Group #26026222 — subgroups

Subgroups — #26026222

Order 11392.

436 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 149 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	#19151540	C₂	1	4	yes	maximal, complemented
2	#19151532	C₂	1	4	yes	maximal, complemented
3	#19151471	C₂	1	4	yes	maximal
4	#19151458	C₂	1	2	yes	maximal
5	C₂×#95×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
8	C₂×#13083553	C₂²	1	4	yes	—
9	C₂×#13083553	C₂²	1	4	yes	—
10	C₂×C₂²⋊Dic₈₉	C₂²	1	4	yes	—
11	#15667919	C₂²	1	4	yes	—
12	#15667919	C₂²	1	4	yes	—
13	#15667919	C₂²	1	4	yes	—
14	C₂³×C₄×C₈₉	C₂²	1	2	yes	—
15	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
16	#95×C₈₉	C₂²	1	4	yes	—
17	C₂²×Dic₈₉	C₂³	1	4	yes	—
18	C₂²×Dic₈₉	C₂³	1	4	yes	—
19	C₄⋊Dic₈₉	—	4	2	no	—
20	C₄⋊Dic₈₉	—	4	2	no	—
21	#13083553	—	4	4	no	—
22	#13083553	—	4	4	no	—
23	C₂²⋊Dic₈₉	—	2	8	no	—
24	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₈₉	—	2	4	no	—
26	C₂²×C₄×C₈₉	—	4	2	no	—
27	C₂²×C₄×C₈₉	—	4	2	no	—
28	C₂²×C₄×C₈₉	—	4	2	no	—
29	C₂²×C₄×C₈₉	Q₈	1	4	yes	—
30	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
31	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
32	C₂³×C₈₉	—	2	1	no	—
33	C₂³×C₈₉	—	2	2	no	—
34	C₂³×C₈₉	—	2	1	no	—
35	C₂³×C₈₉	C₂×Q₈	1	2	yes	—
36	C₂³×C₈₉	#41	1	4	yes	—
37	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
38	C₂×C₄×C₈₉	—	2	2	no	—
39	C₂×C₄×C₈₉	—	2	2	no	—
40	C₂×C₄×C₈₉	—	2	4	no	—
41	C₂×C₄×C₈₉	—	2	4	no	—
42	C₂×C₄×C₈₉	—	2	4	no	—
43	C₂×C₄×C₈₉	—	4	4	no	—
44	C₂×C₄×C₈₉	—	4	8	no	—
45	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
46	C₂×Dic₈₉	—	2	4	no	—
47	C₂×Dic₈₉	—	2	4	no	—
48	C₂×Dic₈₉	—	2	4	no	—
49	C₂×Dic₈₉	—	2	4	no	—
50	C₂×Dic₈₉	—	2	4	no	—
51	C₂×Dic₈₉	—	2	4	no	—
52	C₂²×C₈₉	—	2	4	no	—
53	C₂²×C₈₉	—	2	4	no	—
54	C₂²×C₈₉	—	2	2	no	—
55	C₂²×C₈₉	—	2	2	no	—
56	C₂²×C₈₉	#125	1	4	yes	—
57	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
58	C₂²×C₈₉	C₂×#41	1	2	yes	—
59	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
60	C₂²×C₈₉	H(5,𝔽₂)	1	2	yes	—
61	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
62	C₄×C₈₉	—	2	4	no	—
63	C₄×C₈₉	—	4	4	no	—
64	Dic₈₉	—	4	4	no	—
65	Dic₈₉	—	4	4	no	—
66	C₂×C₈₉	—	2	4	no	—
67	C₂×C₈₉	C₂×#125	1	1	yes	characteristic
68	C₂×C₈₉	#538	1	2	yes	—
69	C₂×C₈₉	#539	1	2	yes	—
70	C₂×C₈₉	#543	1	1	yes	characteristic
71	C₂×C₈₉	#544	1	1	yes	characteristic
72	#2646	—	89	1	no	maximal, Sylow
73	C₈₉	#2646	1	1	yes	characteristic, complemented, Sylow
74	#400	—	89	4	no	—
75	#398	—	89	4	no	—
76	#385	—	89	4	no	—
77	#380	—	89	2	no	—
78	C₂×#95	D₈₉	1	1	yes	characteristic
79	C₂³×C₄	C₂×D₈₉	1	2	yes	—
80	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
81	C₂×C₄⋊C₄	—	89	2	no	—
82	C₂×C₄⋊C₄	—	89	2	no	—
83	C₂×C₄⋊C₄	—	89	4	no	—
84	C₂×C₄⋊C₄	—	89	4	no	—
85	C₂×C₂²⋊C₄	—	89	4	no	—
86	#95	—	89	4	no	—
87	#95	—	89	4	no	—
88	#95	—	89	4	no	—
89	#95	C₂×D₈₉	1	4	yes	—
90	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
91	C₂²×C₄	—	2	4	no	—
92	C₂²×C₄	—	4	2	no	—
93	C₂²×C₄	—	4	2	no	—
94	C₂²×C₄	—	4	2	no	—
95	C₂²×C₄	—	89	4	no	—
96	C₂²×C₄	—	89	4	no	—
97	C₂²×C₄	Dic₁₇₈	1	4	yes	—
98	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
99	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
100	C₄⋊C₄	—	356	4	no	—
101	C₄⋊C₄	—	356	4	no	—
102	C₄⋊C₄	—	356	2	no	—
103	C₄⋊C₄	—	356	2	no	—
104	C₂²⋊C₄	—	178	8	no	—
105	C₂³	—	2	1	no	—
106	C₂³	—	2	2	no	—
107	C₂³	—	2	1	no	—
108	C₂³	C₂×Dic₁₇₈	1	2	yes	—
109	C₂³	#13083574	1	4	yes	—
110	C₂³	C₂³×D₈₉	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	4	8	no	—
118	C₂×C₄	—	178	4	no	—
119	C₂×C₄	—	178	4	no	—
120	C₂×C₄	—	178	4	no	—
121	C₂×C₄	—	178	4	no	—
122	C₂×C₄	—	178	4	no	—
123	C₂×C₄	—	178	4	no	—
124	C₂×C₄	C₂×Dic₁₇₈	1	4	yes	—
125	C₂²	—	2	4	no	—
126	C₂²	—	2	4	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	#15668004	1	4	yes	—
130	C₂²	C₂²×Dic₁₇₈	1	1	yes	characteristic
131	C₂²	C₂×#13083574	1	2	yes	—
132	C₂²	#15668057	1	1	yes	characteristic
133	C₂²	#15668057	1	2	yes	—
134	C₂²	#15668060	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	4	4	no	—
137	C₄	—	356	4	no	—
138	C₄	—	356	4	no	—
139	C₂	—	2	4	no	—
140	C₂	C₂×#15668004	1	1	yes	characteristic
141	C₂	#19152186	1	2	yes	—
142	C₂	#19152195	1	2	yes	—
143	C₂	#19152224	1	1	yes	characteristic
144	C₂	#19152229	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151540

