Group #36475402 — subgroups

Subgroups — #36475402

Order 22784.

182 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 75 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	#26022505	C₂	1	1	yes	maximal, characteristic
2	#26018432	C₂	1	4	yes	maximal
3	C₄×#15667918	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2252	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₇₁₂	—	2	1	no	—
6	C₂×Dic₇₁₂	C₂²	1	2	yes	—
7	#19151978	—	2	1	no	—
8	C₂×#15667918	C₂²	1	2	yes	—
9	#500×C₈₉	—	2	1	no	—
10	#366×C₈₉	—	2	1	no	—
11	#366×C₈₉	C₂²	1	2	yes	—
12	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₅₆	—	2	2	no	—
14	C₂×Dic₃₅₆	—	4	1	no	—
15	#15668012	—	4	1	no	—
16	Dic₇₁₂	—	2	4	no	—
17	Dic₇₁₂	—	4	2	no	—
18	#15667918	—	2	2	no	—
19	C₄⋊Q₈×C₈₉	—	4	1	no	—
20	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
22	#107×C₈₉	—	2	2	no	—
23	#107×C₈₉	—	4	1	no	—
24	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
25	C₂×Dic₁₇₈	—	4	2	no	—
26	Dic₃₅₆	—	4	2	no	—
27	Dic₃₅₆	—	4	2	no	—
28	Dic₃₅₆	—	8	2	no	—
29	C₄⋊Dic₈₉	—	8	1	no	—
30	C₄⋊Dic₈₉	—	8	1	no	—
31	C₂×Q₈×C₈₉	—	8	1	no	—
32	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
33	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
34	C₄⋊C₄×C₈₉	—	4	2	no	—
35	C₄⋊C₄×C₈₉	—	8	1	no	—
36	C₄²×C₈₉	D₈	1	1	yes	characteristic
37	C₁₆×C₈₉	SD₁₆	1	2	yes	—
38	C₁₆×C₈₉	#41	1	2	yes	—
39	Q₈×C₈₉	—	16	2	no	—
40	C₂×C₄×C₈₉	—	8	2	no	—
41	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
42	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
44	C₂×Dic₈₉	—	8	2	no	—
45	Dic₁₇₈	—	8	2	no	—
46	Dic₁₇₈	—	8	2	no	—
47	C₈×C₈₉	SD₃₂	1	2	yes	—
48	C₈×C₈₉	#124	1	1	yes	characteristic
49	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
50	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
51	C₄×C₈₉	—	16	2	no	—
52	C₄×C₈₉	SD₆₄	1	2	yes	—
53	C₄×C₈₉	Q₆₄	1	2	yes	—
54	C₄×C₈₉	#486	1	1	yes	characteristic
55	C₄×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
56	Dic₈₉	—	16	2	no	—
57	#13720	—	89	1	no	maximal, Sylow
58	C₂×C₈₉	#2240	1	1	yes	characteristic
59	C₂×C₈₉	C₂×SD₆₄	1	1	yes	characteristic
60	C₂×C₈₉	C₂×Q₆₄	1	1	yes	characteristic
61	#2252	D₈₉	1	1	yes	characteristic
62	#2247	—	89	1	no	—
63	#1416	—	89	4	no	—
64	C₄×C₃₂	—	89	1	no	—
65	C₈₉	#13720	1	1	yes	characteristic, complemented, Sylow
66	C₂×Q₃₂	—	89	2	no	—
67	C₂×Q₃₂	—	178	1	no	—
68	#500	—	2	1	no	—
69	#494	—	178	1	no	—
70	C₂×C₃₂	—	89	2	no	—
71	#366	—	2	1	no	—
72	#366	C₂×D₈₉	1	2	yes	—
73	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
74	C₂×Q₁₆	—	356	1	no	—
75	C₂×Q₁₆	—	178	2	no	—
76	C₄⋊Q₈	—	4	1	no	—
77	C₄⋊Q₈	—	356	1	no	—
78	Q₃₂	—	356	2	no	—
79	Q₃₂	—	178	4	no	—
80	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
81	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
82	#107	—	2	2	no	—
83	#107	—	4	1	no	—
84	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
85	C₃₂	—	178	2	no	—
86	C₂×Q₈	—	8	1	no	—
87	C₂×Q₈	—	356	2	no	—
88	Q₁₆	—	356	2	no	—
89	Q₁₆	—	356	2	no	—
90	Q₁₆	—	712	2	no	—
91	C₂×C₈	#13083559	1	1	yes	characteristic
92	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	8	1	no	—
95	C₄⋊C₄	—	712	1	no	—
96	C₄⋊C₄	—	712	1	no	—
97	C₄²	#13083559	1	1	yes	characteristic
98	C₁₆	#13083561	1	2	yes	—
99	C₁₆	#13083575	1	2	yes	—
100	Q₈	—	16	2	no	—
101	Q₈	—	712	2	no	—
102	Q₈	—	712	2	no	—
103	C₂×C₄	—	8	2	no	—
104	C₂×C₄	—	712	2	no	—
105	C₂×C₄	#15667953	1	1	yes	characteristic
106	C₂×C₄	#15667953	1	1	yes	characteristic
107	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
108	C₈	#15667955	1	2	yes	—
109	C₈	#15668001	1	1	yes	characteristic
110	C₈	C₂×#13083561	1	1	yes	characteristic
111	C₂²	C₂×#15667953	1	1	yes	characteristic
112	C₄	—	16	2	no	—
113	C₄	—	1424	2	no	—
114	C₄	#19151436	1	2	yes	—
115	C₄	#19151438	1	2	yes	—
116	C₄	#19151946	1	1	yes	characteristic
117	C₄	C₂×#15667955	1	1	yes	characteristic
118	C₂	#26022477	1	1	yes	characteristic
119	C₂	C₂×#19151436	1	1	yes	characteristic
120	C₂	C₂×#19151438	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022505

