Group #26016191 — subgroups

Subgroups — C₈₉×#1717 #26016191

Order 11392.

208 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 72 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	#493×C₈₉	C₂	1	1	yes	maximal, characteristic
2	#492×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#487×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#486×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#486×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#486×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#322×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×SD₁₆×C₈₉	—	2	2	no	—
9	C₂×D₈×C₈₉	—	2	2	no	—
10	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
12	#127×C₈₉	C₂²	1	1	yes	characteristic
13	#127×C₈₉	C₂²	1	1	yes	characteristic
14	#103×C₈₉	—	2	2	no	—
15	#102×C₈₉	—	2	2	no	—
16	#102×C₈₉	—	2	2	no	—
17	#102×C₈₉	—	2	2	no	—
18	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₈₉	—	2	1	no	—
22	C₂×Q₈×C₈₉	—	2	1	no	—
23	C₂×D₄×C₈₉	—	2	1	no	—
24	C₂×D₄×C₈₉	—	2	1	no	—
25	C₂×D₄×C₈₉	—	2	1	no	—
26	C₂×D₄×C₈₉	—	2	1	no	—
27	C₂×D₄×C₈₉	—	2	1	no	—
28	C₂×D₄×C₈₉	—	2	1	no	—
29	SD₁₆×C₈₉	—	4	4	no	—
30	D₈×C₈₉	—	4	4	no	—
31	C₂×C₈×C₈₉	D₄	1	2	yes	—
32	C₂×C₈×C₈₉	D₄	1	2	yes	—
33	C₂×C₈×C₈₉	D₄	1	2	yes	—
34	C₄⋊C₄×C₈₉	—	2	1	no	—
35	C₄⋊C₄×C₈₉	—	2	1	no	—
36	C₄⋊C₄×C₈₉	—	2	1	no	—
37	C₄⋊C₄×C₈₉	—	2	1	no	—
38	C₄²×C₈₉	C₂³	1	1	yes	characteristic
39	C₂³×C₈₉	—	4	1	no	—
40	C₂³×C₈₉	—	4	1	no	—
41	Q₈×C₈₉	—	4	2	no	—
42	Q₈×C₈₉	—	4	2	no	—
43	D₄×C₈₉	—	4	2	no	—
44	D₄×C₈₉	—	4	2	no	—
45	D₄×C₈₉	—	4	2	no	—
46	D₄×C₈₉	—	4	2	no	—
47	D₄×C₈₉	—	4	2	no	—
48	D₄×C₈₉	—	4	2	no	—
49	C₂×C₄×C₈₉	—	4	1	no	—
50	C₂×C₄×C₈₉	—	4	1	no	—
51	C₂×C₄×C₈₉	C₂×D₄	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	C₈×C₈₉	—	2	2	no	—
55	C₈×C₈₉	—	2	2	no	—
56	C₈×C₈₉	—	2	2	no	—
57	C₂²×C₈₉	—	8	1	no	—
58	C₂²×C₈₉	—	8	1	no	—
59	C₂²×C₈₉	—	8	1	no	—
60	C₂²×C₈₉	—	8	1	no	—
61	C₂²×C₈₉	—	8	1	no	—
62	C₂²×C₈₉	—	8	1	no	—
63	C₂²×C₈₉	#127	1	1	yes	characteristic
64	C₄×C₈₉	—	8	1	no	—
65	C₄×C₈₉	—	8	1	no	—
66	C₄×C₈₉	#135	1	2	yes	—
67	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
68	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×C₈₉	—	16	1	no	—
70	C₂×C₈₉	—	16	1	no	—
71	C₂×C₈₉	#495	1	1	yes	characteristic
72	C₂×C₈₉	#496	1	1	yes	characteristic
73	C₂×C₈₉	#496	1	1	yes	characteristic
74	#1717	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₈₉	#1717	1	1	yes	characteristic, direct product factor, Sylow
76	#493	C₂×C₈₉	1	1	yes	characteristic
77	#492	C₂×C₈₉	1	1	yes	characteristic, complemented
78	#487	C₂×C₈₉	1	1	yes	characteristic, complemented
79	#486	C₂×C₈₉	1	1	yes	characteristic, complemented
80	#486	C₂×C₈₉	1	1	yes	characteristic, complemented
81	#486	C₂×C₈₉	1	1	yes	characteristic, complemented
82	#322	C₂×C₈₉	1	1	yes	characteristic, complemented
83	C₂×SD₁₆	—	2	2	no	—
84	C₂×D₈	—	2	2	no	—
85	C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
86	C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
87	#127	C₂²×C₈₉	1	1	yes	characteristic
88	#127	C₂²×C₈₉	1	1	yes	characteristic
89	#103	—	2	2	no	—
90	#102	—	2	2	no	—
91	#102	—	2	2	no	—
92	#102	—	2	2	no	—
93	C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
94	C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
95	C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	2	1	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	2	1	no	—
104	SD₁₆	—	4	4	no	—
105	D₈	—	4	4	no	—
106	C₂×C₈	D₄×C₈₉	1	2	yes	—
107	C₂×C₈	D₄×C₈₉	1	2	yes	—
108	C₂×C₈	D₄×C₈₉	1	2	yes	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄²	C₂³×C₈₉	1	1	yes	characteristic
114	C₂³	—	4	1	no	—
115	C₂³	—	4	1	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	4	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	4	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
127	C₂×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
128	C₂×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	2	2	no	—
131	C₈	—	2	2	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	8	1	no	—
138	C₂²	#127×C₈₉	1	1	yes	characteristic
139	C₄	—	8	1	no	—
140	C₄	—	8	1	no	—
141	C₄	#135×C₈₉	1	2	yes	—
142	C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
143	C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
144	C₂	—	16	1	no	—
145	C₂	—	16	1	no	—
146	C₂	#495×C₈₉	1	1	yes	characteristic
147	C₂	#496×C₈₉	1	1	yes	characteristic
148	C₂	#496×C₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#493×C₈₉

