Group #36475415 — subgroups

Subgroups — #36475415

Order 22784.

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

normal subgroups only

This group has 194 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	#26022531	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022502	C₂	1	1	yes	maximal, characteristic
3	#26018431	C₂	1	4	yes	maximal, complemented
4	C₈₉×#1397	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₄₂₄	—	2	1	no	—
6	C₂×D₁₄₂₄	C₂²	1	2	yes	—
7	#19152000	—	2	1	no	—
8	#19151975	—	2	1	no	—
9	#19151426	—	2	1	no	—
10	#19151424	C₂²	1	2	yes	—
11	C₂×C₃₂×C₈₉	C₂²	1	2	yes	—
12	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×D₇₁₂	—	2	2	no	—
14	C₂×D₇₁₂	—	4	1	no	—
15	#15668009	—	4	1	no	—
16	#15668004	—	4	1	no	—
17	D₁₄₂₄	—	2	4	no	—
18	D₁₄₂₄	—	4	2	no	—
19	#15667946	—	2	2	no	—
20	#15667944	—	4	1	no	—
21	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₃₂×C₈₉	—	2	2	no	—
25	C₂×D₃₅₆	—	4	2	no	—
26	C₂×D₃₅₆	—	8	2	no	—
27	D₇₁₂	—	4	2	no	—
28	D₇₁₂	—	4	2	no	—
29	D₇₁₂	—	8	2	no	—
30	C₄⋊Dic₈₉	—	4	2	no	—
31	#13083553	—	8	2	no	—
32	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
33	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
34	C₄²×C₈₉	D₈	1	1	yes	characteristic
35	C₁₆×C₈₉	SD₁₆	1	2	yes	—
36	C₁₆×C₈₉	#41	1	2	yes	—
37	C₂²×D₈₉	—	8	2	no	—
38	C₂×C₄×C₈₉	D₁₆	1	2	yes	—
39	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
40	C₂×Dic₈₉	—	8	2	no	—
41	D₃₅₆	—	8	2	no	—
42	D₃₅₆	—	8	2	no	—
43	D₃₅₆	—	16	4	no	—
44	C₈×C₈₉	SD₃₂	1	2	yes	—
45	C₈×C₈₉	#124	1	1	yes	characteristic
46	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
47	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
48	C₂×D₈₉	—	16	2	no	—
49	C₂×D₈₉	—	16	4	no	—
50	C₄×C₈₉	—	2	2	no	—
51	C₄×C₈₉	#486	1	1	yes	characteristic
52	C₄×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
53	Dic₈₉	—	16	2	no	—
54	#13723	—	89	1	no	maximal, Sylow
55	C₂×C₈₉	#2240	1	1	yes	characteristic
56	C₂×C₈₉	#2263	1	2	yes	—
57	D₈₉	—	32	2	no	—
58	#2253	—	89	1	no	—
59	#2246	—	89	1	no	—
60	#1415	—	89	4	no	—
61	#1397	D₈₉	1	1	yes	characteristic, complemented
62	C₈₉	#13723	1	1	yes	characteristic, complemented, Sylow
63	C₂×D₁₆	—	89	2	no	—
64	C₂×D₁₆	—	178	1	no	—
65	#499	—	178	1	no	—
66	#493	—	178	1	no	—
67	C₂×C₃₂	C₂×D₈₉	1	2	yes	—
68	#367	—	178	1	no	—
69	#366	—	89	2	no	—
70	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
71	C₂×D₈	—	356	1	no	—
72	C₂×D₈	—	178	2	no	—
73	#127	—	356	1	no	—
74	#125	—	356	1	no	—
75	D₁₆	—	356	2	no	—
76	D₁₆	—	178	4	no	—
77	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
78	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
79	#107	—	178	2	no	—
80	#106	—	356	1	no	—
81	C₄×C₈	D₃₅₆	1	1	yes	characteristic
82	C₃₂	—	2	2	no	—
83	C₂×D₄	—	356	2	no	—
84	C₂×D₄	—	712	2	no	—
85	D₈	—	356	2	no	—
86	D₈	—	356	2	no	—
87	D₈	—	712	2	no	—
88	C₂×C₈	D₇₁₂	1	1	yes	characteristic
89	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
90	C₄⋊C₄	—	356	2	no	—
91	C₄⋊C₄	—	712	2	no	—
92	C₄²	D₇₁₂	1	1	yes	characteristic
93	C₁₆	#13083563	1	2	yes	—
94	C₁₆	#13083574	1	2	yes	—
95	C₂³	—	712	2	no	—
96	D₄	—	1424	4	no	—
97	D₄	—	712	2	no	—
98	D₄	—	712	2	no	—
99	C₂×C₄	—	712	2	no	—
100	C₂×C₄	D₁₄₂₄	1	2	yes	—
101	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
102	C₈	#15667957	1	2	yes	—
103	C₈	#15668000	1	1	yes	characteristic
104	C₈	C₂×#13083563	1	1	yes	characteristic
105	C₂²	—	1424	2	no	—
106	C₂²	—	1424	4	no	—
107	C₂²	C₂×D₁₄₂₄	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	1424	2	no	—
110	C₄	#19151945	1	1	yes	characteristic
111	C₄	C₂×#15667957	1	1	yes	characteristic
112	C₂	—	2848	2	no	—
113	C₂	#26022476	1	1	yes	characteristic
114	C₂	#26022567	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022531

