Group #36470797 — subgroups

Subgroups — #36470797

Order 22784.

234 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 103 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	#26022011	C₂	1	1	yes	maximal, characteristic
2	#26020007	C₂	1	1	yes	maximal, characteristic
3	#26018342	C₂	1	2	yes	maximal
4	#26018337	C₂	1	2	yes	maximal
5	C₈₉×#2153	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	C₂²	1	1	yes	characteristic
7	C₂²×#13083549	C₂²	1	1	yes	characteristic
8	#19151553	C₂²	1	2	yes	—
9	C₄⋊#13083549	—	2	2	no	—
10	#19151356	C₄	1	1	yes	characteristic
11	#19151355	C₄	1	1	yes	characteristic
12	#19151354	C₄	1	2	yes	—
13	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
14	#405×C₈₉	C₂²	1	1	yes	characteristic
15	#364×C₈₉	—	2	2	no	—
16	C₂×#13083558	C₂×C₄	1	2	yes	—
17	#15667971	C₂×C₄	1	2	yes	—
18	#15667952	—	2	2	no	—
19	C₂×#13083549	—	2	1	no	—
20	C₂×#13083549	—	2	1	no	—
21	C₂×#13083549	—	2	2	no	—
22	#15667922	—	2	2	no	—
23	C₈×Dic₈₉	—	2	2	no	—
24	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
26	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₈₉	—	2	2	no	—
28	M₃₂×C₈₉	—	2	2	no	—
29	C₂×C₁₆×C₈₉	—	2	2	no	—
30	#97×C₈₉	Q₈	1	2	yes	—
31	C₄×C₈×C₈₉	D₄	1	2	yes	—
32	#13083558	—	2	2	no	—
33	#13083558	—	2	2	no	—
34	C₂×#10623494	—	2	2	no	—
35	C₄⋊Dic₈₉	—	2	2	no	—
36	C₂²⋊Dic₈₉	—	2	2	no	—
37	C₄×Dic₈₉	—	2	2	no	—
38	#13083549	—	4	1	no	—
39	#13083549	—	4	1	no	—
40	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
41	M₁₆×C₈₉	—	2	2	no	—
42	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₈₉	#41	1	2	yes	—
46	C₂×C₈×C₈₉	#41	1	2	yes	—
47	C₄⋊C₄×C₈₉	—	2	1	no	—
48	C₂²⋊C₄×C₈₉	—	2	1	no	—
49	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
50	C₁₆×C₈₉	—	4	2	no	—
51	C₂³×C₈₉	#131	1	1	yes	characteristic
52	C₂×C₄×C₈₉	—	2	2	no	—
53	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#117	1	2	yes	—
55	C₂×C₄×C₈₉	#131	1	2	yes	—
56	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
57	C₂×Dic₈₉	—	4	2	no	—
58	C₈×C₈₉	—	2	2	no	—
59	C₈×C₈₉	#124	1	1	yes	characteristic
60	C₈×C₈₉	#124	1	1	yes	characteristic
61	C₈×C₈₉	#125	1	2	yes	—
62	#10623494	—	4	2	no	—
63	C₂²×C₈₉	—	4	1	no	—
64	C₂²×C₈₉	#424	1	1	yes	characteristic
65	C₂²×C₈₉	#433	1	2	yes	—
66	C₄×C₈₉	—	4	2	no	—
67	C₄×C₈₉	#382	1	1	yes	characteristic
68	C₄×C₈₉	#424	1	1	yes	characteristic
69	C₄×C₈₉	#433	1	2	yes	—
70	Dic₈₉	—	8	2	no	—
71	#12885	—	89	1	no	maximal, Sylow
72	C₂×C₈₉	—	2	2	no	—
73	C₂×C₈₉	—	2	1	no	—
74	C₂×C₈₉	#1832	1	1	yes	characteristic
75	#2153	D₈₉	1	1	yes	characteristic
76	#2151	—	89	1	no	—
77	#1747	—	89	1	no	—
78	#1379	—	89	2	no	—
79	#1378	—	89	2	no	—
80	C₈₉	#12885	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₃₂	—	89	1	no	—
82	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
83	C₂²×C₁₆	—	89	1	no	—
84	#405	—	89	2	no	—
85	#405	C₂×D₈₉	1	1	yes	characteristic
86	#364	—	2	2	no	—
87	C₄⋊C₁₆	—	178	2	no	—
88	#338	—	89	1	no	—
89	H(3,ℤ/4)	—	89	1	no	—
90	H(3,ℤ/4)	—	89	2	no	—
91	C₂×M₁₆	—	89	2	no	—
92	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
94	#117	—	89	2	no	—
95	#117	C₄×D₈₉	1	1	yes	characteristic
96	M₃₂	—	2	2	no	—
97	M₃₂	—	2	2	no	—
98	M₃₂	—	178	2	no	—
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	178	1	no	—
101	C₂×C₁₆	—	178	1	no	—
102	C₂×C₁₆	—	178	2	no	—
103	#97	—	178	2	no	—
104	#97	Dic₁₇₈	1	2	yes	—
105	C₄×C₈	—	178	2	no	—
106	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	178	2	no	—
110	M₁₆	—	178	2	no	—
111	C₂×C₈	—	178	2	no	—
112	C₂×C₈	#13083553	1	2	yes	—
113	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
114	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
115	C₂×C₈	#13083573	1	2	yes	—
116	C₂×C₈	#13083575	1	2	yes	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	178	2	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	178	2	no	—
121	C₄²	—	178	2	no	—
122	C₄²	#13083553	1	2	yes	—
123	C₁₆	—	4	2	no	—
124	C₁₆	—	356	1	no	—
125	C₁₆	—	356	1	no	—
126	C₂³	#15668020	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	356	2	no	—
129	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
130	C₂×C₄	#15667970	1	2	yes	—
131	C₂×C₄	#15668018	1	2	yes	—
132	C₂×C₄	#15668020	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	356	2	no	—
135	C₈	#15667999	1	1	yes	characteristic
136	C₈	#15668001	1	1	yes	characteristic
137	C₈	#15668003	1	2	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	#19151626	1	1	yes	characteristic
140	C₂²	#19151658	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	712	2	no	—
143	C₄	#19151463	1	1	yes	characteristic
144	C₄	#19151624	1	1	yes	characteristic
145	C₄	#19151658	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	#26020371	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022011

