Group #36474910 — subgroups

Subgroups — #36474910

Order 22784.

218 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₂×#19151432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₈₉	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	C₂²×#13083549	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
7	#19151432	—	2	2	no	—
8	C₂×#15667918	C₂²	1	2	yes	—
9	C₁₆×Dic₈₉	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	C₂²×#10623494	C₈	1	1	yes	characteristic
15	C₂×C₄×Dic₈₉	C₈	1	1	yes	characteristic
16	C₂×#13083549	—	2	2	no	—
17	C₂×#13083549	C₂×C₄	1	2	yes	—
18	C₈×Dic₈₉	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	C₂²×Dic₈₉	C₁₆	1	2	yes	—
28	C₂×#10623494	—	2	2	no	—
29	C₂×#10623494	C₂×C₈	1	2	yes	—
30	C₄×Dic₈₉	C₁₆	1	1	yes	characteristic
31	C₄×Dic₈₉	C₁₆	1	1	yes	characteristic
32	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
33	#13083549	—	2	2	no	—
34	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
38	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
39	C₁₆×C₈₉	C₄⋊C₄	1	2	yes	—
40	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₈₉	C₂×Q₈	1	1	yes	characteristic
42	C₂³×C₈₉	C₂×C₁₆	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
44	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₈₉	M₃₂	1	2	yes	—
46	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
47	C₂×Dic₈₉	—	2	4	no	—
48	C₂×Dic₈₉	C₂×C₁₆	1	2	yes	—
49	C₂×Dic₈₉	C₂×C₁₆	1	2	yes	—
50	C₈×C₈₉	C₄⋊C₈	1	2	yes	—
51	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
53	#10623494	—	2	2	no	—
54	C₂²×C₈₉	—	2	2	no	—
55	C₂²×C₈₉	C₄⋊C₁₆	1	2	yes	—
56	C₂²×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
57	C₄×C₈₉	C₄⋊C₁₆	1	2	yes	—
58	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₄×C₈₉	C₂×M₃₂	1	1	yes	characteristic
60	Dic₈₉	—	2	2	no	—
61	Dic₈₉	M₆₄	1	4	yes	complemented
62	#13627	—	89	1	no	maximal, Sylow
63	C₂×C₈₉	—	2	2	no	—
64	C₂×C₈₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
65	C₂×C₈₉	C₂×M₆₄	1	2	yes	—
66	C₂×M₆₄	—	89	1	no	—
67	C₂×M₆₄	D₈₉	1	1	yes	characteristic
68	C₂×C₄×C₁₆	—	89	1	no	—
69	C₄⋊C₃₂	—	89	4	no	—
70	C₈₉	#13627	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₁₆	—	89	1	no	—
72	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
73	C₂×C₄×C₈	—	89	1	no	—
74	M₆₄	—	2	2	no	—
75	M₆₄	—	178	2	no	—
76	C₂×C₃₂	—	89	2	no	—
77	C₂×C₃₂	C₂×D₈₉	1	2	yes	—
78	C₄×C₁₆	—	89	2	no	—
79	C₄×C₁₆	—	89	2	no	—
80	C₂²×C₈	—	89	1	no	—
81	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
82	C₂×C₄²	—	89	1	no	—
83	C₂×C₁₆	—	89	2	no	—
84	C₂×C₁₆	—	178	2	no	—
85	C₂×C₁₆	Dic₁₇₈	1	2	yes	—
86	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
87	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
88	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
89	C₄×C₈	—	89	2	no	—
90	C₄×C₈	—	89	2	no	—
91	C₃₂	—	2	2	no	—
92	C₃₂	—	178	2	no	—
93	C₂²×C₄	—	89	2	no	—
94	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
95	C₂×C₈	—	89	2	no	—
96	C₂×C₈	—	178	2	no	—
97	C₂×C₈	#13083553	1	2	yes	—
98	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
99	C₂×C₈	#13083557	1	2	yes	—
100	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
101	C₄²	—	89	2	no	—
102	C₄²	—	89	1	no	—
103	C₄²	—	89	1	no	—
104	C₁₆	—	178	2	no	—
105	C₁₆	#13083553	1	2	yes	—
106	C₁₆	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
107	C₁₆	C₂×Dic₁₇₈	1	1	yes	characteristic
108	C₂³	C₁₆×D₈₉	1	1	yes	characteristic
109	C₂×C₄	—	89	2	no	—
110	C₂×C₄	—	89	2	no	—
111	C₂×C₄	—	178	4	no	—
112	C₂×C₄	#15667941	1	2	yes	—
113	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
114	C₂×C₄	#15667951	1	2	yes	—
115	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
116	C₈	—	178	2	no	—
117	C₈	#15667941	1	2	yes	—
118	C₈	C₂×#13083553	1	1	yes	characteristic
119	C₈	C₂×#13083557	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	#19151416	1	2	yes	—
122	C₂²	C₂×C₁₆×D₈₉	1	1	yes	characteristic
123	C₄	—	89	4	no	—
124	C₄	—	178	2	no	—
125	C₄	#19151416	1	2	yes	—
126	C₄	C₂×#15667941	1	1	yes	characteristic
127	C₄	C₂×#15667951	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	C₂×#19151416	1	1	yes	characteristic
130	C₂	M₆₄×D₈₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151432

