Group #36469186 — subgroups

Subgroups — #36469186

Order 22784.

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

normal subgroups only

This group has 296 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	#26022509	C₂	1	2	yes	maximal
2	#26022479	C₂	1	2	yes	maximal, complemented
3	#26019844	C₂	1	1	yes	maximal, characteristic
4	#26018170	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1713	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15667955	—	2	4	no	—
7	#19152003	C₂²	1	1	yes	characteristic
8	#19151975	C₂²	1	1	yes	characteristic
9	#19151971	—	2	1	no	—
10	#19151945	—	2	1	no	—
11	#19151402	—	2	4	no	—
12	C₄×#13083549	C₂²	1	2	yes	—
13	#500×C₈₉	—	2	1	no	—
14	#500×C₈₉	—	2	1	no	—
15	#494×C₈₉	C₂²	1	2	yes	—
16	C₈²×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#13083563	—	4	2	no	—
18	C₂×D₇₁₂	—	2	2	no	—
19	#15668012	—	2	1	no	—
20	#15668009	—	2	1	no	—
21	#15667955	—	4	8	no	—
22	C₂×#13083549	D₄	1	4	yes	—
23	#15667946	—	2	2	no	—
24	#15667934	—	4	2	no	—
25	C₂×Q₁₆×C₈₉	—	2	2	no	—
26	C₂×Q₁₆×C₈₉	—	2	2	no	—
27	C₄⋊Q₈×C₈₉	—	2	2	no	—
28	#107×C₈₉	—	4	2	no	—
29	#107×C₈₉	—	4	2	no	—
30	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₇₈	—	4	1	no	—
34	C₂×D₃₅₆	—	4	1	no	—
35	C₂×D₃₅₆	—	4	1	no	—
36	C₂×D₃₅₆	—	4	1	no	—
37	#13083563	—	8	4	no	—
38	D₇₁₂	—	4	4	no	—
39	C₄⋊Dic₈₉	—	4	1	no	—
40	C₄⋊Dic₈₉	—	4	1	no	—
41	#13083549	—	2	4	no	—
42	C₂×Q₈×C₈₉	—	4	2	no	—
43	Q₁₆×C₈₉	—	4	4	no	—
44	Q₁₆×C₈₉	—	4	4	no	—
45	C₂×C₈×C₈₉	D₈	1	2	yes	—
46	C₂×C₈×C₈₉	D₈	1	2	yes	—
47	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
48	C₄⋊C₄×C₈₉	—	4	2	no	—
49	C₄⋊C₄×C₈₉	—	4	2	no	—
50	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×D₈₉	—	8	1	no	—
52	Q₈×C₈₉	—	8	2	no	—
53	Q₈×C₈₉	—	8	2	no	—
54	C₂×C₄×C₈₉	—	8	2	no	—
55	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₈₉	—	8	1	no	—
59	D₃₅₆	—	8	2	no	—
60	D₃₅₆	—	8	2	no	—
61	D₃₅₆	—	8	2	no	—
62	Dic₁₇₈	—	8	2	no	—
63	C₈×C₈₉	—	2	2	no	—
64	C₈×C₈₉	SD₃₂	1	4	yes	complemented
65	C₈×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
66	C₂²×C₈₉	#493	1	1	yes	characteristic
67	C₂×D₈₉	—	16	1	no	—
68	C₂×D₈₉	—	16	1	no	—
69	C₂×D₈₉	—	16	1	no	—
70	C₄×C₈₉	—	16	2	no	—
71	C₄×C₈₉	#496	1	2	yes	—
72	C₄×C₈₉	C₂×SD₃₂	1	2	yes	—
73	C₄×C₈₉	#510	1	2	yes	—
74	Dic₈₉	—	16	1	no	—
75	#12526	—	89	1	no	maximal, Sylow
76	C₂×C₈₉	#1721	1	1	yes	characteristic
