Group #36470826 — subgroups

Subgroups — #36470826

Order 22784.

234 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 95 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	#26021851	C₂	1	1	yes	maximal, characteristic
2	#26020377	C₂	1	2	yes	maximal
3	#26018354	C₂	1	1	yes	maximal, characteristic
4	#26018349	C₂	1	2	yes	maximal
5	C₈₉×#1385	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083549	C₂²	1	1	yes	characteristic
7	#19151636	C₂²	1	2	yes	—
8	#19151553	C₂²	1	1	yes	characteristic
9	#19151373	C₄	1	2	yes	complemented
10	C₁₆×Dic₈₉	C₄	1	2	yes	complemented
11	#19151370	—	2	2	no	—
12	#19151355	—	2	2	no	—
13	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#108×C₈₉	C₂²	1	2	yes	—
15	C₂×#13083558	D₄	1	1	yes	characteristic
16	#15667971	—	2	2	no	—
17	#15667971	D₄	1	1	yes	characteristic
18	C₂×#13083549	—	2	2	no	—
19	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
20	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
21	#15667946	—	2	2	no	—
22	#15667944	—	2	2	no	—
23	#15667922	Q₈	1	2	yes	—
24	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₈₉	—	2	2	no	—
26	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
27	M₃₂×C₈₉	—	2	2	no	—
28	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
29	#108×C₈₉	—	2	2	no	—
30	#108×C₈₉	—	2	2	no	—
31	#13083558	—	2	2	no	—
32	C₂×#10623494	C₄⋊C₄	1	2	yes	—
33	C₄⋊Dic₈₉	—	2	1	no	—
34	C₄⋊Dic₈₉	—	4	2	no	—
35	C₂²⋊Dic₈₉	—	2	1	no	—
36	C₂²⋊Dic₈₉	—	4	2	no	—
37	C₄×Dic₈₉	—	4	2	no	—
38	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
39	#13083549	—	2	1	no	—
40	#13083549	—	2	1	no	—
41	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
42	M₁₆×C₈₉	—	4	2	no	—
43	M₁₆×C₈₉	—	4	2	no	—
44	C₂×C₈×C₈₉	—	4	2	no	—
45	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₈₉	#41	1	2	yes	—
48	C₂×C₈×C₈₉	#41	1	2	yes	—
49	C₁₆×C₈₉	—	2	2	no	—
50	C₂³×C₈₉	#135	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₈₉	#124	1	2	yes	—
53	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#135	1	2	yes	—
55	C₂×Dic₈₉	—	2	2	no	—
56	C₂×Dic₈₉	—	8	2	no	—
57	C₈×C₈₉	—	8	2	no	—
58	C₈×C₈₉	#117	1	1	yes	characteristic
59	C₈×C₈₉	#117	1	1	yes	characteristic
60	C₈×C₈₉	#125	1	2	yes	—
61	#10623494	—	2	2	no	—
62	C₂²×C₈₉	—	4	1	no	—
63	C₂²×C₈₉	#427	1	1	yes	characteristic
64	C₂²×C₈₉	#488	1	2	yes	—
65	C₄×C₈₉	#382	1	1	yes	characteristic
66	C₄×C₈₉	#427	1	1	yes	characteristic
67	C₄×C₈₉	#488	1	2	yes	—
68	Dic₈₉	—	4	2	no	—
69	Dic₈₉	—	16	2	no	—
70	#12893	—	89	1	no	maximal, Sylow
71	C₂×C₈₉	—	2	1	no	—
72	C₂×C₈₉	—	2	2	no	—
73	C₂×C₈₉	#1835	1	1	yes	characteristic
74	#2108	—	89	1	no	—
75	#1833	—	89	2	no	—
76	#1385	D₈₉	1	1	yes	characteristic
77	#1384	—	89	1	no	—
78	#1382	—	89	2	no	—
79	C₈₉	#12893	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
81	C₂²×C₁₆	—	89	1	no	—
82	C₂×#108	C₂×D₈₉	1	2	yes	—
83	#427	—	89	2	no	—
84	#405	—	89	1	no	—
85	#346	—	89	2	no	—
86	C₄×C₁₆	—	89	2	no	—
87	#344	—	178	2	no	—
88	H(3,ℤ/4)	—	178	2	no	—
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	89	1	no	—
91	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
92	#117	—	89	1	no	—
93	#117	—	178	2	no	—
94	M₃₂	—	2	2	no	—
95	C₂×C₁₆	—	89	1	no	—
96	C₂×C₁₆	—	89	1	no	—
97	C₂×C₁₆	—	178	2	no	—
98	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
99	#108	—	2	2	no	—
100	#108	—	2	2	no	—
101	#107	—	178	2	no	—
102	#106	—	178	2	no	—
103	#97	—	89	2	no	—
104	C₄×C₈	—	89	2	no	—
105	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
106	M₁₆	—	4	2	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	—	178	2	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	89	2	no	—
111	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
112	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
113	C₂×C₈	#13083573	1	2	yes	—
114	C₂×C₈	#13083574	1	2	yes	—
115	C₄⋊C₄	—	356	2	no	—
116	C₄⋊C₄	—	178	1	no	—
117	C₂²⋊C₄	—	356	2	no	—
118	C₂²⋊C₄	—	178	1	no	—
119	C₄²	—	89	2	no	—
120	C₄²	—	356	2	no	—
121	C₁₆	—	2	2	no	—
122	C₁₆	—	178	1	no	—
123	C₁₆	—	178	1	no	—
124	C₂³	#15668035	1	1	yes	characteristic
125	C₂×C₄	—	178	2	no	—
126	C₂×C₄	—	712	2	no	—
127	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
128	C₂×C₄	#15667998	1	2	yes	—
129	C₂×C₄	#15668031	1	1	yes	characteristic
130	C₂×C₄	#15668033	1	2	yes	—
131	C₈	—	8	2	no	—
132	C₈	—	178	2	no	—
133	C₈	#15667968	1	1	yes	characteristic
134	C₈	#15667969	1	1	yes	characteristic
135	C₈	#15668002	1	2	yes	—
136	C₂²	—	4	1	no	—
137	C₂²	#19151635	1	1	yes	characteristic
138	C₂²	#19151954	1	2	yes	—
139	C₄	—	356	2	no	—
140	C₄	—	1424	2	no	—
141	C₄	#19151462	1	1	yes	characteristic
142	C₄	#19151633	1	1	yes	characteristic
143	C₄	#19151951	1	2	yes	—
144	C₂	—	2	1	no	—
145	C₂	—	2	2	no	—
146	C₂	#26020381	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26021851