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₈₉

C₂

yes

maximal, characteristic

#26018447

C₂

yes

maximal, complemented

C₂×M₆₄×C₈₉

C₂

yes

maximal, characteristic

C₂²×#13083549

C₄

yes

characteristic

C₂×C₈×Dic₈₉

C₄

yes

characteristic

#19151432

—

C₂×#15667918

C₂²

yes

—

C₁₆×Dic₈₉

C₄

yes

—

C₁₆×Dic₈₉

C₂²

yes

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

M₆₄×C₈₉

—

C₂×C₃₂×C₈₉

C₂²

yes

—

C₂²×#10623494

C₈

yes

characteristic

C₂×C₄×Dic₈₉

C₈

yes

characteristic

C₂×#13083549

—

C₂×#13083549

C₂×C₄

yes

—

C₈×Dic₈₉

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

—

C₂²×Dic₈₉

C₁₆

yes

—

C₂×#10623494

—

C₂×#10623494

C₂×C₈

yes

—

C₄×Dic₈₉

C₁₆

yes

characteristic

C₄×Dic₈₉

C₁₆

yes

characteristic

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×Dic₈₉

C₂×C₁₆

yes

—

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

—

Dic₈₉

M₆₄

yes

complemented

#13627

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₈₉

C₂×M₆₄

yes

—

C₂×M₆₄

—

C₂×M₆₄

D₈₉

yes

characteristic

C₂×C₄×C₁₆

—

C₄⋊C₃₂

—

C₈₉

#13627

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

C₂×C₄×C₈

—

M₆₄

—

M₆₄

—

178

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₈₉

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

C₄×D₈₉

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

178

—

C₂×C₁₆

Dic₁₇₈

yes

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

C₂²⋊D₈₉

yes

—

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₃₂

—

178

—

C₂²×C₄

—

C₂²×C₄

C₈×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

#13083553

yes

—

C₂×C₈

C₈×D₈₉

yes

characteristic

C₂×C₈

#13083557

yes

—

100

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₁₆

—

178

—

105

C₁₆

#13083553

yes

—

106

C₁₆

C₂×C₂²⋊D₈₉

yes

characteristic

107

C₁₆

C₂×Dic₁₇₈

yes

characteristic

108

C₂³

C₁₆×D₈₉

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

178

—

112

C₂×C₄

#15667941

yes

—

113

C₂×C₄

C₁₆×D₈₉

yes

characteristic

114

C₂×C₄

#15667951

yes

—

115

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

116

C₈

—

178

—

117

C₈

#15667941

yes

—

118

C₈

C₂×#13083553

yes

characteristic

119

C₈

C₂×#13083557

yes

characteristic

120

C₂²

—

121

C₂²

#19151416

yes

—

122

C₂²

C₂×C₁₆×D₈₉

yes

characteristic

123

C₄

—

124

C₄

—

178

—

125

C₄

#19151416

yes

—

126

C₄

C₂×#15667941

yes

characteristic

127

C₄

C₂×#15667951

yes

characteristic

128

C₂

—

129

C₂

C₂×#19151416

yes

characteristic

130

C₂

M₆₄×D₈₉

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	#13627
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
61. Dic₈₉	M₆₄	semidirect
70. C₈₉	#13627	semidirect

H ≅ G/N

type

3. #26018447

C₂

semidirect

61. Dic₈₉

M₆₄

semidirect

70. C₈₉

#13627

semidirect

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

subgroup	index
C₂×#19151432	2
C₂×C₁₆×Dic₈₉	2
#26018447	2
C₂×M₆₄×C₈₉	2
#13627	89

subgroup

index

C₂×#19151432

C₂×C₁₆×Dic₈₉

#26018447

C₂×M₆₄×C₈₉

#13627

Subgroups — #36474910

Conjugacy classes of subgroups

Complements

Maximal subgroups