Group #36470810 — subgroups

Subgroups — #36470810

Order 22784.

310 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 155 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×C₄×#13083549	C₂	1	1	yes	maximal, characteristic
2	#26020416	C₂	1	1	yes	maximal, characteristic
3	#26018343	C₂	1	4	yes	maximal
4	C₈₉×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083549	C₂²	1	2	yes	—
6	C₂×#15667946	C₂²	1	2	yes	—
7	C₂×#15667944	—	2	1	no	—
8	#19151464	—	2	1	no	—
9	C₄×#13083549	C₄	1	4	yes	complemented
10	C₂×#107×C₈₉	C₂²	1	2	yes	—
11	C₂×#106×C₈₉	—	2	1	no	—
12	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	#382×C₈₉	—	2	1	no	—
14	C₂×C₄⋊Dic₈₉	—	2	2	no	—
15	C₂×#13083549	—	2	2	no	—
16	C₂×#13083549	—	2	2	no	—
17	C₂×#13083549	C₂×C₄	1	4	yes	—
18	#15667946	—	2	4	no	—
19	#15667944	—	4	2	no	—
20	#15667919	—	4	2	no	—
21	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
24	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
25	#107×C₈₉	—	2	4	no	—
26	#106×C₈₉	—	4	2	no	—
27	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₈₉	Q₈	1	2	yes	—
29	#95×C₈₉	—	4	2	no	—
30	C₂²×Dic₈₉	—	4	2	no	—
31	C₄⋊Dic₈₉	—	4	2	no	—
32	C₄⋊Dic₈₉	—	4	2	no	—
33	#13083549	—	2	4	no	—
34	C₂²×C₄×C₈₉	—	4	2	no	—
35	C₂²×C₄×C₈₉	D₈	1	2	yes	—
36	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
39	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
40	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₈₉	#41	1	2	yes	—
43	C₂×C₈×C₈₉	#41	1	2	yes	—
44	C₄⋊C₄×C₈₉	—	4	2	no	—
45	C₄⋊C₄×C₈₉	—	4	2	no	—
46	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
47	C₄²×C₈₉	Q₁₆	1	2	yes	—
48	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₈₉	—	2	4	no	—
50	C₂×C₄×C₈₉	—	8	2	no	—
51	C₂×C₄×C₈₉	—	8	4	no	—
52	C₂×C₄×C₈₉	#107	1	4	yes	—
53	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
55	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₈₉	—	8	2	no	—
60	C₂×Dic₈₉	—	8	4	no	—
61	C₈×C₈₉	#106	1	4	yes	—
62	C₈×C₈₉	#117	1	2	yes	—
63	C₈×C₈₉	#125	1	2	yes	—
64	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
65	C₂²×C₈₉	#486	1	1	yes	characteristic
66	C₂²×C₈₉	#486	1	1	yes	characteristic
67	C₂²×C₈₉	#508	1	4	yes	—
68	C₄×C₈₉	—	2	4	no	—
69	C₄×C₈₉	—	16	2	no	—
70	C₄×C₈₉	#382	1	1	yes	characteristic
71	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
72	C₄×C₈₉	#487	1	1	yes	characteristic
73	C₄×C₈₉	#487	1	1	yes	characteristic
74	Dic₈₉	—	16	2	no	—
75	#12889	—	89	1	no	maximal, Sylow
76	C₂×C₈₉	#1834	1	1	yes	characteristic
77	C₂×C₈₉	#2158	1	2	yes	—
78	C₂×C₈₉	#2242	1	2	yes	—
79	C₂×C₈₉	#2242	1	2	yes	—
80	C₂×C₄×C₁₆	—	89	1	no	—
81	#1846	—	89	1	no	—
82	#1846	D₈₉	1	1	yes	characteristic
83	#1380	—	89	4	no	—
84	C₈₉	#12889	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₁₆	—	89	2	no	—
86	C₂×#107	—	89	2	no	—
87	C₂×#107	C₂×D₈₉	1	2	yes	—
88	C₂×#106	—	2	1	no	—
89	C₂×#106	—	178	1	no	—
90	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
91	#382	—	2	1	no	—
92	#382	—	178	1	no	—
93	C₄×C₁₆	—	89	4	no	—
94	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	178	2	no	—
98	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
99	C₂×C₁₆	—	89	4	no	—
100	C₂×C₁₆	—	178	2	no	—
101	C₂×C₁₆	—	178	2	no	—
102	#107	—	2	4	no	—
103	#107	—	178	4	no	—
104	#106	—	4	2	no	—
105	#106	—	356	2	no	—
106	C₄×C₈	Dic₁₇₈	1	2	yes	—
107	C₄×C₈	C₄×D₈₉	1	2	yes	—
108	#95	—	4	2	no	—
109	#95	—	356	2	no	—
110	C₂²×C₄	—	4	2	no	—
111	C₂²×C₄	—	356	2	no	—
112	C₂²×C₄	#13083559	1	2	yes	—
113	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
114	C₂×C₈	#13083553	1	2	yes	—
115	C₂×C₈	#13083561	1	2	yes	—
116	C₂×C₈	#13083562	1	2	yes	—
117	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
118	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
119	C₂×C₈	#13083573	1	2	yes	—
120	C₂×C₈	#13083575	1	2	yes	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	356	2	no	—
124	C₄⋊C₄	—	356	2	no	—
125	C₄²	#13083553	1	2	yes	—
126	C₄²	#13083564	1	2	yes	—
127	C₁₆	—	178	4	no	—
128	C₂³	C₂×#13083559	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	8	2	no	—
131	C₂×C₄	—	8	4	no	—
132	C₂×C₄	—	712	2	no	—
133	C₂×C₄	—	712	4	no	—
134	C₂×C₄	#15667945	1	4	yes	—
135	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
136	C₂×C₄	#15667999	1	1	yes	characteristic
137	C₂×C₄	#15668001	1	1	yes	characteristic
138	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
139	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
140	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
141	C₈	#15667943	1	4	yes	—
142	C₈	#15667970	1	2	yes	—
143	C₈	#15668003	1	2	yes	—
144	C₂²	C₂×#15667945	1	1	yes	characteristic
145	C₂²	#19151944	1	1	yes	characteristic
146	C₂²	#19151946	1	1	yes	characteristic
147	C₂²	#19152028	1	4	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	16	2	no	—
150	C₄	—	1424	2	no	—
151	C₄	#19151463	1	1	yes	characteristic
152	C₄	C₂×#15667943	1	1	yes	characteristic
153	C₄	#19151948	1	1	yes	characteristic
154	C₄	#19151950	1	1	yes	characteristic
155	C₂	#26020379	1	1	yes	characteristic
156	C₂	#26022034	1	2	yes	—
157	C₂	#26022483	1	2	yes	—
158	C₂	#26022486	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#13083549

