Group #36441527 — subgroups

Subgroups — #36441527

Order 22784.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#26019011	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151622	—	2	2	no	—
4	#19151351	—	2	2	no	—
5	#19151351	C₄	1	4	yes	—
6	C₂²×C₁₆×C₈₉	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆×C₈₉	—	2	2	no	—
9	C₂×#13083558	—	4	2	no	—
10	C₄⋊#10623494	—	2	4	no	—
11	C₄⋊#10623494	—	4	2	no	—
12	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₈₉	—	2	2	no	—
16	C₂×C₁₆×C₈₉	—	4	2	no	—
17	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₈₉	D₄	1	2	yes	—
19	#13083558	—	8	2	no	—
20	#13083558	—	8	2	no	—
21	C₂×#10623494	—	4	4	no	—
22	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
24	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
25	C₂×C₈×C₈₉	—	2	1	no	—
26	C₂×C₈×C₈₉	—	2	1	no	—
27	C₂×C₈×C₈₉	—	4	1	no	—
28	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
29	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
31	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
32	C₄²×C₈₉	SD₁₆	1	2	yes	—
33	C₁₆×C₈₉	—	4	2	no	—
34	C₂³×C₈₉	#107	1	1	yes	characteristic
35	C₂×C₄×C₈₉	—	2	1	no	—
36	C₂×C₄×C₈₉	—	2	1	no	—
37	C₂×C₄×C₈₉	—	4	1	no	—
38	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
39	C₂×C₄×C₈₉	#102	1	2	yes	—
40	C₂×C₄×C₈₉	#103	1	2	yes	—
41	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
42	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
43	C₂×C₄×C₈₉	SD₃₂	1	2	yes	—
44	C₂×C₄×C₈₉	Q₃₂	1	1	yes	characteristic
45	C₈×C₈₉	—	2	2	no	—
46	C₈×C₈₉	—	2	1	no	—
47	C₈×C₈₉	#100	1	1	yes	characteristic
48	C₈×C₈₉	#101	1	1	yes	characteristic
49	#10623494	—	8	4	no	—
50	C₂²×C₈₉	—	2	1	no	—
51	C₂²×C₈₉	—	2	1	no	—
52	C₂²×C₈₉	#340	1	1	yes	characteristic
53	C₂²×C₈₉	#366	1	1	yes	characteristic
54	C₂²×C₈₉	#367	1	1	yes	characteristic
55	C₄×C₈₉	—	2	1	no	—
56	C₄×C₈₉	—	2	2	no	—
57	C₄×C₈₉	#338	1	1	yes	characteristic
58	C₄×C₈₉	#343	1	1	yes	characteristic
59	C₄×C₈₉	#357	1	2	yes	—
60	C₄×C₈₉	#358	1	2	yes	—
61	#7241	—	89	1	no	maximal, Sylow
62	C₂×C₈₉	—	2	1	no	—
63	C₂×C₈₉	—	2	1	no	—
64	C₂×C₈₉	#1292	1	1	yes	characteristic
65	C₂×C₈₉	#1380	1	1	yes	characteristic
66	C₂×C₈₉	#1385	1	1	yes	characteristic
67	C₂×C₄⋊C₁₆	D₈₉	1	1	yes	characteristic
68	#1567	—	89	2	no	—
69	C₈₉	#7241	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₁₆	Dic₈₉	1	2	yes	—
71	#423	—	178	2	no	—
72	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
73	C₄⋊C₁₆	—	2	2	no	—
74	#335	—	89	4	no	—
75	#335	—	178	2	no	—
76	C₂×M₁₆	—	356	2	no	—
77	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
78	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
79	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	4	2	no	—
82	C₄⋊C₈	—	356	2	no	—
83	C₄⋊C₈	—	178	4	no	—
84	C₄×C₈	C₄×D₈₉	1	2	yes	—
85	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
86	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
87	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
88	C₂²×C₄	Dic₃₅₆	1	1	yes	characteristic
89	M₁₆	—	712	2	no	—
90	M₁₆	—	712	2	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	356	4	no	—
95	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
96	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
97	C₂×C₈	#13083553	1	2	yes	—
98	C₄²	#13083551	1	2	yes	—
99	C₄²	#13083563	1	2	yes	—
100	C₁₆	—	4	2	no	—
101	C₂³	#15667946	1	1	yes	characteristic
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	#15667919	1	1	yes	characteristic
106	C₂×C₄	#15667934	1	2	yes	—
107	C₂×C₄	#15667937	1	2	yes	—
108	C₂×C₄	#15667944	1	1	yes	characteristic
109	C₂×C₄	D₁₄₂₄	1	1	yes	characteristic
110	C₂×C₄	#15667957	1	2	yes	—
111	C₂×C₄	Dic₇₁₂	1	1	yes	characteristic
112	C₈	—	2	1	no	—
113	C₈	—	2	2	no	—
114	C₈	—	712	4	no	—
115	C₈	#15667929	1	1	yes	characteristic
116	C₈	#15667931	1	1	yes	characteristic
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#19151361	1	1	yes	characteristic
120	C₂²	#19151424	1	1	yes	characteristic
121	C₂²	#19151426	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	1	no	—
124	C₄	#19151356	1	1	yes	characteristic
125	C₄	#19151367	1	1	yes	characteristic
126	C₄	#19151400	1	2	yes	—
127	C₄	#19151403	1	2	yes	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#26018127	1	1	yes	characteristic
131	C₂	#26018344	1	1	yes	characteristic
132	C₂	#26018356	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26019011