C₂

yes

maximal, characteristic

#492×C₈₉

C₂

yes

maximal, characteristic, complemented

#487×C₈₉

C₂

yes

maximal, characteristic, complemented

#486×C₈₉

C₂

yes

maximal, characteristic, complemented

#486×C₈₉

C₂

yes

maximal, characteristic, complemented

#486×C₈₉

C₂

yes

maximal, characteristic, complemented

#322×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₈₉

—

C₂×D₈×C₈₉

—

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#127×C₈₉

C₂²

yes

characteristic

#127×C₈₉

C₂²

yes

characteristic

#103×C₈₉

—

#102×C₈₉

—

#102×C₈₉

—

#102×C₈₉

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

SD₁₆×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#127

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#135

yes

—

C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₈₉

#1717

yes

characteristic, direct product factor, Sylow

#493

C₂×C₈₉

yes

characteristic

#492

C₂×C₈₉

yes

characteristic, complemented

#487

C₂×C₈₉

yes

characteristic, complemented

#486

C₂×C₈₉

yes

characteristic, complemented

#486

C₂×C₈₉

yes

characteristic, complemented

#486

C₂×C₈₉

yes

characteristic, complemented

#322

C₂×C₈₉

yes

characteristic, complemented

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂²×C₈₉

yes

characteristic

C₄⋊Q₈

C₂²×C₈₉

yes

characteristic

#127

C₂²×C₈₉

yes

characteristic

#127

C₂²×C₈₉

yes

—

—

—

—

C₄×C₈

C₂²×C₈₉

yes

characteristic

C₄×C₈

C₂²×C₈₉

yes

characteristic

C₄×C₈

C₂²×C₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

SD₁₆

—

105

D₈

—

106

C₂×C₈

D₄×C₈₉

yes

—

107

C₂×C₈

D₄×C₈₉

yes

—

108

C₂×C₈

D₄×C₈₉

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

C₂³×C₈₉

yes

characteristic

114

C₂³

—

115

C₂³

—

116

Q₈

—

117

Q₈

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂×D₄×C₈₉

yes

characteristic

127

C₂×C₄

C₂×D₄×C₈₉

yes

characteristic

128

C₂×C₄

C₂×D₄×C₈₉

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#127×C₈₉

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

#135×C₈₉

yes

—

142

C₄

AGL(1,ℤ/8)×C₈₉

yes

—

143

C₄

AGL(1,ℤ/8)×C₈₉

yes

—

144

C₂

—

145

C₂

—

146

C₂

#495×C₈₉

yes

characteristic

147

C₂

#496×C₈₉

yes

characteristic

148

C₂

#496×C₈₉

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₈₉
prime radical	#493×C₈₉	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₄²×C₈₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₈₉

#127

commutator, Frattini, cosocle

C₄²

C₂³×C₈₉

prime radical

#493×C₈₉

C₂

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₈₉ ◁ C₄²×C₈₉ ◁ G

N	H ≅ G/N	type
2. #492×C₈₉	C₂	semidirect
3. #487×C₈₉	C₂	semidirect
4. #486×C₈₉	C₂	semidirect
5. #486×C₈₉	C₂	semidirect
6. #486×C₈₉	C₂	semidirect
7. #322×C₈₉	C₂	semidirect
74. #1717	C₈₉	direct (N × H)
75. C₈₉	#1717	direct (N × H)
77. #492	C₂×C₈₉	semidirect
78. #487	C₂×C₈₉	semidirect
79. #486	C₂×C₈₉	semidirect
80. #486	C₂×C₈₉	semidirect
81. #486	C₂×C₈₉	semidirect
82. #322	C₂×C₈₉	semidirect

H ≅ G/N

type

2. #492×C₈₉

C₂

semidirect

3. #487×C₈₉

C₂

semidirect

4. #486×C₈₉

C₂

semidirect

5. #486×C₈₉

C₂

semidirect

6. #486×C₈₉

C₂

semidirect

7. #322×C₈₉

C₂

semidirect

74. #1717

C₈₉

direct (N × H)

75. C₈₉

#1717

direct (N × H)

77. #492

C₂×C₈₉

semidirect

78. #487

C₂×C₈₉

semidirect

79. #486

C₂×C₈₉

semidirect

80. #486

C₂×C₈₉

semidirect

81. #486

C₂×C₈₉

semidirect

82. #322

C₂×C₈₉

semidirect

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

subgroup	index
#493×C₈₉	2
#492×C₈₉	2
#487×C₈₉	2
#486×C₈₉	2
#322×C₈₉	2
#1717	89

subgroup

index

#493×C₈₉

#492×C₈₉

#487×C₈₉

#486×C₈₉

#322×C₈₉

#1717

Subgroups — C₈₉×#1717 #26016191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#1717 #26016191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#1717 #26016191