Group #11786104 — subgroups

Subgroups — #11786104

Order 1152.

270 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 102 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	#10588802	C₂	1	2	yes	maximal
2	C₂×#63375	C₂	1	1	yes	maximal, characteristic
3	C₉×#341	C₂	1	1	yes	maximal, characteristic
4	#10588231	C₂	1	2	yes	maximal
5	#10588227	C₂	1	1	yes	maximal, characteristic
6	#68929	—	3	1	no	maximal
7	C₂×C₉×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
10	C₂×#3667	C₂²	1	2	yes	—
11	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
12	#63375	C₄	1	4	yes	—
13	Dic₉⋊C₈	—	2	4	no	—
14	#5118	—	3	2	no	—
15	C₂×#815	—	3	1	no	—
16	C₃×#341	D₃	1	1	yes	characteristic
17	#4558	—	3	2	no	—
18	#4552	—	3	1	no	—
19	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
20	C₂²×Dic₉	Q₈	1	2	yes	—
21	C₉×M₁₆	—	2	2	no	—
22	C₉×M₁₆	—	2	2	no	—
23	C₂×C₈×C₉	—	2	2	no	—
24	C₂×C₈×C₉	—	2	2	no	—
25	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
26	C₄×Dic₉	C₂×C₄	1	2	yes	—
27	C₄×Dic₉	D₄	1	2	yes	—
28	#3667	—	2	2	no	—
29	#3667	—	2	2	no	—
30	C₂×C₉⋊C₈	—	2	2	no	—
31	C₂×C₉⋊C₈	—	2	2	no	—
32	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
33	#1838	—	9	1	no	Sylow
34	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
35	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
36	C₂×C₄×Dic₃	—	3	1	no	—
37	C₂×#208	—	3	2	no	—
38	C₂²×C₃⋊C₈	—	3	1	no	—
39	#815	—	3	4	no	—
40	Dic₃⋊C₈	—	6	4	no	—
41	C₂³×C₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₉	#41	1	2	yes	—
45	C₂×C₄×C₉	#41	1	2	yes	—
46	C₂×Dic₉	—	4	2	no	—
47	C₂×Dic₉	C₄⋊C₄	1	4	yes	—
48	C₈×C₉	—	2	2	no	—
49	C₈×C₉	—	4	2	no	—
50	C₉⋊C₈	—	2	2	no	—
51	C₉⋊C₈	—	4	2	no	—
52	#423	—	9	2	no	—
53	C₂×#97	—	9	1	no	—
54	#341	—	9	2	no	—
55	#341	—	9	1	no	—
56	#341	D₉	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
58	C₂²×Dic₃	—	3	2	no	—
59	C₃×M₁₆	—	2	2	no	—
60	C₃×M₁₆	—	2	2	no	—
61	C₂×C₃×C₈	—	2	2	no	—
62	C₂×C₃×C₈	—	2	2	no	—
63	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
64	C₄×Dic₃	—	3	2	no	—
65	C₄×Dic₃	—	3	2	no	—
66	#208	—	6	2	no	—
67	#208	—	6	2	no	—
68	C₂×C₃⋊C₈	—	3	2	no	—
69	C₂×C₃⋊C₈	—	6	2	no	—
70	C₂×C₃⋊C₈	—	6	2	no	—
71	C₂²×C₉	—	2	2	no	—
