Group #26018673 — subgroups

Subgroups — #26018673

Order 11392.

218 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 85 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	#19151651	C₂	1	1	yes	maximal, characteristic
2	C₄×#13083558	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#10623494	C₂	1	4	yes	maximal, complemented
4	#533×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13083558	—	2	1	no	—
6	C₄⋊#10623494	C₂²	1	2	yes	—
7	C₂²⋊#10623494	—	2	1	no	—
8	#15667923	—	2	1	no	—
9	C₄×#10623494	—	2	1	no	—
10	C₄×#10623494	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₈₉	C₄	1	1	yes	characteristic
12	C₄⋊Q₈×C₈₉	C₄	1	1	yes	characteristic
13	#125×C₈₉	—	2	1	no	—
14	C₂²⋊Q₈×C₈₉	—	2	1	no	—
15	C₂²⋊Q₈×C₈₉	C₄	1	2	yes	—
16	C₄×Q₈×C₈₉	—	2	1	no	—
17	C₄×Q₈×C₈₉	C₂²	1	2	yes	—
18	#117×C₈₉	—	2	1	no	—
19	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
20	#13083558	—	2	4	no	—
21	C₂×#10623494	—	2	2	no	—
22	C₂×#10623494	—	2	2	no	—
23	C₂×Q₈×C₈₉	—	2	1	no	—
24	C₂×Q₈×C₈₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₈₉	—	2	1	no	—
29	C₄⋊C₄×C₈₉	—	2	1	no	—
30	C₄⋊C₄×C₈₉	—	2	1	no	—
31	C₄⋊C₄×C₈₉	—	2	2	no	—
32	C₄⋊C₄×C₈₉	—	2	2	no	—
33	C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
34	C₂²⋊C₄×C₈₉	—	2	2	no	—
35	C₄²×C₈₉	—	2	2	no	—
36	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₈₉	D₄	1	1	yes	characteristic
38	C₄²×C₈₉	D₄	1	1	yes	characteristic
39	C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	C₂³×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
41	Q₈×C₈₉	—	2	4	no	—
42	Q₈×C₈₉	—	4	2	no	—
43	C₂×C₄×C₈₉	—	2	2	no	—
44	C₂×C₄×C₈₉	—	2	2	no	—
45	C₂×C₄×C₈₉	—	2	2	no	—
46	C₂×C₄×C₈₉	—	2	2	no	—
47	C₂×C₄×C₈₉	—	2	2	no	—
48	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	#10623494	—	2	4	no	—
55	#10623494	—	4	2	no	—
56	C₂²×C₈₉	—	2	1	no	—
57	C₂²×C₈₉	—	2	1	no	—
58	C₂²×C₈₉	—	2	1	no	—
59	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₈₉	—	2	1	no	—
61	C₄×C₈₉	—	2	1	no	—
62	C₄×C₈₉	—	2	1	no	—
63	C₄×C₈₉	—	4	2	no	—
64	C₄×C₈₉	—	4	2	no	—
65	C₄×C₈₉	#104	1	2	yes	—
66	C₄×C₈₉	#131	1	2	yes	—
67	C₄×C₈₉	#137	1	2	yes	—
68	C₂×C₈₉	—	4	1	no	—
69	C₂×C₈₉	#408	1	1	yes	characteristic
70	C₂×C₈₉	#419	1	1	yes	characteristic
71	C₂×C₈₉	C₂×#104	1	1	yes	characteristic
72	#1485	—	89	1	no	maximal, Sylow
73	C₈₉	#1485	1	1	yes	characteristic, complemented, Sylow
74	#533	D₈₉	1	1	yes	characteristic
75	#431	—	89	1	no	—
76	C₄×M₁₆	—	89	1	no	—
77	Q₈⋊C₈	—	89	4	no	—
78	C₂×M₁₆	—	178	1	no	—
79	C₄⋊Q₈	Dic₈₉	1	1	yes	characteristic
80	C₄⋊Q₈	Dic₈₉	1	1	yes	characteristic
81	#125	—	2	1	no	—
82	C₂²⋊Q₈	—	2	1	no	—
83	C₂²⋊Q₈	Dic₈₉	1	2	yes	—
84	C₄×Q₈	—	2	1	no	—
85	C₄×Q₈	C₂×D₈₉	1	2	yes	—
86	#117	—	2	1	no	—
87	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
88	C₄⋊C₈	—	89	2	no	—
89	C₂²⋊C₈	—	178	1	no	—
90	#97	—	178	1	no	—
91	C₄×C₈	—	89	2	no	—
92	C₄×C₈	—	178	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	C₂×Dic₈₉	1	2	yes	—
95	C₂²×C₄	C₂×Dic₈₉	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
98	M₁₆	—	178	4	no	—
99	C₂×C₈	—	178	2	no	—
100	C₂×C₈	—	178	2	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
107	C₂²⋊C₄	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
110	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
111	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
112	C₄²	C₂²×D₈₉	1	1	yes	characteristic
113	C₂³	C₂²⋊Dic₈₉	1	1	yes	characteristic
114	Q₈	—	2	4	no	—
115	Q₈	—	4	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
123	C₂×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
124	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
127	C₈	—	356	2	no	—
128	C₈	—	178	4	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	4	2	no	—
137	C₄	—	4	2	no	—
138	C₄	#15667939	1	2	yes	—
139	C₄	#15668019	1	2	yes	—
140	C₄	#15668043	1	2	yes	—
141	C₂	—	4	1	no	—
142	C₂	#19151565	1	1	yes	characteristic
143	C₂	#19151607	1	1	yes	characteristic
144	C₂	C₂×#15667939	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151651

