Group #36470816 — subgroups

Subgroups — #36470816

Order 22784.

282 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 119 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×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
2	#26020422	C₂	1	2	yes	maximal
3	#26018346	C₂	1	1	yes	maximal, characteristic
4	#26018345	C₂	1	2	yes	maximal
5	C₈₉×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083549	C₂²	1	1	yes	characteristic
7	C₂×#15667948	C₂²	1	2	yes	—
8	C₂×#15667947	—	2	2	no	—
9	#19151621	—	2	2	no	—
10	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₈₉	C₄	1	2	yes	—
12	C₁₆×Dic₈₉	C₄	1	2	yes	—
13	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#108×C₈₉	C₂²	1	2	yes	—
15	C₂×#13083558	—	2	2	no	—
16	C₂²×#10623494	D₄	1	1	yes	characteristic
17	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
18	C₂×#13083549	—	2	2	no	—
19	C₂×#13083549	C₂×C₄	1	2	yes	—
20	#15667948	—	2	2	no	—
21	#15667948	—	2	2	no	—
22	#15667947	—	4	4	no	—
23	#15667941	—	4	4	no	—
24	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
25	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₈₉	Q₈	1	1	yes	characteristic
27	C₈×Dic₈₉	Q₈	1	1	yes	characteristic
28	C₂×M₁₆×C₈₉	—	2	2	no	—
29	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₈₉	—	2	2	no	—
31	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
32	#108×C₈₉	—	2	2	no	—
33	#108×C₈₉	—	2	2	no	—
34	C₂²×Dic₈₉	Q₁₆	1	2	yes	—
35	#13083558	—	4	2	no	—
36	#13083558	—	4	2	no	—
37	C₂×#10623494	—	4	2	no	—
38	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
39	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
40	C₂×#10623494	SD₁₆	1	4	yes	—
41	C₄×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
42	C₄×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
43	C₄×Dic₈₉	D₈	1	2	yes	—
44	#13083549	—	2	2	no	—
45	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	M₁₆×C₈₉	—	4	2	no	—
47	M₁₆×C₈₉	—	4	2	no	—
48	C₂×C₈×C₈₉	—	4	2	no	—
49	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₈₉	#41	1	2	yes	—
52	C₂×C₈×C₈₉	#41	1	2	yes	—
53	C₁₆×C₈₉	—	2	2	no	—
54	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#124	1	2	yes	—
57	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
59	C₂×Dic₈₉	—	2	4	no	—
60	C₂×Dic₈₉	#107	1	2	yes	—
61	C₂×Dic₈₉	#107	1	2	yes	—
62	C₈×C₈₉	—	8	2	no	—
63	C₈×C₈₉	#117	1	1	yes	characteristic
64	C₈×C₈₉	#117	1	1	yes	characteristic
65	C₈×C₈₉	#125	1	2	yes	—
66	#10623494	—	8	2	no	—
67	#10623494	#106	1	2	yes	—
68	#10623494	#106	1	2	yes	—
69	C₂²×C₈₉	—	2	2	no	—
70	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
71	C₂²×C₈₉	#487	1	2	yes	—
72	C₄×C₈₉	#382	1	1	yes	characteristic
