Group #36466297 — subgroups

Subgroups — #36466297

Order 22784.

236 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 87 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	#26022085	C₂	1	2	yes	maximal, complemented
2	C₈×D₇₁₂	C₂	1	1	yes	maximal, characteristic
3	#26018316	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018208	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15667951	—	2	2	no	—
7	C₄×D₇₁₂	C₄	1	1	yes	characteristic
8	C₈×D₃₅₆	C₂²	1	2	yes	—
9	C₄⋊#13083549	C₂²	1	2	yes	—
10	#19151377	—	2	2	no	—
11	#19151351	C₄	1	1	yes	characteristic
12	#19151327	C₄	1	2	yes	—
13	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₈²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×D₇₁₂	C₈	1	1	yes	characteristic
17	C₂×C₈×D₈₉	—	2	2	no	—
18	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
19	#15667951	—	4	4	no	—
20	C₂×#13083549	—	2	2	no	—
21	#15667946	C₈	1	1	yes	characteristic
22	C₄⋊#10623494	C₂×C₄	1	2	yes	—
23	#15667934	C₈	1	2	yes	—
24	#15667924	—	2	2	no	—
25	C₂×C₁₆×C₈₉	—	2	1	no	—
26	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
27	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
31	C₂×C₄×D₈₉	—	2	2	no	—
32	D₇₁₂	—	2	2	no	—
33	C₂×#10623494	—	2	2	no	—
34	C₈×D₈₉	—	4	2	no	—
35	C₈×D₈₉	—	4	2	no	—
36	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
37	#13083551	—	2	2	no	—
38	#13083549	—	4	2	no	—
39	C₂×C₈×C₈₉	—	2	1	no	—
40	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
44	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₁₆×C₈₉	—	2	2	no	—
46	C₁₆×C₈₉	—	4	1	no	—
47	C₂²×D₈₉	—	2	2	no	—
48	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
51	C₂×Dic₈₉	—	2	2	no	—
52	D₃₅₆	—	2	2	no	—
53	D₃₅₆	—	2	2	no	—
54	C₄×D₈₉	—	4	2	no	—
55	C₄×D₈₉	—	4	2	no	—
56	C₈×C₈₉	—	2	1	no	—
57	C₈×C₈₉	—	2	1	no	—
58	C₈×C₈₉	—	4	1	no	—
59	C₈×C₈₉	M₃₂	1	2	yes	complemented
60	C₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
61	#10623494	—	4	2	no	—
62	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
63	C₂×D₈₉	—	4	2	no	—
64	C₂×D₈₉	—	4	2	no	—
65	C₂×D₈₉	—	4	2	no	—
66	C₄×C₈₉	—	2	1	no	—
67	C₄×C₈₉	#442	1	1	yes	characteristic
68	C₄×C₈₉	#444	1	1	yes	characteristic
69	C₄×C₈₉	C₂×M₃₂	1	1	yes	characteristic
70	C₄×C₈₉	#504	1	1	yes	characteristic
71	Dic₈₉	—	4	2	no	—
72	#12064	—	89	1	no	maximal, Sylow
73	C₂×C₈₉	#1580	1	1	yes	characteristic
74	C₂×C₈₉	#2169	1	1	yes	characteristic
75	C₂×C₈₉	#2171	1	1	yes	characteristic
76	D₈₉	—	8	2	no	—
77	#2169	—	89	2	no	—
78	C₈×D₈	—	89	1	no	—
79	#1367	—	89	1	no	—
80	D₄⋊C₁₆	—	89	2	no	—
81	#1312	D₈₉	1	1	yes	characteristic, complemented
82	C₈₉	#12064	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₃₂	—	178	2	no	—
84	C₄×D₈	—	89	1	no	—
85	C₈×D₄	—	89	2	no	—
86	C₄⋊C₁₆	—	89	2	no	—
87	C₂²⋊C₁₆	—	178	2	no	—
88	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
89	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
90	#335	—	89	1	no	—
91	D₄⋊C₈	—	89	2	no	—
92	C₈²	C₂×D₈₉	1	1	yes	characteristic
93	C₂×D₈	—	89	1	no	—
94	C₂²×C₈	—	178	2	no	—
95	C₄×D₄	—	89	2	no	—
96	M₃₂	—	356	4	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	178	2	no	—
99	C₂×C₁₆	D₃₅₆	1	2	yes	—
100	#107	—	89	1	no	—
101	C₄⋊C₈	—	89	2	no	—
102	#102	—	89	2	no	—
103	C₂²⋊C₈	—	178	2	no	—
104	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
105	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
107	C₂×D₄	—	89	2	no	—
108	C₂²×C₄	—	178	2	no	—
109	D₈	—	178	2	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	356	2	no	—
112	C₂×C₈	—	356	2	no	—
113	C₂×C₈	—	178	2	no	—
114	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
115	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
116	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
117	C₂×C₈	#13083574	1	1	yes	characteristic
118	C₄⋊C₄	—	89	2	no	—
119	C₂²⋊C₄	—	178	2	no	—
120	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
121	C₁₆	—	2	2	no	—
122	C₁₆	—	4	1	no	—
123	C₁₆	—	356	2	no	—
124	C₂³	—	178	2	no	—
125	D₄	—	178	2	no	—
126	D₄	—	178	2	no	—
127	C₂×C₄	—	356	2	no	—
128	C₂×C₄	—	356	2	no	—
129	C₂×C₄	—	178	2	no	—
130	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
131	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
132	C₂×C₄	#15668018	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	356	2	no	—
137	C₈	#15667951	1	2	yes	complemented
138	C₈	#15668040	1	2	yes	—
139	C₂²	—	356	2	no	—
140	C₂²	—	356	2	no	—
141	C₂²	—	356	2	no	—
142	C₂²	C₈×D₃₅₆	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	356	2	no	—
145	C₄	#19151711	1	1	yes	characteristic
146	C₄	#19151721	1	1	yes	characteristic
147	C₄	C₂×#15667951	1	1	yes	characteristic
148	C₄	#19152014	1	1	yes	characteristic
149	C₂	—	712	2	no	—
150	C₂	#26019077	1	1	yes	characteristic
151	C₂	#26022085	1	1	yes	characteristic
152	C₂	#26022095	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022085