C₂

yes

maximal, characteristic

#26020416

C₂

yes

maximal, characteristic

#26018343

C₂

yes

maximal

C₈₉×#1846

C₂

yes

maximal, characteristic

C₂²×#13083549

C₂²

yes

—

C₂×#15667946

C₂²

yes

—

C₂×#15667944

—

#19151464

—

C₄×#13083549

C₄

yes

complemented

C₂×#107×C₈₉

C₂²

yes

—

C₂×#106×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#382×C₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×#13083549

—

C₂×#13083549

—

C₂×#13083549

C₂×C₄

yes

—

#15667946

—

#15667944

—

#15667919

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

#107×C₈₉

—

#106×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

#95×C₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083549

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₈

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

Q₁₆

yes

—

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#107

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

#106

yes

—

C₈×C₈₉

#117

yes

—

C₈×C₈₉

#125

yes

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#2242

yes

—

C₂×C₈₉

#2242

yes

—

C₂×C₄×C₁₆

—

#1846

—

#1846

D₈₉

yes

characteristic

#1380

—

C₈₉

#12889

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

C₂×D₈₉

yes

—

C₂×#106

—

C₂×#106

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#382

—

#382

—

178

—

C₄×C₁₆

—

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

C₂²×C₈

C₂²×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

C₂×C₁₆

—

100

C₂×C₁₆

—

178

—

101

C₂×C₁₆

—

178

—

102

#107

—

103

#107

—

178

—

104

#106

—

105

#106

—

356

—

106

C₄×C₈

Dic₁₇₈

yes

—

107

C₄×C₈

C₄×D₈₉

yes

—

108

#95

—

109

#95

—

356

—

110

C₂²×C₄

—

111

C₂²×C₄

—

356

—

112

C₂²×C₄

#13083559

yes

—

113

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

114

C₂×C₈

#13083553

yes

—

115

C₂×C₈

#13083561

yes

—

116

C₂×C₈

#13083562

yes

—

117

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

118

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

119

C₂×C₈

#13083573

yes

—

120

C₂×C₈

#13083575

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

356

—

124

C₄⋊C₄

—

356

—

125

C₄²

#13083553

yes

—

126

C₄²

#13083564

yes

—

127

C₁₆

—

178

—

128

C₂³

C₂×#13083559

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

712

—

133

C₂×C₄

—

712

—

134

C₂×C₄

#15667945

yes

—

135

C₂×C₄

C₂×#13083553

yes

characteristic

136

C₂×C₄

#15667999

yes

characteristic

137

C₂×C₄

#15668001

yes

characteristic

138

C₂×C₄

C₂×#13083561

yes

characteristic

139

C₂×C₄

C₂×#13083562

yes

characteristic

140

C₂×C₄

C₂×#13083564

yes

characteristic

141

C₈

#15667943

yes

—

142

C₈

#15667970

yes

—

143

C₈

#15668003

yes

—

144

C₂²

C₂×#15667945

yes

characteristic

145

C₂²

#19151944

yes

characteristic

146

C₂²

#19151946

yes

characteristic

147

C₂²

#19152028

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

1424

—

151

C₄

#19151463

yes

characteristic

152

C₄

C₂×#15667943

yes

characteristic

153

C₄

#19151948

yes

characteristic

154

C₄

#19151950

yes

characteristic

155

C₂

#26020379

yes

characteristic

156

C₂

#26022034

yes

—

157

C₂

#26022483

yes

—

158

C₂

#26022486

yes

—

	subgroup	G/H
center	C₂³	C₂×#13083559
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1846	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1846	D₈₉
hypocenter	C₈₉	#12889
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#13083559

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1846

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₈

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
9. C₄×#13083549	C₄	semidirect
84. C₈₉	#12889	semidirect

H ≅ G/N

type

9. C₄×#13083549

C₄

semidirect

84. C₈₉

#12889

semidirect

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

subgroup	index
C₂×C₄×#13083549	2
#26020416	2
#26018343	2
C₈₉×#1846	2
#12889	89

subgroup

index

C₈₉×#1846

Subgroups — #36470810

Conjugacy classes of subgroups

Complements

Maximal subgroups