Group #36469081 — subgroups

Subgroups — #36469081

Order 22784.

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	#26022527	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022502	C₂	1	1	yes	maximal, characteristic
3	#26022476	C₂	1	2	yes	maximal, complemented
4	#26019844	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₄₂₄	—	2	2	no	—
7	#19152003	C₂²	1	2	yes	—
8	#19151975	C₂²	1	2	yes	—
9	#19151971	—	2	2	no	—
10	#19151945	—	2	2	no	—
11	#19151424	—	2	2	no	—
12	#19151400	—	2	4	no	—
13	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₈²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#13083563	—	4	4	no	—
17	C₂×D₇₁₂	—	2	2	no	—
18	C₂×D₇₁₂	—	2	2	no	—
19	#15668012	—	2	2	no	—
20	#15668009	—	2	2	no	—
21	D₁₄₂₄	—	4	4	no	—
22	#15667946	—	2	2	no	—
23	#15667946	—	2	2	no	—
24	#15667934	—	4	4	no	—
25	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
26	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
27	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
28	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₇₈	—	4	2	no	—
31	C₂×D₃₅₆	—	4	2	no	—
32	C₂×D₃₅₆	—	4	2	no	—
33	C₂×D₃₅₆	—	4	2	no	—
34	#13083563	—	8	8	no	—
35	D₇₁₂	—	4	4	no	—
36	D₇₁₂	—	4	4	no	—
37	C₄⋊Dic₈₉	—	4	2	no	—
38	C₄⋊Dic₈₉	—	4	2	no	—
39	C₂×C₈×C₈₉	D₈	1	2	yes	—
40	C₂×C₈×C₈₉	D₈	1	2	yes	—
41	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₈₉	—	2	2	no	—
45	C₁₆×C₈₉	SD₁₆	1	4	yes	complemented
46	C₂²×D₈₉	—	8	2	no	—
47	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₈₉	—	8	2	no	—
51	D₃₅₆	—	8	2	no	—
52	D₃₅₆	—	8	2	no	—
53	D₃₅₆	—	8	4	no	—
54	D₃₅₆	—	8	4	no	—
55	Dic₁₇₈	—	8	4	no	—
56	C₈×C₈₉	—	2	2	no	—
57	C₈×C₈₉	D₁₆	1	4	yes	complemented
58	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
59	C₈×C₈₉	#135	1	2	yes	—
60	C₂²×C₈₉	#493	1	1	yes	characteristic
61	C₂×D₈₉	—	16	2	no	—
62	C₂×D₈₉	—	16	2	no	—
63	C₂×D₈₉	—	16	2	no	—
64	C₄×C₈₉	#492	1	1	yes	characteristic
65	C₄×C₈₉	#495	1	1	yes	characteristic
66	C₄×C₈₉	C₂×D₁₆	1	2	yes	—
67	C₄×C₈₉	#508	1	2	yes	—
68	Dic₈₉	—	16	2	no	—
69	#12503	—	89	1	no	maximal, Sylow
70	C₂×C₈₉	#1706	1	1	yes	characteristic
71	C₂×C₈₉	#2246	1	1	yes	characteristic
72	C₂×C₈₉	#2249	1	1	yes	characteristic
73	D₈₉	—	32	2	no	—
74	#2252	—	89	1	no	—
