Group #36441529 — subgroups

Subgroups — #36441529

Order 22784.

190 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 87 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	#26019015	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151655	—	2	2	no	—
4	#19151351	—	2	2	no	—
5	#19151349	C₄	1	4	yes	—
6	C₂²×C₁₆×C₈₉	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆×C₈₉	—	2	2	no	—
9	C₄⋊#10623494	—	2	4	no	—
10	C₂²⋊#10623494	—	4	2	no	—
11	#15667923	—	4	2	no	—
12	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₈₉	—	2	2	no	—
16	C₂×C₁₆×C₈₉	—	4	2	no	—
17	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₈₉	D₄	1	2	yes	—
19	C₂×#10623494	—	4	4	no	—
20	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
22	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
23	C₂×C₈×C₈₉	—	2	1	no	—
24	C₂×C₈×C₈₉	—	2	1	no	—
25	C₂×C₈×C₈₉	—	4	1	no	—
26	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
27	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
29	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
30	C₄²×C₈₉	SD₁₆	1	2	yes	—
31	C₁₆×C₈₉	—	4	2	no	—
32	C₂³×C₈₉	#107	1	1	yes	characteristic
33	C₂×C₄×C₈₉	—	2	2	no	—
34	C₂×C₄×C₈₉	—	2	1	no	—
35	C₂×C₄×C₈₉	—	2	1	no	—
36	C₂×C₄×C₈₉	—	4	1	no	—
37	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
38	C₂×C₄×C₈₉	#102	1	2	yes	—
39	C₂×C₄×C₈₉	#103	1	2	yes	—
40	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
41	C₈×C₈₉	—	2	2	no	—
42	C₈×C₈₉	—	2	1	no	—
43	C₈×C₈₉	#100	1	1	yes	characteristic
44	C₈×C₈₉	#101	1	1	yes	characteristic
45	#10623494	—	8	4	no	—
46	C₂²×C₈₉	—	2	1	no	—
47	C₂²×C₈₉	—	2	1	no	—
48	C₂²×C₈₉	#340	1	1	yes	characteristic
49	C₂²×C₈₉	#368	1	2	yes	—
50	C₄×C₈₉	—	2	1	no	—
51	C₄×C₈₉	—	2	2	no	—
52	C₄×C₈₉	—	2	2	no	—
53	C₄×C₈₉	#338	1	1	yes	characteristic
54	C₄×C₈₉	#343	1	1	yes	characteristic
55	#7242	—	89	1	no	maximal, Sylow
56	C₂×C₈₉	—	2	2	no	—
57	C₂×C₈₉	#1292	1	1	yes	characteristic
58	C₂×C₈₉	#1381	1	1	yes	characteristic
59	C₂×C₈₉	#1384	1	1	yes	characteristic
60	C₂×C₄⋊C₁₆	D₈₉	1	1	yes	characteristic
61	#1568	—	89	2	no	—
62	C₈₉	#7242	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₁₆	Dic₈₉	1	2	yes	—
64	#432	—	178	2	no	—
65	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
66	C₄⋊C₁₆	—	2	2	no	—
67	#335	—	178	2	no	—
68	#334	—	89	4	no	—
69	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
70	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
71	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	4	2	no	—
74	C₄⋊C₈	—	178	4	no	—
75	C₂²⋊C₈	—	356	2	no	—
76	#97	—	356	2	no	—
77	C₄×C₈	C₄×D₈₉	1	2	yes	—
78	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
79	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
80	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
81	C₂²×C₄	Dic₃₅₆	1	1	yes	characteristic
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	356	4	no	—
86	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
87	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
88	C₂×C₈	#13083553	1	2	yes	—
89	C₄²	#13083551	1	2	yes	—
90	C₄²	#13083563	1	2	yes	—
91	C₁₆	—	4	2	no	—
92	C₂³	#15667946	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	#15667919	1	1	yes	characteristic
98	C₂×C₄	#15667934	1	2	yes	—
99	C₂×C₄	#15667937	1	2	yes	—
100	C₂×C₄	#15667944	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	2	no	—
103	C₈	—	712	4	no	—
104	C₈	#15667929	1	1	yes	characteristic
105	C₈	#15667931	1	1	yes	characteristic
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#19151361	1	1	yes	characteristic
109	C₂²	#19151428	1	2	yes	—
110	C₄	—	2	2	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	#19151356	1	1	yes	characteristic
114	C₄	#19151367	1	1	yes	characteristic
115	C₂	—	2	2	no	—
116	C₂	#26018127	1	1	yes	characteristic
117	C₂	#26018346	1	1	yes	characteristic
118	C₂	#26018354	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26019015

