Group #36475339 — subgroups

Subgroups — #36475339

Order 22784.

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

normal subgroups only

This group has 158 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	Q₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26018453	C₂	1	1	yes	maximal, characteristic
4	#26018452	C₂	1	1	yes	maximal, characteristic
5	#26018447	C₂	1	1	yes	maximal, characteristic
6	#26018433	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1416	C₂	1	1	yes	maximal, characteristic
8	#19151999	—	2	1	no	—
9	Q₁₆×Dic₈₉	—	2	1	no	—
10	C₂×#15667918	C₂²	1	1	yes	characteristic
11	#19151425	—	2	1	no	—
12	#19151424	C₂²	1	1	yes	characteristic
13	#19151424	C₂²	1	1	yes	characteristic
14	#19151402	—	2	1	no	—
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	#15668003	—	4	1	no	—
20	Q₈×Dic₈₉	—	4	1	no	—
21	C₂×#13083549	D₄	1	1	yes	characteristic
22	#15667946	—	2	1	no	—
23	#15667946	—	2	1	no	—
24	#15667943	—	4	1	no	—
25	#15667936	—	4	1	no	—
26	C₈×Dic₈₉	D₄	1	1	yes	characteristic
27	#15667918	—	2	1	no	—
28	C₂×Q₁₆×C₈₉	—	2	1	no	—
29	Q₃₂×C₈₉	—	2	1	no	—
30	Q₃₂×C₈₉	Q₈	1	2	yes	—
31	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
32	#107×C₈₉	—	2	1	no	—
33	C₃₂×C₈₉	—	2	1	no	—
34	C₂×#10623494	D₈	1	1	yes	characteristic
35	C₄⋊Dic₈₉	—	4	1	no	—
36	C₄⋊Dic₈₉	—	4	1	no	—
37	C₄⋊Dic₈₉	—	8	1	no	—
38	C₄⋊Dic₈₉	—	8	1	no	—
39	#13083553	—	8	1	no	—
40	C₄×Dic₈₉	—	8	1	no	—
41	C₄×Dic₈₉	D₈	1	1	yes	characteristic
42	#13083549	—	2	1	no	—
43	C₂×Q₈×C₈₉	—	4	1	no	—
44	Q₁₆×C₈₉	—	2	2	no	—
45	Q₁₆×C₈₉	—	4	1	no	—
46	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₈₉	—	4	1	no	—
48	C₁₆×C₈₉	C₂×Q₈	1	1	yes	characteristic
49	C₁₆×C₈₉	#41	1	1	yes	characteristic
50	Q₈×C₈₉	—	4	2	no	—
51	Q₈×C₈₉	—	8	1	no	—
52	C₂×C₄×C₈₉	—	8	1	no	—
53	C₂×C₄×C₈₉	—	8	1	no	—
54	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₈₉	—	8	1	no	—
56	C₂×Dic₈₉	—	8	1	no	—
57	C₂×Dic₈₉	D₁₆	1	1	yes	characteristic
58	C₂×Dic₈₉	D₁₆	1	1	yes	characteristic
59	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
60	C₈×C₈₉	#137	1	1	yes	characteristic
61	#10623494	—	2	1	no	—
62	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
63	C₄×C₈₉	—	8	2	no	—
64	C₄×C₈₉	—	16	1	no	—
65	C₄×C₈₉	#474	1	1	yes	characteristic
66	C₄×C₈₉	#510	1	1	yes	characteristic
67	Dic₈₉	—	2	1	no	—
68	Dic₈₉	—	2	1	no	—
69	Dic₈₉	—	16	1	no	—
70	Dic₈₉	—	16	1	no	—
71	#13708	—	89	1	no	maximal, Sylow
72	C₂×C₈₉	#2224	1	1	yes	characteristic
73	C₂×C₈₉	#2262	1	1	yes	characteristic
74	C₂×C₈₉	#2264	1	1	yes	characteristic
75	#2253	—	89	1	no	—
76	C₄×Q₃₂	—	89	1	no	—
77	#1424	—	89	1	no	—
78	#1423	—	89	1	no	—
79	C₄⋊C₃₂	—	89	1	no	—
80	#1416	—	89	1	no	—
81	#1416	D₈₉	1	1	yes	characteristic
82	C₈₉	#13708	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
84	#499	—	178	1	no	—
85	C₄×Q₁₆	—	178	1	no	—
86	C₂×C₃₂	—	89	1	no	—
87	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
88	#367	—	178	1	no	—
89	#366	—	89	1	no	—
90	#366	—	89	1	no	—
91	#366	C₂×D₈₉	1	1	yes	characteristic
92	#358	—	178	1	no	—
93	C₄×C₁₆	—	89	1	no	—
94	C₂×Q₁₆	—	2	1	no	—
95	#125	—	356	1	no	—
96	C₄×Q₈	—	356	1	no	—
97	Q₃₂	—	2	1	no	—
98	Q₃₂	Dic₁₇₈	1	2	yes	—
99	C₂×C₁₆	—	89	1	no	—
100	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
101	#107	—	2	1	no	—
102	#107	—	178	1	no	—
103	#107	—	178	1	no	—
104	#106	—	356	1	no	—
105	#103	—	356	1	no	—
106	C₄×C₈	—	89	1	no	—
107	C₃₂	—	2	1	no	—
108	C₃₂	—	178	1	no	—
109	C₂×Q₈	—	4	1	no	—
110	Q₁₆	—	2	2	no	—
111	Q₁₆	—	4	1	no	—
112	C₂×C₈	—	89	1	no	—
113	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	356	1	no	—
116	C₄⋊C₄	—	356	1	no	—
117	C₄⋊C₄	—	712	1	no	—
118	C₄⋊C₄	—	712	1	no	—
119	C₄⋊C₄	—	712	1	no	—
120	C₄²	—	89	1	no	—
121	C₄²	—	712	1	no	—
122	C₁₆	—	178	1	no	—
123	C₁₆	C₂×Dic₁₇₈	1	1	yes	characteristic
124	C₁₆	#13083573	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	Q₈	—	8	1	no	—
127	C₂×C₄	—	8	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	89	1	no	—
130	C₂×C₄	—	89	1	no	—
131	C₂×C₄	—	712	1	no	—
132	C₂×C₄	—	712	1	no	—
133	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
134	C₈	—	178	1	no	—
135	C₈	#15667989	1	1	yes	characteristic
136	C₈	#15668041	1	1	yes	characteristic
137	C₂²	D₁₆×D₈₉	1	1	yes	characteristic
138	C₄	—	8	2	no	—
139	C₄	—	16	1	no	—
140	C₄	—	1424	1	no	—
141	C₄	—	1424	1	no	—
142	C₄	—	178	1	no	—
143	C₄	—	178	1	no	—
144	C₄	#19151876	1	1	yes	characteristic
145	C₄	#19152037	1	1	yes	characteristic
146	C₂	#26022387	1	1	yes	characteristic
147	C₂	#26022562	1	1	yes	characteristic
148	C₂	#26022568	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022529