C₂

yes

maximal, complemented

C₈×D₇₁₂

C₂

yes

maximal, characteristic

#26018316

C₂

yes

maximal, characteristic, complemented

#26018208

C₂

yes

maximal, complemented

C₈₉×#1312

C₂

yes

maximal, characteristic, complemented

C₂×#15667951

—

C₄×D₇₁₂

C₄

yes

characteristic

C₈×D₃₅₆

C₂²

yes

—

C₄⋊#13083549

C₂²

yes

—

#19151377

—

#19151351

C₄

yes

characteristic

#19151327

C₄

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₈²×C₈₉

C₂²

yes

characteristic

C₂×D₇₁₂

C₈

yes

characteristic

C₂×C₈×D₈₉

—

C₄×D₃₅₆

C₂×C₄

yes

—

#15667951

—

C₂×#13083549

—

#15667946

C₈

yes

characteristic

C₄⋊#10623494

C₂×C₄

yes

—

#15667934

C₈

yes

—

#15667924

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×D₃₅₆

C₂×C₈

yes

—

C₂×C₄×D₈₉

—

D₇₁₂

—

C₂×#10623494

—

C₈×D₈₉

—

C₈×D₈₉

—

C₄⋊Dic₈₉

C₂×C₈

yes

—

#13083551

—

#13083549

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₉

#131

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

M₃₂

yes

complemented

C₈×C₈₉

AGL(1,ℤ/8)

