Group #36474875 — subgroups

Subgroups — #36474875

Order 22784.

290 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 115 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×#19151432	C₂	1	1	yes	maximal, characteristic
2	#26022036	C₂	1	1	yes	maximal, characteristic
3	#26018433	C₂	1	4	yes	maximal, complemented
4	C₂²×Q₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151636	—	2	1	no	—
6	#19151432	—	2	2	no	—
7	C₂×#15667918	C₂²	1	2	yes	—
8	#19151426	—	2	1	no	—
9	#19151424	C₂²	1	2	yes	—
10	C₂²×Q₁₆×C₈₉	—	2	1	no	—
11	C₂×Q₃₂×C₈₉	—	2	1	no	—
12	C₂×Q₃₂×C₈₉	—	2	2	no	—
13	C₂×Q₃₂×C₈₉	C₄	1	4	yes	complemented
14	C₂×Q₃₂×C₈₉	C₂²	1	2	yes	—
15	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	#15667971	—	4	1	no	—
17	#15667946	—	2	2	no	—
18	#15667944	—	4	1	no	—
19	#15667918	—	2	2	no	—
20	C₂²×Q₈×C₈₉	—	4	1	no	—
21	C₂×Q₁₆×C₈₉	—	2	4	no	—
22	C₂×Q₁₆×C₈₉	—	2	2	no	—
23	C₂×Q₁₆×C₈₉	—	4	1	no	—
24	C₂×Q₁₆×C₈₉	—	4	2	no	—
25	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
26	Q₃₂×C₈₉	—	2	4	no	—
27	Q₃₂×C₈₉	—	2	2	no	—
28	Q₃₂×C₈₉	C₂×C₄	1	4	yes	—
29	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
32	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₈₉	—	4	2	no	—
34	C₂²⋊Dic₈₉	—	8	1	no	—
35	C₄×Dic₈₉	—	8	1	no	—
36	C₂×Q₈×C₈₉	—	4	4	no	—
37	C₂×Q₈×C₈₉	—	4	2	no	—
38	C₂×Q₈×C₈₉	—	8	1	no	—
39	C₂×Q₈×C₈₉	—	8	2	no	—
40	C₂²×C₄×C₈₉	—	8	1	no	—
41	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
42	Q₁₆×C₈₉	—	2	4	no	—
43	Q₁₆×C₈₉	—	4	4	no	—
44	Q₁₆×C₈₉	—	4	2	no	—
45	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
47	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
48	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₈₉	C₂²⋊C₄	1	2	yes	—
50	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂³×C₈₉	D₁₆	1	1	yes	characteristic
53	Q₈×C₈₉	—	4	4	no	—
54	Q₈×C₈₉	—	8	4	no	—
55	Q₈×C₈₉	—	8	2	no	—
56	C₂×C₄×C₈₉	—	8	4	no	—
57	C₂×C₄×C₈₉	—	8	2	no	—
58	C₂×C₄×C₈₉	#102	1	2	yes	—
59	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
60	C₂×C₄×C₈₉	SD₃₂	1	2	yes	—
61	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
62	C₂×Dic₈₉	—	8	2	no	—
63	C₈×C₈₉	#102	1	2	yes	—
64	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₈₉	—	2	2	no	—
67	C₂²×C₈₉	#357	1	2	yes	—
68	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
69	C₄×C₈₉	—	8	4	no	—
70	C₄×C₈₉	#357	1	2	yes	—
71	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
72	C₄×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
73	Dic₈₉	—	16	2	no	—
74	#13618	—	89	1	no	maximal, Sylow
75	C₂×C₈₉	—	2	2	no	—