72	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₉	#125	1	2	yes	—
74	C₄×C₉	#117	1	1	yes	characteristic
75	C₄×C₉	#117	1	1	yes	characteristic
76	C₄×C₉	#124	1	2	yes	—
77	Dic₉	—	2	4	no	—
78	C₂×M₁₆	—	9	2	no	—
79	C₂×M₁₆	C₂×D₉	1	2	yes	—
80	C₂²×C₈	—	9	1	no	—
81	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
82	C₂×C₄²	—	9	1	no	—
83	C₄⋊C₈	—	18	4	no	—
84	#97	—	9	4	no	—
85	C₂³×C₃	D₃×Q₈	1	1	yes	characteristic
86	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
87	C₂×C₃×C₄	#235	1	2	yes	—
88	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄	#237	1	2	yes	—
90	C₂×Dic₃	—	3	4	no	—
91	C₂×Dic₃	—	12	2	no	—
92	C₃×C₈	—	2	2	no	—
93	C₃×C₈	—	4	2	no	—
94	C₃⋊C₈	—	6	2	no	—
95	C₃⋊C₈	—	12	2	no	—
96	C₂×C₉	—	2	2	no	—
97	C₂×C₉	#382	1	1	yes	characteristic
98	C₂×C₉	#430	1	2	yes	—
99	C₂²×C₄	—	9	2	no	—
100	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
101	M₁₆	—	2	2	no	—
102	M₁₆	—	2	2	no	—
103	M₁₆	—	18	2	no	—
104	M₁₆	—	18	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	9	2	no	—
108	C₂×C₈	—	18	2	no	—
109	C₂×C₈	—	18	2	no	—
110	C₂×C₈	C₄×D₉	1	2	yes	—
111	C₄²	—	9	2	no	—
112	C₄²	—	9	2	no	—
113	C₂²×C₃	—	2	2	no	—
114	C₂²×C₃	#889	1	2	yes	—
115	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
116	C₃×C₄	#872	1	1	yes	characteristic
117	C₃×C₄	#877	1	1	yes	characteristic
118	C₃×C₄	#885	1	2	yes	—
119	Dic₃	—	6	4	no	—
120	C₉	#1838	1	1	yes	characteristic, complemented, Sylow
121	C₂³	Q₈×D₉	1	1	yes	characteristic
122	C₂×C₄	—	9	4	no	—
123	C₂×C₄	—	36	2	no	—
124	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
125	C₂×C₄	#3697	1	2	yes	—
126	C₂×C₄	D₄×D₉	1	1	yes	characteristic
127	C₂×C₄	#3699	1	2	yes	—
128	C₈	—	2	2	no	—
129	C₈	—	4	2	no	—
130	C₈	—	18	2	no	—
131	C₈	—	36	2	no	—
132	C₂×C₃	—	2	2	no	—
133	C₂×C₃	#4650	1	1	yes	characteristic
134	C₂×C₃	#4893	1	2	yes	—
135	C₂²	—	2	2	no	—
136	C₂²	#63451	1	2	yes	—
137	C₂²	C₄⋊C₄×D₉	1	1	yes	characteristic
138	C₄	—	18	4	no	—
139	C₄	#63434	1	1	yes	characteristic
140	C₄	#63439	1	1	yes	characteristic
141	C₄	#63447	1	2	yes	—
142	C₃	#68929	1	1	yes	characteristic
143	C₂	—	2	2	no	—
144	C₂	#10588330	1	1	yes	characteristic
145	C₂	#10588573	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10588802