C₂

yes

maximal, characteristic

C₄×#13083558

C₂

yes

maximal, characteristic

Q₈⋊#10623494

C₂

yes

maximal, complemented

#533×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

—

C₄⋊#10623494

C₂²

yes

—

—

—

—

C₂²

yes

—

C₄⋊Q₈×C₈₉

C₄

yes

characteristic

C₄⋊Q₈×C₈₉

C₄

yes

characteristic

#125×C₈₉

—

C₂²⋊Q₈×C₈₉

—

C₂²⋊Q₈×C₈₉

C₄

yes

—

C₄×Q₈×C₈₉

—

C₄×Q₈×C₈₉

C₂²

yes

—

#117×C₈₉

—

C₂×C₄²×C₈₉

C₂²

yes

—

—

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₂²⋊C₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#104

yes

—

C₄×C₈₉

#131

yes

—

C₄×C₈₉

#137

yes

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#533

D₈₉

yes

characteristic

#431

—

C₄×M₁₆

—

Q₈⋊C₈

—

C₂×M₁₆

—

178

—

C₄⋊Q₈

Dic₈₉

yes

characteristic

C₄⋊Q₈

Dic₈₉

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

Dic₈₉

yes

—

C₄×Q₈

—

C₄×Q₈

C₂×D₈₉

yes

—

#117

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

178

—

#97

—

178

—

C₄×C₈

—

C₄×C₈

—

178

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂×Dic₈₉

yes

characteristic

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

M₁₆

—

178

—

C₂×C₈

—

178

—

100

C₂×C₈

—

178

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂×Dic₈₉

yes

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₄²

C₂×Dic₈₉

yes

characteristic

110

C₄²

C₂²⋊D₈₉

yes

characteristic

111

C₄²

C₂²⋊D₈₉

yes

characteristic

112

C₄²

C₂²×D₈₉

yes

characteristic

113

C₂³

C₂²⋊Dic₈₉

yes

characteristic

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²⋊Dic₈₉

yes

characteristic

122

C₂×C₄

C₂²⋊Dic₈₉

yes

characteristic

123

C₂×C₄

C₂²⋊Dic₈₉

yes

characteristic

124

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

127

C₈

—

356

—

128

C₈

—

178

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×C₂²⋊Dic₈₉

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#15667939

yes

—

139

C₄

#15668019

yes

—

140

C₄

#15668043

yes

—

141

C₂

—

142

C₂

#19151565

yes

characteristic

143

C₂

#19151607

yes

characteristic

144

C₂

C₂×#15667939

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₄²	C₂²×D₈₉
Fitting	#533×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₈₉	C₂²⋊C₄
cosocle	C₄²×C₈₉	C₂³
hypercenter	#533	D₈₉
hypocenter	C₈₉	#1485
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₄²

C₂²×D₈₉

Fitting

#533×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₈₉

C₂²⋊C₄

cosocle

C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. Q₈⋊#10623494	C₂	semidirect
73. C₈₉	#1485	semidirect

H ≅ G/N

type

3. Q₈⋊#10623494

C₂

semidirect

73. C₈₉

#1485

semidirect

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

subgroup	index
#19151651	2
C₄×#13083558	2
Q₈⋊#10623494	2
#533×C₈₉	2
#1485	89

subgroup

index

#533×C₈₉

Subgroups — #26018673

Conjugacy classes of subgroups

Complements

Maximal subgroups