Group #36475278 — subgroups

Subgroups — #36475278

Order 22784.

190 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 57 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×#19151431	C₂	1	1	yes	maximal, characteristic
2	#26022340	C₂	1	2	yes	maximal
3	#26018454	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018433	C₂	1	2	yes	maximal, complemented
5	C₂×Q₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
7	#19151836	—	2	2	no	—
8	#19151431	—	2	2	no	—
9	C₂×#15667918	C₂²	1	1	yes	characteristic
10	#19151424	C₂²	1	2	yes	—
11	#19151402	—	2	2	no	—
12	C₂×Q₃₂×C₈₉	C₂²	1	2	yes	—
13	Q₆₄×C₈₉	—	2	2	no	—
14	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
16	#15667988	—	4	2	no	—
17	C₂×#13083549	D₄	1	1	yes	characteristic
18	C₁₆×D₈₉	—	2	1	no	—
19	C₁₆×D₈₉	—	2	1	no	—
20	#15667946	—	2	2	no	—
21	#15667936	—	4	2	no	—
22	#15667918	—	2	1	no	—
23	C₂×Q₁₆×C₈₉	—	2	2	no	—
24	Q₃₂×C₈₉	—	2	2	no	—
25	Q₃₂×C₈₉	—	2	2	no	—
26	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
27	C₃₂×C₈₉	D₄	1	2	yes	—
28	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
29	C₂×#10623494	D₈	1	1	yes	characteristic
30	C₈×D₈₉	—	2	1	no	—
31	C₈×D₈₉	—	2	1	no	—
32	C₄⋊Dic₈₉	—	4	2	no	—
33	#13083553	—	8	2	no	—
34	#13083551	—	8	2	no	—
35	#13083549	—	2	1	no	—
36	C₂×Q₈×C₈₉	—	4	2	no	—
37	Q₁₆×C₈₉	—	4	2	no	—
38	Q₁₆×C₈₉	—	4	2	no	—
39	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₈₉	#41	1	1	yes	characteristic
42	C₂²×D₈₉	D₁₆	1	1	yes	characteristic
43	Q₈×C₈₉	—	8	2	no	—
44	Q₈×C₈₉	—	8	2	no	—
45	C₂×C₄×C₈₉	—	8	2	no	—
46	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
47	C₂×Dic₈₉	—	8	2	no	—
48	C₂×Dic₈₉	D₁₆	1	1	yes	characteristic
49	C₄×D₈₉	—	2	1	no	—
50	C₄×D₈₉	—	2	1	no	—
51	C₈×C₈₉	#121	1	1	yes	characteristic
52	C₈×C₈₉	#135	1	1	yes	characteristic
53	#10623494	—	2	1	no	—
54	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
55	C₂×D₈₉	—	2	1	no	—
56	C₂×D₈₉	—	2	2	no	—
57	C₄×C₈₉	—	16	2	no	—
58	C₄×C₈₉	#466	1	1	yes	characteristic
59	C₄×C₈₉	#508	1	1	yes	characteristic
60	Dic₈₉	—	2	1	no	—
61	Dic₈₉	—	16	2	no	—
62	#13698	—	89	1	no	maximal, Sylow
63	C₂×C₈₉	#2215	1	1	yes	characteristic
64	C₂×C₈₉	#2264	1	2	yes	—
65	D₈₉	—	4	1	no	—
66	C₂×Q₆₄	D₈₉	1	1	yes	characteristic, complemented
67	C₂×M₆₄	—	89	1	no	—
68	#2216	—	89	2	no	—
69	#1424	—	89	1	no	—
70	#1416	—	89	2	no	—
71	C₈₉	#13698	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₃₂	C₂×D₈₉	1	2	yes	—
73	C₂²×C₁₆	—	89	1	no	—
74	#467	—	178	2	no	—
75	Q₆₄	—	2	2	no	—
76	M₆₄	—	178	2	no	—
77	C₂×C₃₂	—	89	1	no	—
78	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
79	#366	—	89	2	no	—
80	#358	—	178	2	no	—
81	C₂×Q₁₆	—	2	2	no	—
82	C₂²×C₈	—	89	1	no	—
83	C₂²⋊Q₈	—	356	2	no	—
84	Q₃₂	—	2	2	no	—
85	Q₃₂	—	2	2	no	—
86	C₂×C₁₆	—	89	1	no	—
87	C₂×C₁₆	—	178	1	no	—
88	C₂×C₁₆	—	178	1	no	—
89	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
90	#107	—	178	2	no	—
91	#103	—	356	2	no	—
92	C₃₂	—	178	1	no	—
93	C₃₂	C₂²⋊D₈₉	1	2	yes	—
94	C₂×Q₈	—	4	2	no	—
95	C₂²×C₄	—	89	1	no	—
96	Q₁₆	—	4	2	no	—
97	Q₁₆	—	4	2	no	—
98	C₂×C₈	—	89	1	no	—
99	C₂×C₈	—	178	1	no	—
100	C₂×C₈	—	178	1	no	—
101	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
102	C₄⋊C₄	—	356	2	no	—
103	C₄⋊C₄	—	712	2	no	—
104	C₂²⋊C₄	—	712	2	no	—
105	C₁₆	—	178	1	no	—
106	C₁₆	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
107	C₁₆	#13083573	1	1	yes	characteristic
108	C₂³	—	89	1	no	—
109	Q₈	—	8	2	no	—
110	Q₈	—	8	2	no	—
111	C₂×C₄	—	8	2	no	—
112	C₂×C₄	—	89	1	no	—
113	C₂×C₄	—	178	1	no	—
114	C₂×C₄	—	178	1	no	—
115	C₂×C₄	—	712	2	no	—
116	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
117	C₈	—	178	1	no	—
118	C₈	#15667983	1	1	yes	characteristic
119	C₈	#15668031	1	1	yes	characteristic
120	C₂²	—	178	2	no	—
121	C₂²	—	178	1	no	—
122	C₂²	D₁₆×D₈₉	1	1	yes	characteristic
123	C₄	—	16	2	no	—
124	C₄	—	1424	2	no	—
125	C₄	—	178	1	no	—
126	C₄	#19151831	1	1	yes	characteristic
127	C₄	#19152027	1	1	yes	characteristic
128	C₂	—	356	1	no	—
129	C₂	#26022335	1	1	yes	characteristic
130	C₂	#26022571	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151431

