Group #36469195 — subgroups

Subgroups — #36469195

Order 22784.

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

normal subgroups only

This group has 274 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	#26022504	C₂	1	2	yes	maximal
2	#26022481	C₂	1	2	yes	maximal
3	#26019870	C₂	1	1	yes	maximal, characteristic
4	#26018170	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1707	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667958	—	2	4	no	—
7	#19152003	—	2	1	no	—
8	#19152003	—	2	1	no	—
9	#19151978	C₂²	1	2	yes	—
10	#19151401	—	2	4	no	—
11	C₄×#13083549	C₂²	1	2	yes	—
12	#500×C₈₉	C₂²	1	1	yes	characteristic
13	#498×C₈₉	—	2	1	no	—
14	#494×C₈₉	C₂²	1	1	yes	characteristic
15	#487×C₈₉	—	2	1	no	—
16	C₈²×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₅₆	—	2	2	no	—
18	C₂×Dic₃₅₆	—	2	2	no	—
19	#15668012	—	2	2	no	—
20	#15667958	—	4	8	no	—
21	C₂×#13083549	D₄	1	4	yes	—
22	#15667946	—	4	2	no	—
23	#15667946	—	4	2	no	—
24	C₂×Q₁₆×C₈₉	—	2	2	no	—
25	C₄⋊Q₈×C₈₉	—	2	1	no	—
26	C₄⋊Q₈×C₈₉	—	2	1	no	—
27	#107×C₈₉	—	2	2	no	—
28	#106×C₈₉	—	4	2	no	—
29	#103×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	2	no	—
34	Dic₃₅₆	—	4	4	no	—
35	Dic₃₅₆	—	4	4	no	—
36	C₄⋊Dic₈₉	—	4	2	no	—
37	C₄⋊Dic₈₉	—	4	2	no	—
38	#13083549	—	2	4	no	—
39	C₂×Q₈×C₈₉	—	4	1	no	—
40	C₂×Q₈×C₈₉	—	4	1	no	—
41	Q₁₆×C₈₉	—	4	4	no	—
42	C₂×C₈×C₈₉	D₈	1	2	yes	—
43	C₂×C₈×C₈₉	D₈	1	2	yes	—
44	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
45	C₄⋊C₄×C₈₉	—	4	1	no	—
46	C₄⋊C₄×C₈₉	—	4	1	no	—
47	C₄⋊C₄×C₈₉	—	4	1	no	—
48	C₄⋊C₄×C₈₉	—	4	1	no	—
49	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₈₉	—	8	2	no	—
51	Q₈×C₈₉	—	8	2	no	—
52	C₂×C₄×C₈₉	—	8	1	no	—
53	C₂×C₄×C₈₉	—	8	1	no	—
54	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
57	C₂×Dic₈₉	—	8	2	no	—
58	Dic₁₇₈	—	8	2	no	—
59	Dic₁₇₈	—	8	2	no	—
60	C₈×C₈₉	—	2	2	no	—
61	C₈×C₈₉	Q₃₂	1	4	yes	complemented
62	C₈×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
63	C₂²×C₈₉	#493	1	1	yes	characteristic
64	C₄×C₈₉	—	16	1	no	—
65	C₄×C₈₉	—	16	1	no	—
66	C₄×C₈₉	#496	1	2	yes	—
67	C₄×C₈₉	C₂×Q₃₂	1	2	yes	—
68	C₄×C₈₉	#510	1	2	yes	—
69	Dic₈₉	—	16	2	no	—
70	#12528	—	89	1	no	maximal, Sylow
71	C₂×C₈₉	#1721	1	1	yes	characteristic
72	C₂×C₈₉	#2247	1	1	yes	characteristic
73	C₂×C₈₉	#2251	1	1	yes	characteristic
74	#2247	—	89	2	no	—
75	#2241	—	89	2	no	—
76	#1713	—	89	1	no	—
77	#1707	D₈₉	1	1	yes	characteristic
78	#1312	—	89	1	no	—
79	C₈₉	#12528	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₃₂	—	178	4	no	—
81	#500	—	178	1	no	—
82	#500	—	178	1	no	—
83	#500	C₂×D₈₉	1	1	yes	characteristic
84	#498	—	2	1	no	—
85	#494	—	89	2	no	—
86	#494	C₂×D₈₉	1	1	yes	characteristic
87	#487	—	2	1	no	—
88	#358	—	178	4	no	—
89	C₄×C₁₆	—	89	2	no	—
90	C₈²	C₂×D₈₉	1	1	yes	characteristic
91	C₂×Q₁₆	—	2	2	no	—
92	C₂×Q₁₆	—	178	2	no	—
93	C₂×Q₁₆	—	178	2	no	—
94	C₄⋊Q₈	—	2	1	no	—
95	C₄⋊Q₈	—	2	1	no	—
96	C₄⋊Q₈	—	178	2	no	—
97	Q₃₂	—	356	8	no	—
98	C₂×C₁₆	—	89	4	no	—
99	#107	—	2	2	no	—
100	#107	—	356	2	no	—
101	#107	—	356	2	no	—
102	#106	—	4	2	no	—
103	#103	—	4	2	no	—
104	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
107	C₂×Q₈	—	4	1	no	—
108	C₂×Q₈	—	4	1	no	—
109	C₂×Q₈	—	356	2	no	—
110	Q₁₆	—	4	4	no	—
111	Q₁₆	—	356	4	no	—
112	Q₁₆	—	356	4	no	—
113	C₂×C₈	#13083559	1	2	yes	—
114	C₂×C₈	#13083559	1	2	yes	—
115	C₂×C₈	D₄×D₈₉	1	2	yes	—
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	4	1	no	—
118	C₄⋊C₄	—	4	1	no	—
119	C₄⋊C₄	—	4	1	no	—
120	C₄⋊C₄	—	356	2	no	—
121	C₄⋊C₄	—	356	2	no	—
122	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
123	C₁₆	—	178	4	no	—
124	Q₈	—	8	2	no	—
125	Q₈	—	8	2	no	—
126	Q₈	—	712	2	no	—
127	Q₈	—	712	2	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	712	2	no	—
131	C₂×C₄	#15668008	1	1	yes	characteristic
132	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
133	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	#15667958	1	4	yes	complemented
136	C₈	#15668039	1	4	yes	—
137	C₂²	#19151974	1	1	yes	characteristic
138	C₄	—	16	1	no	—
139	C₄	—	16	1	no	—
140	C₄	—	1424	2	no	—
141	C₄	#19151984	1	2	yes	—
142	C₄	C₂×#15667958	1	2	yes	—
143	C₄	#19152038	1	2	yes	—
144	C₂	#26019916	1	1	yes	characteristic
145	C₂	#26022504	1	1	yes	characteristic
146	C₂	#26022524	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022504

