Group #26021730 — subgroups

Subgroups — #26021730

Order 11392.

230 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 95 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×#15667944	C₂	1	2	yes	maximal
2	#19151496	C₂	1	1	yes	maximal, characteristic
3	#19151468	C₂	1	1	yes	maximal, characteristic
4	#19151361	C₂	1	2	yes	maximal
5	#336×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₈₉	—	2	1	no	—
7	C₂×C₄⋊Dic₈₉	—	2	1	no	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
11	C₂×#13083553	—	2	1	no	—
12	#15667944	—	2	4	no	—
13	#15667919	—	2	1	no	—
14	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×Dic₈₉	—	2	1	no	—
17	C₂²×Dic₈₉	—	2	1	no	—
18	C₂²×Dic₈₉	—	2	2	no	—
19	C₄⋊Dic₈₉	—	2	1	no	—
20	C₄⋊Dic₈₉	—	2	2	no	—
21	C₄⋊Dic₈₉	—	2	2	no	—
22	C₄⋊Dic₈₉	—	2	2	no	—
23	C₄⋊Dic₈₉	—	4	2	no	—
24	C₄⋊Dic₈₉	—	4	2	no	—
25	C₄⋊Dic₈₉	D₄	1	2	yes	—
26	#13083553	—	4	2	no	—
27	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₈₉	—	2	2	no	—
31	C₂×C₈×C₈₉	—	2	2	no	—
32	C₂×C₈×C₈₉	Q₈	1	4	yes	—
33	C₄²×C₈₉	—	2	2	no	—
34	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₈₉	—	2	1	no	—
36	C₂×C₄×C₈₉	—	2	1	no	—
37	C₂×C₄×C₈₉	—	2	1	no	—
38	C₂×C₄×C₈₉	—	2	1	no	—
39	C₂×C₄×C₈₉	SD₁₆	1	2	yes	—
40	C₂×C₄×C₈₉	SD₁₆	1	2	yes	—
41	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₈₉	#41	1	2	yes	—
44	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
45	C₂×Dic₈₉	—	2	2	no	—
46	C₂×Dic₈₉	—	4	1	no	—
47	C₂×Dic₈₉	—	4	1	no	—
48	C₂×Dic₈₉	—	4	2	no	—
49	C₂×Dic₈₉	—	4	1	no	—
50	C₂×Dic₈₉	—	4	1	no	—
51	C₂×Dic₈₉	—	4	2	no	—
52	C₂×Dic₈₉	—	4	2	no	—
53	C₂×Dic₈₉	—	4	1	no	—
54	C₈×C₈₉	—	2	4	no	—
55	C₂²×C₈₉	#121	1	1	yes	characteristic
56	C₂²×C₈₉	#123	1	1	yes	characteristic
57	C₂²×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₈₉	#137	1	1	yes	characteristic
62	C₄×C₈₉	—	2	2	no	—
63	C₄×C₈₉	—	2	2	no	—
64	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₈₉	#125	1	2	yes	—
66	Dic₈₉	—	4	2	no	—
67	Dic₈₉	—	8	2	no	—
68	Dic₈₉	—	8	1	no	—
69	C₂×C₈₉	#398	1	1	yes	characteristic
70	C₂×C₈₉	#460	1	1	yes	characteristic
71	C₂×C₈₉	#461	1	1	yes	characteristic
72	C₂×C₈₉	#481	1	1	yes	characteristic
73	C₂×C₈₉	#483	1	1	yes	characteristic
74	C₂×C₈₉	#498	1	1	yes	characteristic
75	C₂×C₈₉	#501	1	1	yes	characteristic
76	#2084	—	89	1	no	maximal, Sylow
77	C₈₉	#2084	1	1	yes	characteristic, complemented, Sylow
78	C₂×#106	—	89	2	no	—
79	#389	—	89	1	no	—
80	#384	—	89	1	no	—
81	#340	—	89	2	no	—
82	#336	D₈₉	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₈₉	1	2	yes	—
84	C₂×C₄⋊C₄	—	89	1	no	—
85	C₂×C₄⋊C₄	—	89	1	no	—
86	C₂×C₄⋊C₄	—	89	2	no	—
87	C₂×C₄⋊C₄	—	178	1	no	—
88	C₂×C₄⋊C₄	—	178	1	no	—
89	C₂×C₄⋊C₄	—	178	1	no	—
90	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
91	#106	—	178	4	no	—
92	#95	—	178	1	no	—
93	C₂²×C₄	—	178	1	no	—
94	C₂²×C₄	—	178	2	no	—
95	C₂²×C₄	—	178	1	no	—
96	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
97	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	Dic₁₇₈	1	4	yes	—
102	C₄⋊C₄	—	89	2	no	—
103	C₄⋊C₄	—	356	2	no	—
104	C₄⋊C₄	—	356	2	no	—
105	C₄⋊C₄	—	356	2	no	—
106	C₄⋊C₄	—	178	2	no	—
107	C₄⋊C₄	—	178	2	no	—
108	C₄⋊C₄	—	178	2	no	—
109	C₄⋊C₄	—	178	1	no	—
110	C₄²	—	2	2	no	—
111	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	356	1	no	—
117	C₂×C₄	—	356	1	no	—
118	C₂×C₄	—	356	2	no	—
119	C₂×C₄	—	356	1	no	—
120	C₂×C₄	—	356	1	no	—
121	C₂×C₄	—	356	2	no	—
122	C₂×C₄	—	356	2	no	—
123	C₂×C₄	—	356	1	no	—
124	C₂×C₄	—	178	2	no	—
125	C₂×C₄	#13083563	1	2	yes	—
126	C₂×C₄	#13083563	1	2	yes	—
127	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
128	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
129	C₂×C₄	#13083573	1	2	yes	—
130	C₂×C₄	#13083575	1	1	yes	characteristic
131	C₈	—	2	4	no	—
132	C₂²	#15667987	1	1	yes	characteristic
133	C₂²	#15667996	1	1	yes	characteristic
134	C₂²	#15668012	1	1	yes	characteristic
135	C₂²	C₂×#13083563	1	1	yes	characteristic
136	C₂²	C₂×#13083563	1	1	yes	characteristic
137	C₂²	#15668040	1	1	yes	characteristic
138	C₂²	#15668045	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	356	2	no	—
142	C₄	—	712	1	no	—
143	C₄	—	712	2	no	—
144	C₄	#15667989	1	2	yes	—
145	C₄	#15668003	1	2	yes	—
146	C₂	#19151530	1	1	yes	characteristic
147	C₂	#19151799	1	1	yes	characteristic
148	C₂	#19151804	1	1	yes	characteristic
149	C₂	#19151917	1	1	yes	characteristic
150	C₂	#19151929	1	1	yes	characteristic
151	C₂	#19151996	1	1	yes	characteristic
152	C₂	#19152007	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667944

