Group #26026366 — subgroups

Subgroups — #26026366

Order 11392.

524 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 181 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×#15667991	C₂	1	2	yes	maximal
2	#19151530	C₂	1	4	yes	maximal, complemented
3	#19151519	C₂	1	4	yes	maximal, complemented
4	#19151458	C₂	1	4	yes	maximal
5	C₂×#95×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
7	#15667991	—	2	8	no	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
11	C₂×#13083553	C₂²	1	4	yes	—
12	C₂×C₂²⋊Dic₈₉	C₂²	1	4	yes	—
13	#15667919	C₂²	1	8	yes	—
14	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₈₉	C₂²	1	2	yes	—
16	#95×C₈₉	C₂²	1	4	yes	—
17	C₂×Dic₁₇₈	—	2	4	no	—
18	C₂×Dic₁₇₈	—	4	4	no	—
19	C₂²×Dic₈₉	C₂³	1	4	yes	—
20	C₂²×Dic₈₉	C₂³	1	4	yes	—
21	C₄⋊Dic₈₉	—	2	4	no	—
22	C₄⋊Dic₈₉	—	4	4	no	—
23	C₄⋊Dic₈₉	—	4	4	no	—
24	#13083553	—	2	8	no	—
25	C₂²⋊Dic₈₉	—	2	8	no	—
26	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₈₉	—	4	2	no	—
28	C₂²×C₄×C₈₉	—	4	2	no	—
29	C₂²×C₄×C₈₉	—	4	2	no	—
30	C₂²×C₄×C₈₉	D₄	1	4	yes	—
31	C₂²×C₄×C₈₉	Q₈	1	8	yes	complemented
32	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
33	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
34	C₂³×C₈₉	—	2	2	no	—
35	C₂³×C₈₉	—	2	1	no	—
36	C₂³×C₈₉	—	2	1	no	—
37	C₂³×C₈₉	C₂×D₄	1	2	yes	—
38	C₂³×C₈₉	C₂×Q₈	1	4	yes	—
39	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
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×D₄	1	4	yes	—
46	C₂×C₄×C₈₉	C₂×Q₈	1	8	yes	—
47	C₂×Dic₈₉	—	2	4	no	—
48	C₂×Dic₈₉	—	2	4	no	—
49	C₂×Dic₈₉	—	2	4	no	—
50	C₂×Dic₈₉	—	2	8	no	—
51	C₂×Dic₈₉	—	2	4	no	—
52	Dic₁₇₈	—	4	8	no	—
53	C₂²×C₈₉	—	2	2	no	—
54	C₂²×C₈₉	—	2	2	no	—
55	C₂²×C₈₉	—	2	4	no	—
56	C₂²×C₈₉	—	2	4	no	—
57	C₂²×C₈₉	C₄⋊Q₈	1	4	yes	—
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	H(5,𝔽₂)	1	1	yes	characteristic
62	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
63	C₄×C₈₉	—	2	4	no	—
64	C₄×C₈₉	—	2	8	no	—
65	Dic₈₉	—	4	4	no	—
66	Dic₈₉	—	4	4	no	—
67	C₂×C₈₉	—	2	4	no	—
68	C₂×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₈₉	#534	1	1	yes	characteristic
70	C₂×C₈₉	#537	1	1	yes	characteristic
71	C₂×C₈₉	#543	1	2	yes	—
72	C₂×C₈₉	#544	1	2	yes	—
73	#2659	—	89	1	no	maximal, Sylow
74	C₈₉	#2659	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₂²⋊Q₈	—	89	2	no	—
76	#398	—	89	4	no	—
77	#395	—	89	4	no	—
78	#380	—	89	4	no	—
79	C₂×#95	D₈₉	1	1	yes	characteristic
80	C₂²×Q₈	—	89	2	no	—
81	C₂³×C₄	C₂×D₈₉	1	2	yes	—
82	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
83	C₂²⋊Q₈	—	178	8	no	—
84	C₂×C₄⋊C₄	—	89	4	no	—
85	C₂×C₄⋊C₄	—	89	4	no	—
86	C₂×C₄⋊C₄	—	89	4	no	—
87	C₂×C₄⋊C₄	—	89	2	no	—
88	C₂×C₂²⋊C₄	—	89	4	no	—
89	#95	—	89	8	no	—
90	#95	C₂×D₈₉	1	4	yes	—
91	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
92	C₂×Q₈	—	356	4	no	—
93	C₂×Q₈	—	178	4	no	—
94	C₂²×C₄	—	4	2	no	—
95	C₂²×C₄	—	4	2	no	—
96	C₂²×C₄	—	4	2	no	—
97	C₂²×C₄	—	89	4	no	—
98	C₂²×C₄	—	89	4	no	—
99	C₂²×C₄	Dic₁₇₈	1	8	yes	complemented
100	C₂²×C₄	D₃₅₆	1	4	yes	—
101	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
102	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
103	C₄⋊C₄	—	356	4	no	—
104	C₄⋊C₄	—	356	4	no	—
105	C₄⋊C₄	—	178	8	no	—
106	C₄⋊C₄	—	178	4	no	—
107	C₂²⋊C₄	—	178	8	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	1	no	—
110	C₂³	—	2	1	no	—
111	C₂³	C₂×D₃₅₆	1	2	yes	—
112	C₂³	C₂×Dic₁₇₈	1	4	yes	—
113	C₂³	C₂³×D₈₉	1	1	yes	characteristic
114	Q₈	—	356	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	8	no	—
119	C₂×C₄	—	4	4	no	—
120	C₂×C₄	—	178	4	no	—
121	C₂×C₄	—	178	4	no	—
122	C₂×C₄	—	178	4	no	—
123	C₂×C₄	—	178	8	no	—
124	C₂×C₄	—	178	4	no	—
125	C₂×C₄	C₂×D₃₅₆	1	4	yes	—
126	C₂×C₄	C₂×Dic₁₇₈	1	8	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	4	no	—
131	C₂²	#15668012	1	4	yes	—
132	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
133	C₂²	C₂²×Dic₁₇₈	1	2	yes	—
134	C₂²	#15668057	1	2	yes	—
135	C₂²	#15668057	1	1	yes	characteristic
136	C₂²	#15668060	1	1	yes	characteristic
137	C₄	—	2	4	no	—
138	C₄	—	2	8	no	—
139	C₄	—	356	4	no	—
140	C₄	—	356	4	no	—
141	C₂	—	2	4	no	—
142	C₂	C₂×#15668012	1	1	yes	characteristic
143	C₂	#19152160	1	1	yes	characteristic
144	C₂	#19152179	1	1	yes	characteristic
145	C₂	#19152224	1	2	yes	—
146	C₂	#19152229	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667991