C₂

yes

maximal, characteristic, complemented

#26022502

C₂

yes

maximal, characteristic

#26018431

C₂

yes

maximal, complemented

C₈₉×#1397

C₂

yes

maximal, characteristic, complemented

C₂×D₁₄₂₄

—

C₂×D₁₄₂₄

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×C₃₂×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×D₇₁₂

—

C₂×D₇₁₂

—

#15668009

—

#15668004

—

D₁₄₂₄

—

D₁₄₂₄

—

#15667946

—

#15667944

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₃₂×C₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

C₄⋊Dic₈₉

—

#13083553

—

C₂×C₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

D₈

yes

characteristic

C₁₆×C₈₉

SD₁₆

yes

—

C₁₆×C₈₉

#41

yes

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

D₁₆

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

SD₃₂

yes

—

C₈×C₈₉

#124

yes

characteristic

C₈×C₈₉

C₂×SD₁₆

yes

characteristic

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#486

yes

characteristic

C₄×C₈₉

C₂×SD₃₂

yes

characteristic

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

—

—

—

D₈₉

yes

characteristic, complemented

C₈₉

#13723

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₂×D₁₆

—

178

—

#499

—

178

—

#493

—

178

—

C₂×C₃₂

C₂×D₈₉

yes

—

#367

—

178

—

#366

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₂×D₈

—

356

—

C₂×D₈

—

178

—

#127

—

356

—

#125

—

356

—

D₁₆

—

356

—

D₁₆

—

178

—

C₂×C₁₆

D₃₅₆

yes

characteristic

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#107

—

178

—

#106

—

356

—

C₄×C₈

D₃₅₆

yes

characteristic

C₃₂

—

C₂×D₄

—

356

—

C₂×D₄

—

712

—

D₈

—

356

—

D₈

—

356

—

D₈

—

712

—

C₂×C₈

D₇₁₂

yes

characteristic

C₂×C₈

C₂×D₃₅₆

yes

characteristic

C₄⋊C₄

—

356

—

C₄⋊C₄

—

712

—

C₄²

D₇₁₂

yes

characteristic

C₁₆

#13083563

yes

—

C₁₆

#13083574

yes

—

C₂³

—

712

—

D₄

—

1424

—

D₄

—

712

—

D₄

—

712

—

C₂×C₄

—

712

—

100

C₂×C₄

D₁₄₂₄

yes

—

101

C₂×C₄

C₂×D₇₁₂

yes

characteristic

102

C₈

#15667957

yes

—

103

C₈

#15668000

yes

characteristic

104

C₈

C₂×#13083563

yes

characteristic

105

C₂²

—

1424

—

106

C₂²

—

1424

—

107

C₂²

C₂×D₁₄₂₄

yes

characteristic

108

C₄

—

109

C₄

—

1424

—

110

C₄

#19151945

yes

characteristic

111

C₄

C₂×#15667957

yes

characteristic

112

C₂

—

2848

—

113

C₂

#26022476

yes

characteristic

114

C₂

#26022567

yes

—

	subgroup	G/H
center	C₂²	C₂×D₁₄₂₄
commutator, cosocle	C₂×C₁₆×C₈₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#1397	C₂
socle	C₂²×C₈₉	C₂×D₁₆
prime radical	#26022502	C₂
hypercenter	#1397	D₈₉
hypocenter	C₈₉	#13723
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

subgroup

G/H

center

C₂²

C₂×D₁₄₂₄

commutator, cosocle

C₂×C₁₆×C₈₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₈₉×#1397

C₂

socle

C₂²×C₈₉

C₂×D₁₆

C₂

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

N	H ≅ G/N	type
1. #26022531	C₂	semidirect
3. #26018431	C₂	semidirect
4. C₈₉×#1397	C₂	semidirect
61. #1397	D₈₉	semidirect
62. C₈₉	#13723	semidirect

H ≅ G/N

type

1. #26022531

C₂

semidirect

3. #26018431

C₂

semidirect

4. C₈₉×#1397

C₂

semidirect

61. #1397

D₈₉

semidirect

62. C₈₉

#13723

semidirect

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

subgroup	index
#26022531	2
#26022502	2
#26018431	2
C₈₉×#1397	2
#13723	89

subgroup

index

C₈₉×#1397

Subgroups — #36475415

Conjugacy classes of subgroups

Complements

Maximal subgroups