Group #36475392 — subgroups

Subgroups — #36475392

Order 22784.

160 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 55 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	#26022360	C₂	1	1	yes	maximal, characteristic
2	#26018459	C₂	1	2	yes	maximal, complemented
3	#26018444	C₂	1	2	yes	maximal, complemented
4	#26018392	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2161	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
7	#19151839	—	2	1	no	—
8	#19151432	C₂²	1	2	yes	—
9	#19151426	C₂²	1	1	yes	characteristic
10	#19151403	—	2	1	no	—
11	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×#108×C₈₉	—	2	1	no	—
13	#368×C₈₉	—	2	1	no	—
14	#368×C₈₉	C₂²	1	2	yes	—
15	C₂×Dic₃₅₆	—	2	1	no	—
16	#15667991	—	4	1	no	—
17	Dic₇₁₂	—	2	2	no	—
18	#15667946	—	2	1	no	—
19	#15667937	—	4	1	no	—
20	#15667918	—	2	2	no	—
21	C₂×M₁₆×C₈₉	—	4	1	no	—
22	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
23	M₃₂×C₈₉	—	2	2	no	—
24	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
26	#108×C₈₉	—	2	2	no	—
27	#108×C₈₉	—	4	1	no	—
28	C₂×Dic₁₇₈	—	4	1	no	—
29	Dic₃₅₆	—	4	1	no	—
30	Dic₃₅₆	—	4	1	no	—
31	C₄⋊Dic₈₉	—	4	1	no	—
32	#13083553	—	8	1	no	—
33	C₂²⋊Dic₈₉	—	8	1	no	—
34	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
35	M₁₆×C₈₉	—	4	2	no	—
36	M₁₆×C₈₉	—	8	1	no	—
37	C₂×C₈×C₈₉	—	2	1	no	—
38	C₂×C₈×C₈₉	—	2	1	no	—
39	C₂×C₈×C₈₉	—	8	1	no	—
40	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₈₉	—	2	1	no	—
43	C₁₆×C₈₉	#41	1	2	yes	—
44	C₂³×C₈₉	SD₃₂	1	1	yes	characteristic
45	C₂×C₄×C₈₉	—	2	1	no	—
46	C₂×C₄×C₈₉	—	2	1	no	—
47	C₂×C₄×C₈₉	SD₃₂	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
49	C₂×Dic₈₉	—	8	1	no	—
50	C₂×Dic₈₉	—	8	1	no	—
51	Dic₁₇₈	—	8	1	no	—
52	Dic₁₇₈	—	8	1	no	—
53	C₈×C₈₉	—	2	1	no	—
54	C₈×C₈₉	—	8	2	no	—
55	C₈×C₈₉	#123	1	1	yes	characteristic
56	C₈×C₈₉	#137	1	1	yes	characteristic
57	C₂²×C₈₉	—	2	1	no	—
58	C₂²×C₈₉	—	4	1	no	—
59	C₂²×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
60	C₄×C₈₉	—	2	1	no	—
61	C₄×C₈₉	#480	1	1	yes	characteristic
62	C₄×C₈₉	#509	1	1	yes	characteristic
63	Dic₈₉	—	16	1	no	—
64	Dic₈₉	—	16	1	no	—
65	#13718	—	89	1	no	maximal, Sylow
66	C₂×C₈₉	—	2	1	no	—
67	C₂×C₈₉	—	4	1	no	—
68	C₂×C₈₉	#2233	1	1	yes	characteristic
69	#2219	—	89	1	no	—
70	#2161	D₈₉	1	1	yes	characteristic
71	#1426	—	89	2	no	—
72	#1420	—	89	2	no	—
73	#1400	—	89	1	no	—
74	C₈₉	#13718	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₃₂	—	89	1	no	—
76	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
77	#467	—	178	1	no	—
78	C₂×#108	—	2	1	no	—
79	M₆₄	—	89	2	no	—
80	#368	—	2	1	no	—
81	#368	C₂×D₈₉	1	2	yes	—
82	#367	—	89	1	no	—
83	#358	—	178	1	no	—
84	C₂×Q₁₆	—	178	1	no	—
85	C₂×M₁₆	—	4	1	no	—
86	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
87	C₂²⋊Q₈	—	356	1	no	—
88	Q₃₂	—	178	2	no	—
89	M₃₂	—	2	2	no	—
90	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
91	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
92	#108	—	2	2	no	—
93	#108	—	4	1	no	—
94	#107	—	178	1	no	—
95	#103	—	356	1	no	—
96	C₃₂	—	178	2	no	—
97	C₂×Q₈	—	356	1	no	—
98	C₂²×C₄	#13083559	1	1	yes	characteristic
99	Q₁₆	—	356	1	no	—
100	Q₁₆	—	356	1	no	—
101	M₁₆	—	4	2	no	—
102	M₁₆	—	8	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	8	1	no	—
106	C₂×C₈	#13083559	1	1	yes	characteristic
107	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
108	C₄⋊C₄	—	356	1	no	—
109	C₄⋊C₄	—	712	1	no	—
110	C₂²⋊C₄	—	712	1	no	—
111	C₁₆	—	2	1	no	—
112	C₁₆	#13083574	1	2	yes	—
113	C₂³	#15667955	1	1	yes	characteristic
114	Q₈	—	712	1	no	—
115	Q₈	—	712	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	712	1	no	—
119	C₂×C₄	—	712	1	no	—
120	C₂×C₄	#15667955	1	1	yes	characteristic
121	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
122	C₈	—	2	1	no	—
123	C₈	—	8	2	no	—
124	C₈	#15667995	1	1	yes	characteristic
125	C₈	#15668043	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	4	1	no	—
128	C₂²	C₂×#15667955	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	1424	1	no	—
131	C₄	—	1424	1	no	—
132	C₄	#19151911	1	1	yes	characteristic
133	C₄	#19152034	1	1	yes	characteristic
134	C₂	—	2	1	no	—
135	C₂	—	4	1	no	—
136	C₂	#26022437	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022360