C₂

yes

maximal

C₂×#63375

C₂

yes

maximal, characteristic

C₉×#341

C₂

yes

maximal, characteristic

#10588231

C₂

yes

maximal

#10588227

C₂

yes

maximal, characteristic

#68929

—

maximal

C₂×C₉×M₁₆

C₂²

yes

—

C₂²×C₈×C₉

C₂²

yes

characteristic

C₂×C₄×Dic₉

C₂²

yes

characteristic

C₂×#3667

C₂²

yes

—

C₂²×C₉⋊C₈

C₂²

yes

characteristic

#63375

C₄

yes

—

Dic₉⋊C₈

—

#5118

—

C₂×#815

—

C₃×#341

D₃

yes

characteristic

#4558

—

#4552

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₂²×Dic₉

Q₈

yes

—

C₉×M₁₆

—

C₉×M₁₆

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂×C₄

yes

—

C₄×Dic₉

C₂×C₄

yes

—

C₄×Dic₉

D₄

yes

—

#3667

—

#3667

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

C₂×C₄

yes

—

#1838

—

Sylow

C₂×C₃×M₁₆

C₂×D₃

yes

—

C₂²×C₃×C₈

C₂×D₃

yes

characteristic

C₂×C₄×Dic₃

—

C₂×#208

—

C₂²×C₃⋊C₈

—

#815

—

Dic₃⋊C₈

—

C₂³×C₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

#41

yes

—

C₂×C₄×C₉

#41

yes

—

C₂×Dic₉

—

C₂×Dic₉

C₄⋊C₄

yes

—

C₈×C₉

—

C₈×C₉

—

C₉⋊C₈

—

C₉⋊C₈

—

#423

—

C₂×#97

—

#341

—

#341

—

#341

D₉

yes

characteristic

C₂²×C₃×C₄

C₂²×D₃

yes

characteristic

C₂²×Dic₃

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄×D₃

yes

—

C₄×Dic₃

—

C₄×Dic₃

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₉

—

C₂²×C₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉

#125

yes

—

C₄×C₉

yes

C₄×C₉

yes

C₄×C₉

yes

—

Dic₉

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₉

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

#97

—

C₂³×C₃

D₃×Q₈

yes

characteristic

C₂×C₃×C₄

C₂×C₄×D₃

yes

characteristic

C₂×C₃×C₄

#235

yes

—

C₂×C₃×C₄

D₃×D₄

yes

characteristic

C₂×C₃×C₄

#237

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₉

—

C₂×C₉

#382

yes

characteristic

C₂×C₉

#430

yes

—

C₂²×C₄

—

100

C₂²×C₄

C₂²×D₉

yes

characteristic

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₄×D₉

yes

—

111

C₄²

—

112

C₄²

—

113

C₂²×C₃

—

114

C₂²×C₃

#889

yes

—

115

C₂²×C₃

D₃×C₄⋊C₄

yes

characteristic

116

C₃×C₄

#872

yes

characteristic

117

C₃×C₄

#877

yes

characteristic

118

C₃×C₄

#885

yes

—

119

Dic₃

—

120

C₉

#1838

yes

characteristic, complemented, Sylow

121

C₂³

Q₈×D₉

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂×C₄×D₉

yes

characteristic

125

C₂×C₄

#3697

yes

—

126

C₂×C₄

D₄×D₉

yes

characteristic

127

C₂×C₄

#3699

yes

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂×C₃

—

133

C₂×C₃

#4650

yes

characteristic

134

C₂×C₃

#4893

yes

—

135

C₂²

—

136

C₂²

#63451

yes

—

137

C₂²

C₄⋊C₄×D₉

yes

characteristic

138

C₄

—

139

C₄

#63434

yes

characteristic

140

C₄

#63439

yes

characteristic

141

C₄

#63447

yes

—

142

C₃

#68929

yes

characteristic

143

C₂

—

144

C₂

#10588330

yes

characteristic

145

C₂

#10588573

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₉
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#341	C₂
socle	C₂²×C₃	D₃×C₄⋊C₄
prime radical	C₂³×C₃	D₃×Q₈
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#341	D₉
hypocenter	C₉	#1838
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₉

commutator

C₂×C₄×C₉

C₂²×C₄

Frattini

C₂²×C₃×C₄

C₂²×D₃

Fitting

C₉×#341

C₂

socle

C₂²×C₃

D₃×C₄⋊C₄

prime radical

C₂³×C₃

D₃×Q₈

cosocle

C₂²×C₄×C₉

C₂³

D₉

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
120. C₉	#1838	semidirect

H ≅ G/N

type

120. C₉

#1838

semidirect

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

subgroup	index
#10588802	2
C₂×#63375	2
C₉×#341	2
#10588231	2
#10588227	2
#68929	3

subgroup

index

Subgroups — #11786104

Conjugacy classes of subgroups

Complements

Maximal subgroups