Group #26030762 — subgroups

Subgroups — #26030762

Order 11392.

356 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 157 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	#19152155	C₂	1	1	yes	maximal, characteristic
2	#19152001	C₂	1	2	yes	maximal, complemented
3	#19151997	C₂	1	4	yes	maximal, complemented
4	#19151994	C₂	1	2	yes	maximal, complemented
5	#19151634	C₂	1	4	yes	maximal
6	C₂×C₄×#10623494	C₂	1	1	yes	maximal, characteristic
7	#533×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10623494	C₂²	1	2	yes	—
9	#15668012	C₂²	1	2	yes	—
10	#15668004	C₂²	1	2	yes	—
11	#15667991	—	2	2	no	—
12	C₄×Dic₁₇₈	—	2	2	no	—
13	#15667971	C₂²	1	2	yes	—
14	#15667945	C₂²	1	8	yes	—
15	#15667943	C₂²	1	8	yes	—
16	C₄×#10623494	C₂²	1	1	yes	characteristic
17	C₄×#10623494	C₂²	1	2	yes	—
18	C₄×#10623494	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
20	#125×C₈₉	C₂²	1	2	yes	—
21	C₂²⋊Q₈×C₈₉	—	2	2	no	—
22	C₄×Q₈×C₈₉	—	2	2	no	—
23	#117×C₈₉	C₂²	1	2	yes	—
24	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₇₈	—	2	2	no	—
26	C₂×#10623494	Q₈	1	8	yes	complemented
27	C₂×#10623494	C₂³	1	2	yes	—
28	C₂×#10623494	C₂³	1	2	yes	—
29	C₄⋊Dic₈₉	—	2	2	no	—
30	C₄⋊Dic₈₉	C₂³	1	4	yes	—
31	#13083553	—	2	4	no	—
32	C₂²⋊Dic₈₉	—	2	2	no	—
33	C₄×Dic₈₉	—	2	2	no	—
34	C₂×Q₈×C₈₉	—	2	2	no	—
35	C₂²×C₄×C₈₉	D₄	1	2	yes	—
36	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₈₉	—	2	4	no	—
38	C₄⋊C₄×C₈₉	—	2	2	no	—
39	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
40	C₂²⋊C₄×C₈₉	—	2	2	no	—
41	C₄²×C₈₉	—	2	2	no	—
42	C₄²×C₈₉	D₄	1	2	yes	—
43	C₄²×C₈₉	C₂³	1	1	yes	characteristic
44	C₄²×C₈₉	C₂³	1	1	yes	characteristic
45	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₈₉	—	4	4	no	—
47	C₂×C₄×C₈₉	—	2	4	no	—
48	C₂×C₄×C₈₉	—	2	4	no	—
49	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₈₉	—	2	4	no	—
55	Dic₁₇₈	—	4	4	no	—
56	#10623494	C₂×Q₈	1	4	yes	—
57	#10623494	C₂×Q₈	1	4	yes	—
58	C₂²×C₈₉	—	2	2	no	—
59	C₂²×C₈₉	C₄⋊Q₈	1	2	yes	—
60	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
61	C₄×C₈₉	—	2	2	no	—
62	C₄×C₈₉	—	4	4	no	—
63	C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
64	C₄×C₈₉	#135	1	4	yes	—
65	C₄×C₈₉	C₂²×Q₈	1	2	yes	—
66	Dic₈₉	—	4	4	no	—
67	C₂×C₈₉	—	2	2	no	—
68	C₂×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₈₉	C₂×#135	1	2	yes	—
70	#3160	—	89	1	no	maximal, Sylow
71	C₈₉	#3160	1	1	yes	characteristic, complemented, Sylow
72	#533	—	89	1	no	—
73	#533	D₈₉	1	1	yes	characteristic
74	#500	—	89	2	no	—
75	#499	—	89	4	no	—
76	#498	—	89	2	no	—
77	#427	—	89	4	no	—
78	C₂×C₄×C₈	—	89	1	no	—
79	C₂²×C₈	—	89	2	no	—
80	C₄⋊Q₈	—	89	2	no	—
81	C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
82	#125	—	89	2	no	—
83	#125	C₂×D₈₉	1	2	yes	—
84	C₂²⋊Q₈	—	2	2	no	—
85	C₂²⋊Q₈	—	178	2	no	—
86	C₄×Q₈	—	2	2	no	—
87	C₄×Q₈	—	178	2	no	—
88	#117	—	89	2	no	—
89	#117	C₂×D₈₉	1	2	yes	—
90	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
91	#107	—	89	8	no	—
92	#106	—	89	8	no	—
93	C₄×C₈	—	89	1	no	—
94	C₄×C₈	—	89	2	no	—
95	C₄×C₈	—	89	1	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	178	2	no	—
98	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
99	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
100	C₂×C₈	—	89	8	no	—
101	C₂×C₈	—	89	2	no	—
102	C₂×C₈	—	89	2	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	89	4	no	—
106	C₄⋊C₄	—	178	4	no	—
107	C₄⋊C₄	—	178	2	no	—
108	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₂²⋊C₄	—	178	2	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	178	2	no	—
113	C₄²	C₂²⋊D₈₉	1	2	yes	—
114	C₄²	C₂²×D₈₉	1	1	yes	characteristic
115	C₄²	C₂²×D₈₉	1	1	yes	characteristic
116	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
117	Q₈	—	4	4	no	—
118	Q₈	—	356	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	178	4	no	—
122	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
123	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₈₉	1	4	yes	—
126	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
127	C₈	—	89	4	no	—
128	C₈	—	89	4	no	—
129	C₂²	—	2	2	no	—
130	C₂²	#15668010	1	2	yes	—
131	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	4	4	no	—
134	C₄	—	356	4	no	—
135	C₄	#15668010	1	2	yes	—
136	C₄	#15668032	1	4	yes	—
137	C₄	C₂×Q₈×D₈₉	1	2	yes	—
138	C₂	—	2	2	no	—
139	C₂	C₂×#15668010	1	1	yes	characteristic
140	C₂	C₂×#15668032	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152155

