Group #26026373 — subgroups

Subgroups — #26026373

Order 11392.

448 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15668012	C₂	1	1	yes	maximal, characteristic
2	#19151528	C₂	1	4	yes	maximal
3	#19151518	C₂	1	4	yes	maximal
4	#19151467	C₂	1	2	yes	maximal
5	#19151463	C₂	1	2	yes	maximal
6	#19151444	C₂	1	1	yes	maximal, characteristic
7	C₂×#125×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
9	#15668012	—	2	4	no	—
10	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
12	C₂×#13083553	C₂²	1	8	yes	—
13	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
14	#15667919	C₂²	1	2	yes	—
15	#15667919	C₂²	1	2	yes	—
16	#15667919	C₂²	1	8	yes	—
17	#125×C₈₉	—	2	4	no	—
18	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₇₈	—	2	4	no	—
24	C₂×Dic₁₇₈	—	4	4	no	—
25	C₂²×Dic₈₉	C₂³	1	4	yes	—
26	C₂²×Dic₈₉	C₂³	1	4	yes	—
27	C₄⋊Dic₈₉	—	2	4	no	—
28	C₄⋊Dic₈₉	—	2	4	no	—
29	#13083553	—	4	8	no	—
30	C₄×Dic₈₉	Q₈	1	8	yes	complemented
31	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
32	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
33	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₈₉	—	2	4	no	—
37	C₄⋊C₄×C₈₉	—	2	4	no	—
38	C₄⋊C₄×C₈₉	—	2	2	no	—
39	C₄⋊C₄×C₈₉	—	2	2	no	—
40	C₄²×C₈₉	D₄	1	4	yes	—
41	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₈₉	—	2	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	4	no	—
47	C₂×C₄×C₈₉	—	2	4	no	—
48	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
52	C₂×Dic₈₉	—	2	4	no	—
53	C₂×Dic₈₉	—	2	4	no	—
54	C₂×Dic₈₉	—	2	4	no	—
55	C₂×Dic₈₉	—	2	8	no	—
56	C₂×Dic₈₉	C₂×Q₈	1	8	yes	—
57	Dic₁₇₈	—	4	8	no	—
58	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
59	C₂²×C₈₉	C₂²×Q₈	1	2	yes	—
60	C₂²×C₈₉	H(5,𝔽₂)	1	2	yes	—
61	C₂²×C₈₉	2₋¹⁺⁴	1	2	yes	—
62	C₄×C₈₉	—	2	2	no	—
63	C₄×C₈₉	—	2	2	no	—
64	C₄×C₈₉	—	4	2	no	—
65	C₄×C₈₉	—	4	2	no	—
66	C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
67	Dic₈₉	—	2	8	no	—
68	Dic₈₉	—	4	4	no	—
69	C₂×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₈₉	#535	1	1	yes	characteristic
71	C₂×C₈₉	#536	1	1	yes	characteristic
72	C₂×C₈₉	#544	1	4	yes	—
73	#2660	—	89	1	no	maximal, Sylow
74	C₈₉	#2660	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₄⋊Q₈	—	89	1	no	—
76	C₂×#125	D₈₉	1	1	yes	characteristic
77	#398	—	89	4	no	—
78	#395	—	89	4	no	—
79	#384	—	89	2	no	—
80	#382	—	89	2	no	—
81	#376	—	89	1	no	—
82	C₂²×Q₈	—	89	2	no	—
83	C₄⋊Q₈	—	178	4	no	—
84	#125	—	2	4	no	—
85	C₂×C₄⋊C₄	—	89	8	no	—
86	C₂×C₄⋊C₄	—	89	2	no	—
87	C₂×C₄⋊C₄	—	89	2	no	—
88	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
91	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
92	C₂×C₄²	—	89	2	no	—
93	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
94	#95	—	89	2	no	—
95	#95	—	89	2	no	—
96	#95	—	89	8	no	—
97	C₂×Q₈	—	356	4	no	—
98	C₂×Q₈	—	178	4	no	—
99	C₂²×C₄	—	89	4	no	—
100	C₂²×C₄	—	89	4	no	—
101	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
102	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
103	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	2	4	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	356	8	no	—
111	C₄⋊C₄	—	178	4	no	—
112	C₄⋊C₄	—	178	4	no	—
113	C₄²	—	89	8	no	—
114	C₄²	C₂²⋊D₈₉	1	4	yes	—
115	C₂³	C₂³×D₈₉	1	1	yes	characteristic
116	Q₈	—	356	8	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	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	89	8	no	—
124	C₂×C₄	—	178	4	no	—
125	C₂×C₄	—	178	8	no	—
126	C₂×C₄	—	178	4	no	—
127	C₂×C₄	—	178	4	no	—
128	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
131	C₂×C₄	Q₈×D₈₉	1	4	yes	—
132	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
133	C₂²	C₂×Q₈×D₈₉	1	2	yes	—
134	C₂²	#15668058	1	2	yes	—
135	C₂²	#15668059	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	2	no	—
139	C₄	—	4	2	no	—
140	C₄	—	356	4	no	—
141	C₄	—	178	8	no	—
142	C₄	#15668010	1	4	yes	—
143	C₂	C₂×#15668010	1	1	yes	characteristic
144	C₂	#19152162	1	1	yes	characteristic
145	C₂	#19152169	1	1	yes	characteristic
146	C₂	#19152228	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668012

