Group #36475042 — subgroups

Subgroups — #36475042

Order 22784.

184 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 67 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	#26022561	C₂	1	1	yes	maximal, characteristic
2	#26022134	C₂	1	2	yes	maximal, complemented
3	#26018456	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018437	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152030	C₂²	1	2	yes	—
7	#19151716	—	2	2	no	—
8	Dic₁₄₂₄	C₄	1	1	yes	characteristic, complemented
9	#19151437	C₄	1	2	yes	complemented
10	D₂₈₄₈	C₄	1	1	yes	characteristic, complemented
11	#19151428	C₂²	1	2	yes	—
12	#19151406	—	2	2	no	—
13	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	#15668034	—	2	2	no	—
17	#15668018	—	4	2	no	—
18	Dic₇₁₂	C₂×C₄	1	2	yes	—
19	#15667957	—	2	2	no	—
20	D₁₄₂₄	C₂×C₄	1	2	yes	—
21	#15667948	—	2	2	no	—
22	#15667940	—	4	2	no	—
23	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
25	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
26	C₃₂×C₈₉	—	2	1	no	—
27	C₃₂×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₃₂×C₈₉	C₂×C₄	1	1	yes	characteristic
29	#13083574	—	4	2	no	—
30	Dic₃₅₆	—	2	2	no	—
31	#13083563	—	4	2	no	—
32	D₇₁₂	—	2	2	no	—
33	#13083558	—	4	2	no	—
34	#13083557	—	8	2	no	—
35	C₈×D₈₉	—	8	2	no	—
36	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
37	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₄²×C₈₉	D₈	1	1	yes	characteristic
39	C₁₆×C₈₉	—	2	1	no	—
40	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
41	C₁₆×C₈₉	#41	1	1	yes	characteristic
42	C₂×C₄×C₈₉	—	2	1	no	—
43	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
44	C₂²⋊D₈₉	—	8	2	no	—
45	D₃₅₆	—	4	2	no	—
46	C₄×D₈₉	—	8	2	no	—
47	Dic₁₇₈	—	4	2	no	—
48	C₈×C₈₉	D₁₆	1	2	yes	—
49	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
50	C₈×C₈₉	#135	1	1	yes	characteristic
51	#10623494	—	8	2	no	—
52	C₂²×C₈₉	#508	1	1	yes	characteristic
53	C₂×D₈₉	—	8	2	no	—
54	C₄×C₈₉	—	4	1	no	—
55	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
56	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
57	Dic₈₉	—	8	2	no	—
58	#13656	—	89	1	no	maximal, Sylow
59	C₂×C₈₉	—	2	1	no	—
60	C₂×C₈₉	C₄×D₁₆	1	1	yes	characteristic
61	D₈₉	—	16	2	no	—
62	#2262	—	89	1	no	—
63	#2178	—	89	2	no	—
64	#1425	—	89	1	no	—
65	#1417	—	89	2	no	—
66	#1397	D₈₉	1	1	yes	characteristic, complemented
67	C₈₉	#13656	1	1	yes	characteristic, complemented, Sylow
68	#508	—	89	2	no	—
69	#443	—	178	2	no	—
70	Q₆₄	—	89	1	no	—
71	SD₆₄	—	89	2	no	—
72	D₃₂	—	89	1	no	—
73	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
74	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
75	#368	—	89	2	no	—
76	#359	—	178	2	no	—
77	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
78	#135	—	178	2	no	—
79	#131	—	356	2	no	—
80	Q₃₂	—	89	2	no	—
81	SD₃₂	—	178	2	no	—
82	D₁₆	—	89	2	no	—
83	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
84	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
85	#108	—	178	2	no	—
86	#104	—	356	2	no	—
87	C₄×C₈	D₃₅₆	1	1	yes	characteristic
88	C₃₂	—	2	1	no	—
89	C₃₂	C₄×D₈₉	1	1	yes	characteristic
90	C₃₂	C₄×D₈₉	1	1	yes	characteristic
91	#41	—	356	2	no	—
92	Q₁₆	—	178	2	no	—
93	SD₁₆	—	356	2	no	—
94	D₈	—	178	2	no	—
95	M₁₆	—	356	2	no	—
96	M₁₆	—	712	2	no	—
97	C₂×C₈	—	712	2	no	—
98	C₂×C₈	D₇₁₂	1	1	yes	characteristic
99	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
100	C₄²	D₇₁₂	1	1	yes	characteristic
101	C₁₆	—	2	1	no	—
102	C₁₆	C₂×C₄×D₈₉	1	1	yes	characteristic
103	C₁₆	#13083574	1	1	yes	characteristic
104	Q₈	—	356	2	no	—
105	D₄	—	356	2	no	—
106	D₄	—	712	2	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	712	2	no	—
109	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
110	C₈	—	712	2	no	—
111	C₈	D₁₄₂₄	1	2	yes	—
112	C₈	C₄×D₃₅₆	1	1	yes	characteristic
113	C₈	#15668034	1	1	yes	characteristic
114	C₂²	—	712	2	no	—
115	C₂²	#19152030	1	1	yes	characteristic
116	C₄	—	4	1	no	—
117	C₄	—	712	2	no	—
118	C₄	C₄×D₇₁₂	1	1	yes	characteristic
119	C₄	C₂×D₁₄₂₄	1	1	yes	characteristic
120	C₂	—	2	1	no	—
121	C₂	—	1424	2	no	—
122	C₂	C₄×D₁₄₂₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022561

