Group #34202922 — subgroups

Subgroups — #34202922

Order 20224.

306 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 99 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, complemented
2	#23765650	C₂	1	1	yes	maximal, characteristic
3	#23765624	C₂	1	2	yes	maximal, complemented
4	#23762992	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₂₆₄	—	2	2	no	—
7	#19098968	C₂²	1	2	yes	—
8	#19098940	C₂²	1	2	yes	—
9	#19098936	—	2	2	no	—
10	#19098910	—	2	2	no	—
11	#19098389	—	2	2	no	—
12	#19098365	—	2	4	no	—
13	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₈²×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×#11929613	—	4	4	no	—
17	C₂×D₆₃₂	—	2	2	no	—
18	C₂×D₆₃₂	—	2	2	no	—
19	#14579161	—	2	2	no	—
20	#14579158	—	2	2	no	—
21	D₁₂₆₄	—	4	4	no	—
22	#14579095	—	2	2	no	—
23	#14579095	—	2	2	no	—
24	#14579083	—	4	4	no	—
25	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
26	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
27	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₅₈	—	4	2	no	—
31	C₂×D₃₁₆	—	4	2	no	—
32	C₂×D₃₁₆	—	4	2	no	—
33	C₂×D₃₁₆	—	4	2	no	—
34	#11929613	—	8	8	no	—
35	D₆₃₂	—	4	4	no	—
36	D₆₃₂	—	4	4	no	—
37	C₄⋊Dic₇₉	—	4	2	no	—
38	C₄⋊Dic₇₉	—	4	2	no	—
39	C₂×C₈×C₇₉	D₈	1	2	yes	—
40	C₂×C₈×C₇₉	D₈	1	2	yes	—
41	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₉	—	2	2	no	—
45	C₁₆×C₇₉	SD₁₆	1	4	yes	complemented
46	C₂²×D₇₉	—	8	2	no	—
47	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₇₉	—	8	2	no	—
51	D₃₁₆	—	8	2	no	—
52	D₃₁₆	—	8	2	no	—
53	D₃₁₆	—	8	4	no	—
54	D₃₁₆	—	8	4	no	—
55	Dic₁₅₈	—	8	4	no	—
56	C₈×C₇₉	—	2	2	no	—
57	C₈×C₇₉	D₁₆	1	4	yes	complemented
58	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
59	C₈×C₇₉	#135	1	2	yes	—
60	C₂²×C₇₉	#493	1	1	yes	characteristic
61	C₂×D₇₉	—	16	2	no	—
62	C₂×D₇₉	—	16	2	no	—
63	C₂×D₇₉	—	16	2	no	—
64	C₄×C₇₉	#492	1	1	yes	characteristic
65	C₄×C₇₉	#495	1	1	yes	characteristic
66	C₄×C₇₉	C₂×D₁₆	1	2	yes	—
67	C₄×C₇₉	#508	1	2	yes	—
68	Dic₇₉	—	16	2	no	—
69	#12503	—	79	1	no	maximal, Sylow
70	C₂×C₇₉	#1706	1	1	yes	characteristic
71	C₂×C₇₉	#2246	1	1	yes	characteristic
72	C₂×C₇₉	#2249	1	1	yes	characteristic
73	D₇₉	—	32	2	no	—
74	#2252	—	79	1	no	—
75	#2246	—	79	1	no	—