73	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
74	C₄×C₈₉	#486	1	2	yes	—
75	Dic₈₉	—	2	2	no	—
76	Dic₈₉	#368	1	4	yes	—
77	#12891	—	89	1	no	maximal, Sylow
78	C₂×C₈₉	—	2	2	no	—
79	C₂×C₈₉	#1834	1	1	yes	characteristic
80	C₂×C₈₉	C₂×#368	1	2	yes	—
81	C₂×C₄×C₁₆	—	89	1	no	—
82	#1848	—	89	2	no	—
83	#1381	—	89	1	no	—
84	#1381	—	89	2	no	—
85	#1381	D₈₉	1	1	yes	characteristic
86	C₈₉	#12891	1	1	yes	characteristic, complemented, Sylow
87	C₂²×C₁₆	—	89	1	no	—
88	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
89	C₂×#108	—	89	2	no	—
90	C₂×#108	—	178	2	no	—
91	C₂×#108	C₂×D₈₉	1	2	yes	—
92	#423	—	178	2	no	—
93	C₂×C₄×C₈	—	89	1	no	—
94	C₄×C₁₆	—	89	2	no	—
95	C₄×C₁₆	—	89	2	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	178	2	no	—
98	C₂²×C₈	—	89	1	no	—
99	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
100	C₂×C₄²	—	89	1	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	89	2	no	—
103	C₂×C₁₆	—	178	2	no	—
104	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
105	#108	—	2	2	no	—
106	#108	—	2	2	no	—
107	#108	—	356	4	no	—
108	#108	—	178	2	no	—
109	#108	—	178	2	no	—
110	C₄⋊C₈	—	356	4	no	—
111	C₄×C₈	—	89	1	no	—
112	C₄×C₈	—	89	1	no	—
113	C₄×C₈	—	89	1	no	—
114	C₄×C₈	—	89	1	no	—
115	C₂²×C₄	—	89	2	no	—
116	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
117	M₁₆	—	4	2	no	—
118	M₁₆	—	4	2	no	—
119	M₁₆	—	356	2	no	—
120	M₁₆	—	356	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	89	4	no	—
123	C₂×C₈	—	89	1	no	—
124	C₂×C₈	—	89	1	no	—
125	C₂×C₈	—	356	2	no	—
126	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
127	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
128	C₂×C₈	#13083573	1	2	yes	—
129	C₂×C₈	#13083574	1	2	yes	—
130	C₄²	—	89	2	no	—
131	C₄²	—	89	1	no	—
132	C₄²	—	89	1	no	—
133	C₁₆	—	2	2	no	—
134	C₁₆	—	178	2	no	—
135	C₂³	Q₁₆×D₈₉	1	1	yes	characteristic
136	C₂×C₄	—	89	2	no	—
137	C₂×C₄	—	89	2	no	—
138	C₂×C₄	—	178	4	no	—
139	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
140	C₂×C₄	#15667998	1	2	yes	—
141	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
142	C₂×C₄	SD₁₆×D₈₉	1	2	yes	—
143	C₈	—	8	2	no	—
144	C₈	—	89	2	no	—
145	C₈	—	89	2	no	—
146	C₈	—	712	2	no	—
147	C₈	#15667968	1	1	yes	characteristic
148	C₈	#15667969	1	1	yes	characteristic
149	C₈	#15668002	1	2	yes	—
150	C₂²	—	2	2	no	—
151	C₂²	#107×D₈₉	1	1	yes	characteristic
152	C₂²	#19151947	1	2	yes	—
153	C₄	—	89	4	no	—
154	C₄	—	178	2	no	—
155	C₄	#19151462	1	1	yes	characteristic
156	C₄	#106×D₈₉	1	1	yes	characteristic
157	C₄	#19151943	1	2	yes	—
158	C₂	—	2	2	no	—
159	C₂	#26020378	1	1	yes	characteristic
160	C₂	#368×D₈₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×Dic₈₉