C₂

yes

maximal, characteristic

#26018432

C₂

yes

maximal

C₄×#15667918

C₂

yes

maximal, characteristic

C₈₉×#2252

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

—

C₂×Dic₇₁₂

C₂²

yes

—

#19151978

—

C₂×#15667918

C₂²

yes

—

#500×C₈₉

—

#366×C₈₉

—

#366×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

#15668012

—

Dic₇₁₂

—

Dic₇₁₂

—

#15667918

—

C₄⋊Q₈×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

#107×C₈₉

—

#107×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

D₈

yes

characteristic

C₁₆×C₈₉

SD₁₆

yes

—

C₁₆×C₈₉

#41

yes

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

D₁₆

yes

characteristic

C₂×C₄×C₈₉

D₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

SD₃₂

yes

—

C₈×C₈₉

#124

yes

characteristic

C₈×C₈₉

C₂×SD₁₆

yes

characteristic

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

SD₆₄

yes

—

C₄×C₈₉

Q₆₄

yes

—

C₄×C₈₉

#486

yes

characteristic

C₄×C₈₉

C₂×SD₃₂

yes

characteristic

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

C₂×SD₆₄

yes

characteristic

C₂×C₈₉

C₂×Q₆₄

yes

characteristic

#2252

D₈₉

yes

characteristic

#2247

—

#1416

—

C₄×C₃₂

—

C₈₉

#13720

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

178

—

#500

—

#494

—

178

—

C₂×C₃₂

—

#366

—

#366

C₂×D₈₉

yes

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

356

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

C₄⋊Q₈

—

356

—

Q₃₂

—

356

—

Q₃₂

—

178

—

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#107

—

#107

—

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

C₃₂

—

178

—

C₂×Q₈

—

C₂×Q₈

—

356

—

Q₁₆

—

356

—

Q₁₆

—

356

—

Q₁₆

—

712

—

C₂×C₈

#13083559

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

712

—

C₄⋊C₄

—

712

—

C₄²

#13083559

yes

characteristic

C₁₆

#13083561

yes

—

C₁₆

#13083575

yes

—

100

Q₈

—

101

Q₈

—

712

—

102

Q₈

—

712

—

103

C₂×C₄

—

104

C₂×C₄

—

712

—

105

C₂×C₄

#15667953

yes

characteristic

106

C₂×C₄

#15667953

yes

characteristic

107

C₂×C₄

C₂×#13083559

yes

characteristic

108

C₈

#15667955

yes

—

109

C₈

#15668001

yes

characteristic

110

C₈

C₂×#13083561

yes

characteristic

111

C₂²

C₂×#15667953

yes

characteristic

112

C₄

—

113

C₄

—

1424

—

114

C₄

#19151436

yes

—

115

C₄

#19151438

yes

—

116

C₄

#19151946

yes

characteristic

117

C₄

C₂×#15667955

yes

characteristic

118

C₂

#26022477

yes

characteristic

119

C₂

C₂×#19151436

yes

characteristic

120

C₂

C₂×#19151438

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#15667953
commutator, cosocle	C₂×C₁₆×C₈₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#2252	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₁₆
hypercenter	#2252	D₈₉
hypocenter	C₈₉	#13720
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

subgroup

G/H

center

C₂²

C₂×#15667953

commutator, cosocle

C₂×C₁₆×C₈₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₈₉×#2252

C₂

socle, prime radical

C₂²×C₈₉

C₂×D₁₆

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

N	H ≅ G/N	type
65. C₈₉	#13720	semidirect

H ≅ G/N

type

65. C₈₉

#13720

semidirect

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

subgroup	index
#26022505	2
#26018432	2
C₄×#15667918	2
C₈₉×#2252	2
#13720	89

subgroup

index

C₈₉×#2252

Subgroups — #36475402

Conjugacy classes of subgroups

Complements

Maximal subgroups