C₂

yes

maximal

C₂×C₄⋊C₁₆×C₈₉

C₂

yes

maximal, characteristic

#19151622

—

#19151351

—

#19151351

C₄

yes

—

C₂²×C₁₆×C₈₉

C₄

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

—

C₂×#13083558

—

C₄⋊#10623494

—

C₄⋊#10623494

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

Q₈

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

#13083558

—

#13083558

—

C₂×#10623494

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₈₉

D₈

yes

characteristic

C₂²×C₄×C₈₉

Q₁₆

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄²

yes

characteristic

C₂×C₈×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

SD₁₆

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

#107

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#95

yes

characteristic

C₂×C₄×C₈₉

#102

yes

—

C₂×C₄×C₈₉

#103

yes

—

C₂×C₄×C₈₉

#106

yes

characteristic

C₂×C₄×C₈₉

D₁₆

yes

characteristic

C₂×C₄×C₈₉

SD₃₂

yes

—

C₂×C₄×C₈₉

Q₃₂

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₄×C₈₉

#358

yes

—

#7241

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₄⋊C₁₆

D₈₉

yes

characteristic

#1567

—

C₈₉

#7241

yes

characteristic, complemented, Sylow

C₂²×C₁₆

Dic₈₉

yes

—

#423

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₄⋊C₁₆

—

#335

—

#335

—

178

—

C₂×M₁₆

—

356

—

C₂²×C₈

Dic₁₇₈

yes

characteristic

C₂²×C₈

C₂×Dic₈₉

yes

characteristic

C₂×C₄²

D₃₅₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

356

—

C₄⋊C₈

—

178

—

C₄×C₈

C₄×D₈₉

yes

—

C₄×C₈

C₂²⋊D₈₉

yes

—

C₂²×C₄

C₄⋊Dic₈₉

yes

characteristic

C₂²×C₄

D₇₁₂

yes

characteristic

C₂²×C₄

Dic₃₅₆

yes

characteristic

M₁₆

—

712

—

M₁₆

—

712

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

356

—

C₂×C₈

C₄×Dic₈₉

yes

characteristic

C₂×C₈

C₂²⋊Dic₈₉

yes

characteristic

C₂×C₈

#13083553

yes

—

C₄²

#13083551

yes

—

C₄²

#13083563

yes

—

100

C₁₆

—

101

C₂³

#15667946

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

#15667919

yes

characteristic

106

C₂×C₄

#15667934

yes

—

107

C₂×C₄

#15667937

yes

—

108

C₂×C₄

#15667944

yes

characteristic

109

C₂×C₄

D₁₄₂₄

yes

characteristic

110

C₂×C₄

#15667957

yes

—

111

C₂×C₄

Dic₇₁₂

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

712

—

115

C₈

#15667929

yes

characteristic

116

C₈

#15667931

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

#19151361

yes

characteristic

120

C₂²

#19151424

yes

characteristic

121

C₂²

#19151426

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

#19151356

yes

characteristic

125

C₄

#19151367

yes

characteristic

126

C₄

#19151400

yes

—

127

C₄

#19151403

yes

—

128

C₂

—

129

C₂

—

130

C₂

#26018127

yes

characteristic

131

C₂

#26018344

yes

characteristic

132

C₂

#26018356

yes

characteristic

	subgroup	G/H
center	C₂²	#19151361
commutator	C₂×C₈×C₈₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₈₉
Fitting	C₂×C₄⋊C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	#340
prime radical	C₂³×C₈₉	#107
cosocle	C₂×C₄×C₈×C₈₉	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₈₉
hypocenter	C₈₉	#7241
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂²

#19151361

commutator

C₂×C₈×C₈₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₈₉

Fitting

C₂×C₄⋊C₁₆×C₈₉

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄×C₈×C₈₉

C₂²

hypercenter

C₂×C₄⋊C₁₆

D₈₉

hypocenter

C₈₉

#7241

LCS

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
69. C₈₉	#7241	semidirect

H ≅ G/N

type

69. C₈₉

#7241

semidirect

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

subgroup	index
#26019011	2
C₂×C₄⋊C₁₆×C₈₉	2
#7241	89

subgroup

index

#26019011

C₂×C₄⋊C₁₆×C₈₉

#7241

Subgroups — #36441527

Conjugacy classes of subgroups

Complements

Maximal subgroups