C₂

yes

maximal

#26022481

C₂

yes

maximal

#26019870

C₂

yes

maximal, characteristic

#26018170

C₂

yes

maximal, characteristic

C₈₉×#1707

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

#19151401

—

C₄×#13083549

C₂²

yes

—

#500×C₈₉

C₂²

yes

characteristic

#498×C₈₉

—

#494×C₈₉

C₂²

yes

characteristic

#487×C₈₉

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

#15668012

—

#15667958

—

C₂×#13083549

D₄

yes

—

#15667946

—

#15667946

—

C₂×Q₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

#107×C₈₉

—

#106×C₈₉

—

#103×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083549

—

C₂×Q₈×C₈₉

—

C₂×Q₈×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

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×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

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

Q₃₂

yes

complemented

C₈×C₈₉

AGL(1,ℤ/8)

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#496

yes

—

C₄×C₈₉

C₂×Q₃₂

yes

—

C₄×C₈₉

#510

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

—

—

D₈₉

yes

characteristic

#1312

—

C₈₉

#12528

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

178

—

#500

—

178

—

#500

—

178

—

#500

C₂×D₈₉

yes

—

—

C₂×D₈₉

yes

characteristic

#487

—

#358

—

178

—

C₄×C₁₆

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

Q₃₂

—

356

—

C₂×C₁₆

—

#107

—

100

#107

—

356

—

101

#107

—

356

—

102

#106

—

103

#103

—

104

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

105

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

106

C₄×C₈

C₂²×D₈₉

yes

characteristic

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

356

—

110

Q₁₆

—

111

Q₁₆

—

356

—

112

Q₁₆

—

356

—

113

C₂×C₈

#13083559

yes

—

114

C₂×C₈

#13083559

yes

—

115

C₂×C₈

D₄×D₈₉

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

356

—

121

C₄⋊C₄

—

356

—

122

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

123

C₁₆

—

178

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

712

—

127

Q₈

—

712

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

712

—

131

C₂×C₄

#15668008

yes

characteristic

132

C₂×C₄

C₂×#13083559

yes

characteristic

133

C₂×C₄

C₂×#13083559

yes

characteristic

134

C₈

—

135

C₈

#15667958

yes

complemented

136

C₈

#15668039

yes

—

137

C₂²

#19151974

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

1424

—

141

C₄

#19151984

yes

—

142

C₄

C₂×#15667958

yes

—

143

C₄

#19152038

yes

—

144

C₂

#26019916

yes

characteristic

145

C₂

#26022504

yes

characteristic

146

C₂

#26022524

yes

characteristic

	subgroup	G/H
center	C₂²	#19151974
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1707	C₂
socle, prime radical	C₂²×C₈₉	#493
hypercenter	#1707	D₈₉
hypocenter	C₈₉	#12528
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1707

subgroup

G/H

center

C₂²

#19151974

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1707

C₂

C₂²×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1707

N	H ≅ G/N	type
61. C₈×C₈₉	Q₃₂	semidirect
79. C₈₉	#12528	semidirect
135. C₈	#15667958	semidirect

H ≅ G/N

type

61. C₈×C₈₉

Q₃₂

semidirect

79. C₈₉

#12528

semidirect

135. C₈

#15667958

semidirect

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

subgroup	index
#26022504	2
#26022481	2
#26019870	2
#26018170	2
C₈₉×#1707	2
#12528	89

subgroup

index

C₈₉×#1707

Subgroups — #36469195

Conjugacy classes of subgroups

Complements

Maximal subgroups