C₂

yes

maximal, characteristic

#26020007

C₂

yes

maximal, characteristic

#26018342

C₂

yes

maximal

#26018337

C₂

yes

maximal

C₈₉×#2153

C₂

yes

maximal, characteristic

C₂×#15667952

C₂²

yes

characteristic

C₂²×#13083549

C₂²

yes

characteristic

#19151553

C₂²

yes

—

C₄⋊#13083549

—

#19151356

C₄

yes

characteristic

#19151355

C₄

yes

characteristic

#19151354

C₄

yes

—

C₂×M₃₂×C₈₉

C₂²

yes

—

#405×C₈₉

C₂²

yes

characteristic

#364×C₈₉

—

C₂×#13083558

C₂×C₄

yes

—

#15667971

C₂×C₄

yes

—

—

—

—

—

—

C₈×Dic₈₉

—

C₂×M₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

#117×C₈₉

C₂×C₄

yes

characteristic

M₃₂×C₈₉

—

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

—

#97×C₈₉

Q₈

yes

—

C₄×C₈×C₈₉

D₄

yes

—

#13083558

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

#13083549

—

#13083549

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

C₄⋊C₄

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

#131

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#117

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₂×C₄×C₈₉

#131

yes

characteristic

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

yes

C₈×C₈₉

yes

C₈×C₈₉

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#424

yes

characteristic

C₂²×C₈₉

#433

yes

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

#12885

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1832

yes

characteristic

#2153

D₈₉

yes

—

—

—

—

C₈₉

#12885

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₈₉

yes

—

C₂²×C₁₆

—

#405

—

#405

C₂×D₈₉

yes

characteristic

#364

—

C₄⋊C₁₆

—

178

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₈₉

yes

characteristic

C₂²×C₈

C₂²×D₈₉

yes

characteristic

#117

—

#117

C₄×D₈₉

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

178

—

C₂×C₁₆

—

100

C₂×C₁₆

—

178

—

101

C₂×C₁₆

—

178

—

102

C₂×C₁₆

—

178

—

103

#97

—

178

—

104

#97

Dic₁₇₈

yes

—

105

C₄×C₈

—

178

—

106

C₄×C₈

C₂²⋊D₈₉

yes

—

107

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

108

M₁₆

—

109

M₁₆

—

178

—

110

M₁₆

—

178

—

111

C₂×C₈

—

178

—

112

C₂×C₈

#13083553

yes

—

113

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

114

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

115

C₂×C₈

#13083573

yes

—

116

C₂×C₈

#13083575

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

178

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

178

—

121

C₄²

—

178

—

122

C₄²

#13083553

yes

—

123

C₁₆

—

124

C₁₆

—

356

—

125

C₁₆

—

356

—

126

C₂³

#15668020

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

356

—

129

C₂×C₄

C₂×#13083553

yes

characteristic

130

C₂×C₄

#15667970

yes

—

131

C₂×C₄

#15668018

yes

—

132

C₂×C₄

#15668020

yes

characteristic

133

C₈

—

134

C₈

—

356

—

135

C₈

#15667999

yes

characteristic

136

C₈

#15668001

yes

characteristic

137

C₈

#15668003

yes

—

138

C₂²

—

139

C₂²

#19151626

yes

characteristic

140

C₂²

#19151658

yes

—

141

C₄

—

142

C₄

—

712

—

143

C₄

#19151463

yes

characteristic

144

C₄

#19151624

yes

characteristic

145

C₄

#19151658

yes

—

146

C₂

—

147

C₂

—

148

C₂

#26020371

yes

characteristic

	subgroup	G/H
center	C₈	#15667999
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2153	C₂
socle	C₂×C₈₉	#1832
prime radical	C₂³×C₈₉	#131
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2153	D₈₉
hypocenter	C₈₉	#12885
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

subgroup

G/H

center

C₈

#15667999

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#2153

C₂

C₂×C₈₉

C₂³×C₈₉

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

N	H ≅ G/N	type
80. C₈₉	#12885	semidirect

H ≅ G/N

type

80. C₈₉

#12885

semidirect

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

subgroup	index
#26022011	2
#26020007	2
#26018342	2
#26018337	2
C₈₉×#2153	2
#12885	89

subgroup

index

C₈₉×#2153

Subgroups — #36470797

Conjugacy classes of subgroups

Complements

Maximal subgroups