Group #36474907 — subgroups

Subgroups — C₂×#26018447 #36474907

Order 22784.

230 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 147 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	C₂×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26018447	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₃₂×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	C₂×#15667918	—	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	C₂×C₃₂×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	1	yes	characteristic
28	C₂²×Dic₈₉	C₁₆	1	1	yes	characteristic
29	C₂×#10623494	—	2	2	no	—
30	C₂×#10623494	C₂×C₈	1	2	yes	—
31	C₄×Dic₈₉	C₁₆	1	2	yes	—
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	2	no	—
48	C₂×Dic₈₉	C₃₂	1	4	yes	complemented
49	C₂×Dic₈₉	C₂×C₁₆	1	2	yes	—
50	C₂×Dic₈₉	C₂×C₁₆	1	2	yes	—
51	C₈×C₈₉	C₄⋊C₈	1	2	yes	—
52	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
54	#10623494	—	2	2	no	—
55	C₂²×C₈₉	C₄⋊C₁₆	1	2	yes	—
56	C₂²×C₈₉	C₂×C₃₂	1	2	yes	—
57	C₂²×C₈₉	M₆₄	1	2	yes	—
58	C₂²×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
59	C₄×C₈₉	C₄⋊C₁₆	1	2	yes	—
60	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
61	C₄×C₈₉	C₂×M₃₂	1	1	yes	characteristic
62	Dic₈₉	—	2	2	no	—
63	Dic₈₉	C₂×C₃₂	1	4	yes	complemented
64	C₂×C₄⋊C₃₂	—	89	1	no	maximal, Sylow
65	C₂×C₈₉	C₄⋊C₃₂	1	4	yes	complemented
66	C₂×C₈₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
67	C₂×C₈₉	C₂²×C₃₂	1	1	yes	characteristic
68	C₂×C₈₉	C₂×M₆₄	1	1	yes	characteristic
69	C₂²×C₃₂	—	89	1	no	—
70	C₂²×C₃₂	D₈₉	1	1	yes	characteristic
71	C₂×C₄×C₁₆	—	89	1	no	—
72	C₄⋊C₃₂	—	89	4	no	—
73	C₈₉	C₂×C₄⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₁₆	—	89	1	no	—
75	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
76	C₂×C₄×C₈	—	89	1	no	—
77	C₂×C₃₂	—	2	2	no	—
78	C₂×C₃₂	—	89	2	no	—
79	C₂×C₃₂	—	178	2	no	—
80	C₂×C₃₂	C₂×D₈₉	1	2	yes	—
81	C₄×C₁₆	—	89	2	no	—
82	C₄×C₁₆	—	89	2	no	—
83	C₂²×C₈	—	89	1	no	—
84	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
85	C₂×C₄²	—	89	1	no	—
86	C₂×C₁₆	—	89	2	no	—
87	C₂×C₁₆	—	178	2	no	—
88	C₂×C₁₆	Dic₁₇₈	1	2	yes	—
89	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
90	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
91	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
92	C₄×C₈	—	89	2	no	—
93	C₄×C₈	—	89	2	no	—
94	C₃₂	—	2	2	no	—
95	C₃₂	—	178	2	no	—
96	C₂²×C₄	—	89	1	no	—
97	C₂²×C₄	—	89	1	no	—
98	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
99	C₂×C₈	—	89	2	no	—
100	C₂×C₈	—	178	2	no	—
101	C₂×C₈	#13083553	1	2	yes	—
102	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
103	C₂×C₈	#13083557	1	2	yes	—
104	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
105	C₄²	—	89	2	no	—
106	C₄²	—	89	2	no	—
107	C₁₆	—	178	2	no	—
108	C₁₆	#13083553	1	2	yes	—
109	C₁₆	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
110	C₁₆	C₂×Dic₁₇₈	1	1	yes	characteristic
111	C₂³	C₁₆×D₈₉	1	1	yes	characteristic
112	C₂×C₄	—	89	4	no	—
113	C₂×C₄	—	89	2	no	—
114	C₂×C₄	—	89	2	no	—
115	C₂×C₄	—	178	2	no	—
116	C₂×C₄	#15667941	1	2	yes	—
117	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
118	C₂×C₄	#15667951	1	2	yes	—
119	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
120	C₈	—	178	2	no	—
121	C₈	#15667941	1	2	yes	—
122	C₈	C₂×#13083553	1	1	yes	characteristic
123	C₈	C₂×#13083557	1	1	yes	characteristic
124	C₂²	#19151416	1	2	yes	—
125	C₂²	C₃₂×D₈₉	1	2	yes	—
126	C₂²	#19151431	1	2	yes	—
127	C₂²	C₂×C₁₆×D₈₉	1	1	yes	characteristic
128	C₄	—	89	4	no	—
129	C₄	—	178	2	no	—
130	C₄	#19151416	1	2	yes	—
131	C₄	C₂×#15667941	1	1	yes	characteristic
132	C₄	C₂×#15667951	1	1	yes	characteristic
133	C₂	#26018447	1	4	yes	direct product factor
134	C₂	C₂×#19151416	1	1	yes	characteristic
135	C₂	C₂×C₃₂×D₈₉	1	1	yes	characteristic
136	C₂	C₂×#19151431	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#15667918

