Group #26022277 — subgroups

Subgroups — #26022277

Order 11392.

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

normal subgroups only

This group has 236 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	#19152314	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151771	C₂	1	2	yes	maximal, complemented
3	#19151397	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151394	C₂	1	2	yes	maximal, complemented
5	#456×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15668037	—	2	2	no	—
7	#15668001	—	2	1	no	—
8	#15668001	C₂²	1	2	yes	—
9	#15667987	—	2	2	no	—
10	C₄×D₃₅₆	—	2	1	no	—
11	#15667970	—	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×D₃₅₆	—	2	1	no	—
20	C₂×D₃₅₆	—	2	1	no	—
21	C₂×D₃₅₆	D₄	1	2	yes	—
22	C₂×C₄×D₈₉	—	2	2	no	—
23	#13083561	—	4	2	no	—
24	#13083559	—	4	2	no	—
25	#13083558	—	2	2	no	—
26	C₄⋊Dic₈₉	—	2	1	no	—
27	#13083551	—	2	2	no	—
28	#13083551	—	2	2	no	—
29	#13083551	—	2	2	no	—
30	C₄×Dic₈₉	—	2	1	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	C₂²×D₈₉	—	2	2	no	—
44	Q₈×C₈₉	—	2	1	no	—
45	Q₈×C₈₉	—	2	2	no	—
46	Q₈×C₈₉	—	4	1	no	—
47	Q₈×C₈₉	—	4	1	no	—
48	D₄×C₈₉	—	2	1	no	—
49	D₄×C₈₉	—	2	2	no	—
50	D₄×C₈₉	—	4	1	no	—
51	C₂×C₄×C₈₉	—	2	1	no	—
52	C₂×C₄×C₈₉	—	2	1	no	—
53	C₂×C₄×C₈₉	—	4	2	no	—
54	C₂×C₄×C₈₉	—	4	1	no	—
55	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₈₉	—	2	2	no	—
58	D₃₅₆	—	4	2	no	—
59	D₃₅₆	—	4	2	no	—
60	D₃₅₆	—	8	1	no	—
61	C₄×D₈₉	—	4	4	no	—
62	C₈×C₈₉	—	4	1	no	—
63	#10623494	—	4	2	no	—
64	C₂²×C₈₉	—	4	1	no	—
65	C₂²×C₈₉	#120	1	1	yes	characteristic
66	C₂×D₈₉	—	4	2	no	—
67	C₂×D₈₉	—	4	2	no	—
68	C₂×D₈₉	—	4	2	no	—
69	C₄×C₈₉	—	2	2	no	—
70	C₄×C₈₉	—	2	2	no	—
71	C₄×C₈₉	—	2	1	no	—
72	C₄×C₈₉	—	4	1	no	—
73	C₄×C₈₉	—	4	1	no	—
74	Dic₈₉	—	4	2	no	—
75	C₂×C₈₉	—	2	1	no	—
76	C₂×C₈₉	—	8	1	no	—
77	C₂×C₈₉	B(4,𝔽₂)	1	1	yes	characteristic
78	D₈₉	—	8	2	no	—
79	#2204	—	89	1	no	maximal, Sylow
80	C₈₉	#2204	1	1	yes	characteristic, complemented, Sylow
81	#555	—	89	1	no	—
82	#456	D₈₉	1	1	yes	characteristic, complemented
83	#454	—	89	2	no	—
84	#356	—	89	1	no	—
85	#355	—	89	2	no	—
86	2₋¹⁺⁴	C₂×D₈₉	1	1	yes	characteristic
87	#137	—	2	1	no	—
88	AGL(1,ℤ/8)	—	178	2	no	—
89	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
90	#124	—	89	2	no	—
91	#124	—	178	1	no	—
92	#121	—	178	2	no	—
93	C₄×D₄	—	178	1	no	—
94	#117	—	178	2	no	—
95	#104	—	2	1	no	—
96	#104	—	178	2	no	—
97	#101	—	89	2	no	—
98	#101	C₂×D₈₉	1	1	yes	characteristic
99	#41	—	2	2	no	—
100	#41	—	2	1	no	—
101	#41	—	4	1	no	—
102	C₂×Q₈	—	4	1	no	—
103	C₂×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
104	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
105	C₂×D₄	—	89	2	no	—
106	C₂×D₄	—	178	1	no	—
107	C₂×D₄	—	178	1	no	—
108	C₂²×C₄	—	178	2	no	—
109	Q₁₆	—	4	1	no	—
110	SD₁₆	—	4	1	no	—
111	SD₁₆	—	356	2	no	—
112	D₈	—	356	2	no	—
113	M₁₆	—	2	1	no	—
114	M₁₆	—	178	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	178	1	no	—
117	C₂²⋊C₄	—	178	2	no	—
118	C₂²⋊C₄	—	178	2	no	—
119	C₂²⋊C₄	—	178	2	no	—
120	C₄²	—	89	2	no	—
121	C₄²	—	178	1	no	—
122	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
123	C₂³	—	178	2	no	—
124	Q₈	—	2	2	no	—
125	Q₈	—	2	1	no	—
126	Q₈	—	4	1	no	—
127	Q₈	—	4	1	no	—
128	D₄	—	2	2	no	—
129	D₄	—	2	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	356	2	no	—
132	D₄	—	356	2	no	—
133	D₄	—	712	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	356	4	no	—
139	C₂×C₄	—	178	2	no	—
140	C₂×C₄	D₄×D₈₉	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
142	C₈	—	4	1	no	—
143	C₈	—	356	2	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	356	2	no	—
146	C₂²	—	356	2	no	—
147	C₂²	—	356	2	no	—
148	C₂²	#15667980	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	2	no	—
152	C₄	—	4	1	no	—
153	C₄	—	4	1	no	—
154	C₄	—	356	2	no	—
155	C₂	—	2	1	no	—
156	C₂	—	8	1	no	—
157	C₂	—	712	2	no	—
158	C₂	#19151784	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152314