77	C₂×C₈₉	#2248	1	1	yes	characteristic
78	C₂×C₈₉	#2251	1	1	yes	characteristic
79	D₈₉	—	32	1	no	—
80	#2248	—	89	2	no	—
81	#2241	—	89	2	no	—
82	#1713	D₈₉	1	1	yes	characteristic, complemented
83	#1706	—	89	1	no	—
84	#1312	—	89	1	no	—
85	C₈₉	#12526	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₃₂	—	178	4	no	—
87	#500	—	2	1	no	—
88	#500	—	2	1	no	—
89	#500	—	89	1	no	—
90	#494	C₂×D₈₉	1	2	yes	—
91	#493	—	89	1	no	—
92	#492	—	178	1	no	—
93	#486	—	178	1	no	—
94	#358	—	178	4	no	—
95	C₄×C₁₆	—	89	2	no	—
96	C₈²	C₂×D₈₉	1	1	yes	characteristic
97	C₂×Q₁₆	—	2	2	no	—
98	C₂×Q₁₆	—	2	2	no	—
99	C₂×SD₁₆	—	356	2	no	—
100	C₂×D₈	—	178	2	no	—
101	C₄⋊Q₈	—	2	2	no	—
102	C₄⋊Q₈	—	178	1	no	—
103	#127	—	178	1	no	—
104	SD₃₂	—	356	8	no	—
105	C₂×C₁₆	—	89	4	no	—
106	#107	—	4	2	no	—
107	#107	—	4	2	no	—
108	#107	—	178	2	no	—
109	#102	—	356	2	no	—
110	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
111	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
112	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
113	C₂×Q₈	—	4	2	no	—
114	C₂×Q₈	—	356	1	no	—
115	C₂×D₄	—	356	1	no	—
116	C₂×D₄	—	356	1	no	—
117	C₂×D₄	—	356	1	no	—
118	Q₁₆	—	4	4	no	—
119	Q₁₆	—	4	4	no	—
120	SD₁₆	—	712	4	no	—
121	D₈	—	356	4	no	—
122	C₂×C₈	#13083559	1	2	yes	—
123	C₂×C₈	#13083559	1	2	yes	—
124	C₂×C₈	D₄×D₈₉	1	2	yes	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	356	1	no	—
128	C₄⋊C₄	—	356	1	no	—
129	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
130	C₁₆	—	178	4	no	—
131	C₂³	—	712	1	no	—
132	Q₈	—	8	2	no	—
133	Q₈	—	8	2	no	—
134	Q₈	—	712	2	no	—
135	D₄	—	712	2	no	—
136	D₄	—	712	2	no	—
137	D₄	—	712	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	712	1	no	—
140	C₂×C₄	#15668008	1	1	yes	characteristic
141	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
142	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	#15667955	1	4	yes	complemented
145	C₈	#15668036	1	4	yes	—
146	C₂²	—	1424	1	no	—
147	C₂²	—	1424	1	no	—
148	C₂²	—	1424	1	no	—
149	C₂²	#19151974	1	1	yes	characteristic
150	C₄	—	16	2	no	—
151	C₄	—	1424	1	no	—
152	C₄	#19151986	1	2	yes	—
153	C₄	C₂×#15667955	1	2	yes	—
154	C₄	#19152039	1	2	yes	—
155	C₂	—	2848	1	no	—
156	C₂	#26019919	1	1	yes	characteristic
157	C₂	#26022509	1	1	yes	characteristic
158	C₂	#26022521	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022509