C₂

yes

maximal, characteristic

Q₃₂×Dic₈₉

C₂

yes

maximal, characteristic

#26018453

C₂

yes

maximal, characteristic

#26018452

C₂

yes

maximal, characteristic

#26018447

C₂

yes

maximal, characteristic

#26018433

C₂

yes

maximal, characteristic

C₈₉×#1416

C₂

yes

maximal, characteristic

#19151999

—

Q₁₆×Dic₈₉

—

C₂×#15667918

C₂²

yes

characteristic

#19151425

—

#19151424

C₂²

yes

characteristic

#19151424

C₂²

yes

characteristic

#19151402

—

C₁₆×Dic₈₉

C₂²

yes

characteristic

C₂×Q₃₂×C₈₉

C₂²

yes

characteristic

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

#366×C₈₉

C₂²

yes

characteristic

#15668003

—

Q₈×Dic₈₉

—

C₂×#13083549

D₄

yes

—

—

—

—

C₈×Dic₈₉

D₄

yes

characteristic

#15667918

—

C₂×Q₁₆×C₈₉

—

Q₃₂×C₈₉

—

Q₃₂×C₈₉

Q₈

yes

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

#107×C₈₉

—

C₃₂×C₈₉

—

C₂×#10623494

D₈

yes

characteristic

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

D₈

yes

characteristic

#13083549

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

C₂×Q₈

yes

characteristic

C₁₆×C₈₉

#41

yes

characteristic

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

C₈×C₈₉

C₂²⋊Q₈

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₄×Q₃₂

—

#1424

—

#1423

—

C₄⋊C₃₂

—

#1416

—

#1416

D₈₉

yes

characteristic

C₈₉

#13708

yes

characteristic, complemented, Sylow

C₂×Q₃₂

C₂×D₈₉

yes

characteristic

#499

—

178

—

C₄×Q₁₆

—

178

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₈₉

yes

characteristic

#367

—

178

—

#366

—

#366

—

#366

C₂×D₈₉

yes

characteristic

#358

—

178

—

C₄×C₁₆

—

C₂×Q₁₆

—

#125

—

356

—

C₄×Q₈

—

356

—

Q₃₂

—

Q₃₂

Dic₁₇₈

yes

—

C₂×C₁₆

—

100

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

101

#107

—

102

#107

—

178

—

103

#107

—

178

—

104

#106

—

356

—

105

#103

—

356

—

106

C₄×C₈

—

107

C₃₂

—

108

C₃₂

—

178

—

109

C₂×Q₈

—

110

Q₁₆

—

111

Q₁₆

—

112

C₂×C₈

—

113

C₂×C₈

D₄×D₈₉

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

356

—

116

C₄⋊C₄

—

356

—

117

C₄⋊C₄

—

712

—

118

C₄⋊C₄

—

712

—

119

C₄⋊C₄

—

712

—

120

C₄²

—

121

C₄²

—

712

—

122

C₁₆

—

178

—

123

C₁₆

C₂×Dic₁₇₈

yes

characteristic

124

C₁₆

#13083573

yes

characteristic

125

Q₈

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

712

—

132

C₂×C₄

—

712

—

133

C₂×C₄

D₈×D₈₉

yes

characteristic

134

C₈

—

178

—

135

C₈

#15667989

yes

characteristic

136

C₈

#15668041

yes

characteristic

137

C₂²

D₁₆×D₈₉

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

1424

—

141

C₄

—

1424

—

142

C₄

—

178

—

143

C₄

—

178

—

144

C₄

#19151876

yes

characteristic

145

C₄

#19152037

yes

characteristic

146

C₂

#26022387

yes

characteristic

147

C₂

#26022562

yes

characteristic

148

C₂

#26022568

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	#13708
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
82. C₈₉	#13708	semidirect

H ≅ G/N

type

82. C₈₉

#13708

semidirect

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

subgroup	index
#26022529	2
Q₃₂×Dic₈₉	2
#26018453	2
#26018452	2
#26018447	2
#26018433	2
C₈₉×#1416	2
#13708	89

subgroup

index

Q₃₂×Dic₈₉

C₈₉×#1416

Subgroups — #36475339

Conjugacy classes of subgroups

Complements

Maximal subgroups