C₂

yes

maximal, characteristic, complemented

#19151771

C₂

yes

maximal, complemented

#19151397

C₂

yes

maximal, characteristic, complemented

#19151394

C₂

yes

maximal, complemented

#456×C₈₉

C₂

yes

maximal, characteristic, complemented

#15668037

—

#15668001

—

#15668001

C₂²

yes

—

#15667987

—

C₄×D₃₅₆

—

#15667970

—

#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×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

D₄

yes

—

C₂×C₄×D₈₉

—

#13083561

—

#13083559

—

#13083558

—

C₄⋊Dic₈₉

—

#13083551

—

#13083551

—

#13083551

—

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

C₂²×D₈₉

—

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

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#120

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

B(4,𝔽₂)

yes

characteristic

D₈₉

—

#2204

—

maximal, Sylow

C₈₉

#2204

yes

characteristic, complemented, Sylow

#555

—

#456

D₈₉

yes

characteristic, complemented

#454

—

#356

—

#355

—

2₋¹⁺⁴

C₂×D₈₉

yes

characteristic

#137

—

AGL(1,ℤ/8)

—

178

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

#124

—

#124

—

178

—

#121

—

178

—

C₄×D₄

—

178

—

#117

—

178

—

#104

—

#104

—

178

—

#101

—

#101

C₂×D₈₉

yes

characteristic

#41

—

100

#41

—

101

#41

—

102

C₂×Q₈

—

103

C₂×Q₈

C₂²⋊D₈₉

yes

characteristic

104

C₂×Q₈

C₂²×D₈₉

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

178

—

107

C₂×D₄

—

178

—

108

C₂²×C₄

—

178

—

109

Q₁₆

—

110

SD₁₆

—

111

SD₁₆

—

356

—

112

D₈

—

356

—

113

M₁₆

—

114

M₁₆

—

178

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

178

—

117

C₂²⋊C₄

—

178

—

118

C₂²⋊C₄

—

178

—

119

C₂²⋊C₄

—

178

—

120

C₄²

—

121

C₄²

—

178

—

122

C₄²

C₂²⋊D₈₉

yes

characteristic

123

C₂³

—

178

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

131

D₄

—

356

—

132

D₄

—

356

—

133

D₄

—

712

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

356

—

139

C₂×C₄

—

178

—

140

C₂×C₄

D₄×D₈₉

yes

—

141

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

142

C₈

—

143

C₈

—

356

—

144

C₂²

—

145

C₂²

—

356

—

146

C₂²

—

356

—

147

C₂²

—

356

—

148

C₂²

#15667980

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

356

—

155

C₂

—

156

C₂

—

157

C₂

—

712

—

158

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)
hypercenter	#456	D₈₉
hypocenter	C₈₉	#2204
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

C₂×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ #456

N	H ≅ G/N	type
1. #19152314	C₂	semidirect
2. #19151771	C₂	semidirect
3. #19151397	C₂	semidirect
4. #19151394	C₂	semidirect
5. #456×C₈₉	C₂	semidirect
80. C₈₉	#2204	semidirect
82. #456	D₈₉	semidirect

H ≅ G/N

type

1. #19152314

C₂

semidirect

2. #19151771

C₂

semidirect

3. #19151397

C₂

semidirect

4. #19151394

C₂

semidirect

5. #456×C₈₉

C₂

semidirect

80. C₈₉

#2204

semidirect

82. #456

D₈₉

semidirect

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

subgroup	index
#19152314	2
#19151771	2
#19151397	2
#19151394	2
#456×C₈₉	2
#2204	89

subgroup

index

#456×C₈₉

Subgroups — #26022277

Conjugacy classes of subgroups

Complements

Maximal subgroups