C₂

yes

maximal

#26022479

C₂

yes

maximal, complemented

#26019844

C₂

yes

maximal, characteristic

#26018170

C₂

yes

maximal, characteristic, complemented

C₈₉×#1713

C₂

yes

maximal, characteristic, complemented

C₂×#15667955

—

#19152003

C₂²

yes

characteristic

#19151975

C₂²

yes

—

—

—

C₂²

yes

—

#500×C₈₉

—

#500×C₈₉

—

#494×C₈₉

C₂²

yes

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×#13083563

—

C₂×D₇₁₂

—

—

—

—

D₄

yes

—

#15667946

—

#15667934

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

—

#107×C₈₉

—

#107×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

#13083563

—

D₇₁₂

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083549

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

SD₃₂

yes

complemented

C₈×C₈₉

AGL(1,ℤ/8)

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#496

yes

—

C₄×C₈₉

C₂×SD₃₂

yes

—

C₄×C₈₉

#510

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#2248

—

#2241

—

#1713

D₈₉

yes

characteristic, complemented

#1706

—

#1312

—

C₈₉

#12526

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

178

—

—

—

—

C₂×D₈₉

yes

—

#493

—

#492

—

178

—

#486

—

178

—

#358

—

178

—

C₄×C₁₆

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

356

—

100

C₂×D₈

—

178

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

178

—

103

#127

—

178

—

104

SD₃₂

—

356

—

105

C₂×C₁₆

—

106

#107

—

107

#107

—

108

#107

—

178

—

109

#102

—

356

—

110

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

111

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

112

C₄×C₈

C₂²×D₈₉

yes

characteristic

113

C₂×Q₈

—

114

C₂×Q₈

—

356

—

115

C₂×D₄

—

356

—

116

C₂×D₄

—

356

—

117

C₂×D₄

—

356

—

118

Q₁₆

—

119

Q₁₆

—

120

SD₁₆

—

712

—

121

D₈

—

356

—

122

C₂×C₈

#13083559

yes

—

123

C₂×C₈

#13083559

yes

—

124

C₂×C₈

D₄×D₈₉

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

356

—

128

C₄⋊C₄

—

356

—

129

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

130

C₁₆

—

178

—

131

C₂³

—

712

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

712

—

135

D₄

—

712

—

136

D₄

—

712

—

137

D₄

—

712

—

138

C₂×C₄

—

139

C₂×C₄

—

712

—

140

C₂×C₄

#15668008

yes

characteristic

141

C₂×C₄

C₂×#13083559

yes

characteristic

142

C₂×C₄

C₂×#13083559

yes

characteristic

143

C₈

—

144

C₈

#15667955

yes

complemented

145

C₈

#15668036

yes

—

146

C₂²

—

1424

—

147

C₂²

—

1424

—

148

C₂²

—

1424

—

149

C₂²

#19151974

yes

characteristic

150

C₄

—

151

C₄

—

1424

—

152

C₄

#19151986

yes

—

153

C₄

C₂×#15667955

yes

—

154

C₄

#19152039

yes

—

155

C₂

—

2848

—

156

C₂

#26019919

yes

characteristic

157

C₂

#26022509

yes

characteristic

158

C₂

#26022521

yes

characteristic

	subgroup	G/H
center	C₂²	#19151974
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1713	C₂
socle	C₂²×C₈₉	#493
prime radical	#19151975	C₂²
hypercenter	#1713	D₈₉
hypocenter	C₈₉	#12526
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#19151974

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1713

C₂

C₂²×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
2. #26022479	C₂	semidirect
4. #26018170	C₂	semidirect
5. C₈₉×#1713	C₂	semidirect
64. C₈×C₈₉	SD₃₂	semidirect
82. #1713	D₈₉	semidirect
85. C₈₉	#12526	semidirect
144. C₈	#15667955	semidirect

H ≅ G/N

type

2. #26022479

C₂

semidirect

4. #26018170

C₂

semidirect

5. C₈₉×#1713

C₂

semidirect

64. C₈×C₈₉

SD₃₂

semidirect

82. #1713

D₈₉

semidirect

85. C₈₉

#12526

semidirect

144. C₈

#15667955

semidirect

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

subgroup	index
#26022509	2
#26022479	2
#26019844	2
#26018170	2
C₈₉×#1713	2
#12526	89

subgroup

index

C₈₉×#1713

Subgroups — #36469186

Conjugacy classes of subgroups

Complements

Maximal subgroups