Group #26022281 — subgroups

Subgroups — #26022281

Order 11392.

224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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	#19152328	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151781	C₂	1	2	yes	maximal
3	#19151397	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151396	C₂	1	2	yes	maximal, complemented
5	#456×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#15668042	—	2	2	no	—
7	#15668011	—	2	2	no	—
8	#15668010	—	2	1	no	—
9	#15668010	C₂²	1	2	yes	—
10	C₄×Dic₁₇₈	—	2	1	no	—
11	#15667977	—	2	2	no	—
12	#15667939	—	2	2	no	—
13	#15667931	C₂²	1	2	yes	—
14	2₋¹⁺⁴×C₈₉	C₂²	1	1	yes	characteristic
15	#137×C₈₉	—	2	1	no	—
16	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
17	#104×C₈₉	—	2	1	no	—
18	#101×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₇₈	—	2	1	no	—
20	C₂×Dic₁₇₈	—	2	1	no	—
21	C₂×Dic₁₇₈	D₄	1	2	yes	—
22	#13083564	—	4	2	no	—
23	#13083562	—	4	2	no	—
24	#13083558	—	2	2	no	—
25	C₄⋊Dic₈₉	—	2	1	no	—
26	#13083553	—	2	2	no	—
27	#13083553	—	2	2	no	—
28	#13083553	—	2	2	no	—
29	C₄×Dic₈₉	—	2	1	no	—
30	C₄×Dic₈₉	—	2	2	no	—
31	C₄×Dic₈₉	D₄	1	2	yes	—
32	#41×C₈₉	—	2	2	no	—
33	#41×C₈₉	—	2	1	no	—
34	#41×C₈₉	—	4	1	no	—
35	C₂×Q₈×C₈₉	—	4	1	no	—
36	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
37	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
38	Q₁₆×C₈₉	—	4	1	no	—
39	SD₁₆×C₈₉	—	4	1	no	—
40	M₁₆×C₈₉	—	2	1	no	—
41	C₄⋊C₄×C₈₉	—	2	2	no	—
42	C₄²×C₈₉	D₄	1	1	yes	characteristic
43	Q₈×C₈₉	—	2	1	no	—
44	Q₈×C₈₉	—	2	2	no	—
45	Q₈×C₈₉	—	4	1	no	—
46	Q₈×C₈₉	—	4	1	no	—
47	D₄×C₈₉	—	2	1	no	—
48	D₄×C₈₉	—	2	2	no	—
49	D₄×C₈₉	—	4	1	no	—
50	C₂×C₄×C₈₉	—	2	1	no	—
51	C₂×C₄×C₈₉	—	2	1	no	—
52	C₂×C₄×C₈₉	—	4	2	no	—
53	C₂×C₄×C₈₉	—	4	1	no	—
54	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₈₉	—	2	2	no	—
57	C₂×Dic₈₉	—	2	2	no	—
58	Dic₁₇₈	—	2	4	no	—
59	Dic₁₇₈	—	4	2	no	—
60	Dic₁₇₈	—	8	1	no	—
61	C₈×C₈₉	—	4	1	no	—
62	#10623494	—	4	2	no	—
63	C₂²×C₈₉	—	4	1	no	—
64	C₂²×C₈₉	#120	1	1	yes	characteristic
65	C₄×C₈₉	—	2	2	no	—
66	C₄×C₈₉	—	2	2	no	—
67	C₄×C₈₉	—	2	1	no	—
68	C₄×C₈₉	—	4	1	no	—
69	C₄×C₈₉	—	4	1	no	—
70	Dic₈₉	—	2	4	no	—
71	Dic₈₉	—	4	2	no	—
72	C₂×C₈₉	—	2	1	no	—
73	C₂×C₈₉	—	8	1	no	—
74	C₂×C₈₉	B(4,𝔽₂)	1	1	yes	characteristic
75	#2205	—	89	1	no	maximal, Sylow
76	C₈₉	#2205	1	1	yes	characteristic, complemented, Sylow
77	Q₈²	—	89	1	no	—
78	#456	—	89	2	no	—
79	#456	D₈₉	1	1	yes	characteristic
80	#356	—	89	1	no	—
81	#356	—	89	2	no	—
82	2₋¹⁺⁴	C₂×D₈₉	1	1	yes	characteristic
83	#137	—	2	1	no	—
84	#137	—	178	2	no	—
85	C₄⋊Q₈	—	89	2	no	—
86	C₄⋊Q₈	—	178	2	no	—
87	C₄⋊Q₈	—	178	1	no	—
88	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
89	C₄×Q₈	—	178	2	no	—
90	C₄×Q₈	—	178	1	no	—
91	#104	—	2	1	no	—
92	#104	—	178	2	no	—
93	#101	—	89	2	no	—
94	#101	C₂×D₈₉	1	1	yes	characteristic
95	#41	—	2	2	no	—
96	#41	—	2	1	no	—
97	#41	—	4	1	no	—
98	C₂×Q₈	—	4	1	no	—
99	C₂×Q₈	—	89	2	no	—
100	C₂×Q₈	—	178	1	no	—
101	C₂×Q₈	—	178	1	no	—
102	C₂×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
103	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
104	Q₁₆	—	4	1	no	—
105	Q₁₆	—	356	2	no	—
106	SD₁₆	—	4	1	no	—
107	SD₁₆	—	356	2	no	—
108	M₁₆	—	2	1	no	—
109	M₁₆	—	178	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	178	2	no	—
112	C₄⋊C₄	—	178	1	no	—
113	C₄⋊C₄	—	178	2	no	—
114	C₄⋊C₄	—	178	2	no	—
115	C₄²	—	89	2	no	—
116	C₄²	—	178	2	no	—
117	C₄²	—	178	1	no	—
118	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
119	Q₈	—	2	1	no	—
120	Q₈	—	2	2	no	—
121	Q₈	—	4	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	356	2	no	—
124	Q₈	—	178	4	no	—
125	Q₈	—	712	1	no	—
126	D₄	—	2	1	no	—
127	D₄	—	2	2	no	—
128	D₄	—	4	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	178	2	no	—
134	C₂×C₄	—	178	2	no	—
135	C₂×C₄	D₄×D₈₉	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
137	C₈	—	4	1	no	—
138	C₈	—	356	2	no	—
139	C₂²	—	4	1	no	—
140	C₂²	#15667980	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	2	no	—
144	C₄	—	4	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	356	2	no	—
147	C₄	—	178	4	no	—
148	C₂	—	2	1	no	—
149	C₂	—	8	1	no	—
150	C₂	#19151784	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152328