C₂

yes

maximal, characteristic

#19151528

C₂

yes

maximal

#19151518

C₂

yes

maximal

#19151467

C₂

yes

maximal

#19151463

C₂

yes

maximal

#19151444

C₂

yes

maximal, characteristic

C₂×#125×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

#15668012

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#125×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

Q₈

yes

complemented

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×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

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×Q₈

yes

—

Dic₁₇₈

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂²×Q₈

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄⋊Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

C₂×C₄⋊Q₈

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

#2660

—

maximal, Sylow

C₈₉

#2660

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#125

D₈₉

yes

—

—

—

—

—

C₂²×Q₈

—

C₄⋊Q₈

—

178

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₉

yes

—

—

—

C₂×Q₈

—

356

—

C₂×Q₈

—

178

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂²×D₈₉

yes

—

102

C₂²×C₄

C₂²×D₈₉

yes

—

103

C₂²×C₄

C₂²×D₈₉

yes

characteristic

104

C₂²×C₄

C₂²×D₈₉

yes

characteristic

105

C₂²×C₄

C₂²×D₈₉

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

356

—

111

C₄⋊C₄

—

178

—

112

C₄⋊C₄

—

178

—

113

C₄²

—

114

C₄²

C₂²⋊D₈₉

yes

—

115

C₂³

C₂³×D₈₉

yes

characteristic

116

Q₈

—

356

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

178

—

125

C₂×C₄

—

178

—

126

C₂×C₄

—

178

—

127

C₂×C₄

—

178

—

128

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

129

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

130

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

131

C₂×C₄

Q₈×D₈₉

yes

—

132

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

133

C₂²

C₂×Q₈×D₈₉

yes

—

134

C₂²

#15668058

yes

—

135

C₂²

#15668059

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

356

—

141

C₄

—

178

—

142

C₄

#15668010

yes

—

143

C₂

C₂×#15668010

yes

characteristic

144

C₂

#19152162

yes

characteristic

145

C₂

#19152169

yes

characteristic

146

C₂

#19152228

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₂×#125×C₈₉	C₂
hypercenter	C₂×#125	D₈₉
hypocenter	C₈₉	#2660
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₂×#125×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
30. C₄×Dic₈₉	Q₈	semidirect
74. C₈₉	#2660	semidirect

H ≅ G/N

type

30. C₄×Dic₈₉

Q₈

semidirect

74. C₈₉

#2660

semidirect

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

subgroup	index
C₂×#15668012	2
#19151528	2
#19151518	2
#19151467	2
#19151463	2
#19151444	2
C₂×#125×C₈₉	2
#2660	89

subgroup

index

C₂×#125×C₈₉

Subgroups — #26026373

Conjugacy classes of subgroups

Complements

Maximal subgroups