C₂

yes

maximal, complemented

#19151532

C₂

yes

maximal, complemented

#19151471

C₂

yes

maximal

#19151458

C₂

yes

maximal

C₂×#95×C₈₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂³×C₄×C₈₉

C₂²

yes

—

C₂³×C₄×C₈₉

C₂²

yes

characteristic

#95×C₈₉

C₂²

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂⁴×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

Q₈

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×Q₈

yes

—

C₂³×C₈₉

#41

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×C₄×C₈₉

C₂×Q₈

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#125

yes

—

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#125

yes

characteristic

C₂×C₈₉

#538

yes

—

C₂×C₈₉

#539

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×#95

D₈₉

yes

characteristic

C₂³×C₄

C₂×D₈₉

yes

—

C₂³×C₄

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

—

—

—

C₂×D₈₉

yes

—

C₂⁴

C₂²×D₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₇₈

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

100

C₄⋊C₄

—

356

—

101

C₄⋊C₄

—

356

—

102

C₄⋊C₄

—

356

—

103

C₄⋊C₄

—

356

—

104

C₂²⋊C₄

—

178

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₂×Dic₁₇₈

yes

—

109

C₂³

#13083574

yes

—

110

C₂³

C₂³×D₈₉

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

178

—

119

C₂×C₄

—

178

—

120

C₂×C₄

—

178

—

121

C₂×C₄

—

178

—

122

C₂×C₄

—

178

—

123

C₂×C₄

—

178

—

124

C₂×C₄

C₂×Dic₁₇₈

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#15668004

yes

—

130

C₂²

C₂²×Dic₁₇₈

yes

characteristic

131

C₂²

C₂×#13083574

yes

—

132

C₂²

#15668057

yes

characteristic

133

C₂²

#15668057

yes

—

134

C₂²

#15668060

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

356

—

138

C₄

—

356

—

139

C₂

—

140

C₂

C₂×#15668004

yes

characteristic

141

C₂

#19152186

yes

—

142

C₂

#19152195

yes

—

143

C₂

#19152224

yes

characteristic

144

C₂

#19152229

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₂×#95×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₂³
hypercenter	C₂×#95	D₈₉
hypocenter	C₈₉	#2646
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×#95

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₂×#95×C₈₉

C₂

prime radical

C₂⁴×C₈₉

C₂³

hypercenter

C₂×#95

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₂×#95

N	H ≅ G/N	type
1. #19151540	C₂	semidirect
2. #19151532	C₂	semidirect
73. C₈₉	#2646	semidirect

H ≅ G/N

type

1. #19151540

C₂

semidirect

2. #19151532

C₂

semidirect

73. C₈₉

#2646

semidirect

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

subgroup	index
#19151540	2
#19151532	2
#19151471	2
#19151458	2
C₂×#95×C₈₉	2
#2646	89

subgroup

index

C₂×#95×C₈₉

Subgroups — #26026222

Conjugacy classes of subgroups

Complements

Maximal subgroups