76	#2240	—	79	2	no	—
77	#1706	—	79	2	no	—
78	C₈×C₁₆	D₇₉	1	1	yes	characteristic, complemented
79	C₇₉	#12503	1	1	yes	characteristic, complemented, Sylow
80	C₂×D₁₆	—	158	2	no	—
81	#500	—	79	2	no	—
82	#493	—	79	2	no	—
83	#492	—	158	2	no	—
84	#486	—	158	2	no	—
85	#366	—	158	2	no	—
86	#357	—	158	4	no	—
87	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
88	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
89	C₈²	C₂×D₇₉	1	1	yes	characteristic
90	C₂×SD₁₆	—	316	4	no	—
91	C₂×D₈	—	158	2	no	—
92	C₂×D₈	—	158	2	no	—
93	C₄⋊Q₈	—	158	2	no	—
94	#127	—	158	2	no	—
95	D₁₆	—	316	4	no	—
96	C₂×C₁₆	D₃₁₆	1	2	yes	—
97	C₂×C₁₆	D₃₁₆	1	2	yes	—
98	#107	—	158	2	no	—
99	#107	—	158	2	no	—
100	#102	—	316	4	no	—
101	C₄×C₈	D₃₁₆	1	1	yes	characteristic
102	C₄×C₈	D₃₁₆	1	1	yes	characteristic
103	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
104	C₂×Q₈	—	316	2	no	—
105	C₂×D₄	—	316	2	no	—
106	C₂×D₄	—	316	2	no	—
107	C₂×D₄	—	316	2	no	—
108	SD₁₆	—	632	8	no	—
109	D₈	—	316	4	no	—
110	D₈	—	316	4	no	—
111	C₂×C₈	D₆₃₂	1	2	yes	—
112	C₂×C₈	D₆₃₂	1	2	yes	—
113	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
114	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
115	C₄⋊C₄	—	316	2	no	—
116	C₄⋊C₄	—	316	2	no	—
117	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
118	C₁₆	—	2	2	no	—
119	C₁₆	#11929613	1	4	yes	complemented
120	C₂³	—	632	2	no	—
121	Q₈	—	632	4	no	—
122	D₄	—	632	2	no	—
123	D₄	—	632	2	no	—
124	D₄	—	632	4	no	—
125	D₄	—	632	4	no	—
126	C₂×C₄	—	632	2	no	—
127	C₂×C₄	#14579158	1	1	yes	characteristic
128	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
129	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	D₁₂₆₄	1	4	yes	complemented
132	C₈	C₂×#11929613	1	2	yes	—
133	C₈	#14579183	1	2	yes	—
134	C₂²	—	1264	2	no	—
135	C₂²	—	1264	2	no	—
136	C₂²	—	1264	2	no	—
137	C₂²	#19098940	1	1	yes	characteristic
138	C₄	—	1264	2	no	—
139	C₄	#19098936	1	1	yes	characteristic
140	C₄	#19098947	1	1	yes	characteristic
141	C₄	C₂×D₁₂₆₄	1	2	yes	—
142	C₄	#19098995	1	2	yes	—
143	C₂	—	2528	2	no	—
144	C₂	#23762992	1	1	yes	characteristic
145	C₂	#23765650	1	1	yes	characteristic
146	C₂	#23765661	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765675

