Group #26019558 — subgroups

Subgroups — #26019558

Order 11392.

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

normal subgroups only

This group has 228 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	#19151993	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151957	C₂	1	2	yes	maximal, complemented
3	#19151338	C₂	1	2	yes	maximal, complemented
4	#19151338	C₂	1	1	yes	maximal, characteristic, complemented
5	H(3,𝔽₄)×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083564	—	2	1	no	—
7	C₂×#13083562	—	2	1	no	—
8	#15668012	C₂²	1	1	yes	characteristic, complemented
9	#15667936	—	2	2	no	—
10	#15667933	—	2	2	no	—
11	#15667923	C₂²	1	2	yes	complemented
12	#15667923	C₂²	1	1	yes	characteristic, complemented
13	#124×C₈₉	—	2	1	no	—
14	#124×C₈₉	—	2	2	no	—
15	#124×C₈₉	C₂²	1	2	yes	—
16	#124×C₈₉	C₂²	1	1	yes	characteristic
17	#120×C₈₉	—	2	1	no	—
18	#120×C₈₉	—	2	2	no	—
19	C₂×Dic₁₇₈	—	2	1	no	—
20	#13083564	—	4	2	no	—
21	#13083562	—	4	2	no	—
22	C₂×#10623494	—	2	1	no	—
23	C₂×#10623494	—	2	2	no	—
24	C₄⋊Dic₈₉	—	2	2	no	—
25	C₂⁴×C₈₉	—	2	1	no	—
26	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
27	C₂×Q₈×C₈₉	D₄	1	2	yes	—
28	C₂×D₄×C₈₉	—	2	1	no	—
29	C₂×D₄×C₈₉	—	2	2	no	—
30	C₂×D₄×C₈₉	D₄	1	1	yes	characteristic
31	C₂×D₄×C₈₉	D₄	1	2	yes	—
32	C₂²⋊C₄×C₈₉	—	2	2	no	—
33	C₂²⋊C₄×C₈₉	—	2	1	no	—
34	C₂²⋊C₄×C₈₉	—	2	2	no	—
35	C₂²⋊C₄×C₈₉	—	2	2	no	—
36	C₂²⋊C₄×C₈₉	—	2	2	no	—
37	C₄²×C₈₉	—	2	1	no	—
38	C₄²×C₈₉	C₂³	1	1	yes	characteristic
39	C₂³×C₈₉	—	2	1	no	—
40	C₂³×C₈₉	—	2	2	no	—
41	C₂³×C₈₉	—	8	1	no	—
42	C₂³×C₈₉	—	8	2	no	—
43	Q₈×C₈₉	—	4	2	no	—
44	Q₈×C₈₉	—	4	1	no	—
45	D₄×C₈₉	—	4	2	no	—
46	D₄×C₈₉	—	4	1	no	—
47	D₄×C₈₉	—	8	1	no	—
48	D₄×C₈₉	—	8	2	no	—
49	C₂×C₄×C₈₉	—	2	1	no	—
50	C₂×C₄×C₈₉	—	2	2	no	—
51	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₈₉	—	4	1	no	—
54	Dic₁₇₈	—	4	1	no	—
55	Dic₁₇₈	—	4	1	no	—
56	#10623494	—	2	2	no	—
57	#10623494	—	4	2	no	—
58	C₂²×C₈₉	—	4	1	no	—
59	C₂²×C₈₉	—	4	2	no	—
60	C₂²×C₈₉	—	4	2	no	—
61	C₂²×C₈₉	—	4	2	no	—
62	C₂²×C₈₉	—	4	2	no	—
63	C₂²×C₈₉	—	8	4	no	—
64	C₂²×C₈₉	#120	1	1	yes	characteristic
65	C₄×C₈₉	—	2	1	no	—
66	C₄×C₈₉	—	2	2	no	—
67	C₄×C₈₉	—	4	1	no	—
68	C₄×C₈₉	—	4	2	no	—
69	Dic₈₉	—	8	1	no	—
70	C₂×C₈₉	—	8	1	no	—
71	C₂×C₈₉	—	8	2	no	—
72	C₂×C₈₉	#453	1	1	yes	characteristic
73	C₂×C₈₉	#455	1	2	yes	—
74	#1659	—	89	1	no	maximal, Sylow
75	C₈₉	#1659	1	1	yes	characteristic, complemented, Sylow
76	H(3,𝔽₄)	D₈₉	1	1	yes	characteristic
77	#497	—	89	1	no	—
78	#489	—	89	2	no	—
79	#329	—	89	1	no	—
80	#329	—	89	2	no	—
81	C₂×Q₁₆	—	178	1	no	—
82	C₂×SD₁₆	—	178	1	no	—
83	C₄⋊Q₈	—	89	1	no	—
84	#124	—	2	2	no	—
85	#124	—	2	1	no	—
86	#124	C₂×D₈₉	1	2	yes	—
87	#124	C₂×D₈₉	1	1	yes	characteristic
88	#120	—	2	2	no	—
89	#120	—	2	1	no	—
90	#103	—	178	2	no	—
91	#102	—	178	2	no	—
92	#97	—	89	2	no	—
93	#97	—	89	1	no	—
94	C₂⁴	—	2	1	no	—
95	C₂×Q₈	—	178	1	no	—
96	C₂×Q₈	C₂²⋊D₈₉	1	2	yes	—
97	C₂×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	2	2	no	—
100	C₂×D₄	C₂²⋊D₈₉	1	2	yes	—
101	C₂×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
102	Q₁₆	—	356	2	no	—
103	SD₁₆	—	356	2	no	—
104	C₂×C₈	—	178	2	no	—
105	C₂×C₈	—	178	1	no	—
106	C₄⋊C₄	—	178	2	no	—
107	C₂²⋊C₄	—	2	2	no	—
108	C₂²⋊C₄	—	2	1	no	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₂²⋊C₄	—	2	2	no	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	C₂²×D₈₉	1	1	yes	characteristic
114	C₂³	—	2	1	no	—
115	C₂³	—	2	2	no	—
116	C₂³	—	8	1	no	—
117	C₂³	—	8	2	no	—
118	Q₈	—	4	1	no	—
119	Q₈	—	4	2	no	—
120	Q₈	—	356	1	no	—
121	Q₈	—	356	1	no	—
122	D₄	—	4	1	no	—
123	D₄	—	4	2	no	—
124	D₄	—	8	1	no	—
125	D₄	—	8	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	356	1	no	—
129	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
131	C₈	—	356	2	no	—
132	C₈	—	178	2	no	—
133	C₂²	—	4	1	no	—
134	C₂²	—	4	2	no	—
135	C₂²	—	4	2	no	—
136	C₂²	—	4	2	no	—
137	C₂²	—	4	2	no	—
138	C₂²	—	8	4	no	—
139	C₂²	#15667981	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	—	4	2	no	—
144	C₄	—	712	1	no	—
145	C₂	—	8	1	no	—
146	C₂	—	8	2	no	—
147	C₂	#19151766	1	1	yes	characteristic
148	C₂	#19151776	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151993