C₂

yes

maximal

#19151530

C₂

yes

maximal, complemented

#19151519

C₂

yes

maximal, complemented

#19151458

C₂

yes

maximal

C₂×#95×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

#15667991

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

C₂³×C₄×C₈₉

C₂²

yes

characteristic

C₂³×C₄×C₈₉

C₂²

yes

—

#95×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂⁴×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

Q₈

yes

complemented

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

C₂×Q₈

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×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄⋊Q₈

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂²×Q₈

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₈₉

2₋¹⁺⁴

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×C₄⋊Q₈

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#544

yes

—

#2659

—

maximal, Sylow

C₈₉

#2659

yes

characteristic, complemented, Sylow

C₂×C₂²⋊Q₈

—

#398

—

#395

—

#380

—

C₂×#95

D₈₉

yes

characteristic

C₂²×Q₈

—

C₂³×C₄

C₂×D₈₉

yes

—

C₂³×C₄

C₂×D₈₉

yes

characteristic

C₂²⋊Q₈

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

#95

—

#95

C₂×D₈₉

yes

—

C₂⁴

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

356

—

C₂×Q₈

—

178

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₇₈

yes

complemented

100

C₂²×C₄

D₃₅₆

yes

—

101

C₂²×C₄

C₂²×D₈₉

yes

—

102

C₂²×C₄

C₂²×D₈₉

yes

—

103

C₄⋊C₄

—

356

—

104

C₄⋊C₄

—

356

—

105

C₄⋊C₄

—

178

—

106

C₄⋊C₄

—

178

—

107

C₂²⋊C₄

—

178

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

C₂×D₃₅₆

yes

—

112

C₂³

C₂×Dic₁₇₈

yes

—

113

C₂³

C₂³×D₈₉

yes

characteristic

114

Q₈

—

356

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

178

—

121

C₂×C₄

—

178

—

122

C₂×C₄

—

178

—

123

C₂×C₄

—

178

—

124

C₂×C₄

—

178

—

125

C₂×C₄

C₂×D₃₅₆

yes

—

126

C₂×C₄

C₂×Dic₁₇₈

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

#15668012

yes

—

132

C₂²

C₂²×D₃₅₆

yes

characteristic

133

C₂²

C₂²×Dic₁₇₈

yes

—

134

C₂²

#15668057

yes

—

135

C₂²

#15668057

yes

characteristic

136

C₂²

#15668060

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

356

—

140

C₄

—

356

—

141

C₂

—

142

C₂

C₂×#15668012

yes

characteristic

143

C₂

#19152160

yes

characteristic

144

C₂

#19152179

yes

characteristic

145

C₂

#19152224

yes

—

146

C₂

#19152229

yes

—

	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	#2659
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
2. #19151530	C₂	semidirect
3. #19151519	C₂	semidirect
31. C₂²×C₄×C₈₉	Q₈	semidirect
74. C₈₉	#2659	semidirect
99. C₂²×C₄	Dic₁₇₈	semidirect

H ≅ G/N

type

2. #19151530

C₂

semidirect

3. #19151519

C₂

semidirect

31. C₂²×C₄×C₈₉

Q₈

semidirect

74. C₈₉

#2659

semidirect

99. C₂²×C₄

Dic₁₇₈

semidirect

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

subgroup	index
C₂×#15667991	2
#19151530	2
#19151519	2
#19151458	2
C₂×#95×C₈₉	2
#2659	89

subgroup

index

C₂×#95×C₈₉

Subgroups — #26026366

Conjugacy classes of subgroups

Complements

Maximal subgroups