Group #36474915 — subgroups

Subgroups — #36474915

Order 22784.

206 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 123 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	C₂²×#15667918	C₂	1	1	yes	maximal, characteristic
2	#26021848	C₂	1	1	yes	maximal, characteristic
3	#26018447	C₂	1	4	yes	maximal, complemented
4	C₂×M₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667952	C₄	1	1	yes	characteristic
6	#19151553	C₄	1	1	yes	characteristic
7	C₂×#15667918	—	2	2	no	—
8	C₂×#15667918	C₂²	1	2	yes	—
9	#19151373	C₄	1	2	yes	—
10	C₁₆×Dic₈₉	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
12	M₆₄×C₈₉	—	2	2	no	—
13	C₂×C₃₂×C₈₉	C₂²	1	2	yes	—
14	C₂×#13083558	C₈	1	1	yes	characteristic
15	#15667971	C₈	1	1	yes	characteristic
16	#15667952	—	2	2	no	—
17	C₂×#13083549	C₂×C₄	1	2	yes	—
18	#15667922	C₈	1	2	yes	—
19	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
20	#15667918	—	2	2	no	—
21	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
24	C₂×C₁₆×C₈₉	Q₈	1	2	yes	—
25	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
26	C₃₂×C₈₉	—	2	2	no	—
27	#13083558	—	2	2	no	—
28	C₂×#10623494	C₂×C₈	1	2	yes	—
29	C₄⋊Dic₈₉	C₁₆	1	2	yes	—
30	C₂²⋊Dic₈₉	C₁₆	1	2	yes	—
31	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
32	#13083549	—	2	2	no	—
33	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
37	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
38	C₁₆×C₈₉	C₄⋊C₄	1	2	yes	—
39	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₈₉	C₂×Q₈	1	1	yes	characteristic
41	C₂³×C₈₉	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
43	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₈₉	M₃₂	1	2	yes	—
45	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
46	C₂×Dic₈₉	C₂×C₁₆	1	4	yes	—
47	C₈×C₈₉	C₄⋊C₈	1	2	yes	—
48	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
50	#10623494	—	2	2	no	—
51	C₂²×C₈₉	—	2	2	no	—
52	C₂²×C₈₉	C₄⋊C₁₆	1	2	yes	—
53	C₂²×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
54	C₄×C₈₉	C₄⋊C₁₆	1	2	yes	—
55	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
56	C₄×C₈₉	C₂×M₃₂	1	1	yes	characteristic
57	Dic₈₉	—	2	4	no	—
58	#13628	—	89	1	no	maximal, Sylow
59	C₂×C₈₉	—	2	2	no	—
60	C₂×C₈₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
61	C₂×C₈₉	#2258	1	2	yes	—
62	C₂×M₆₄	D₈₉	1	1	yes	characteristic
63	C₂²×C₃₂	—	89	1	no	—
64	#2108	—	89	1	no	—
65	C₄⋊C₃₂	—	89	4	no	—
66	C₈₉	#13628	1	1	yes	characteristic, complemented, Sylow
67	C₂×M₃₂	—	89	1	no	—
68	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
69	#405	—	89	1	no	—
70	M₆₄	—	2	2	no	—
71	C₂×C₃₂	—	89	2	no	—
72	C₂×C₃₂	—	178	2	no	—
73	C₂×C₃₂	C₂×D₈₉	1	2	yes	—
74	#346	—	89	2	no	—
75	C₄×C₁₆	—	89	2	no	—
76	C₂×M₁₆	—	89	1	no	—
77	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
78	#117	—	89	1	no	—
79	M₃₂	—	178	2	no	—
80	C₂×C₁₆	—	89	2	no	—
81	C₂×C₁₆	Dic₁₇₈	1	2	yes	—
82	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
83	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
84	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
85	#97	—	89	2	no	—
86	C₄×C₈	—	89	2	no	—
87	C₃₂	—	2	2	no	—
88	C₃₂	—	178	2	no	—
89	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
90	M₁₆	—	178	2	no	—
91	C₂×C₈	—	89	2	no	—
92	C₂×C₈	#13083553	1	2	yes	—
93	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
94	C₂×C₈	#13083557	1	2	yes	—
95	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
96	C₄⋊C₄	—	89	2	no	—
97	C₂²⋊C₄	—	89	2	no	—
98	C₄²	—	89	2	no	—
99	C₁₆	—	178	2	no	—
100	C₁₆	#13083553	1	2	yes	—
101	C₁₆	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
102	C₁₆	C₂×Dic₁₇₈	1	1	yes	characteristic
103	C₂³	C₁₆×D₈₉	1	1	yes	characteristic
104	C₂×C₄	—	89	4	no	—
105	C₂×C₄	#15667941	1	2	yes	—
106	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
107	C₂×C₄	#15667951	1	2	yes	—
108	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
109	C₈	—	178	2	no	—
110	C₈	#15667941	1	2	yes	—
111	C₈	C₂×#13083553	1	1	yes	characteristic
112	C₈	C₂×#13083557	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	#19151416	1	2	yes	—
115	C₂²	C₂×C₁₆×D₈₉	1	1	yes	characteristic
116	C₄	—	178	4	no	—
117	C₄	#19151416	1	2	yes	—
118	C₄	C₂×#15667941	1	1	yes	characteristic
119	C₄	C₂×#15667951	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×#19151416	1	1	yes	characteristic
122	C₂	#26022547	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#15667918

