Group #36469127 — subgroups

Subgroups — #36469127

Order 22784.

268 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 83 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	#26022480	C₂	1	2	yes	maximal, complemented
2	#26022476	C₂	1	2	yes	maximal
3	#26019870	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019858	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152003	—	2	1	no	—
7	#19152003	C₂²	1	2	yes	—
8	#19151978	—	2	1	no	—
9	#19151978	C₂²	1	1	yes	characteristic
10	#19151975	C₂²	1	1	yes	characteristic
11	#19151945	—	2	1	no	—
12	#19151945	—	2	1	no	—
13	#19151403	—	2	4	no	—
14	#19151400	—	2	4	no	—
15	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
16	C₈²×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₅₆	—	2	2	no	—
18	C₂×Dic₃₅₆	—	4	2	no	—
19	C₂×D₇₁₂	—	2	2	no	—
20	#15668012	—	2	1	no	—
21	#15668012	—	2	2	no	—
22	#15668009	—	2	1	no	—
23	#15667946	—	2	2	no	—
24	#15667946	—	2	2	no	—
25	#15667946	—	4	2	no	—
26	#15667934	—	4	2	no	—
27	#15667934	—	4	2	no	—
28	C₂×C₁₆×C₈₉	D₄	1	4	yes	—
29	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₇₈	—	4	1	no	—
33	C₂×Dic₁₇₈	—	4	2	no	—
34	C₂×D₃₅₆	—	4	1	no	—
35	C₂×D₃₅₆	—	4	1	no	—
36	C₂×D₃₅₆	—	4	1	no	—
37	Dic₃₅₆	—	4	4	no	—
38	Dic₃₅₆	—	8	4	no	—
39	D₇₁₂	—	4	4	no	—
40	C₄⋊Dic₈₉	—	4	2	no	—
41	C₄⋊Dic₈₉	—	4	1	no	—
42	C₄⋊Dic₈₉	—	4	1	no	—
43	C₄⋊Dic₈₉	—	4	2	no	—
44	C₂×C₈×C₈₉	D₈	1	2	yes	—
45	C₂×C₈×C₈₉	D₈	1	2	yes	—
46	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
47	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₈₉	—	2	4	no	—
49	C₂²×D₈₉	—	8	1	no	—
50	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₈₉	—	8	1	no	—
54	C₂×Dic₈₉	—	8	2	no	—
55	D₃₅₆	—	8	2	no	—
56	D₃₅₆	—	8	2	no	—
57	D₃₅₆	—	8	2	no	—
58	Dic₁₇₈	—	8	2	no	—
59	Dic₁₇₈	—	8	4	no	—
60	C₈×C₈₉	—	2	2	no	—
61	C₈×C₈₉	—	2	2	no	—
62	C₈×C₈₉	#135	1	4	yes	—
63	C₂²×C₈₉	#493	1	1	yes	characteristic
64	C₂×D₈₉	—	16	1	no	—
65	C₂×D₈₉	—	16	1	no	—
66	C₂×D₈₉	—	16	1	no	—
67	C₄×C₈₉	#495	1	2	yes	—
68	C₄×C₈₉	#509	1	2	yes	—
69	C₄×C₈₉	#510	1	2	yes	—
70	Dic₈₉	—	16	1	no	—
71	Dic₈₉	—	16	2	no	—
72	#12512	—	89	1	no	maximal, Sylow
73	C₂×C₈₉	#1709	1	1	yes	characteristic
74	C₂×C₈₉	#2250	1	1	yes	characteristic
75	C₂×C₈₉	#2251	1	1	yes	characteristic
76	D₈₉	—	32	1	no	—
77	#2241	—	89	2	no	—
78	#2240	—	89	2	no	—
79	#1713	—	89	1	no	—
80	#1709	—	89	1	no	—
81	#1311	D₈₉	1	1	yes	characteristic, complemented
82	C₈₉	#12512	1	1	yes	characteristic, complemented, Sylow
83	#500	—	89	2	no	—
84	#500	—	178	1	no	—
85	#494	—	89	1	no	—
86	#494	—	178	1	no	—
87	#493	—	89	1	no	—
88	#486	—	178	1	no	—
89	#486	—	178	1	no	—
90	#358	—	178	4	no	—
91	#357	—	178	4	no	—
92	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
93	C₈²	C₂×D₈₉	1	1	yes	characteristic
94	C₂×Q₁₆	—	356	2	no	—
95	C₂×Q₁₆	—	178	2	no	—
96	C₂×D₈	—	178	2	no	—
97	C₄⋊Q₈	—	178	1	no	—
98	C₄⋊Q₈	—	178	2	no	—
99	#127	—	178	1	no	—
100	C₂×C₁₆	D₃₅₆	1	4	yes	—
101	#107	—	356	2	no	—
102	#107	—	178	2	no	—
103	#107	—	178	2	no	—
104	#102	—	356	2	no	—
105	#102	—	356	2	no	—
106	C₄×C₈	D₃₅₆	1	1	yes	characteristic
107	C₄×C₈	D₃₅₆	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
109	C₂×Q₈	—	356	1	no	—
110	C₂×Q₈	—	356	2	no	—
111	C₂×D₄	—	356	1	no	—
112	C₂×D₄	—	356	1	no	—
113	C₂×D₄	—	356	1	no	—
114	Q₁₆	—	356	4	no	—
115	Q₁₆	—	712	4	no	—
116	D₈	—	356	4	no	—
117	C₂×C₈	D₇₁₂	1	2	yes	—
118	C₂×C₈	D₇₁₂	1	2	yes	—
119	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
120	C₄⋊C₄	—	356	2	no	—
121	C₄⋊C₄	—	356	1	no	—
122	C₄⋊C₄	—	356	1	no	—
123	C₄⋊C₄	—	356	2	no	—
124	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
125	C₁₆	—	2	4	no	—
126	C₂³	—	712	1	no	—
127	Q₈	—	712	2	no	—
128	Q₈	—	712	4	no	—
129	D₄	—	712	2	no	—
130	D₄	—	712	2	no	—
131	D₄	—	712	2	no	—
132	C₂×C₄	—	712	1	no	—
133	C₂×C₄	—	712	2	no	—
134	C₂×C₄	#15668009	1	1	yes	characteristic
135	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
136	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	#15668034	1	4	yes	—
140	C₂²	—	1424	1	no	—
141	C₂²	—	1424	1	no	—
142	C₂²	—	1424	1	no	—
143	C₂²	#19151975	1	1	yes	characteristic
144	C₄	—	1424	1	no	—
145	C₄	—	1424	2	no	—
146	C₄	#19151982	1	2	yes	—
147	C₄	#19152036	1	2	yes	—
148	C₄	#19152041	1	2	yes	—
149	C₂	—	2848	1	no	—
150	C₂	#26019858	1	1	yes	characteristic
151	C₂	#26022518	1	1	yes	characteristic
152	C₂	#26022523	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022480

