Group #36468775 — subgroups

Subgroups — #36468775

Order 22784.

206 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 71 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	#26019651	C₂	1	2	yes	maximal
3	#26018326	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018309	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2169	C₂	1	1	yes	maximal, characteristic
6	#19152003	C₂²	1	2	yes	—
7	#19151888	—	2	2	no	—
8	#19151424	—	2	1	no	—
9	#19151424	—	2	1	no	—
10	#19151351	C₂²	1	2	yes	—
11	#19151346	—	2	2	no	—
12	C₂×M₃₂×C₈₉	—	2	1	no	—
13	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²⋊C₁₆×C₈₉	—	2	1	no	—
16	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	#15668012	—	2	2	no	—
18	#15667946	—	2	2	no	—
19	#15667946	—	2	2	no	—
20	#15667943	—	4	2	no	—
21	C₄⋊#10623494	—	2	2	no	—
22	#15667933	—	4	2	no	—
23	C₂²×C₈×C₈₉	—	2	1	no	—
24	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
25	M₃₂×C₈₉	—	4	2	no	—
26	C₂×C₁₆×C₈₉	—	2	1	no	—
27	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₂²⋊C₈×C₈₉	—	2	1	no	—
31	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₇₈	—	4	2	no	—
33	C₂×#10623494	—	4	2	no	—
34	C₄⋊Dic₈₉	—	4	2	no	—
35	C₄⋊Dic₈₉	—	4	2	no	—
36	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	—	2	1	no	—
38	C₂×C₈×C₈₉	—	2	1	no	—
39	C₂×C₈×C₈₉	—	4	1	no	—
40	C₂×C₈×C₈₉	—	4	1	no	—
41	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
43	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
44	C₂²⋊C₄×C₈₉	—	2	1	no	—
45	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₁₆×C₈₉	—	2	1	no	—
47	C₁₆×C₈₉	—	4	1	no	—
48	C₁₆×C₈₉	SD₁₆	1	2	yes	—
49	C₂³×C₈₉	—	2	1	no	—
50	D₄×C₈₉	—	2	2	no	—
51	C₂×C₄×C₈₉	—	2	1	no	—
52	C₂×C₄×C₈₉	—	4	1	no	—
53	C₂×C₄×C₈₉	—	4	1	no	—
54	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
57	C₂×Dic₈₉	—	8	2	no	—
58	Dic₁₇₈	—	16	2	no	—
59	C₈×C₈₉	—	2	1	no	—
60	C₈×C₈₉	—	4	1	no	—
61	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₈×C₈₉	#137	1	1	yes	characteristic
63	#10623494	—	8	2	no	—
64	C₂²×C₈₉	—	4	1	no	—
65	C₂²×C₈₉	—	4	1	no	—
66	C₂²×C₈₉	—	4	1	no	—
67	C₂²×C₈₉	#466	1	1	yes	characteristic
68	C₄×C₈₉	—	2	1	no	—
69	C₄×C₈₉	—	4	1	no	—
70	C₄×C₈₉	#461	1	1	yes	characteristic
71	C₄×C₈₉	#473	1	1	yes	characteristic
72	C₄×C₈₉	#510	1	2	yes	—
73	Dic₈₉	—	16	2	no	—
74	#12455	—	89	1	no	maximal, Sylow
75	C₂×C₈₉	—	8	1	no	—
76	C₂×C₈₉	#1673	1	1	yes	characteristic
77	C₂×C₈₉	#2219	1	1	yes	characteristic
78	C₂×C₈₉	#2221	1	1	yes	characteristic
79	#2252	—	89	1	no	—
80	#2169	D₈₉	1	1	yes	characteristic
81	#1675	—	89	2	no	—
82	#1372	—	89	1	no	—
83	#1365	—	89	2	no	—
84	C₈₉	#12455	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₃₂	—	2	1	no	—
86	#500	—	89	2	no	—
87	#476	—	178	2	no	—
88	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
89	#366	—	178	1	no	—
90	#366	—	178	1	no	—
91	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
92	C₂²⋊C₁₆	—	2	1	no	—
93	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
94	#335	—	89	2	no	—
95	#333	—	178	2	no	—
96	C₂²×C₈	—	2	1	no	—
97	C₄⋊Q₈	—	178	2	no	—
98	C₄×D₄	D₃₅₆	1	1	yes	characteristic
99	M₃₂	—	4	2	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
103	#107	—	178	2	no	—
104	#107	—	178	2	no	—
105	#106	—	356	2	no	—
106	C₄⋊C₈	—	178	2	no	—
107	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
108	#102	—	356	2	no	—
109	C₂²⋊C₈	—	2	1	no	—
110	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
111	C₂×Q₈	—	356	2	no	—
112	C₂×D₄	D₇₁₂	1	1	yes	characteristic
113	C₂²×C₄	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	356	2	no	—
118	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
119	C₂×C₈	#13083574	1	1	yes	characteristic
120	C₄⋊C₄	—	356	2	no	—
121	C₄⋊C₄	—	356	2	no	—
122	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
123	C₂²⋊C₄	—	2	1	no	—
124	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	4	1	no	—
127	C₁₆	#13083562	1	2	yes	—
128	C₂³	—	2	1	no	—
129	Q₈	—	1424	2	no	—
130	D₄	—	2	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	712	2	no	—
135	C₂×C₄	#15667986	1	1	yes	characteristic
136	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
137	C₂×C₄	#15668034	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	4	1	no	—
140	C₈	—	712	2	no	—
141	C₈	C₂×#13083562	1	1	yes	characteristic
142	C₈	#15668042	1	1	yes	characteristic
143	C₂²	—	4	1	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	#19151834	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	4	1	no	—
149	C₄	—	1424	2	no	—
150	C₄	#19151803	1	1	yes	characteristic
151	C₄	#19151873	1	1	yes	characteristic
152	C₄	#19152041	1	2	yes	—
153	C₂	—	8	1	no	—
154	C₂	#26019641	1	1	yes	characteristic
155	C₂	#26022360	1	1	yes	characteristic
156	C₂	#26022372	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022527