C₂

yes

maximal, characteristic

#19152001

C₂

yes

maximal, complemented

#19151997

C₂

yes

maximal, complemented

#19151994

C₂

yes

maximal, complemented

#19151634

C₂

yes

maximal

C₂×C₄×#10623494

C₂

yes

maximal, characteristic

#533×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

C₂²

yes

—

#15668012

C₂²

yes

—

#15668004

C₂²

yes

—

#15667991

—

C₄×Dic₁₇₈

—

#15667971

C₂²

yes

—

#15667945

C₂²

yes

—

#15667943

C₂²

yes

—

C₄×#10623494

C₂²

yes

characteristic

C₄×#10623494

C₂²

yes

—

C₄×#10623494

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

—

#125×C₈₉

C₂²

yes

—

C₂²⋊Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

#117×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×#10623494

Q₈

yes

complemented

C₂×#10623494

C₂³

yes

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

#10623494

C₂×Q₈

yes

—

#10623494

C₂×Q₈

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄⋊Q₈

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄⋊Q₈

yes

—

C₄×C₈₉

#135

yes

—

C₄×C₈₉

C₂²×Q₈

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₈₉

C₂×#135

yes

—

#3160

—

maximal, Sylow

C₈₉

#3160

yes

characteristic, complemented, Sylow

#533

—

#533

D₈₉

yes

—

—

—

—

C₂×C₄×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₉

yes

—

#125

—

#125

C₂×D₈₉

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

178

—

C₄×Q₈

—

C₄×Q₈

—

178

—

#117

—

#117

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#107

—

#106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂²×C₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

178

—

107

C₄⋊C₄

—

178

—

108

C₄⋊C₄

C₂²×D₈₉

yes

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

178

—

111

C₄²

—

112

C₄²

—

178

—

113

C₄²

C₂²⋊D₈₉

yes

—

114

C₄²

C₂²×D₈₉

yes

characteristic

115

C₄²

C₂²×D₈₉

yes

characteristic

116

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

117

Q₈

—

118

Q₈

—

356

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

178

—

122

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

123

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

124

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

125

C₂×C₄

Q₈×D₈₉

yes

—

126

C₂×C₄

C₂³×D₈₉

yes

characteristic

127

C₈

—

128

C₈

—

129

C₂²

—

130

C₂²

#15668010

yes

—

131

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

356

—

135

C₄

#15668010

yes

—

136

C₄

#15668032

yes

—

137

C₄

C₂×Q₈×D₈₉

yes

—

138

C₂

—

139

C₂

C₂×#15668010

yes

characteristic

140

C₂

C₂×#15668032

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#533×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#533	D₈₉
hypocenter	C₈₉	#3160
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, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#533×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #19152001	C₂	semidirect
3. #19151997	C₂	semidirect
4. #19151994	C₂	semidirect
26. C₂×#10623494	Q₈	semidirect
71. C₈₉	#3160	semidirect

H ≅ G/N

type

2. #19152001

C₂

semidirect

3. #19151997

C₂

semidirect

4. #19151994

C₂

semidirect

26. C₂×#10623494

Q₈

semidirect

71. C₈₉

#3160

semidirect

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

subgroup	index
#19152155	2
#19152001	2
#19151997	2
#19151994	2
#19151634	2
C₂×C₄×#10623494	2
#533×C₈₉	2
#3160	89

subgroup

index

#533×C₈₉

Subgroups — #26030762

Conjugacy classes of subgroups

Complements

Maximal subgroups