Group #26022296 — subgroups

Subgroups — #26022296

Order 11392.

166 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 49 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×#15667957	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667955	C₂	1	1	yes	maximal, characteristic
3	#19151976	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151680	C₂	1	1	yes	maximal, characteristic
5	#19151416	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151401	C₂	1	1	yes	maximal, characteristic, complemented
7	#358×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
9	C₂×#13083564	—	2	1	no	—
10	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
11	#15668010	—	2	1	no	—
12	#15667974	—	2	1	no	—
13	#15667957	—	2	2	no	—
14	#15667955	—	2	2	no	—
15	C₂×#13083549	C₂²	1	1	yes	characteristic
16	#15667932	—	2	1	no	—
17	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
20	#107×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₇₈	—	2	1	no	—
22	C₂×D₃₅₆	—	2	1	no	—
23	C₂×C₄×D₈₉	—	4	1	no	—
24	Dic₃₅₆	—	2	1	no	—
25	Dic₃₅₆	—	2	1	no	—
26	#13083564	—	4	2	no	—
27	D₇₁₂	—	2	1	no	—
28	D₇₁₂	D₄	1	2	yes	—
29	C₂×#10623494	D₄	1	1	yes	characteristic
30	#13083553	—	4	1	no	—
31	#13083553	—	4	1	no	—
32	#13083551	—	4	1	no	—
33	C₄×Dic₈₉	D₄	1	1	yes	characteristic
34	#13083549	—	2	1	no	—
35	C₂×Q₈×C₈₉	—	2	1	no	—
36	Q₁₆×C₈₉	—	2	1	no	—
37	Q₁₆×C₈₉	—	2	1	no	—
38	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₈₉	—	2	1	no	—
40	C₁₆×C₈₉	—	2	1	no	—
41	C₂²×D₈₉	—	4	1	no	—
42	Q₈×C₈₉	—	4	1	no	—
43	Q₈×C₈₉	—	4	1	no	—
44	C₂×C₄×C₈₉	—	4	1	no	—
45	C₂×C₄×C₈₉	—	4	1	no	—
46	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₈₉	—	4	1	no	—
48	C₂×Dic₈₉	D₈	1	1	yes	characteristic
49	C₂×Dic₈₉	D₈	1	1	yes	characteristic
50	D₃₅₆	—	2	2	no	—
51	D₃₅₆	—	4	1	no	—
52	C₄×D₈₉	—	8	2	no	—
53	Dic₁₇₈	—	4	1	no	—
54	Dic₁₇₈	—	4	1	no	—
55	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₈×C₈₉	#41	1	1	yes	characteristic
57	#10623494	—	2	1	no	—
58	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
59	C₂×D₈₉	—	4	2	no	—
60	C₂×D₈₉	—	8	1	no	—
61	C₂×D₈₉	—	8	1	no	—
62	C₄×C₈₉	—	8	1	no	—
63	C₄×C₈₉	—	8	1	no	—
64	C₄×C₈₉	#121	1	1	yes	characteristic
65	C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
66	Dic₈₉	—	2	1	no	—
67	Dic₈₉	—	8	1	no	—
68	Dic₈₉	SD₃₂	1	2	yes	complemented
69	C₂×C₈₉	#459	1	1	yes	characteristic
70	C₂×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
71	C₂×C₈₉	#510	1	1	yes	characteristic
72	D₈₉	—	8	2	no	—
73	#2208	—	89	1	no	maximal, Sylow
74	C₈₉	#2208	1	1	yes	characteristic, complemented, Sylow
75	C₂×SD₃₂	—	89	1	no	—
76	C₂×SD₃₂	—	89	1	no	—
77	#494	—	89	1	no	—
78	C₄×D₈	—	89	1	no	—
79	C₄⋊C₁₆	—	89	1	no	—
80	#358	—	89	1	no	—
81	#358	D₈₉	1	1	yes	characteristic, complemented
82	C₂×Q₁₆	—	89	1	no	—
83	C₂×Q₁₆	—	178	1	no	—
84	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
85	C₂×D₈	—	89	1	no	—
86	C₄⋊Q₈	—	178	1	no	—
87	C₄×D₄	—	178	1	no	—
88	SD₃₂	—	178	2	no	—
89	SD₃₂	—	178	2	no	—
90	C₂×C₁₆	—	89	1	no	—
91	C₂×C₁₆	C₂×D₈₉	1	1	yes	characteristic
92	#107	C₂×D₈₉	1	1	yes	characteristic
93	#102	—	178	1	no	—
94	C₄×C₈	—	89	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	178	1	no	—
97	C₂×D₄	—	178	1	no	—
98	C₂²×C₄	—	356	1	no	—
99	Q₁₆	—	2	1	no	—
100	Q₁₆	—	2	1	no	—
101	Q₁₆	—	356	2	no	—
102	Q₁₆	—	178	1	no	—
103	Q₁₆	—	178	1	no	—
104	D₈	—	89	2	no	—
105	D₈	—	178	1	no	—
106	C₂×C₈	—	89	1	no	—
107	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	356	1	no	—
110	C₄⋊C₄	—	356	1	no	—
111	C₂²⋊C₄	—	356	1	no	—
112	C₄²	—	89	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	—	178	1	no	—
115	C₂³	—	356	1	no	—
116	Q₈	—	4	1	no	—
117	Q₈	—	4	1	no	—
118	Q₈	—	356	1	no	—
119	Q₈	—	356	1	no	—
120	D₄	—	356	1	no	—
121	D₄	—	178	2	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	89	1	no	—
125	C₂×C₄	—	89	1	no	—
126	C₂×C₄	—	356	1	no	—
127	C₂×C₄	—	712	2	no	—
128	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
129	C₈	—	178	1	no	—
130	C₈	D₄×D₈₉	1	1	yes	characteristic
131	C₈	#13083574	1	1	yes	characteristic
132	C₂²	—	356	2	no	—
133	C₂²	—	712	1	no	—
134	C₂²	—	712	1	no	—
135	C₂²	D₈×D₈₉	1	1	yes	characteristic
136	C₄	—	8	1	no	—
137	C₄	—	8	1	no	—
138	C₄	—	89	2	no	—
139	C₄	—	178	1	no	—
140	C₄	—	712	1	no	—
141	C₄	#15667985	1	1	yes	characteristic
142	C₄	#15668039	1	1	yes	characteristic
143	C₂	—	712	2	no	—
144	C₂	#19151792	1	1	yes	characteristic
145	C₂	SD₃₂×D₈₉	1	1	yes	characteristic
146	C₂	#19152040	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667957