C₂

yes

maximal

C₂×C₄⋊C₁₆×C₈₉

C₂

yes

maximal, characteristic

#19151655

—

#19151351

—

#19151349

C₄

yes

—

C₂²×C₁₆×C₈₉

C₄

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

—

C₄⋊#10623494

—

C₂²⋊#10623494

—

#15667923

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

Q₈

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₂×#10623494

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₈₉

D₈

yes

characteristic

C₂²×C₄×C₈₉

Q₁₆

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄²

yes

characteristic

C₂×C₈×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

SD₁₆

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

#107

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#95

yes

characteristic

C₂×C₄×C₈₉

#102

yes

—

C₂×C₄×C₈₉

#103

yes

—

C₂×C₄×C₈₉

#106

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#340

yes

characteristic

C₂²×C₈₉

#368

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₄⋊C₁₆

D₈₉

yes

characteristic

#1568

—

C₈₉

#7242

yes

characteristic, complemented, Sylow

C₂²×C₁₆

Dic₈₉

yes

—

#432

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₄⋊C₁₆

—

#335

—

178

—

#334

—

C₂²×C₈

Dic₁₇₈

yes

characteristic

C₂²×C₈

C₂×Dic₈₉

yes

characteristic

C₂×C₄²

D₃₅₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

178

—

C₂²⋊C₈

—

356

—

#97

—

356

—

C₄×C₈

C₄×D₈₉

yes

—

C₄×C₈

C₂²⋊D₈₉

yes

—

C₂²×C₄

C₄⋊Dic₈₉

yes

characteristic

C₂²×C₄

D₇₁₂

yes

characteristic

C₂²×C₄

Dic₃₅₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

356

—

C₂×C₈

C₄×Dic₈₉

yes

characteristic

C₂×C₈

C₂²⋊Dic₈₉

yes

characteristic

C₂×C₈

#13083553

yes

—

C₄²

#13083551

yes

—

C₄²

#13083563

yes

—

C₁₆

—

C₂³

#15667946

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#15667919

yes

characteristic

C₂×C₄

#15667934

yes

—

C₂×C₄

#15667937

yes

—

100

C₂×C₄

#15667944

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

712

—

104

C₈

#15667929

yes

characteristic

105

C₈

#15667931

yes

characteristic

106

C₂²

—

107

C₂²

—

108

C₂²

#19151361

yes

characteristic

109

C₂²

#19151428

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#19151356

yes

characteristic

114

C₄

#19151367

yes

characteristic

115

C₂

—

116

C₂

#26018127

yes

characteristic

117

C₂

#26018346

yes

characteristic

118

C₂

#26018354

yes

characteristic

	subgroup	G/H
center	C₂²	#19151361
commutator	C₂×C₈×C₈₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₈₉
Fitting	C₂×C₄⋊C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	#340
prime radical	C₂³×C₈₉	#107
cosocle	C₂×C₄×C₈×C₈₉	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₈₉
hypocenter	C₈₉	#7242
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂²

#19151361

commutator

C₂×C₈×C₈₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₈₉

Fitting

C₂×C₄⋊C₁₆×C₈₉

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄×C₈×C₈₉

C₂²

hypercenter

C₂×C₄⋊C₁₆

D₈₉

hypocenter

C₈₉

#7242

LCS

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
62. C₈₉	#7242	semidirect

H ≅ G/N

type

62. C₈₉

#7242

semidirect

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

subgroup	index
#26019015	2
C₂×C₄⋊C₁₆×C₈₉	2
#7242	89

subgroup

index

#26019015

C₂×C₄⋊C₁₆×C₈₉

#7242

Subgroups — #36441529

Conjugacy classes of subgroups

Complements

Maximal subgroups