C₂

yes

maximal, characteristic

#26020422

C₂

yes

maximal

#26018346

C₂

yes

maximal, characteristic

#26018345

C₂

yes

maximal

C₈₉×#1381

C₂

yes

maximal, characteristic

C₂²×#13083549

C₂²

yes

characteristic

C₂×#15667948

C₂²

yes

—

C₂×#15667947

—

#19151621

—

C₂×C₈×Dic₈₉

C₂²

yes

characteristic

C₁₆×Dic₈₉

C₄

yes

—

C₁₆×Dic₈₉

C₄

yes

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#108×C₈₉

C₂²

yes

—

C₂×#13083558

—

C₂²×#10623494

D₄

yes

characteristic

C₂×C₄×Dic₈₉

D₄

yes

characteristic

C₂×#13083549

—

C₂×#13083549

C₂×C₄

yes

—

—

—

—

—

C₈×Dic₈₉

C₂×C₄

yes

characteristic

C₈×Dic₈₉

C₂×C₄

yes

characteristic

C₈×Dic₈₉

Q₈

yes

characteristic

C₈×Dic₈₉

Q₈

yes

characteristic

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

#108×C₈₉

—

#108×C₈₉

—

C₂²×Dic₈₉

Q₁₆

yes

—

—

—

—

C₄⋊C₄

yes

characteristic

C₂×#10623494

C₄⋊C₄

yes

characteristic

C₂×#10623494

SD₁₆

yes

—

C₄×Dic₈₉

C₄⋊C₄

yes

characteristic

C₄×Dic₈₉

C₄⋊C₄

yes

characteristic

C₄×Dic₈₉

D₈

yes

—

#13083549

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

#107

yes

—

C₂×Dic₈₉

#107

yes

—

C₈×C₈₉

—

C₈×C₈₉

yes

C₈×C₈₉

yes

C₈×C₈₉

yes

—

#10623494

—

#10623494

#106

yes

—

#10623494

#106

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×#107

yes

characteristic

C₂²×C₈₉

#487

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

Dic₈₉

#368

yes

—

#12891

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1834

yes

characteristic

C₂×C₈₉

C₂×#368

yes

—

C₂×C₄×C₁₆

—

—

—

—

D₈₉

yes

characteristic

C₈₉

#12891

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

C₂×#108

—

C₂×#108

—

178

—

C₂×#108

C₂×D₈₉

yes

—

#423

—

178

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

178

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₈₉

yes

characteristic

100

C₂×C₄²

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

178

—

104

C₂×C₁₆

C₄×D₈₉

yes

—

105

#108

—

106

#108

—

107

#108

—

356

—

108

#108

—

178

—

109

#108

—

178

—

110

C₄⋊C₈

—

356

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₂²×C₄

—

116

C₂²×C₄

D₄×D₈₉

yes

characteristic

117

M₁₆

—

118

M₁₆

—

119

M₁₆

—

356

—

120

M₁₆

—

356

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

356

—

126

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

127

C₂×C₈

Q₈×D₈₉

yes

characteristic

128

C₂×C₈

#13083573

yes

—

129

C₂×C₈

#13083574

yes

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₁₆

—

134

C₁₆

—

178

—

135

C₂³

Q₁₆×D₈₉

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

178

—

139

C₂×C₄

C₄⋊C₄×D₈₉

yes

characteristic

140

C₂×C₄

#15667998

yes

—

141

C₂×C₄

D₈×D₈₉

yes

characteristic

142

C₂×C₄

SD₁₆×D₈₉

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

712

—

147

C₈

#15667968

yes

characteristic

148

C₈

#15667969

yes

characteristic

149

C₈

#15668002

yes

—

150

C₂²

—

151

C₂²

#107×D₈₉

yes

characteristic

152

C₂²

#19151947

yes

—

153

C₄

—

154

C₄

—

178

—

155

C₄

#19151462

yes

characteristic

156

C₄

#106×D₈₉

yes

characteristic

157

C₄

#19151943

yes

—

158

C₂

—

159

C₂

#26020378

yes

characteristic

160

C₂

#368×D₈₉

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₈₉
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1381	C₂
socle	C₂²×C₈₉	C₂×#107
prime radical	C₂³×C₈₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1381	D₈₉
hypocenter	C₈₉	#12891
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₈₉

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1381

C₂

socle

C₂²×C₈₉

C₂×#107

prime radical

C₂³×C₈₉

C₂×Q₁₆

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
86. C₈₉	#12891	semidirect

H ≅ G/N

type

86. C₈₉

#12891

semidirect

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

subgroup	index
C₂×C₁₆×Dic₈₉	2
#26020422	2
#26018346	2
#26018345	2
C₈₉×#1381	2
#12891	89

subgroup

index

C₂×C₁₆×Dic₈₉

C₈₉×#1381

Subgroups — #36470816

Conjugacy classes of subgroups

Complements

Maximal subgroups