Group #36475417 — subgroups

Subgroups — #36475417

Order 22784.

174 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 57 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	#26022529	C₂	1	1	yes	maximal, characteristic
2	#26022503	C₂	1	1	yes	maximal, characteristic
3	#26018434	C₂	1	1	yes	maximal, characteristic
4	#26018433	C₂	1	1	yes	maximal, characteristic
5	#26018432	C₂	1	1	yes	maximal, characteristic
6	#26018385	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1416	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
9	C₂×#15667958	—	2	1	no	—
10	#19151999	—	2	1	no	—
11	#19151976	—	2	1	no	—
12	C₂×#15667918	C₂²	1	1	yes	characteristic
13	#19151425	—	2	1	no	—
14	#19151424	C₂²	1	1	yes	characteristic
15	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
18	#366×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₅₆	—	2	1	no	—
20	C₂×#13083564	—	4	1	no	—
21	#15668010	—	4	1	no	—
22	#15668003	—	4	1	no	—
23	Dic₇₁₂	—	2	1	no	—
24	Dic₇₁₂	—	2	1	no	—
25	#15667958	—	4	2	no	—
26	C₂×#13083549	D₄	1	1	yes	characteristic
27	#15667946	—	2	1	no	—
28	#15667943	—	4	1	no	—
29	C₈×Dic₈₉	D₄	1	1	yes	characteristic
30	#15667918	—	2	1	no	—
31	C₂×Q₁₆×C₈₉	—	2	1	no	—
32	Q₃₂×C₈₉	—	2	1	no	—
33	Q₃₂×C₈₉	—	2	1	no	—
34	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
35	#107×C₈₉	—	2	1	no	—
36	C₃₂×C₈₉	—	2	1	no	—
37	C₂×Dic₁₇₈	—	4	1	no	—
38	Dic₃₅₆	—	4	1	no	—
39	Dic₃₅₆	—	4	1	no	—
40	#13083564	—	8	2	no	—
41	C₂×#10623494	D₈	1	1	yes	characteristic
42	C₄⋊Dic₈₉	—	4	1	no	—
43	C₄⋊Dic₈₉	—	8	1	no	—
44	#13083553	—	8	1	no	—
45	#13083553	—	8	1	no	—
46	#13083553	—	8	1	no	—
47	C₄×Dic₈₉	D₈	1	1	yes	characteristic
48	#13083549	SD₁₆	1	2	yes	—
49	C₂×Q₈×C₈₉	—	4	1	no	—
50	Q₁₆×C₈₉	—	4	1	no	—
51	Q₁₆×C₈₉	—	4	1	no	—
52	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₄⋊C₄×C₈₉	—	4	1	no	—
54	C₁₆×C₈₉	#41	1	1	yes	characteristic
55	C₁₆×C₈₉	#41	1	1	yes	characteristic
56	Q₈×C₈₉	—	8	1	no	—
57	Q₈×C₈₉	—	8	1	no	—
58	C₂×C₄×C₈₉	—	8	1	no	—
59	C₂×C₄×C₈₉	—	8	1	no	—
60	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
61	C₂×Dic₈₉	—	8	1	no	—
62	C₂×Dic₈₉	—	8	1	no	—
63	C₂×Dic₈₉	D₁₆	1	1	yes	characteristic
64	C₂×Dic₈₉	D₁₆	1	1	yes	characteristic
65	Dic₁₇₈	—	8	1	no	—
66	Dic₁₇₈	—	8	1	no	—
67	C₈×C₈₉	#124	1	1	yes	characteristic
68	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
69	#10623494	SD₃₂	1	2	yes	—
70	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
71	C₄×C₈₉	—	16	1	no	—
72	C₄×C₈₉	—	16	1	no	—
73	C₄×C₈₉	#486	1	1	yes	characteristic
74	C₄×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