C₂

yes

maximal, characteristic

#26018459

C₂

yes

maximal, complemented

#26018444

C₂

yes

maximal, complemented

#26018392

C₂

yes

maximal, characteristic

C₈₉×#2161

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

C₂²

yes

characteristic

#19151839

—

#19151432

C₂²

yes

—

#19151426

C₂²

yes

characteristic

#19151403

—

C₂×M₃₂×C₈₉

C₂²

yes

characteristic

C₂×#108×C₈₉

—

#368×C₈₉

—

#368×C₈₉

C₂²

yes

—

C₂×Dic₃₅₆

—

#15667991

—

Dic₇₁₂

—

#15667946

—

#15667937

—

#15667918

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

#108×C₈₉

—

#108×C₈₉

—

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂²×C₄×C₈₉

D₈

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

#41

yes

—

C₂³×C₈₉

SD₃₂

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

SD₃₂

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

yes

C₈×C₈₉

yes

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×SD₃₂

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

#13718

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

D₈₉

yes

—

—

—

C₈₉

#13718

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×M₃₂

C₂×D₈₉

yes

characteristic

#467

—

178

—

C₂×#108

—

M₆₄

—

#368

—

#368

C₂×D₈₉

yes

—

#367

—

#358

—

178

—

C₂×Q₁₆

—

178

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

C₂²⋊Q₈

—

356

—

Q₃₂

—

178

—

M₃₂

—

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

C₂×C₁₆

C₂²×D₈₉

yes

—

—

—

178

—

#103

—

356

—

C₃₂

—

178

—

C₂×Q₈

—

356

—

C₂²×C₄

#13083559

yes

characteristic

Q₁₆

—

356

—

100

Q₁₆

—

356

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

#13083559

yes

characteristic

107

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

108

C₄⋊C₄

—

356

—

109

C₄⋊C₄

—

712

—

110

C₂²⋊C₄

—

712

—

111

C₁₆

—

112

C₁₆

#13083574

yes

—

113

C₂³

#15667955

yes

characteristic

114

Q₈

—

712

—

115

Q₈

—

712

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

712

—

119

C₂×C₄

—

712

—

120

C₂×C₄

#15667955

yes

characteristic

121

C₂×C₄

C₂×#13083559

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

#15667995

yes

characteristic

125

C₈

#15668043

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

C₂×#15667955

yes

characteristic

129

C₄

—

130

C₄

—

1424

—

131

C₄

—

1424

—

132

C₄

#19151911

yes

characteristic

133

C₄

#19152034

yes

characteristic

134

C₂

—

135

C₂

—

136

C₂

#26022437

yes

characteristic

	subgroup	G/H
center	C₂	#26022437
commutator, cosocle	C₂×C₁₆×C₈₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#2161	C₂
socle	C₂×C₈₉	#2233
prime radical	C₂³×C₈₉	SD₃₂
hypercenter	#2161	D₈₉
hypocenter	C₈₉	#13718
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂ ◁ #2161

subgroup

G/H

center

C₂

#26022437

commutator, cosocle

C₂×C₁₆×C₈₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₈₉×#2161

C₂

socle

C₂×C₈₉

#2233

prime radical

C₂³×C₈₉

SD₃₂

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂ ◁ #2161

N	H ≅ G/N	type
2. #26018459	C₂	semidirect
3. #26018444	C₂	semidirect
74. C₈₉	#13718	semidirect

H ≅ G/N

type

2. #26018459

C₂

semidirect

3. #26018444

C₂

semidirect

74. C₈₉

#13718

semidirect

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

subgroup	index
#26022360	2
#26018459	2
#26018444	2
#26018392	2
C₈₉×#2161	2
#13718	89

subgroup

index

C₈₉×#2161

Subgroups — #36475392

Conjugacy classes of subgroups

Complements

Maximal subgroups