Group #26020247 — subgroups

Subgroups — #26020247

Order 11392.

272 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 87 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×#15667977	C₂	1	1	yes	maximal, characteristic
2	#19151621	C₂	1	2	yes	maximal
3	C₂×#15667930	C₂	1	2	yes	maximal
4	#19151367	C₂	1	1	yes	maximal, characteristic
5	#343×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
7	C₂×#13083558	C₂²	1	2	yes	—
8	#15667977	—	2	2	no	—
9	#15667977	—	2	2	no	—
10	C₂×#13083553	C₄	1	2	yes	—
11	C₂×C₄×Dic₈₉	C₄	1	2	yes	—
12	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
13	#15667941	—	2	2	no	—
14	#15667941	—	2	2	no	—
15	#15667930	—	2	4	no	—
16	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₇₈	—	2	2	no	—
19	C₂×Dic₁₇₈	—	2	2	no	—
20	C₂×Dic₁₇₈	D₄	1	2	yes	—
21	C₂×Dic₁₇₈	Q₈	1	2	yes	—
22	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
24	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
25	#13083558	—	2	2	no	—
26	#13083558	—	2	2	no	—
27	C₂×#10623494	—	2	2	no	—
28	#13083553	—	2	4	no	—
29	C₄×Dic₈₉	—	2	2	no	—
30	C₄×Dic₈₉	—	2	2	no	—
31	C₄×Dic₈₉	D₄	1	2	yes	—
32	C₄×Dic₈₉	D₄	1	2	yes	—
33	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
35	M₁₆×C₈₉	—	2	2	no	—
36	M₁₆×C₈₉	—	2	2	no	—
37	C₂×C₈×C₈₉	—	2	2	no	—
38	C₄⋊C₄×C₈₉	—	2	2	no	—
39	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈₉	—	2	2	no	—
41	C₂×C₄×C₈₉	—	4	2	no	—
42	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₈₉	#41	1	2	yes	—
46	C₂×Dic₈₉	—	2	2	no	—
47	C₂×Dic₈₉	—	2	2	no	—
48	C₂×Dic₈₉	—	2	2	no	—
49	C₂×Dic₈₉	—	4	1	no	—
50	C₂×Dic₈₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₈₉	C₂²⋊C₄	1	2	yes	—
52	C₂×Dic₈₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₈₉	C₄⋊C₄	1	4	yes	—
54	Dic₁₇₈	—	2	4	no	—
55	Dic₁₇₈	—	2	4	no	—
56	C₈×C₈₉	—	4	2	no	—
57	#10623494	—	4	2	no	—
58	C₂²×C₈₉	—	2	1	no	—
59	C₂²×C₈₉	—	2	1	no	—
60	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₈₉	#117	1	1	yes	characteristic
63	C₄×C₈₉	—	4	2	no	—
64	C₄×C₈₉	#121	1	2	yes	—
65	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
66	Dic₈₉	—	2	4	no	—
67	Dic₈₉	—	2	1	no	—
68	Dic₈₉	—	2	2	no	—
69	Dic₈₉	#101	1	2	yes	complemented
70	C₂×C₈₉	—	2	1	no	—
71	C₂×C₈₉	—	2	1	no	—
72	C₂×C₈₉	#380	1	1	yes	characteristic
73	C₂×C₈₉	C₂×#101	1	1	yes	characteristic
74	C₂×C₈₉	#413	1	1	yes	characteristic
75	#1801	—	89	1	no	maximal, Sylow
76	C₈₉	#1801	1	1	yes	characteristic, complemented, Sylow
77	C₂×C₄×Q₈	—	89	1	no	—
78	#423	—	89	2	no	—
79	C₂×#101	—	89	2	no	—
80	#343	—	89	1	no	—
81	#343	D₈₉	1	1	yes	characteristic
82	C₂²×Q₈	—	89	1	no	—
83	C₂×M₁₆	—	89	2	no	—
84	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
85	C₄×Q₈	—	178	2	no	—
86	C₄×Q₈	—	178	2	no	—
87	C₂×C₄⋊C₄	—	89	2	no	—
88	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
89	C₂×C₄²	—	89	1	no	—
90	C₂×C₄²	—	89	2	no	—
91	C₄⋊C₈	—	178	2	no	—
92	C₄⋊C₈	—	178	2	no	—
93	#101	—	178	4	no	—
94	C₂×Q₈	—	89	2	no	—
95	C₂×Q₈	—	89	2	no	—
96	C₂×Q₈	—	178	2	no	—
97	C₂×Q₈	—	178	2	no	—
98	C₂²×C₄	—	89	1	no	—
99	C₂²×C₄	—	89	1	no	—
100	C₂²×C₄	—	89	2	no	—
101	C₂²×C₄	C₄×D₈₉	1	2	yes	—
102	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
103	M₁₆	—	2	2	no	—
104	M₁₆	—	2	2	no	—
105	M₁₆	—	178	2	no	—
106	M₁₆	—	178	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	178	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	178	4	no	—
111	C₄²	—	89	2	no	—
112	C₄²	—	89	2	no	—
113	C₄²	—	178	2	no	—
114	C₄²	—	178	2	no	—
115	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
116	Q₈	—	178	4	no	—
117	Q₈	—	178	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	89	1	no	—
121	C₂×C₄	—	89	1	no	—
122	C₂×C₄	—	89	2	no	—
123	C₂×C₄	—	89	4	no	—
124	C₂×C₄	—	356	1	no	—
125	C₂×C₄	—	178	2	no	—
126	C₂×C₄	—	178	2	no	—
127	C₂×C₄	—	178	2	no	—
128	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
129	C₂×C₄	D₄×D₈₉	1	2	yes	—
130	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
131	C₂×C₄	#13083574	1	2	yes	—
132	C₈	—	4	2	no	—
133	C₈	—	356	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
137	C₂²	C₄⋊C₄×D₈₉	1	1	yes	characteristic
138	C₂²	#15667969	1	1	yes	characteristic
139	C₄	—	4	2	no	—
140	C₄	—	89	2	no	—
141	C₄	—	178	4	no	—
142	C₄	—	178	1	no	—
143	C₄	—	178	2	no	—
144	C₄	#15667985	1	2	yes	—
145	C₄	#15667990	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	1	no	—
148	C₂	#19151457	1	1	yes	characteristic
149	C₂	#101×D₈₉	1	1	yes	characteristic
150	C₂	#19151579	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667977