76	C₂×C₈₉	C₂×#357	1	1	yes	characteristic
77	C₂×C₈₉	#2264	1	2	yes	—
78	C₂²×Q₃₂	D₈₉	1	1	yes	characteristic
79	C₂×M₆₄	—	89	1	no	—
80	#2158	—	89	1	no	—
81	#1416	—	89	4	no	—
82	C₈₉	#13618	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₁₆	—	2	1	no	—
84	C₂×Q₃₂	—	2	1	no	—
85	C₂×Q₃₂	—	2	2	no	—
86	C₂×Q₃₂	Dic₈₉	1	4	yes	complemented
87	C₂×Q₃₂	C₂×D₈₉	1	2	yes	—
88	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
89	#427	—	178	1	no	—
90	M₆₄	—	178	2	no	—
91	C₂×C₃₂	—	89	2	no	—
92	#367	—	178	1	no	—
93	#366	—	89	2	no	—
94	C₂²×Q₈	—	4	1	no	—
95	C₂×Q₁₆	—	2	4	no	—
96	C₂×Q₁₆	—	2	2	no	—
97	C₂×Q₁₆	—	4	1	no	—
98	C₂×Q₁₆	—	4	2	no	—
99	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
100	#117	—	356	1	no	—
101	Q₃₂	—	2	4	no	—
102	Q₃₂	—	2	2	no	—
103	Q₃₂	C₂×Dic₈₉	1	4	yes	—
104	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
105	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
106	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
107	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
108	#107	—	178	2	no	—
109	#106	—	356	1	no	—
110	C₃₂	—	178	2	no	—
111	C₂×Q₈	—	4	2	no	—
112	C₂×Q₈	—	4	4	no	—
113	C₂×Q₈	—	8	1	no	—
114	C₂×Q₈	—	8	2	no	—
115	C₂²×C₄	—	8	1	no	—
116	C₂²×C₄	#13083559	1	1	yes	characteristic
117	Q₁₆	—	2	4	no	—
118	Q₁₆	—	4	2	no	—
119	Q₁₆	—	4	4	no	—
120	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
121	C₂×C₈	#13083559	1	1	yes	characteristic
122	C₂×C₈	#13083562	1	2	yes	—
123	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
124	C₄⋊C₄	—	356	2	no	—
125	C₂²⋊C₄	—	712	1	no	—
126	C₄²	—	712	1	no	—
127	C₁₆	C₂²⋊Dic₈₉	1	2	yes	—
128	C₁₆	C₂²×Dic₈₉	1	1	yes	characteristic
129	C₁₆	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
130	C₂³	#15667953	1	1	yes	characteristic
131	Q₈	—	4	4	no	—
132	Q₈	—	8	4	no	—
133	Q₈	—	8	2	no	—
134	C₂×C₄	—	8	4	no	—
135	C₂×C₄	—	8	2	no	—
136	C₂×C₄	—	712	2	no	—
137	C₂×C₄	#15667933	1	2	yes	—
138	C₂×C₄	#15667953	1	1	yes	characteristic
139	C₂×C₄	#15667956	1	2	yes	—
140	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
141	C₈	#15667933	1	2	yes	—
142	C₈	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
143	C₈	C₂×#13083562	1	1	yes	characteristic
144	C₂²	—	2	2	no	—
145	C₂²	#19151399	1	2	yes	—
146	C₂²	C₂×#15667953	1	1	yes	characteristic
147	C₄	—	8	4	no	—
148	C₄	—	1424	2	no	—
149	C₄	#19151399	1	2	yes	—
150	C₄	C₂×#15667933	1	1	yes	characteristic
151	C₄	C₂×#15667956	1	1	yes	characteristic
152	C₂	—	2	2	no	—
153	C₂	C₂×#19151399	1	1	yes	characteristic
154	C₂	#26022570	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151432

