Group #11800069 — subgroups

Subgroups — #11800069

Order 1152.

210 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₉×#500	C₂	1	1	yes	maximal, characteristic
2	#10588365	C₂	1	1	yes	maximal, characteristic
3	#10588169	C₂	1	4	yes	maximal
4	C₄×C₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#71382	—	3	1	no	maximal
6	C₉×C₄⋊Q₈	—	2	1	no	—
7	C₂×Dic₃₆	—	2	1	no	—
8	C₂×Dic₃₆	C₂²	1	2	yes	—
9	#63431	—	2	1	no	—
10	C₉×#107	—	2	1	no	—
11	C₉×#107	C₂²	1	2	yes	—
12	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₉⋊C₁₆	C₂²	1	2	yes	—
14	C₃×#500	D₃	1	1	yes	characteristic
15	#4699	—	3	1	no	—
16	#4492	—	3	4	no	—
17	C₄×C₃⋊C₁₆	—	3	1	no	—
18	C₂×Q₈×C₉	—	4	1	no	—
19	C₂×Dic₁₈	—	2	2	no	—
20	C₂×C₈×C₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
22	C₉×C₄⋊C₄	—	2	2	no	—
23	C₉×C₄⋊C₄	—	4	1	no	—
24	C₄²×C₉	D₄	1	1	yes	characteristic
25	C₄⋊Dic₉	—	4	1	no	—
26	C₄⋊Dic₉	—	4	1	no	—
27	Dic₃₆	—	2	4	no	—
28	Dic₃₆	—	4	2	no	—
29	C₉⋊C₁₆	—	2	2	no	—
30	#2241	—	9	1	no	Sylow
31	C₃×C₄⋊Q₈	—	2	1	no	—
32	C₂×Dic₁₂	—	3	2	no	—
33	C₂×Dic₁₂	—	6	1	no	—
34	#869	—	6	1	no	—
35	C₃×#107	—	2	1	no	—
36	C₃×#107	C₂×D₃	1	2	yes	—
37	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
38	C₂×C₃⋊C₁₆	—	3	2	no	—
39	Q₈×C₉	—	8	2	no	—
40	C₂×C₄×C₉	—	4	2	no	—
41	C₂×C₄×C₉	D₈	1	1	yes	characteristic
42	C₂×C₄×C₉	D₈	1	1	yes	characteristic
43	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₉	—	4	2	no	—
45	Dic₁₈	—	4	2	no	—
46	Dic₁₈	—	4	2	no	—
47	C₈×C₉	SD₁₆	1	2	yes	—
48	C₈×C₉	#41	1	2	yes	—
49	#500	D₉	1	1	yes	characteristic
50	#494	—	9	1	no	—
51	#358	—	9	4	no	—
52	C₄×C₁₆	—	9	1	no	—
53	C₂×C₃×Q₈	—	4	1	no	—
54	C₂×Dic₆	—	6	2	no	—
55	C₂×C₃×C₈	#67	1	1	yes	characteristic
56	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
57	C₃×C₄⋊C₄	—	2	2	no	—
58	C₃×C₄⋊C₄	—	4	1	no	—
59	C₃×C₄²	#67	1	1	yes	characteristic
60	C₄⋊Dic₃	—	12	1	no	—
61	C₄⋊Dic₃	—	12	1	no	—
62	Dic₁₂	—	6	4	no	—
63	Dic₁₂	—	12	2	no	—
64	C₃⋊C₁₆	—	6	2	no	—
65	C₂²×C₉	C₂×D₈	1	1	yes	characteristic
66	C₄×C₉	—	8	2	no	—
67	C₄×C₉	SD₃₂	1	2	yes	—
68	C₄×C₉	Q₃₂	1	2	yes	—