yes

—

#10623494

—

C₂²×C₈₉

C₈×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₂×M₃₂

yes

characteristic

C₄×C₈₉

#504

yes

characteristic

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#2169

—

C₈×D₈

—

#1367

—

D₄⋊C₁₆

—

#1312

D₈₉

yes

characteristic, complemented

C₈₉

#12064

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

178

—

C₄×D₈

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₄×C₁₆

C₂×D₈₉

yes

characteristic

#335

—

D₄⋊C₈

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×D₈

—

C₂²×C₈

—

178

—

C₄×D₄

—

M₃₂

—

356

—

C₂×C₁₆

—

C₂×C₁₆

—

178

—

C₂×C₁₆

D₃₅₆

yes

—

100

#107

—

101

C₄⋊C₈

—

102

#102

—

103

C₂²⋊C₈

—

178

—

104

C₄×C₈

C₄×D₈₉

yes

characteristic

105

C₄×C₈

C₄×D₈₉

yes

characteristic

106

C₄×C₈

C₂²×D₈₉

yes

characteristic

107

C₂×D₄

—

108

C₂²×C₄

—

178

—

109

D₈

—

178

—

110

C₂×C₈

—

111

C₂×C₈

—

356

—

112

C₂×C₈

—

356

—

113

C₂×C₈

—

178

—

114

C₂×C₈

C₈×D₈₉

yes

characteristic

115

C₂×C₈

C₈×D₈₉

yes

characteristic

116

C₂×C₈

C₂×D₃₅₆

yes

characteristic

117

C₂×C₈

#13083574

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

178

—

120

C₄²

C₂×C₄×D₈₉

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

356

—

124

C₂³

—

178

—

125

D₄

—

178

—

126

D₄

—

178

—

127

C₂×C₄

—

356

—

128

C₂×C₄

—

356

—

129

C₂×C₄

—

178

—

130

C₂×C₄

C₄×D₃₅₆

yes

characteristic

131

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

132

C₂×C₄

#15668018

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

356

—

137

C₈

#15667951

yes

complemented

138

C₈

#15668040

yes

—

139

C₂²

—

356

—

140

C₂²

—

356

—

141

C₂²

—

356

—

142

C₂²

C₈×D₃₅₆

yes

characteristic

143

C₄

—

144

C₄

—

356

—

145

C₄

#19151711

yes

characteristic

146

C₄

#19151721

yes

characteristic

147

C₄

C₂×#15667951

yes

characteristic

148

C₄

#19152014

yes

characteristic

149

C₂

—

712

—

150

C₂

#26019077

yes

characteristic

151

C₂

#26022085

yes

characteristic

152

C₂

#26022095

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1312	C₂
socle	C₂²×C₈₉	C₈×D₄
prime radical	C₂×D₇₁₂	C₈
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#1312	D₈₉
hypocenter	C₈₉	#12064
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

subgroup

G/H

center

C₂×C₈

C₂×D₃₅₆

commutator

C₂×C₄×C₈₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1312

C₂

socle

C₂²×C₈₉

C₈×D₄

prime radical

C₂×D₇₁₂

C₈

cosocle

C₄×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

N	H ≅ G/N	type
1. #26022085	C₂	semidirect
3. #26018316	C₂	semidirect
4. #26018208	C₂	semidirect
5. C₈₉×#1312	C₂	semidirect
59. C₈×C₈₉	M₃₂	semidirect
81. #1312	D₈₉	semidirect
82. C₈₉	#12064	semidirect
137. C₈	#15667951	semidirect

H ≅ G/N

type

1. #26022085

C₂

semidirect

3. #26018316

C₂

semidirect

4. #26018208

C₂

semidirect

5. C₈₉×#1312

C₂

semidirect

59. C₈×C₈₉

M₃₂

semidirect

81. #1312

D₈₉

semidirect

82. C₈₉

#12064

semidirect

137. C₈

#15667951

semidirect

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

subgroup	index
#26022085	2
C₈×D₇₁₂	2
#26018316	2
#26018208	2
C₈₉×#1312	2
#12064	89

subgroup

index

C₈×D₇₁₂

C₈₉×#1312

Subgroups — #36466297

Conjugacy classes of subgroups

Complements

Maximal subgroups