C₂

yes

maximal, characteristic

#26021848

C₂

yes

maximal, characteristic

#26018447

C₂

yes

maximal, complemented

C₂×M₆₄×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667952

C₄

yes

characteristic

#19151553

C₄

yes

characteristic

C₂×#15667918

—

C₂×#15667918

C₂²

yes

—

#19151373

C₄

yes

—

C₁₆×Dic₈₉

C₂²

yes

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

M₆₄×C₈₉

—

C₂×C₃₂×C₈₉

C₂²

yes

—

C₂×#13083558

C₈

yes

characteristic

#15667971

C₈

yes

characteristic

#15667952

—

C₂×#13083549

C₂×C₄

yes

—

#15667922

C₈

yes

—

C₈×Dic₈₉

C₂×C₄

yes

—

#15667918

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

—

C₂×C₁₆×C₈₉

Q₈

yes

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₃₂×C₈₉

—

#13083558

—

C₂×#10623494

C₂×C₈

yes

—

C₄⋊Dic₈₉

C₁₆

yes

—

C₂²⋊Dic₈₉

C₁₆

yes

—

C₄×Dic₈₉

C₂×C₈

yes

—

#13083549

—

C₂²×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

M₁₆

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₁₆×C₈₉

C₄⋊C₄

yes

—

C₁₆×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

C₂×Q₈

yes

characteristic

C₂³×C₈₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₉

C₄⋊C₈

yes

—

C₂×C₄×C₈₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₉

M₃₂

yes

—

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×Dic₈₉

C₂×C₁₆

yes

—

C₈×C₈₉

C₄⋊C₈

yes

—

C₈×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₈₉

C₂×M₁₆

yes

characteristic

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄⋊C₁₆

yes

—

C₂²×C₈₉

C₂²×C₁₆

yes

characteristic

C₄×C₈₉

C₄⋊C₁₆

yes

—

C₄×C₈₉

C₂×C₄⋊C₈

yes

characteristic

C₄×C₈₉

C₂×M₃₂

yes

characteristic

Dic₈₉

—

#13628

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₈₉

#2258

yes

—

C₂×M₆₄

D₈₉

yes

characteristic

C₂²×C₃₂

—

#2108

—

C₄⋊C₃₂

—

C₈₉

#13628

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

#405

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

178

—

C₂×C₃₂

C₂×D₈₉

yes

—

#346

—

C₄×C₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₄×D₈₉

yes

characteristic

#117

—

M₃₂

—

178

—

C₂×C₁₆

—

C₂×C₁₆

Dic₁₇₈

yes

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

C₂²⋊D₈₉

yes

—

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#97

—

C₄×C₈

—

C₃₂

—

C₃₂

—

178

—

C₂²×C₄

C₈×D₈₉

yes

characteristic

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

#13083553

yes

—

C₂×C₈

C₈×D₈₉

yes

characteristic

C₂×C₈

#13083557

yes

—

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

178

—

100

C₁₆

#13083553

yes

—

101

C₁₆

C₂×C₂²⋊D₈₉

yes

characteristic

102

C₁₆

C₂×Dic₁₇₈

yes

characteristic

103

C₂³

C₁₆×D₈₉

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

#15667941

yes

—

106

C₂×C₄

C₁₆×D₈₉

yes

characteristic

107

C₂×C₄

#15667951

yes

—

108

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

109

C₈

—

178

—

110

C₈

#15667941

yes

—

111

C₈

C₂×#13083553

yes

characteristic

112

C₈

C₂×#13083557

yes

characteristic

113

C₂²

—

114

C₂²

#19151416

yes

—

115

C₂²

C₂×C₁₆×D₈₉

yes

characteristic

116

C₄

—

178

—

117

C₄

#19151416

yes

—

118

C₄

C₂×#15667941

yes

characteristic

119

C₄

C₂×#15667951

yes

characteristic

120

C₂

—

121

C₂

C₂×#19151416

yes

characteristic

122

C₂

#26022547

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₈₉
commutator, socle	C₂²×C₈₉	C₂²×C₁₆
Fitting	C₂×M₆₄×C₈₉	C₂
prime radical	C₂³×C₈₉	C₂×C₁₆
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	C₂×M₆₄	D₈₉
hypocenter	C₈₉	#13628
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₈₉

commutator, socle

C₂²×C₈₉

C₂²×C₁₆

Fitting

C₂×M₆₄×C₈₉

C₂

prime radical

C₂³×C₈₉

C₂×C₁₆

cosocle

C₂×C₁₆×C₈₉

C₂³

hypercenter

C₂×M₆₄

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

N	H ≅ G/N	type
3. #26018447	C₂	semidirect
66. C₈₉	#13628	semidirect

H ≅ G/N

type

3. #26018447

C₂

semidirect

66. C₈₉

#13628

semidirect

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

subgroup	index
C₂²×#15667918	2
#26021848	2
#26018447	2
C₂×M₆₄×C₈₉	2
#13628	89

subgroup

index

C₂×M₆₄×C₈₉

Subgroups — #36474915

Conjugacy classes of subgroups

Complements

Maximal subgroups