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₈₉

C₂

yes

maximal, characteristic

#26018447

C₂

yes

maximal, direct product factor

C₂²×C₃₂×C₈₉

C₂

yes

maximal, characteristic

C₂²×#13083549

C₄

yes

characteristic

C₂×C₈×Dic₈₉

C₄

yes

characteristic

C₂×#15667918

—

C₂×#15667918

C₂²

yes

—

C₁₆×Dic₈₉

C₄

yes

—

C₁₆×Dic₈₉

C₂²

yes

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×C₃₂×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

characteristic

C₂²×Dic₈₉

C₁₆

yes

characteristic

C₂×#10623494

—

C₂×#10623494

C₂×C₈

yes

—

C₄×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×Dic₈₉

C₃₂

yes

complemented

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

C₄⋊C₁₆

yes

—

C₂²×C₈₉

C₂×C₃₂

yes

—

C₂²×C₈₉

M₆₄

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

C₂×C₃₂

yes

complemented

C₂×C₄⋊C₃₂

—

maximal, Sylow

C₂×C₈₉

C₄⋊C₃₂

yes

complemented

C₂×C₈₉

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₈₉

C₂²×C₃₂

yes

characteristic

C₂×C₈₉

C₂×M₆₄

yes

characteristic

C₂²×C₃₂

—

C₂²×C₃₂

D₈₉

yes

characteristic

C₂×C₄×C₁₆

—

C₄⋊C₃₂

—

C₈₉

C₂×C₄⋊C₃₂

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

C₂×C₄×C₈

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

178

—

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

—

C₂²×C₄

C₈×D₈₉

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

178

—

101

C₂×C₈

#13083553

yes

—

102

C₂×C₈

C₈×D₈₉

yes

characteristic

103

C₂×C₈

#13083557

yes

—

104

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

105

C₄²

—

106

C₄²

—

107

C₁₆

—

178

—

108

C₁₆

#13083553

yes

—

109

C₁₆

C₂×C₂²⋊D₈₉

yes

characteristic

110

C₁₆

C₂×Dic₁₇₈

yes

characteristic

111

C₂³

C₁₆×D₈₉

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

178

—

116

C₂×C₄

#15667941

yes

—

117

C₂×C₄

C₁₆×D₈₉

yes

characteristic

118

C₂×C₄

#15667951

yes

—

119

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

120

C₈

—

178

—

121

C₈

#15667941

yes

—

122

C₈

C₂×#13083553

yes

characteristic

123

C₈

C₂×#13083557

yes

characteristic

124

C₂²

#19151416

yes

—

125

C₂²

C₃₂×D₈₉

yes

—

126

C₂²

#19151431

yes

—

127

C₂²

C₂×C₁₆×D₈₉

yes

characteristic

128

C₄

—

129

C₄

—

178

—

130

C₄

#19151416

yes

—

131

C₄

C₂×#15667941

yes

characteristic

132

C₄

C₂×#15667951

yes

characteristic

133

C₂

#26018447

yes

direct product factor

134

C₂

C₂×#19151416

yes

characteristic

135

C₂

C₂×C₃₂×D₈₉

yes

characteristic

136

C₂

C₂×#19151431

yes

characteristic

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₈₉
commutator	C₂×C₈₉	C₂²×C₃₂
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₂²×C₃₂×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₁₆
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	C₂²×C₃₂	D₈₉
hypocenter	C₈₉	C₂×C₄⋊C₃₂
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

subgroup

G/H

center

C₂²×C₁₆

C₂×D₈₉

commutator

C₂×C₈₉

C₂²×C₃₂

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₂²×C₃₂×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₁₆

cosocle

C₂×C₁₆×C₈₉

C₂³

hypercenter

C₂²×C₃₂

D₈₉

hypocenter

C₈₉

C₂×C₄⋊C₃₂

LCS

G ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

N	H ≅ G/N	type
3. #26018447	C₂	direct (N × H)
48. C₂×Dic₈₉	C₃₂	semidirect
63. Dic₈₉	C₂×C₃₂	semidirect
65. C₂×C₈₉	C₄⋊C₃₂	semidirect
73. C₈₉	C₂×C₄⋊C₃₂	semidirect
133. C₂	#26018447	direct (N × H)

H ≅ G/N

type

3. #26018447

C₂

direct (N × H)

48. C₂×Dic₈₉

C₃₂

semidirect

63. Dic₈₉

C₂×C₃₂

semidirect

65. C₂×C₈₉

C₄⋊C₃₂

semidirect

73. C₈₉

C₂×C₄⋊C₃₂

semidirect

133. C₂

#26018447

direct (N × H)

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

subgroup	index
C₂²×#15667918	2
C₂×C₁₆×Dic₈₉	2
#26018447	2
C₂²×C₃₂×C₈₉	2
C₂×C₄⋊C₃₂	89

subgroup

index

C₂²×#15667918

C₂×C₁₆×Dic₈₉

#26018447

C₂²×C₃₂×C₈₉

C₂×C₄⋊C₃₂

Subgroups — C₂×#26018447 #36474907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#26018447 #36474907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018447 #36474907