C₂

yes

maximal, characteristic

#19151621

C₂

yes

maximal

C₂×#15667930

C₂

yes

maximal

#19151367

C₂

yes

maximal, characteristic

#343×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂×#13083558

C₂²

yes

—

#15667977

—

#15667977

—

C₂×#13083553

C₄

yes

—

C₂×C₄×Dic₈₉

C₄

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

—

—

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

—

C₂×Dic₁₇₈

Q₈

yes

—

C₂²×Dic₈₉

C₂×C₄

yes

characteristic

C₂²×Dic₈₉

C₂×C₄

yes

—

C₂²×Dic₈₉

C₂×C₄

yes

—

—

—

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

D₄

yes

—

C₄×Dic₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₈₉

C₂²⋊C₄

yes

—

C₂×Dic₈₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₈₉

C₄⋊C₄

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₉

#117

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#121

yes

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

#101

yes

complemented

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

—

—

—

D₈₉

yes

characteristic

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₄×Q₈

—

178

—

C₄×Q₈

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

178

—

C₄⋊C₈

—

178

—

#101

—

178

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₄×D₈₉

yes

—

102

C₂²×C₄

C₂²×D₈₉

yes

characteristic

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

178

—

106

M₁₆

—

178

—

107

C₂×C₈

—

108

C₂×C₈

—

178

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

178

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

178

—

114

C₄²

—

178

—

115

C₂³

C₂×C₄×D₈₉

yes

characteristic

116

Q₈

—

178

—

117

Q₈

—

178

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

356

—

125

C₂×C₄

—

178

—

126

C₂×C₄

—

178

—

127

C₂×C₄

—

178

—

128

C₂×C₄

D₄×D₈₉

yes

characteristic

129

C₂×C₄

D₄×D₈₉

yes

—

130

C₂×C₄

Q₈×D₈₉

yes

characteristic

131

C₂×C₄

#13083574

yes

—

132

C₈

—

133

C₈

—

356

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂²⋊C₄×D₈₉

yes

characteristic

137

C₂²

C₄⋊C₄×D₈₉

yes

characteristic

138

C₂²

#15667969

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

178

—

142

C₄

—

178

—

143

C₄

—

178

—

144

C₄

#15667985

yes

—

145

C₄

#15667990

yes

—

146

C₂

—

147

C₂

—

148

C₂

#19151457

yes

characteristic

149

C₂

#101×D₈₉

yes

characteristic

150

C₂

#19151579

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₈₉
commutator, prime radical	C₂³×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#343×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#343	D₈₉
hypocenter	C₈₉	#1801
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₈₉

commutator, prime radical

C₂³×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#343×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
69. Dic₈₉	#101	semidirect
76. C₈₉	#1801	semidirect

H ≅ G/N

type

69. Dic₈₉

#101

semidirect

76. C₈₉

#1801

semidirect

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

subgroup	index
C₂×#15667977	2
#19151621	2
C₂×#15667930	2
#19151367	2
#343×C₈₉	2
#1801	89

subgroup

index

#343×C₈₉

Subgroups — #26020247

Conjugacy classes of subgroups

Complements

Maximal subgroups