69	C₄×C₉	#124	1	1	yes	characteristic
70	C₄×C₉	C₂×SD₁₆	1	1	yes	characteristic
71	Dic₉	—	8	2	no	—
72	C₂×Q₁₆	—	9	2	no	—
73	C₂×Q₁₆	—	18	1	no	—
74	C₄⋊Q₈	—	2	1	no	—
75	C₄⋊Q₈	—	18	1	no	—
76	C₂×C₁₆	—	9	2	no	—
77	#107	—	2	1	no	—
78	#107	C₂×D₉	1	2	yes	—
79	C₄×C₈	C₂×D₉	1	1	yes	characteristic
80	C₃×Q₈	—	8	2	no	—
81	C₂×C₃×C₄	—	4	2	no	—
82	C₂×C₃×C₄	#213	1	1	yes	characteristic
83	C₂×C₃×C₄	#213	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
85	C₂×Dic₃	—	12	2	no	—
86	Dic₆	—	12	2	no	—
87	Dic₆	—	12	2	no	—
88	C₃×C₈	#215	1	2	yes	—
89	C₃×C₈	#239	1	2	yes	—
90	C₂×C₉	#486	1	1	yes	characteristic
91	C₂×C₉	C₂×SD₃₂	1	1	yes	characteristic
92	C₂×C₉	C₂×Q₃₂	1	1	yes	characteristic
93	C₂×Q₈	—	4	1	no	—
94	C₂×Q₈	—	18	2	no	—
95	Q₁₆	—	18	4	no	—
96	Q₁₆	—	36	2	no	—
97	C₂×C₈	#611	1	1	yes	characteristic
98	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	36	1	no	—
102	C₄⋊C₄	—	36	1	no	—
103	C₄²	#611	1	1	yes	characteristic
104	C₁₆	—	18	2	no	—
105	C₂²×C₃	C₂×#213	1	1	yes	characteristic
106	C₃×C₄	—	8	2	no	—
107	C₃×C₄	#827	1	2	yes	—
108	C₃×C₄	#829	1	2	yes	—
109	C₃×C₄	C₂×#215	1	1	yes	characteristic
110	C₃×C₄	#947	1	1	yes	characteristic
111	Dic₃	—	24	2	no	—
112	C₉	#2241	1	1	yes	characteristic, complemented, Sylow
113	Q₈	—	8	2	no	—
114	Q₈	—	36	2	no	—
115	Q₈	—	36	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	36	2	no	—
118	C₂×C₄	#3673	1	1	yes	characteristic
119	C₂×C₄	#3673	1	1	yes	characteristic
120	C₂×C₄	C₂×#611	1	1	yes	characteristic
121	C₈	#3675	1	2	yes	—
122	C₈	#3701	1	2	yes	—
123	C₂×C₃	#5088	1	1	yes	characteristic
124	C₂×C₃	C₂×#827	1	1	yes	characteristic
125	C₂×C₃	C₂×#829	1	1	yes	characteristic
126	C₂²	C₂×#3673	1	1	yes	characteristic
127	C₄	—	8	2	no	—
128	C₄	—	72	2	no	—
129	C₄	#63386	1	2	yes	—
130	C₄	#63388	1	2	yes	—
131	C₄	C₂×#3675	1	1	yes	characteristic
132	C₄	#63509	1	1	yes	characteristic
133	C₃	#71382	1	1	yes	characteristic
134	C₂	#10588773	1	1	yes	characteristic
135	C₂	C₂×#63386	1	1	yes	characteristic
136	C₂	C₂×#63388	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#500