C₂

yes

maximal, characteristic

#26022134

C₂

yes

maximal, complemented

#26018456

C₂

yes

maximal, characteristic, complemented

#26018437

C₂

yes

maximal, complemented

C₈₉×#1397

C₂

yes

maximal, characteristic, complemented

#19152030

C₂²

yes

—

#19151716

—

Dic₁₄₂₄

C₄

yes

characteristic, complemented

#19151437

C₄

yes

complemented

D₂₈₄₈

C₄

yes

characteristic, complemented

#19151428

C₂²

yes

—

#19151406

—

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

#15668034

—

#15668018

—

Dic₇₁₂

C₂×C₄

yes

—

#15667957

—

D₁₄₂₄

C₂×C₄

yes

—

#15667948

—

#15667940

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₃₂×C₈₉

—

C₃₂×C₈₉

C₂×C₄

yes

characteristic

C₃₂×C₈₉

C₂×C₄

yes

characteristic

#13083574

—

Dic₃₅₆

—

#13083563

—

D₇₁₂

—

#13083558

—

#13083557

—

C₈×D₈₉

—

C₂×C₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

D₈

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂²×C₄

yes

characteristic

C₁₆×C₈₉

#41

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂²⋊D₈₉

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

D₁₆

yes

—

C₈×C₈₉

C₄×D₄

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

#508

yes

characteristic

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄×D₈

yes

characteristic

C₄×C₈₉

C₂×D₁₆

yes

characteristic

Dic₈₉

—

#13656

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₄×D₁₆

yes

characteristic

D₈₉

—

—

—

—

—

D₈₉

yes

characteristic, complemented

C₈₉

#13656

yes

characteristic, complemented, Sylow

#508

—

#443

—

178

—

Q₆₄

—

SD₆₄

—

D₃₂

—

C₂×C₃₂

C₂×D₈₉

yes

characteristic

C₂×C₃₂

C₂×D₈₉

yes

characteristic

#368

—

#359

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

#135

—

178

—

#131

—

356

—

Q₃₂

—

SD₃₂

—

178

—

D₁₆

—

C₂×C₁₆

D₃₅₆

yes

characteristic

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#108

—

178

—

#104

—

356

—

C₄×C₈

D₃₅₆

yes

characteristic

C₃₂

—

C₃₂

C₄×D₈₉

yes

characteristic

C₃₂

C₄×D₈₉

yes

characteristic

#41

—

356

—

Q₁₆

—

178

—

SD₁₆

—

356

—

D₈

—

178

—

M₁₆

—

356

—

M₁₆

—

712

—

C₂×C₈

—

712

—

C₂×C₈

D₇₁₂

yes

characteristic

C₂×C₈

C₂×D₃₅₆

yes

characteristic

100

C₄²

D₇₁₂

yes

characteristic

101

C₁₆

—

102

C₁₆

C₂×C₄×D₈₉

yes

characteristic

103

C₁₆

#13083574

yes

characteristic

104

Q₈

—

356

—

105

D₄

—

356

—

106

D₄

—

712

—

107

C₂×C₄

—

108

C₂×C₄

—

712

—

109

C₂×C₄

C₂×D₇₁₂

yes

characteristic

110

C₈

—

712

—

111

C₈

D₁₄₂₄

yes

—

112

C₈

C₄×D₃₅₆

yes

characteristic

113

C₈

#15668034

yes

characteristic

114

C₂²

—

712

—

115

C₂²

#19152030

yes

characteristic

116

C₄

—

117

C₄

—

712

—

118

C₄

C₄×D₇₁₂

yes

characteristic

119

C₄

C₂×D₁₄₂₄

yes

characteristic

120

C₂

—

121

C₂

—

1424

—

122

C₂

C₄×D₁₄₂₄

yes

characteristic

	subgroup	G/H
center	C₄	C₂×D₁₄₂₄
commutator	C₁₆×C₈₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#1397	C₂
socle	C₂×C₈₉	C₄×D₁₆
prime radical	#26022561	C₂
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	#1397	D₈₉
hypocenter	C₈₉	#13656
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

subgroup

G/H

center

C₄

C₂×D₁₄₂₄

commutator

C₁₆×C₈₉

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₈₉×#1397

C₂

socle

C₂×C₈₉

C₄×D₁₆

prime radical

#26022561

C₂

cosocle

C₂×C₁₆×C₈₉

C₂³

D₈₉

C₈₉

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

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

N	H ≅ G/N	type
2. #26022134	C₂	semidirect
3. #26018456	C₂	semidirect
4. #26018437	C₂	semidirect
5. C₈₉×#1397	C₂	semidirect
8. Dic₁₄₂₄	C₄	semidirect
9. #19151437	C₄	semidirect
10. D₂₈₄₈	C₄	semidirect
66. #1397	D₈₉	semidirect
67. C₈₉	#13656	semidirect

H ≅ G/N

type

2. #26022134

C₂

semidirect

3. #26018456

C₂

semidirect

4. #26018437

C₂

semidirect

5. C₈₉×#1397

C₂

semidirect

8. Dic₁₄₂₄

C₄

semidirect

9. #19151437

C₄

semidirect

10. D₂₈₄₈

C₄

semidirect

66. #1397

D₈₉

semidirect

67. C₈₉

#13656

semidirect

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

subgroup	index
#26022561	2
#26022134	2
#26018456	2
#26018437	2
C₈₉×#1397	2
#13656	89

subgroup

index

C₈₉×#1397

Subgroups — #36475042

Conjugacy classes of subgroups

Complements

Maximal subgroups