C₂

yes

maximal, characteristic

C₂×#15667955

C₂

yes

maximal, characteristic

#19151976

C₂

yes

maximal, characteristic, complemented

#19151680

C₂

yes

maximal, characteristic

#19151416

C₂

yes

maximal, characteristic, complemented

#19151401

C₂

yes

maximal, characteristic, complemented

#358×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₅₆

C₂²

yes

characteristic

C₂×#13083564

—

C₂×D₇₁₂

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

#15667932

—

C₈×Dic₈₉

C₂²

yes

characteristic

C₂×Q₁₆×C₈₉

C₂²

yes

characteristic

C₂×C₁₆×C₈₉

C₂²

yes

characteristic

#107×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×C₄×D₈₉

—

Dic₃₅₆

—

Dic₃₅₆

—

#13083564

—

D₇₁₂

—

D₇₁₂

D₄

yes

—

C₂×#10623494

D₄

yes

—

—

—

C₄×Dic₈₉

D₄

yes

characteristic

#13083549

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

—

C₂²×D₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

D₈

yes

characteristic

C₂×Dic₈₉

D₈

yes

characteristic

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

C₂×D₄

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

C₂×D₈

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

Dic₈₉

SD₃₂

yes

complemented

C₂×C₈₉

#459

yes

characteristic

C₂×C₈₉

C₂×SD₃₂

yes

characteristic

C₂×C₈₉

#510

yes

characteristic

D₈₉

—

#2208

—

maximal, Sylow

C₈₉

#2208

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

C₂×SD₃₂

—

#494

—

C₄×D₈

—

C₄⋊C₁₆

—

#358

—

#358

D₈₉

yes

characteristic, complemented

C₂×Q₁₆

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

C₂×D₈₉

yes

characteristic

C₂×D₈

—

C₄⋊Q₈

—

178

—

C₄×D₄

—

178

—

SD₃₂

—

178

—

SD₃₂

—

178

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₈₉

yes

characteristic

#107

C₂×D₈₉

yes

characteristic

#102

—

178

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×D₄

—

178

—

C₂²×C₄

—

356

—

Q₁₆

—

100

Q₁₆

—

101

Q₁₆

—

356

—

102

Q₁₆

—

178

—

103

Q₁₆

—

178

—

104

D₈

—

105

D₈

—

178

—

106

C₂×C₈

—

107

C₂×C₈

C₂²×D₈₉

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

356

—

110

C₄⋊C₄

—

356

—

111

C₂²⋊C₄

—

356

—

112

C₄²

—

113

C₁₆

—

114

C₁₆

—

178

—

115

C₂³

—

356

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

356

—

119

Q₈

—

356

—

120

D₄

—

356

—

121

D₄

—

178

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

356

—

127

C₂×C₄

—

712

—

128

C₂×C₄

D₄×D₈₉

yes

characteristic

129

C₈

—

178

—

130

C₈

D₄×D₈₉

yes

characteristic

131

C₈

#13083574

yes

characteristic

132

C₂²

—

356

—

133

C₂²

—

712

—

134

C₂²

—

712

—

135

C₂²

D₈×D₈₉

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

178

—

140

C₄

—

712

—

141

C₄

#15667985

yes

characteristic

142

C₄

#15668039

yes

characteristic

143

C₂

—

712

—

144

C₂

#19151792

yes

characteristic

145

C₂

SD₃₂×D₈₉

yes

characteristic

146

C₂

#19152040

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂³
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	#358×C₈₉	C₂
socle	C₂²×C₈₉	C₂×D₈
prime radical	C₂×D₇₁₂	C₂²
hypercenter	#358	D₈₉
hypocenter	C₈₉	#2208
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₈₉

commutator, cosocle

C₂×C₈×C₈₉

C₂³

Frattini

C₂×C₈

C₂²×D₈₉

Fitting

#358×C₈₉

C₂

socle

C₂²×C₈₉

C₂×D₈

prime radical

C₂×D₇₁₂

C₂²

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
3. #19151976	C₂	semidirect
5. #19151416	C₂	semidirect
6. #19151401	C₂	semidirect
7. #358×C₈₉	C₂	semidirect
68. Dic₈₉	SD₃₂	semidirect
74. C₈₉	#2208	semidirect
81. #358	D₈₉	semidirect

H ≅ G/N

type

3. #19151976

C₂

semidirect

5. #19151416

C₂

semidirect

6. #19151401

C₂

semidirect

7. #358×C₈₉

C₂

semidirect

68. Dic₈₉

SD₃₂

semidirect

74. C₈₉

#2208

semidirect

81. #358

D₈₉

semidirect

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

subgroup	index
C₂×#15667957	2
C₂×#15667955	2
#19151976	2
#19151680	2
#19151416	2
#19151401	2
#358×C₈₉	2
#2208	89

subgroup

index

#358×C₈₉

Subgroups — #26022296

Conjugacy classes of subgroups

Complements

Maximal subgroups