C₂

yes

maximal, characteristic

#10588365

C₂

yes

maximal, characteristic

#10588169

C₂

yes

maximal

C₄×C₉⋊C₁₆

C₂

yes

maximal, characteristic

#71382

—

maximal

C₉×C₄⋊Q₈

—

C₂×Dic₃₆

—

C₂×Dic₃₆

C₂²

yes

—

#63431

—

C₉×#107

—

C₉×#107

C₂²

yes

—

C₄×C₈×C₉

C₂²

yes

characteristic

C₂×C₉⋊C₁₆

C₂²

yes

—

C₃×#500

D₃

yes

characteristic

#4699

—

#4492

—

C₄×C₃⋊C₁₆

—

C₂×Q₈×C₉

—

C₂×Dic₁₈

—

C₂×C₈×C₉

D₄

yes

characteristic

C₂×C₈×C₉

C₂³

yes

characteristic

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₄²×C₉

D₄

yes

characteristic

C₄⋊Dic₉

—

C₄⋊Dic₉

—

Dic₃₆

—

Dic₃₆

—

C₉⋊C₁₆

—

#2241

—

Sylow

C₃×C₄⋊Q₈

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

#869

—

C₃×#107

—

C₃×#107

C₂×D₃

yes

—

C₃×C₄×C₈

C₂×D₃

yes

characteristic

C₂×C₃⋊C₁₆

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

D₈

yes

characteristic

C₂×C₄×C₉

D₈

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

C₈×C₉

SD₁₆

yes

—

C₈×C₉

#41

yes

—

#500

D₉

yes

characteristic

#494

—

#358

—

C₄×C₁₆

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×C₃×C₈

#67

yes

characteristic

C₂×C₃×C₈

C₂²×D₃

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

#67

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₁₂

—

Dic₁₂

—

C₃⋊C₁₆

—

C₂²×C₉

C₂×D₈

yes

characteristic

C₄×C₉

—

C₄×C₉

SD₃₂

yes

—

C₄×C₉

Q₃₂

yes

—

C₄×C₉

#124

yes

characteristic

C₄×C₉

C₂×SD₁₆

yes

characteristic

Dic₉

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

#107

—

#107

C₂×D₉

yes

—

C₄×C₈

C₂×D₉

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

yes

C₂×C₃×C₄

yes

C₂×C₃×C₄

C₂×#67

yes

characteristic

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

#215

yes

—

C₃×C₈

#239

yes

—

C₂×C₉

#486

yes

characteristic

C₂×C₉

C₂×SD₃₂

yes

characteristic

C₂×C₉

C₂×Q₃₂

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

#611

yes

characteristic

C₂×C₈

C₂²×D₉

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄²

#611

yes

characteristic

104

C₁₆

—

105

C₂²×C₃

C₂×#213

yes

characteristic

106

C₃×C₄

—

107

C₃×C₄

#827

yes

—

108

C₃×C₄

#829

yes

—

109

C₃×C₄

C₂×#215

yes

characteristic

110

C₃×C₄

#947

yes

characteristic

111

Dic₃

—

112

C₉

#2241

yes

characteristic, complemented, Sylow

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#3673

yes

characteristic

119

C₂×C₄

#3673

yes

characteristic

120

C₂×C₄

C₂×#611

yes

characteristic

121

C₈

#3675

yes

—

122

C₈

#3701

yes

—

123

C₂×C₃

#5088

yes

characteristic

124

C₂×C₃

C₂×#827

yes

characteristic

125

C₂×C₃

C₂×#829

yes

characteristic

126

C₂²

C₂×#3673

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

#63386

yes

—

130

C₄

#63388

yes

—

131

C₄

C₂×#3675

yes

characteristic

132

C₄

#63509

yes

characteristic

133

C₃

#71382

yes

characteristic

134

C₂

#10588773

yes

characteristic

135

C₂

C₂×#63386

yes

characteristic

136

C₂

C₂×#63388

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#3673
commutator, cosocle	C₂×C₈×C₉	C₂³
Frattini	C₂×C₃×C₈	C₂²×D₃
Fitting	C₉×#500	C₂
socle, prime radical	C₂²×C₃	C₂×#213
hypercenter	#500	D₉
hypocenter	C₉	#2241
LCS	G ▷ C₂×C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

subgroup

G/H

center

C₂²

C₂×#3673

commutator, cosocle

C₂×C₈×C₉

C₂³

Frattini

C₂×C₃×C₈

C₂²×D₃

C₂

C₂²×C₃

D₉

C₉

G ▷ C₂×C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

N	H ≅ G/N	type
112. C₉	#2241	semidirect

H ≅ G/N

type

112. C₉

#2241

semidirect

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

subgroup	index
C₉×#500	2
#10588365	2
#10588169	2
C₄×C₉⋊C₁₆	2
#71382	3

subgroup

index

C₄×C₉⋊C₁₆

Subgroups — #11800069

Conjugacy classes of subgroups

Complements

Maximal subgroups