C₂

yes

maximal

#19151496

C₂

yes

maximal, characteristic

#19151468

C₂

yes

maximal, characteristic

#19151361

C₂

yes

maximal

#336×C₈₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

—

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

—

#13083553

—

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

Q₈

yes

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

SD₁₆

yes

—

C₂×C₄×C₈₉

SD₁₆

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

C₄⋊Q₈

yes

characteristic

C₂²×C₈₉

C₂×SD₁₆

yes

characteristic

C₂²×C₈₉

C₂×SD₁₆

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

#125

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#106

—

178

—

#95

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

D₃₅₆

yes

characteristic

C₂²×C₄

D₃₅₆

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

Dic₁₇₈

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

356

—

104

C₄⋊C₄

—

356

—

105

C₄⋊C₄

—

356

—

106

C₄⋊C₄

—

178

—

107

C₄⋊C₄

—

178

—

108

C₄⋊C₄

—

178

—

109

C₄⋊C₄

—

178

—

110

C₄²

—

111

C₂³

C₂×D₃₅₆

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

356

—

117

C₂×C₄

—

356

—

118

C₂×C₄

—

356

—

119

C₂×C₄

—

356

—

120

C₂×C₄

—

356

—

121

C₂×C₄

—

356

—

122

C₂×C₄

—

356

—

123

C₂×C₄

—

356

—

124

C₂×C₄

—

178

—

125

C₂×C₄

#13083563

yes

—

126

C₂×C₄

#13083563

yes

—

127

C₂×C₄

D₄×D₈₉

yes

characteristic

128

C₂×C₄

C₂×Dic₁₇₈

yes

—

129

C₂×C₄

#13083573

yes

—

130

C₂×C₄

#13083575

yes

characteristic

131

C₈

—

132

C₂²

#15667987

yes

characteristic

133

C₂²

#15667996

yes

characteristic

134

C₂²

#15668012

yes

characteristic

135

C₂²

C₂×#13083563

yes

characteristic

136

C₂²

C₂×#13083563

yes

characteristic

137

C₂²

#15668040

yes

characteristic

138

C₂²

#15668045

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

356

—

142

C₄

—

712

—

143

C₄

—

712

—

144

C₄

#15667989

yes

—

145

C₄

#15668003

yes

—

146

C₂

#19151530

yes

characteristic

147

C₂

#19151799

yes

characteristic

148

C₂

#19151804

yes

characteristic

149

C₂

#19151917

yes

characteristic

150

C₂

#19151929

yes

characteristic

151

C₂

#19151996

yes

characteristic

152

C₂

#19152007

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₅₆
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#336×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#336	D₈₉
hypocenter	C₈₉	#2084
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₃₅₆

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#336×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
77. C₈₉	#2084	semidirect

H ≅ G/N

type

77. C₈₉

#2084

semidirect

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

subgroup	index
C₂×#15667944	2
#19151496	2
#19151468	2
#19151361	2
#336×C₈₉	2
#2084	89

subgroup

index

#336×C₈₉

Subgroups — #26021730

Conjugacy classes of subgroups

Complements

Maximal subgroups