C₂

yes

maximal, characteristic

#26022340

C₂

yes

maximal

#26018454

C₂

yes

maximal, characteristic, complemented

#26018433

C₂

yes

maximal, complemented

C₂×Q₆₄×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₈₉

C₂²

yes

—

—

C₂²

yes

characteristic

#19151424

C₂²

yes

—

#19151402

—

C₂×Q₃₂×C₈₉

C₂²

yes

—

Q₆₄×C₈₉

—

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

C₂×C₈×D₈₉

D₄

yes

characteristic

#15667988

—

C₂×#13083549

D₄

yes

characteristic

C₁₆×D₈₉

—

C₁₆×D₈₉

—

#15667946

—

#15667936

—

#15667918

—

C₂×Q₁₆×C₈₉

—

Q₃₂×C₈₉

—

Q₃₂×C₈₉

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₃₂×C₈₉

D₄

yes

—

C₂×C₄×D₈₉

D₈

yes

characteristic

C₂×#10623494

D₈

yes

characteristic

C₈×D₈₉

—

C₈×D₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083551

—

#13083549

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

#41

yes

characteristic

C₂²×D₈₉

D₁₆

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

D₁₆

yes

characteristic

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

C₂×Q₆₄

D₈₉

yes

characteristic, complemented

C₂×M₆₄

—

#2216

—

#1424

—

#1416

—

C₈₉

#13698

yes

characteristic, complemented, Sylow

C₂×Q₃₂

C₂×D₈₉

yes

—

C₂²×C₁₆

—

#467

—

178

—

Q₆₄

—

M₆₄

—

178

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₈₉

yes

characteristic

#366

—

#358

—

178

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²⋊Q₈

—

356

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

178

—

C₂×C₁₆

—

178

—

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#107

—

178

—

#103

—

356

—

C₃₂

—

178

—

C₃₂

C₂²⋊D₈₉

yes

—

C₂×Q₈

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

178

—

100

C₂×C₈

—

178

—

101

C₂×C₈

D₄×D₈₉

yes

characteristic

102

C₄⋊C₄

—

356

—

103

C₄⋊C₄

—

712

—

104

C₂²⋊C₄

—

712

—

105

C₁₆

—

178

—

106

C₁₆

C₂×C₂²⋊D₈₉

yes

characteristic

107

C₁₆

#13083573

yes

characteristic

108

C₂³

—

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

178

—

114

C₂×C₄

—

178

—

115

C₂×C₄

—

712

—

116

C₂×C₄

D₈×D₈₉

yes

characteristic

117

C₈

—

178

—

118

C₈

#15667983

yes

characteristic

119

C₈

#15668031

yes

characteristic

120

C₂²

—

178

—

121

C₂²

—

178

—

122

C₂²

D₁₆×D₈₉

yes

characteristic

123

C₄

—

124

C₄

—

1424

—

125

C₄

—

178

—

126

C₄

#19151831

yes

characteristic

127

C₄

#19152027

yes

characteristic

128

C₂

—

356

—

129

C₂

#26022335

yes

characteristic

130

C₂

#26022571

yes

—

	subgroup	G/H
center	C₂²	D₁₆×D₈₉
commutator, cosocle	C₂×C₁₆×C₈₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₂×Q₆₄×C₈₉	C₂
socle	C₂²×C₈₉	C₂×D₁₆
prime radical	C₂²×D₈₉	D₁₆
hypercenter	C₂×Q₆₄	D₈₉
hypocenter	C₈₉	#13698
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

subgroup

G/H

center

C₂²

D₁₆×D₈₉

commutator, cosocle

C₂×C₁₆×C₈₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₂×Q₆₄×C₈₉

C₂

socle

C₂²×C₈₉

C₂×D₁₆

prime radical

C₂²×D₈₉

D₁₆

hypercenter

C₂×Q₆₄

D₈₉

hypocenter

C₈₉

#13698

LCS

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

N	H ≅ G/N	type
3. #26018454	C₂	semidirect
4. #26018433	C₂	semidirect
5. C₂×Q₆₄×C₈₉	C₂	semidirect
66. C₂×Q₆₄	D₈₉	semidirect
71. C₈₉	#13698	semidirect

H ≅ G/N

type

3. #26018454

C₂

semidirect

4. #26018433

C₂

semidirect

5. C₂×Q₆₄×C₈₉

C₂

semidirect

66. C₂×Q₆₄

D₈₉

semidirect

71. C₈₉

#13698

semidirect

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

subgroup	index
C₂×#19151431	2
#26022340	2
#26018454	2
#26018433	2
C₂×Q₆₄×C₈₉	2
#13698	89

subgroup

index

C₂×Q₆₄×C₈₉

Subgroups — #36475278

Conjugacy classes of subgroups

Complements

Maximal subgroups