C₂

yes

maximal, characteristic

#26022036

C₂

yes

maximal, characteristic

#26018433

C₂

yes

maximal, complemented

C₂²×Q₃₂×C₈₉

C₂

yes

maximal, characteristic

#19151636

—

#19151432

—

C₂×#15667918

C₂²

yes

—

#19151426

—

#19151424

C₂²

yes

—

C₂²×Q₁₆×C₈₉

—

C₂×Q₃₂×C₈₉

—

C₂×Q₃₂×C₈₉

—

C₂×Q₃₂×C₈₉

C₄

yes

complemented

C₂×Q₃₂×C₈₉

C₂²

yes

—

C₂²×C₁₆×C₈₉

C₂²

yes

—

—

—

—

C₂²×Q₈×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

Q₃₂×C₈₉

—

Q₃₂×C₈₉

—

Q₃₂×C₈₉

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₈

yes

characteristic

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

C₂²⋊C₄

yes

—

C₁₆×C₈₉

C₂²×C₄

yes

characteristic

C₁₆×C₈₉

C₂×D₄

yes

characteristic

C₂³×C₈₉

D₁₆

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#102

yes

—

C₂×C₄×C₈₉

D₁₆

yes

characteristic

C₂×C₄×C₈₉

SD₃₂

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

C₈×C₈₉

#102

yes

—

C₈×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₈₉

C₂×SD₁₆

yes

characteristic

C₂²×C₈₉

—

C₂²×C₈₉

#357

yes

—

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#357

yes

—

C₄×C₈₉

C₂×#102

yes

characteristic

C₄×C₈₉

C₂×SD₃₂

yes

characteristic

Dic₈₉

—

#13618

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₂×#357

yes

characteristic

C₂×C₈₉

#2264

yes

—

C₂²×Q₃₂

D₈₉

yes

characteristic

C₂×M₆₄

—

#2158

—

#1416

—

C₈₉

#13618

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

Dic₈₉

yes

complemented

C₂×Q₃₂

C₂×D₈₉

yes

—

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

#427

—

178

—

M₆₄

—

178

—

C₂×C₃₂

—

#367

—

178

—

#366

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

100

#117

—

356

—

101

Q₃₂

—

102

Q₃₂

—

103

Q₃₂

C₂×Dic₈₉

yes

—

104

C₂×C₁₆

C₂×Dic₈₉

yes

—

105

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

106

C₂×C₁₆

C₂²⋊D₈₉

yes

—

107

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

108

#107

—

178

—

109

#106

—

356

—

110

C₃₂

—

178

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂²×C₄

—

116

C₂²×C₄

#13083559

yes

characteristic

117

Q₁₆

—

118

Q₁₆

—

119

Q₁₆

—

120

C₂×C₈

C₂²⋊Dic₈₉

yes

—

121

C₂×C₈

#13083559

yes

characteristic

122

C₂×C₈

#13083562

yes

—

123

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

124

C₄⋊C₄

—

356

—

125

C₂²⋊C₄

—

712

—

126

C₄²

—

712

—

127

C₁₆

C₂²⋊Dic₈₉

yes

—

128

C₁₆

C₂²×Dic₈₉

yes

characteristic

129

C₁₆

C₂×C₂²⋊D₈₉

yes

characteristic

130

C₂³

#15667953

yes

characteristic

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

712

—

137

C₂×C₄

#15667933

yes

—

138

C₂×C₄

#15667953

yes

characteristic

139

C₂×C₄

#15667956

yes

—

140

C₂×C₄

C₂×#13083559

yes

characteristic

141

C₈

#15667933

yes

—

142

C₈

C₂×C₂²⋊Dic₈₉

yes

characteristic

143

C₈

C₂×#13083562

yes

characteristic

144

C₂²

—

145

C₂²

#19151399

yes

—

146

C₂²

C₂×#15667953

yes

characteristic

147

C₄

—

148

C₄

—

1424

—

149

C₄

#19151399

yes

—

150

C₄

C₂×#15667933

yes

characteristic

151

C₄

C₂×#15667956

yes

characteristic

152

C₂

—

153

C₂

C₂×#19151399

yes

characteristic

154

C₂

#26022570

yes

—

	subgroup	G/H
center	C₂²	C₂×#15667953
commutator	C₁₆×C₈₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₂²×Q₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂×D₁₆
prime radical	C₂³×C₈₉	D₁₆
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	C₂²×Q₃₂	D₈₉
hypocenter	C₈₉	#13618
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×Q₃₂

subgroup

G/H

center

C₂²

C₂×#15667953

commutator

C₁₆×C₈₉

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₂²×Q₃₂×C₈₉

C₂

socle

C₂²×C₈₉

C₂×D₁₆

prime radical

C₂³×C₈₉

D₁₆

cosocle

C₂×C₁₆×C₈₉

C₂³

hypercenter

C₂²×Q₃₂

D₈₉

hypocenter

C₈₉

#13618

LCS

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×Q₃₂

N	H ≅ G/N	type
3. #26018433	C₂	semidirect
13. C₂×Q₃₂×C₈₉	C₄	semidirect
82. C₈₉	#13618	semidirect
86. C₂×Q₃₂	Dic₈₉	semidirect

H ≅ G/N

type

3. #26018433

C₂

semidirect

13. C₂×Q₃₂×C₈₉

C₄

semidirect

82. C₈₉

#13618

semidirect

86. C₂×Q₃₂

Dic₈₉

semidirect

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

subgroup	index
C₂×#19151432	2
#26022036	2
#26018433	2
C₂²×Q₃₂×C₈₉	2
#13618	89

subgroup

index

C₂²×Q₃₂×C₈₉

Subgroups — #36474875

Conjugacy classes of subgroups

Complements

Maximal subgroups