C₂

yes

maximal, characteristic, complemented

#23765650

C₂

yes

maximal, characteristic

#23765624

C₂

yes

maximal, complemented

#23762992

C₂

yes

maximal, complemented

C₈×C₁₆×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×D₁₂₆₄

—

#19098968

C₂²

yes

—

#19098940

C₂²

yes

—

—

—

—

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₈²×C₇₉

C₂²

yes

characteristic

C₂×#11929613

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

#14579161

—

#14579158

—

D₁₂₆₄

—

#14579095

—

#14579095

—

#14579083

—

C₂×C₁₆×C₇₉

D₄

yes

—

C₂×C₁₆×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

#11929613

—

D₆₃₂

—

D₆₃₂

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

SD₁₆

yes

complemented

C₂²×D₇₉

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

D₁₆

yes

complemented

C₈×C₇₉

C₂×SD₁₆

yes

—

C₈×C₇₉

#135

yes

—

C₂²×C₇₉

#493

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂×D₁₆

yes

—

C₄×C₇₉

#508

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

—

—

—

—

C₈×C₁₆

D₇₉

yes

characteristic, complemented

C₇₉

#12503

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

158

—

#500

—

#493

—

#492

—

158

—

#486

—

158

—

#366

—

158

—

#357

—

158

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₈²

C₂×D₇₉

yes

characteristic

C₂×SD₁₆

—

316

—

C₂×D₈

—

158

—

C₂×D₈

—

158

—

C₄⋊Q₈

—

158

—

#127

—

158

—

D₁₆

—

316

—

C₂×C₁₆

D₃₁₆

yes

—

C₂×C₁₆

D₃₁₆

yes

—

#107

—

158

—

#107

—

158

—

100

#102

—

316

—

101

C₄×C₈

D₃₁₆

yes

characteristic

102

C₄×C₈

D₃₁₆

yes

characteristic

103

C₄×C₈

C₂²×D₇₉

yes

characteristic

104

C₂×Q₈

—

316

—

105

C₂×D₄

—

316

—

106

C₂×D₄

—

316

—

107

C₂×D₄

—

316

—

108

SD₁₆

—

632

—

109

D₈

—

316

—

110

D₈

—

316

—

111

C₂×C₈

D₆₃₂

yes

—

112

C₂×C₈

D₆₃₂

yes

—

113

C₂×C₈

C₂×D₃₁₆

yes

characteristic

114

C₂×C₈

C₂×D₃₁₆

yes

characteristic

115

C₄⋊C₄

—

316

—

116

C₄⋊C₄

—

316

—

117

C₄²

C₂×D₃₁₆

yes

characteristic

118

C₁₆

—

119

C₁₆

#11929613

yes

complemented

120

C₂³

—

632

—

121

Q₈

—

632

—

122

D₄

—

632

—

123

D₄

—

632

—

124

D₄

—

632

—

125

D₄

—

632

—

126

C₂×C₄

—

632

—

127

C₂×C₄

#14579158

yes

characteristic

128

C₂×C₄

C₂×D₆₃₂

yes

characteristic

129

C₂×C₄

C₂×D₆₃₂

yes

characteristic

130

C₈

—

131

C₈

D₁₂₆₄

yes

complemented

132

C₈

C₂×#11929613

yes

—

133

C₈

#14579183

yes

—

134

C₂²

—

1264

—

135

C₂²

—

1264

—

136

C₂²

—

1264

—

137

C₂²

#19098940

yes

characteristic

138

C₄

—

1264

—

139

C₄

#19098936

yes

characteristic

140

C₄

#19098947

yes

characteristic

141

C₄

C₂×D₁₂₆₄

yes

—

142

C₄

#19098995

yes

—

143

C₂

—

2528

—

144

C₂

#23762992

yes

characteristic

145

C₂

#23765650

yes

characteristic

146

C₂

#23765661

yes

characteristic

	subgroup	G/H
center	C₂²	#19098940
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₈×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	#493
prime radical	#23765650	C₂
hypercenter	C₈×C₁₆	D₇₉
hypocenter	C₇₉	#12503
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#19098940

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₈×C₁₆×C₇₉

C₂

C₂²×C₇₉

C₂

C₈×C₁₆

D₇₉

hypocenter

C₇₉

#12503

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #23765675	C₂	semidirect
3. #23765624	C₂	semidirect
4. #23762992	C₂	semidirect
5. C₈×C₁₆×C₇₉	C₂	semidirect
45. C₁₆×C₇₉	SD₁₆	semidirect
57. C₈×C₇₉	D₁₆	semidirect
78. C₈×C₁₆	D₇₉	semidirect
79. C₇₉	#12503	semidirect
119. C₁₆	#11929613	semidirect
131. C₈	D₁₂₆₄	semidirect

H ≅ G/N

type

1. #23765675

C₂

semidirect

3. #23765624

C₂

semidirect

4. #23762992

C₂

semidirect

5. C₈×C₁₆×C₇₉

C₂

semidirect

45. C₁₆×C₇₉

SD₁₆

semidirect

57. C₈×C₇₉

D₁₆

semidirect

78. C₈×C₁₆

D₇₉

semidirect

79. C₇₉

#12503

semidirect

119. C₁₆

#11929613

semidirect

131. C₈

D₁₂₆₄

semidirect

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

subgroup	index
#23765675	2
#23765650	2
#23765624	2
#23762992	2
C₈×C₁₆×C₇₉	2
#12503	79

subgroup

index

C₈×C₁₆×C₇₉

Subgroups — #34202922

Conjugacy classes of subgroups

Complements

Maximal subgroups