C₂

yes

maximal, characteristic

#26020377

C₂

yes

maximal

#26018354

C₂

yes

maximal, characteristic

#26018349

C₂

yes

maximal

C₈₉×#1385

C₂

yes

maximal, characteristic

C₂²×#13083549

C₂²

yes

characteristic

#19151636

C₂²

yes

—

#19151553

C₂²

yes

characteristic

#19151373

C₄

yes

complemented

C₁₆×Dic₈₉

C₄

yes

complemented

#19151370

—

#19151355

—

C₂×M₃₂×C₈₉

C₂²

yes

characteristic

C₂×#108×C₈₉

C₂²

yes

—

C₂×#13083558

D₄

yes

characteristic

#15667971

—

#15667971

D₄

yes

characteristic

C₂×#13083549

—

C₂×#13083549

C₂×C₄

yes

characteristic

C₂×#13083549

C₂×C₄

yes

—

—

Q₈

yes

—

C₈×Dic₈₉

C₂×C₄

yes

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

#108×C₈₉

—

#108×C₈₉

—

#13083558

—

C₂×#10623494

C₄⋊C₄

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₄⋊C₄

yes

—

#13083549

—

#13083549

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

#135

yes

characteristic

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

#135

yes

characteristic

C₂×C₄×C₈₉

#135

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

yes

C₈×C₈₉

yes

C₈×C₈₉

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#427

yes

characteristic

C₂²×C₈₉

#488

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

Dic₈₉

—

#12893

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

—

D₈₉

yes

characteristic

#1384

—

#1382

—

C₈₉

#12893

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₈₉

yes

characteristic

C₂²×C₁₆

—

C₂×#108

C₂×D₈₉

yes

—

#427

—

#405

—

#346

—

C₄×C₁₆

—

#344

—

178

—

H(3,ℤ/4)

—

178

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²×D₈₉

yes

characteristic

#117

—

#117

—

178

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

178

—

C₂×C₁₆

C₄×D₈₉

yes

—

#108

—

100

#108

—

101

#107

—

178

—

102

#106

—

178

—

103

#97

—

104

C₄×C₈

—

105

C₂²×C₄

D₄×D₈₉

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

178

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

112

C₂×C₈

Q₈×D₈₉

yes

characteristic

113

C₂×C₈

#13083573

yes

—

114

C₂×C₈

#13083574

yes

—

115

C₄⋊C₄

—

356

—

116

C₄⋊C₄

—

178

—

117

C₂²⋊C₄

—

356

—

118

C₂²⋊C₄

—

178

—

119

C₄²

—

120

C₄²

—

356

—

121

C₁₆

—

122

C₁₆

—

178

—

123

C₁₆

—

178

—

124

C₂³

#15668035

yes

characteristic

125

C₂×C₄

—

178

—

126

C₂×C₄

—

712

—

127

C₂×C₄

C₄⋊C₄×D₈₉

yes

characteristic

128

C₂×C₄

#15667998

yes

—

129

C₂×C₄

#15668031

yes

characteristic

130

C₂×C₄

#15668033

yes

—

131

C₈

—

132

C₈

—

178

—

133

C₈

#15667968

yes

characteristic

134

C₈

#15667969

yes

characteristic

135

C₈

#15668002

yes

—

136

C₂²

—

137

C₂²

#19151635

yes

characteristic

138

C₂²

#19151954

yes

—

139

C₄

—

356

—

140

C₄

—

1424

—

141

C₄

#19151462

yes

characteristic

142

C₄

#19151633

yes

characteristic

143

C₄

#19151951

yes

—

144

C₂

—

145

C₂

—

146

C₂

#26020381

yes

characteristic

	subgroup	G/H
center	C₄	#19151633
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1385	C₂
socle	C₂×C₈₉	#1835
prime radical	C₂³×C₈₉	#135
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1385	D₈₉
hypocenter	C₈₉	#12893
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#19151633

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1385

C₂

C₂×C₈₉

C₂³×C₈₉

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
9. #19151373	C₄	semidirect
10. C₁₆×Dic₈₉	C₄	semidirect
79. C₈₉	#12893	semidirect

H ≅ G/N

type

9. #19151373

C₄

semidirect

10. C₁₆×Dic₈₉

C₄

semidirect

79. C₈₉

#12893

semidirect

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

subgroup	index
#26021851	2
#26020377	2
#26018354	2
#26018349	2
C₈₉×#1385	2
#12893	89

subgroup

index

C₈₉×#1385

Subgroups — #36470826

Conjugacy classes of subgroups

Complements

Maximal subgroups