C₂

yes

maximal, characteristic, complemented

#26019651

C₂

yes

maximal

#26018326

C₂

yes

maximal, characteristic, complemented

#26018309

C₂

yes

maximal, complemented

C₈₉×#2169

C₂

yes

maximal, characteristic

#19152003

C₂²

yes

—

—

—

—

C₂²

yes

—

#19151346

—

C₂×M₃₂×C₈₉

—

C₈×D₄×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

C₂²

yes

characteristic

C₂²⋊C₁₆×C₈₉

—

C₄×C₁₆×C₈₉

C₂²

yes

—

—

—

—

—

—

C₂²×C₈×C₈₉

—

C₄×D₄×C₈₉

D₄

yes

characteristic

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₄⋊C₈×C₈₉

D₄

yes

characteristic

C₂²⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×D₄×C₈₉

D₈

yes

characteristic

C₂²×C₄×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄⋊C₄×C₈₉

D₈

yes

characteristic

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

—

C₁₆×C₈₉

SD₁₆

yes

—

C₂³×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

#135

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂×SD₁₆

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#466

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

#12455

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

yes

—

—

—

C₈₉

#12455

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

#500

—

#476

—

178

—

C₈×D₄

C₂×D₈₉

yes

characteristic

#366

—

178

—

#366

—

178

—

C₄⋊C₁₆

C₂×D₈₉

yes

characteristic

C₂²⋊C₁₆

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

#335

—

#333

—

178

—

C₂²×C₈

—

C₄⋊Q₈

—

178

—

C₄×D₄

D₃₅₆

yes

characteristic

M₃₂

—

100

C₂×C₁₆

—

101

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

102

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

103

#107

—

178

—

104

#107

—

178

—

105

#106

—

356

—

106

C₄⋊C₈

—

178

—

107

C₄⋊C₈

D₃₅₆

yes

characteristic

108

#102

—

356

—

109

C₂²⋊C₈

—

110

C₄×C₈

C₂²×D₈₉

yes

characteristic

111

C₂×Q₈

—

356

—

112

C₂×D₄

D₇₁₂

yes

characteristic

113

C₂²×C₄

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

356

—

118

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

119

C₂×C₈

#13083574

yes

characteristic

120

C₄⋊C₄

—

356

—

121

C₄⋊C₄

—

356

—

122

C₄⋊C₄

D₇₁₂

yes

characteristic

123

C₂²⋊C₄

—

124

C₄²

C₂×D₃₅₆

yes

characteristic

125

C₁₆

—

126

C₁₆

—

127

C₁₆

#13083562

yes

—

128

C₂³

—

129

Q₈

—

1424

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

712

—

135

C₂×C₄

#15667986

yes

characteristic

136

C₂×C₄

C₂×D₇₁₂

yes

characteristic

137

C₂×C₄

#15668034

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

712

—

141

C₈

C₂×#13083562

yes

characteristic

142

C₈

#15668042

yes

characteristic

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#19151834

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

1424

—

150

C₄

#19151803

yes

characteristic

151

C₄

#19151873

yes

characteristic

152

C₄

#19152041

yes

—

153

C₂

—

154

C₂

#26019641

yes

characteristic

155

C₂

#26022360

yes

characteristic

156

C₂

#26022372

yes

characteristic

	subgroup	G/H
center	C₂²	#19151834
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#2169	C₂
socle	C₂²×C₈₉	#466
prime radical	C₂×D₄×C₈₉	D₈
hypercenter	#2169	D₈₉
hypocenter	C₈₉	#12455
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ #2169

subgroup

G/H

center

C₂²

#19151834

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#2169

C₂

socle

C₂²×C₈₉

#466

prime radical

C₂×D₄×C₈₉

D₈

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ #2169

N	H ≅ G/N	type
1. #26022527	C₂	semidirect
3. #26018326	C₂	semidirect
4. #26018309	C₂	semidirect
84. C₈₉	#12455	semidirect

H ≅ G/N

type

1. #26022527

C₂

semidirect

3. #26018326

C₂

semidirect

4. #26018309

C₂

semidirect

84. C₈₉

#12455

semidirect

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

subgroup	index
#26022527	2
#26019651	2
#26018326	2
#26018309	2
C₈₉×#2169	2
#12455	89

subgroup

index

C₈₉×#2169

Subgroups — #36468775

Conjugacy classes of subgroups

Complements

Maximal subgroups