C₂

yes

maximal, complemented

#26022476

C₂

yes

maximal

#26019870

C₂

yes

maximal, characteristic, complemented

#26019858

C₂

yes

maximal, characteristic

C₈₉×#1311

C₂

yes

maximal, characteristic, complemented

#19152003

—

#19152003

C₂²

yes

—

#19151978

—

#19151978

C₂²

yes

characteristic

#19151975

C₂²

yes

—

—

—

—

C₄×C₁₆×C₈₉

C₂²

yes

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×D₇₁₂

—

—

—

—

—

—

—

—

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

D₇₁₂

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#135

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

#495

yes

—

C₄×C₈₉

#509

yes

—

C₄×C₈₉

#510

yes

—

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

—

—

—

—

D₈₉

yes

characteristic, complemented

C₈₉

#12512

yes

characteristic, complemented, Sylow

#500

—

#500

—

178

—

#494

—

#494

—

178

—

#493

—

#486

—

178

—

#486

—

178

—

#358

—

178

—

#357

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

356

—

C₂×Q₁₆

—

178

—

C₂×D₈

—

178

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

—

178

—

#127

—

178

—

100

C₂×C₁₆

D₃₅₆

yes

—

101

#107

—

356

—

102

#107

—

178

—

103

#107

—

178

—

104

#102

—

356

—

105

#102

—

356

—

106

C₄×C₈

D₃₅₆

yes

characteristic

107

C₄×C₈

D₃₅₆

yes

characteristic

108

C₄×C₈

C₂²×D₈₉

yes

characteristic

109

C₂×Q₈

—

356

—

110

C₂×Q₈

—

356

—

111

C₂×D₄

—

356

—

112

C₂×D₄

—

356

—

113

C₂×D₄

—

356

—

114

Q₁₆

—

356

—

115

Q₁₆

—

712

—

116

D₈

—

356

—

117

C₂×C₈

D₇₁₂

yes

—

118

C₂×C₈

D₇₁₂

yes

—

119

C₂×C₈

C₂×D₃₅₆

yes

—

120

C₄⋊C₄

—

356

—

121

C₄⋊C₄

—

356

—

122

C₄⋊C₄

—

356

—

123

C₄⋊C₄

—

356

—

124

C₄²

C₂×D₃₅₆

yes

characteristic

125

C₁₆

—

126

C₂³

—

712

—

127

Q₈

—

712

—

128

Q₈

—

712

—

129

D₄

—

712

—

130

D₄

—

712

—

131

D₄

—

712

—

132

C₂×C₄

—

712

—

133

C₂×C₄

—

712

—

134

C₂×C₄

#15668009

yes

characteristic

135

C₂×C₄

C₂×D₇₁₂

yes

characteristic

136

C₂×C₄

C₂×D₇₁₂

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

#15668034

yes

—

140

C₂²

—

1424

—

141

C₂²

—

1424

—

142

C₂²

—

1424

—

143

C₂²

#19151975

yes

characteristic

144

C₄

—

1424

—

145

C₄

—

1424

—

146

C₄

#19151982

yes

—

147

C₄

#19152036

yes

—

148

C₄

#19152041

yes

—

149

C₂

—

2848

—

150

C₂

#26019858

yes

characteristic

151

C₂

#26022518

yes

characteristic

152

C₂

#26022523

yes

characteristic

	subgroup	G/H
center	C₂²	#19151975
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1311	C₂
socle	C₂²×C₈₉	#493
prime radical	#19151975	C₂²
hypercenter	#1311	D₈₉
hypocenter	C₈₉	#12512
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#19151975

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1311

C₂

C₂²×C₈₉

C₂²

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #26022480	C₂	semidirect
3. #26019870	C₂	semidirect
5. C₈₉×#1311	C₂	semidirect
81. #1311	D₈₉	semidirect
82. C₈₉	#12512	semidirect

H ≅ G/N

type

1. #26022480

C₂

semidirect

3. #26019870

C₂

semidirect

5. C₈₉×#1311

C₂

semidirect

81. #1311

D₈₉

semidirect

82. C₈₉

#12512

semidirect

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

subgroup	index
#26022480	2
#26022476	2
#26019870	2
#26019858	2
C₈₉×#1311	2
#12512	89

subgroup

index

C₈₉×#1311

Subgroups — #36469127

Conjugacy classes of subgroups

Complements

Maximal subgroups