C₂

yes

maximal, characteristic, complemented

#19151781

C₂

yes

maximal

#19151397

C₂

yes

maximal, characteristic, complemented

#19151396

C₂

yes

maximal, complemented

#456×C₈₉

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₄×Dic₁₇₈

—

#15667977

—

#15667939

—

#15667931

C₂²

yes

—

2₋¹⁺⁴×C₈₉

C₂²

yes

characteristic

#137×C₈₉

—

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#104×C₈₉

—

#101×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

—

#13083564

—

#13083562

—

#13083558

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

D₄

yes

—

#41×C₈₉

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×Q₈×C₈₉

C₂³

yes

characteristic

Q₁₆×C₈₉

—

SD₁₆×C₈₉

—

M₁₆×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#120

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

Q₈²

—

#456

—

#456

D₈₉

yes

characteristic

#356

—

#356

—

2₋¹⁺⁴

C₂×D₈₉

yes

characteristic

#137

—

#137

—

178

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

C₄×Q₈

—

178

—

C₄×Q₈

—

178

—

#104

—

#104

—

178

—

#101

—

#101

C₂×D₈₉

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

178

—

101

C₂×Q₈

—

178

—

102

C₂×Q₈

C₂²⋊D₈₉

yes

characteristic

103

C₂×Q₈

C₂²×D₈₉

yes

characteristic

104

Q₁₆

—

105

Q₁₆

—

356

—

106

SD₁₆

—

107

SD₁₆

—

356

—

108

M₁₆

—

109

M₁₆

—

178

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

178

—

112

C₄⋊C₄

—

178

—

113

C₄⋊C₄

—

178

—

114

C₄⋊C₄

—

178

—

115

C₄²

—

116

C₄²

—

178

—

117

C₄²

—

178

—

118

C₄²

C₂²⋊D₈₉

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

356

—

124

Q₈

—

178

—

125

Q₈

—

712

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

178

—

134

C₂×C₄

—

178

—

135

C₂×C₄

D₄×D₈₉

yes

—

136

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

137

C₈

—

138

C₈

—

356

—

139

C₂²

—

140

C₂²

#15667980

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

356

—

147

C₄

—

178

—

148

C₂

—

149

C₂

—

150

C₂

#19151784

yes

characteristic

	subgroup	G/H
center	C₂	#19151784
commutator, cosocle	C₂×Q₈×C₈₉	C₂³
Frattini	C₂×Q₈	C₂²×D₈₉
Fitting	#456×C₈₉	C₂
socle	C₂×C₈₉	B(4,𝔽₂)
prime radical	2₋¹⁺⁴×C₈₉	C₂²
hypercenter	#456	D₈₉
hypocenter	C₈₉	#2205
LCS	G ▷ C₂×Q₈×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ #456

subgroup

G/H

center

C₂

#19151784

commutator, cosocle

C₂×Q₈×C₈₉

C₂³

Frattini

C₂×Q₈

C₂²×D₈₉

Fitting

#456×C₈₉

C₂

socle

C₂×C₈₉

B(4,𝔽₂)

prime radical

2₋¹⁺⁴×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂×Q₈×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ #456

N	H ≅ G/N	type
1. #19152328	C₂	semidirect
3. #19151397	C₂	semidirect
4. #19151396	C₂	semidirect
76. C₈₉	#2205	semidirect

H ≅ G/N

type

1. #19152328

C₂

semidirect

3. #19151397

C₂

semidirect

4. #19151396

C₂

semidirect

76. C₈₉

#2205

semidirect

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

subgroup	index
#19152328	2
#19151781	2
#19151397	2
#19151396	2
#456×C₈₉	2
#2205	89

subgroup

index

#456×C₈₉

Subgroups — #26022281

Conjugacy classes of subgroups

Complements

Maximal subgroups