75	#2246	—	89	1	no	—
76	#2240	—	89	2	no	—
77	#1706	—	89	2	no	—
78	C₈×C₁₆	D₈₉	1	1	yes	characteristic, complemented
79	C₈₉	#12503	1	1	yes	characteristic, complemented, Sylow
80	C₂×D₁₆	—	178	2	no	—
81	#500	—	89	2	no	—
82	#493	—	89	2	no	—
83	#492	—	178	2	no	—
84	#486	—	178	2	no	—
85	#366	—	178	2	no	—
86	#357	—	178	4	no	—
87	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
88	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
89	C₈²	C₂×D₈₉	1	1	yes	characteristic
90	C₂×SD₁₆	—	356	4	no	—
91	C₂×D₈	—	178	2	no	—
92	C₂×D₈	—	178	2	no	—
93	C₄⋊Q₈	—	178	2	no	—
94	#127	—	178	2	no	—
95	D₁₆	—	356	4	no	—
96	C₂×C₁₆	D₃₅₆	1	2	yes	—
97	C₂×C₁₆	D₃₅₆	1	2	yes	—
98	#107	—	178	2	no	—
99	#107	—	178	2	no	—
100	#102	—	356	4	no	—
101	C₄×C₈	D₃₅₆	1	1	yes	characteristic
102	C₄×C₈	D₃₅₆	1	1	yes	characteristic
103	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
104	C₂×Q₈	—	356	2	no	—
105	C₂×D₄	—	356	2	no	—
106	C₂×D₄	—	356	2	no	—
107	C₂×D₄	—	356	2	no	—
108	SD₁₆	—	712	8	no	—
109	D₈	—	356	4	no	—
110	D₈	—	356	4	no	—
111	C₂×C₈	D₇₁₂	1	2	yes	—
112	C₂×C₈	D₇₁₂	1	2	yes	—
113	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
114	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
115	C₄⋊C₄	—	356	2	no	—
116	C₄⋊C₄	—	356	2	no	—
117	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
118	C₁₆	—	2	2	no	—
119	C₁₆	#13083563	1	4	yes	complemented
120	C₂³	—	712	2	no	—
121	Q₈	—	712	4	no	—
122	D₄	—	712	2	no	—
123	D₄	—	712	2	no	—
124	D₄	—	712	4	no	—
125	D₄	—	712	4	no	—
126	C₂×C₄	—	712	2	no	—
127	C₂×C₄	#15668009	1	1	yes	characteristic
128	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
129	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	D₁₄₂₄	1	4	yes	complemented
132	C₈	C₂×#13083563	1	2	yes	—
133	C₈	#15668034	1	2	yes	—
134	C₂²	—	1424	2	no	—
135	C₂²	—	1424	2	no	—
136	C₂²	—	1424	2	no	—
137	C₂²	#19151975	1	1	yes	characteristic
138	C₄	—	1424	2	no	—
139	C₄	#19151971	1	1	yes	characteristic
140	C₄	#19151982	1	1	yes	characteristic
141	C₄	C₂×D₁₄₂₄	1	2	yes	—
142	C₄	#19152030	1	2	yes	—
143	C₂	—	2848	2	no	—
144	C₂	#26019844	1	1	yes	characteristic
145	C₂	#26022502	1	1	yes	characteristic
146	C₂	#26022513	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022527