C₂

yes

maximal, characteristic, complemented

#19151957

C₂

yes

maximal, complemented

#19151338

C₂

yes

maximal, complemented

#19151338

C₂

yes

maximal, characteristic, complemented

H(3,𝔽₄)×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083564

—

C₂×#13083562

—

#15668012

C₂²

yes

characteristic, complemented

#15667936

—

#15667933

—

#15667923

C₂²

yes

complemented

#15667923

C₂²

yes

characteristic, complemented

#124×C₈₉

—

#124×C₈₉

—

#124×C₈₉

C₂²

yes

—

#124×C₈₉

C₂²

yes

characteristic

#120×C₈₉

—

#120×C₈₉

—

C₂×Dic₁₇₈

—

—

—

—

—

C₄⋊Dic₈₉

—

C₂⁴×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×Q₈×C₈₉

D₄

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

D₄

yes

characteristic

C₂×D₄×C₈₉

D₄

yes

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#120

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#453

yes

characteristic

C₂×C₈₉

#455

yes

—

#1659

—

maximal, Sylow

C₈₉

#1659

yes

characteristic, complemented, Sylow

H(3,𝔽₄)

D₈₉

yes

—

—

—

—

C₂×Q₁₆

—

178

—

C₂×SD₁₆

—

178

—

C₄⋊Q₈

—

#124

—

#124

—

#124

C₂×D₈₉

yes

—

#124

C₂×D₈₉

yes

—

—

—

178

—

#102

—

178

—

#97

—

#97

—

C₂⁴

—

C₂×Q₈

—

178

—

C₂×Q₈

C₂²⋊D₈₉

yes

—

C₂×Q₈

C₂²⋊D₈₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

C₂²⋊D₈₉

yes

—

101

C₂×D₄

C₂²⋊D₈₉

yes

characteristic

102

Q₁₆

—

356

—

103

SD₁₆

—

356

—

104

C₂×C₈

—

178

—

105

C₂×C₈

—

178

—

106

C₄⋊C₄

—

178

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

C₂²×D₈₉

yes

characteristic

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

356

—

121

Q₈

—

356

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

356

—

129

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

130

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

131

C₈

—

356

—

132

C₈

—

178

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

#15667981

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

712

—

145

C₂

—

146

C₂

—

147

C₂

#19151766

yes

characteristic

148

C₂

#19151776

yes

—

	subgroup	G/H
center	C₂²	#15667981
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting, prime radical	H(3,𝔽₄)×C₈₉	C₂
socle	C₂²×C₈₉	#120
hypercenter	H(3,𝔽₄)	D₈₉
hypocenter	C₈₉	#1659
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ H(3,𝔽₄)

subgroup

G/H

center

C₂²

#15667981

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting, prime radical

H(3,𝔽₄)×C₈₉

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ H(3,𝔽₄)

N	H ≅ G/N	type
1. #19151993	C₂	semidirect
2. #19151957	C₂	semidirect
3. #19151338	C₂	semidirect
4. #19151338	C₂	semidirect
8. #15668012	C₂²	semidirect
11. #15667923	C₂²	semidirect
12. #15667923	C₂²	semidirect
75. C₈₉	#1659	semidirect

H ≅ G/N

type

1. #19151993

C₂

semidirect

2. #19151957

C₂

semidirect

3. #19151338

C₂

semidirect

4. #19151338

C₂

semidirect

8. #15668012

C₂²

semidirect

11. #15667923

C₂²

semidirect

12. #15667923

C₂²

semidirect

75. C₈₉

#1659

semidirect

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

subgroup	index
#19151993	2
#19151957	2
#19151338	2
H(3,𝔽₄)×C₈₉	2
#1659	89

subgroup

index

H(3,𝔽₄)×C₈₉

Subgroups — #26019558

Conjugacy classes of subgroups

Complements

Maximal subgroups