75	Dic₈₉	—	2	1	no	—
76	Dic₈₉	—	2	1	no	—
77	Dic₈₉	—	16	1	no	—
78	Dic₈₉	—	16	1	no	—
79	#13724	—	89	1	no	maximal, Sylow
80	C₂×C₈₉	#2240	1	1	yes	characteristic
81	C₂×C₈₉	#2264	1	1	yes	characteristic
82	C₂×C₈₉	#2264	1	1	yes	characteristic
83	#2253	—	89	1	no	—
84	#2247	—	89	1	no	—
85	#1416	—	89	1	no	—
86	#1416	—	89	1	no	—
87	#1416	—	89	1	no	—
88	#1416	D₈₉	1	1	yes	characteristic
89	#1397	—	89	1	no	—
90	C₈₉	#13724	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₃₂	—	89	1	no	—
92	C₂×Q₃₂	—	178	1	no	—
93	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
94	#499	—	178	1	no	—
95	#494	—	178	1	no	—
96	C₂×C₃₂	—	89	1	no	—
97	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
98	#367	—	178	1	no	—
99	#366	—	89	1	no	—
100	#366	C₂×D₈₉	1	1	yes	characteristic
101	C₄×C₁₆	—	89	1	no	—
102	C₂×Q₁₆	—	2	1	no	—
103	C₂×Q₁₆	—	356	1	no	—
104	C₂×Q₁₆	—	178	1	no	—
105	C₄⋊Q₈	—	356	1	no	—
106	#125	—	356	1	no	—
107	Q₃₂	—	2	1	no	—
108	Q₃₂	—	2	1	no	—
109	Q₃₂	—	356	2	no	—
110	Q₃₂	—	178	1	no	—
111	Q₃₂	—	178	1	no	—
112	C₂×C₁₆	—	89	1	no	—
113	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
114	#107	—	2	1	no	—
115	#107	—	178	1	no	—
116	#106	—	356	1	no	—
117	C₄×C₈	—	89	1	no	—
118	C₃₂	—	2	1	no	—
119	C₃₂	—	178	1	no	—
120	C₂×Q₈	—	4	1	no	—
121	C₂×Q₈	—	356	1	no	—
122	Q₁₆	—	4	1	no	—
123	Q₁₆	—	4	1	no	—
124	Q₁₆	—	356	1	no	—
125	Q₁₆	—	356	1	no	—
126	Q₁₆	—	712	2	no	—
127	C₂×C₈	—	89	1	no	—
128	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
129	C₄⋊C₄	—	4	1	no	—
130	C₄⋊C₄	—	356	1	no	—
131	C₄⋊C₄	—	712	1	no	—
132	C₄⋊C₄	—	712	1	no	—
133	C₄⋊C₄	—	712	1	no	—
134	C₄⋊C₄	—	712	1	no	—
135	C₄²	—	89	1	no	—
136	C₁₆	—	89	2	no	—
137	C₁₆	#13083573	1	1	yes	characteristic
138	C₁₆	#13083574	1	1	yes	characteristic
139	Q₈	—	8	1	no	—
140	Q₈	—	8	1	no	—
141	Q₈	—	712	1	no	—
142	Q₈	—	712	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	8	1	no	—
145	C₂×C₄	—	89	1	no	—
146	C₂×C₄	—	89	1	no	—
147	C₂×C₄	—	712	1	no	—
148	C₂×C₄	—	712	1	no	—
149	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
150	C₈	—	89	2	no	—
151	C₈	#15667998	1	1	yes	characteristic
152	C₈	SD₁₆×D₈₉	1	1	yes	characteristic
153	C₂²	D₁₆×D₈₉	1	1	yes	characteristic
154	C₄	—	16	1	no	—
155	C₄	—	16	1	no	—
156	C₄	—	1424	1	no	—
157	C₄	—	1424	1	no	—
158	C₄	—	178	1	no	—
159	C₄	—	178	1	no	—
160	C₄	#19151943	1	1	yes	characteristic
161	C₄	SD₃₂×D₈₉	1	1	yes	characteristic
162	C₂	#26022474	1	1	yes	characteristic
163	C₂	#26022568	1	1	yes	characteristic
164	C₂	#26022571	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022529

