C₂×Q₆₄

—

C₂×SD₆₄

—

#2249

—

#2247

—

#1397

D₇₉

yes

characteristic, complemented

C₇₉

#13730

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

158

—

C₂×SD₃₂

—

158

—

C₂×D₁₆

—

#495

—

158

—

#494

—

158

—

Q₆₄

—

158

—

SD₆₄

—

158

—

C₂×C₃₂

C₂×D₇₉

yes

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

316

—

C₂×Q₁₆

—

158

—

C₂×Q₁₆

—

158

—

C₂×SD₁₆

—

316

—

C₂×D₈

—

158

—

100

C₄⋊Q₈

—

316

—

101

#124

—

316

—

102

Q₃₂

—

316

—

103

Q₃₂

—

158

—

104

Q₃₂

—

158

—

105

Q₃₂

—

158

—

106

SD₃₂

—

316

—

107

D₁₆

—

158

—

108

C₂×C₁₆

D₃₁₆

yes

characteristic

109

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

110

C₄×C₈

D₃₁₆

yes

characteristic

111

C₃₂

D₃₁₆

yes

—

112

C₂×Q₈

—

316

—

113

C₂×Q₈

—

316

—

114

C₂×D₄

—

316

—

115

Q₁₆

—

316

—

116

Q₁₆

—

316

—

117

Q₁₆

—

316

—

118

Q₁₆

—

316

—

119

Q₁₆

—

632

—

120

SD₁₆

—

632

—

121

D₈

—

316

—

122

D₈

—

316

—

123

C₂×C₈

D₆₃₂

yes

characteristic

124

C₂×C₈

C₂×D₃₁₆

yes

characteristic

125

C₄⋊C₄

—

632

—

126

C₂²⋊C₄

—

632

—

127

C₄²

D₆₃₂

yes

characteristic

128

C₁₆

D₆₃₂

yes

—

129

C₁₆

C₂×D₃₁₆

yes

—

130

C₂³

—

632

—

131

Q₈

—

632

—

132

Q₈

—

632

—

133

Q₈

—

632

—

134

Q₈

—

632

—

135

D₄

—

632

—

136

D₄

—

632

—

137

C₂×C₄

—

632

—

138

C₂×C₄

—

632

—

139

C₂×C₄

D₁₂₆₄

yes

—

140

C₂×C₄

C₂×D₆₃₂

yes

characteristic

141

C₈

D₁₂₆₄

yes

—

142

C₈

#14579158

yes

characteristic

143

C₈

C₂×D₆₃₂

yes

characteristic

144

C₂²

—

1264

—

145

C₂²

—

1264

—

146

C₂²

C₂×D₁₂₆₄

yes

characteristic

147

C₄

—

148

C₄

—

1264

—

149

C₄

—

1264

—

150

C₄

#19098940

yes

characteristic

151

C₄

C₂×D₁₂₆₄

yes

characteristic

152

C₂

—

2528

—

153

C₂

#23765650

yes

characteristic

154

C₂

#23765720

yes

—

	subgroup	G/H
center	C₂²	C₂×D₁₂₆₄
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₇₉×#1397	C₂
socle	C₂²×C₇₉	C₂×D₁₆
prime radical	C₂×D₁₂₆₄	C₂²
hypercenter	#1397	D₇₉
hypocenter	C₇₉	#13730
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

subgroup

G/H

center

C₂²

C₂×D₁₂₆₄

commutator, cosocle

C₂×C₁₆×C₇₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₉

Fitting

C₇₉×#1397

C₂

socle

C₂²×C₇₉

C₂×D₁₆

prime radical

C₂×D₁₂₆₄

C₂²

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

N	H ≅ G/N	type
1. C₂×Dic₁₂₆₄	C₂	semidirect
4. #23765653	C₂	semidirect
5. C₇₉×#1397	C₂	semidirect
82. #1397	D₇₉	semidirect
83. C₇₉	#13730	semidirect

H ≅ G/N

type

1. C₂×Dic₁₂₆₄

C₂

semidirect

4. #23765653

C₂

semidirect

5. C₇₉×#1397

C₂

semidirect

82. #1397

D₇₉

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×Dic₁₂₆₄	2
C₂×#19098402	2
#23765661	2
#23765653	2
C₇₉×#1397	2
#13730	79

subgroup

index

C₂×Dic₁₂₆₄

C₇₉×#1397

Subgroups — #34209285

Conjugacy classes of subgroups

Complements

Maximal subgroups