C₂

yes

maximal, characteristic, complemented

#26022502

C₂

yes

maximal, characteristic

#26022476

C₂

yes

maximal, complemented

#26019844

C₂

yes

maximal, complemented

C₈×C₁₆×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×D₁₄₂₄

—

#19152003

C₂²

yes

—

#19151975

C₂²

yes

—

—

—

—

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₈²×C₈₉

C₂²

yes

characteristic

C₂×#13083563

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

#15668012

—

#15668009

—

D₁₄₂₄

—

#15667946

—

#15667946

—

#15667934

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

#13083563

—

D₇₁₂

—

D₇₁₂

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

SD₁₆

yes

complemented

C₂²×D₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

D₁₆

yes

complemented

C₈×C₈₉

C₂×SD₁₆

yes

—

C₈×C₈₉

#135

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₂×D₁₆

yes

—

C₄×C₈₉

#508

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

—

—

—

—

C₈×C₁₆

D₈₉

yes

characteristic, complemented

C₈₉

#12503

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

178

—

#500

—

#493

—

#492

—

178

—

#486

—

178

—

#366

—

178

—

#357

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₈²

C₂×D₈₉

yes

characteristic

C₂×SD₁₆

—

356

—

C₂×D₈

—

178

—

C₂×D₈

—

178

—

C₄⋊Q₈

—

178

—

#127

—

178

—

D₁₆

—

356

—

C₂×C₁₆

D₃₅₆

yes

—

C₂×C₁₆

D₃₅₆

yes

—

#107

—

178

—

#107

—

178

—

100

#102

—

356

—

101

C₄×C₈

D₃₅₆

yes

characteristic

102

C₄×C₈

D₃₅₆

yes

characteristic

103

C₄×C₈

C₂²×D₈₉

yes

characteristic

104

C₂×Q₈

—

356

—

105

C₂×D₄

—

356

—

106

C₂×D₄

—

356

—

107

C₂×D₄

—

356

—

108

SD₁₆

—

712

—

109

D₈

—

356

—

110

D₈

—

356

—

111

C₂×C₈

D₇₁₂

yes

—

112

C₂×C₈

D₇₁₂

yes

—

113

C₂×C₈

C₂×D₃₅₆

yes

characteristic

114

C₂×C₈

C₂×D₃₅₆

yes

characteristic

115

C₄⋊C₄

—

356

—

116

C₄⋊C₄

—

356

—

117

C₄²

C₂×D₃₅₆

yes

characteristic

118

C₁₆

—

119

C₁₆

#13083563

yes

complemented

120

C₂³

—

712

—

121

Q₈

—

712

—

122

D₄

—

712

—

123

D₄

—

712

—

124

D₄

—

712

—

125

D₄

—

712

—

126

C₂×C₄

—

712

—

127

C₂×C₄

#15668009

yes

characteristic

128

C₂×C₄

C₂×D₇₁₂

yes

characteristic

129

C₂×C₄

C₂×D₇₁₂

yes

characteristic

130

C₈

—

131

C₈

D₁₄₂₄

yes

complemented

132

C₈

C₂×#13083563

yes

—

133

C₈

#15668034

yes

—

134

C₂²

—

1424

—

135

C₂²

—

1424

—

136

C₂²

—

1424

—

137

C₂²

#19151975

yes

characteristic

138

C₄

—

1424

—

139

C₄

#19151971

yes

characteristic

140

C₄

#19151982

yes

characteristic

141

C₄

C₂×D₁₄₂₄

yes

—

142

C₄

#19152030

yes

—

143

C₂

—

2848

—

144

C₂

#26019844

yes

characteristic

145

C₂

#26022502

yes

characteristic

146

C₂

#26022513

yes

characteristic

	subgroup	G/H
center	C₂²	#19151975
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	#493
prime radical	#26022502	C₂
hypercenter	C₈×C₁₆	D₈₉
hypocenter	C₈₉	#12503
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#19151975

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈×C₁₆×C₈₉

C₂

C₂²×C₈₉

C₂

C₈×C₁₆

D₈₉

hypocenter

C₈₉

#12503

LCS

G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #26022527	C₂	semidirect
3. #26022476	C₂	semidirect
4. #26019844	C₂	semidirect
5. C₈×C₁₆×C₈₉	C₂	semidirect
45. C₁₆×C₈₉	SD₁₆	semidirect
57. C₈×C₈₉	D₁₆	semidirect
78. C₈×C₁₆	D₈₉	semidirect
79. C₈₉	#12503	semidirect
119. C₁₆	#13083563	semidirect
131. C₈	D₁₄₂₄	semidirect

H ≅ G/N

type

1. #26022527

C₂

semidirect

3. #26022476

C₂

semidirect

4. #26019844

C₂

semidirect

5. C₈×C₁₆×C₈₉

C₂

semidirect

45. C₁₆×C₈₉

SD₁₆

semidirect

57. C₈×C₈₉

D₁₆

semidirect

78. C₈×C₁₆

D₈₉

semidirect

79. C₈₉

#12503

semidirect

119. C₁₆

#13083563

semidirect

131. C₈

D₁₄₂₄

semidirect

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

subgroup	index
#26022527	2
#26022502	2
#26022476	2
#26019844	2
C₈×C₁₆×C₈₉	2
#12503	89

subgroup

index

C₈×C₁₆×C₈₉

Subgroups — #36469081

Conjugacy classes of subgroups

Complements

Maximal subgroups