C₂

yes

maximal, characteristic

#26022503

C₂

yes

maximal, characteristic

#26018434

C₂

yes

maximal, characteristic

#26018433

C₂

yes

maximal, characteristic

#26018432

C₂

yes

maximal, characteristic

#26018385

C₂

yes

maximal, characteristic

C₈₉×#1416

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

C₂²

yes

—

—

—

C₂²

yes

characteristic

#19151425

—

#19151424

C₂²

yes

characteristic

C₁₆×Dic₈₉

C₂²

yes

characteristic

C₂×Q₃₂×C₈₉

C₂²

yes

characteristic

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

#366×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×#13083564

—

#15668010

—

#15668003

—

Dic₇₁₂

—

Dic₇₁₂

—

#15667958

—

C₂×#13083549

D₄

yes

characteristic

#15667946

—

#15667943

—

C₈×Dic₈₉

D₄

yes

characteristic

#15667918

—

C₂×Q₁₆×C₈₉

—

Q₃₂×C₈₉

—

Q₃₂×C₈₉

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

#107×C₈₉

—

C₃₂×C₈₉

—

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

#13083564

—

C₂×#10623494

D₈

yes

characteristic

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

#13083553

—

C₄×Dic₈₉

D₈

yes

characteristic

#13083549

SD₁₆

yes

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

yes

C₁₆×C₈₉

yes

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

D₁₆

yes

characteristic

C₂×Dic₈₉

D₁₆

yes

characteristic

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

#124

yes

characteristic

C₈×C₈₉

C₂×SD₁₆

yes

characteristic

#10623494

SD₃₂

yes

—

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#486

yes

characteristic

C₄×C₈₉

C₂×SD₃₂

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

—

—

—

—

D₈₉

yes

characteristic

#1397

—

C₈₉

#13724

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

178

—

C₂×Q₃₂

C₂×D₈₉

yes

characteristic

#499

—

178

—

#494

—

178

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₈₉

yes

characteristic

#367

—

178

—

#366

—

100

#366

C₂×D₈₉

yes

characteristic

101

C₄×C₁₆

—

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

356

—

104

C₂×Q₁₆

—

178

—

105

C₄⋊Q₈

—

356

—

106

#125

—

356

—

107

Q₃₂

—

108

Q₃₂

—

109

Q₃₂

—

356

—

110

Q₃₂

—

178

—

111

Q₃₂

—

178

—

112

C₂×C₁₆

—

113

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

114

#107

—

115

#107

—

178

—

116

#106

—

356

—

117

C₄×C₈

—

118

C₃₂

—

119

C₃₂

—

178

—

120

C₂×Q₈

—

121

C₂×Q₈

—

356

—

122

Q₁₆

—

123

Q₁₆

—

124

Q₁₆

—

356

—

125

Q₁₆

—

356

—

126

Q₁₆

—

712

—

127

C₂×C₈

—

128

C₂×C₈

D₄×D₈₉

yes

characteristic

129

C₄⋊C₄

—

130

C₄⋊C₄

—

356

—

131

C₄⋊C₄

—

712

—

132

C₄⋊C₄

—

712

—

133

C₄⋊C₄

—

712

—

134

C₄⋊C₄

—

712

—

135

C₄²

—

136

C₁₆

—

137

C₁₆

#13083573

yes

characteristic

138

C₁₆

#13083574

yes

characteristic

139

Q₈

—

140

Q₈

—

141

Q₈

—

712

—

142

Q₈

—

712

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

712

—

148

C₂×C₄

—

712

—

149

C₂×C₄

D₈×D₈₉

yes

characteristic

150

C₈

—

151

C₈

#15667998

yes

characteristic

152

C₈

SD₁₆×D₈₉

yes

characteristic

153

C₂²

D₁₆×D₈₉

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

1424

—

157

C₄

—

1424

—

158

C₄

—

178

—

159

C₄

—

178

—

160

C₄

#19151943

yes

characteristic

161

C₄

SD₃₂×D₈₉

yes

characteristic

162

C₂

#26022474

yes

characteristic

163

C₂

#26022568

yes

characteristic

164

C₂

#26022571

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₈₉
commutator, cosocle	C₂×C₁₆×C₈₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#1416	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₁₆
hypercenter	#1416	D₈₉
hypocenter	C₈₉	#13724
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₆×D₈₉

commutator, cosocle

C₂×C₁₆×C₈₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₈₉×#1416

C₂

socle, prime radical

C₂²×C₈₉

C₂×D₁₆

D₈₉

C₈₉

G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
90. C₈₉	#13724	semidirect

H ≅ G/N

type

90. C₈₉

#13724

semidirect

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

subgroup	index
#26022529	2
#26022503	2
#26018434	2
#26018433	2
#26018432	2
#26018385	2
C₈₉×#1416	2
#13724	89

subgroup

index

C₈₉×#1416

Subgroups — #36475417

Conjugacy classes of subgroups

Complements

Maximal subgroups