Group #11776238 — subgroups

Subgroups — #11776238

Order 1152.

303 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 122 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×#533	C₂	1	1	yes	maximal, characteristic
2	#10588605	C₂	1	2	yes	maximal
3	#10588224	C₂	1	4	yes	maximal, complemented
4	#71829	—	3	1	no	maximal
5	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
6	C₉×#125	C₄	1	2	yes	—
7	C₉×C₂²⋊Q₈	—	2	2	no	—
8	C₄×Q₈×C₉	—	2	2	no	—
9	C₉×#117	C₄	1	2	yes	—
10	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
11	#63389	—	2	2	no	—
12	C₄⋊(C₉⋊C₈)	C₂²	1	4	yes	—
13	C₄×C₉⋊C₈	—	2	2	no	—
14	C₃×#533	D₃	1	1	yes	characteristic
15	#4925	—	3	2	no	—
16	#4547	—	3	4	no	—
17	C₂×Q₈×C₉	—	2	2	no	—
18	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₉	D₄	1	2	yes	—
20	C₉×C₄⋊C₄	—	2	4	no	—
21	C₉×C₄⋊C₄	—	2	2	no	—
22	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
23	C₉×C₂²⋊C₄	—	2	2	no	—
24	C₄²×C₉	—	2	2	no	—
25	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₉	D₄	1	2	yes	—
27	C₄²×C₉	C₂³	1	1	yes	characteristic
28	C₂×C₉⋊C₈	—	2	4	no	—
29	#1544	—	9	1	no	Sylow
30	C₃×C₄⋊Q₈	C₂×D₃	1	2	yes	—
31	C₃×#125	Dic₃	1	2	yes	—
32	C₃×C₂²⋊Q₈	—	2	2	no	—
33	C₃×C₄×Q₈	—	2	2	no	—
34	C₃×#117	Dic₃	1	2	yes	—
35	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
36	#830	—	6	2	no	—
37	C₄⋊(C₃⋊C₈)	—	3	4	no	—
38	C₄×C₃⋊C₈	—	6	2	no	—
39	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
40	Q₈×C₉	—	4	4	no	—
41	C₂×C₄×C₉	—	2	2	no	—
42	C₂×C₄×C₉	—	2	4	no	—
43	C₂×C₄×C₉	—	2	4	no	—
44	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
48	C₉⋊C₈	—	4	4	no	—
49	#533	D₉	1	1	yes	characteristic
50	#431	—	9	2	no	—
51	#332	—	9	4	no	—
52	C₂×C₃×Q₈	—	2	2	no	—
53	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
54	C₂²×C₃×C₄	#67	1	2	yes	—
55	C₃×C₄⋊C₄	—	2	4	no	—
56	C₃×C₄⋊C₄	—	2	2	no	—
57	C₃×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
58	C₃×C₂²⋊C₄	—	2	2	no	—
59	C₃×C₄²	—	2	2	no	—
60	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
61	C₃×C₄²	#67	1	2	yes	—
62	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
63	C₂×C₃⋊C₈	—	6	4	no	—
64	C₂²×C₉	—	2	1	no	—
65	C₂²×C₉	—	2	2	no	—
66	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₉	—	2	1	no	—
68	C₄×C₉	—	2	2	no	—
69	C₄×C₉	—	4	4	no	—
70	C₄×C₉	#101	1	2	yes	—
71	C₄×C₉	#135	1	4	yes	—
72	C₄⋊Q₈	C₂×D₉	1	2	yes	—
73	#125	Dic₉	1	2	yes	—
74	C₂²⋊Q₈	—	2	2	no	—
75	C₄×Q₈	—	2	2	no	—
76	#117	Dic₉	1	2	yes	—
77	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
78	C₄⋊C₈	—	9	4	no	—
79	C₂²⋊C₈	—	18	2	no	—
80	C₄×C₈	—	18	2	no	—
81	C₂³×C₃	#217	1	1	yes	characteristic
82	C₃×Q₈	—	4	4	no	—
83	C₂×C₃×C₄	—	2	2	no	—
84	C₂×C₃×C₄	—	2	4	no	—
85	C₂×C₃×C₄	—	2	4	no	—
86	C₂×C₃×C₄	#217	1	2	yes	—
87	C₂×C₃×C₄	#217	1	1	yes	characteristic
88	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂×#67	1	2	yes	—
90	C₃⋊C₈	—	12	4	no	—
91	C₂×C₉	—	4	1	no	—
92	C₂×C₉	C₂×#101	1	1	yes	characteristic
93	C₂×C₉	#416	1	2	yes	—
94	C₂×Q₈	—	2	2	no	—
95	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
96	C₂²×C₄	#611	1	2	yes	—
97	C₂×C₈	—	18	4	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	C₂×Dic₉	1	4	yes	—
101	C₂²⋊C₄	—	2	2	no	—
102	C₄²	—	2	2	no	—
103	C₄²	C₂×Dic₉	1	1	yes	characteristic
104	C₄²	#611	1	2	yes	—
105	C₄²	C₂²×D₉	1	1	yes	characteristic
106	C₂²×C₃	—	2	1	no	—
107	C₂²×C₃	—	2	2	no	—
108	C₂²×C₃	C₂×#217	1	1	yes	characteristic
109	C₃×C₄	—	2	1	no	—
110	C₃×C₄	—	2	2	no	—
111	C₃×C₄	—	4	4	no	—
112	C₃×C₄	#836	1	2	yes	—
113	C₃×C₄	#950	1	4	yes	—
114	C₉	#1544	1	1	yes	characteristic, complemented, Sylow
115	C₂³	#3676	1	1	yes	characteristic
116	Q₈	—	4	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	#3676	1	2	yes	—
121	C₂×C₄	#3676	1	1	yes	characteristic
122	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
123	C₂×C₄	C₂×#611	1	2	yes	—
124	C₈	—	36	4	no	—
125	C₂×C₃	—	4	1	no	—
126	C₂×C₃	C₂×#836	1	1	yes	characteristic
127	C₂×C₃	#5233	1	2	yes	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	2	no	—
130	C₂²	C₂×#3676	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	2	no	—
133	C₄	—	4	4	no	—
134	C₄	#63394	1	2	yes	—
135	C₄	#63513	1	4	yes	—
136	C₃	#71829	1	1	yes	characteristic
137	C₂	—	4	1	no	—
138	C₂	C₂×#63394	1	1	yes	characteristic
139	C₂	#10588917	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#533

C₂

yes

maximal, characteristic

#10588605

C₂

yes

maximal

#10588224

C₂

yes

maximal, complemented

#71829

—

maximal

C₉×C₄⋊Q₈

C₂²

yes

—

C₉×#125

C₄

yes

—

C₉×C₂²⋊Q₈

—

C₄×Q₈×C₉

—

C₉×#117

C₄

yes

—

C₂×C₄²×C₉

C₂²

yes

characteristic

#63389

—

C₄⋊(C₉⋊C₈)

C₂²

yes

—

C₄×C₉⋊C₈

—

C₃×#533

D₃

yes

characteristic

#4925

—

#4547

—

C₂×Q₈×C₉

—

C₂²×C₄×C₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₉

D₄

yes

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

C₂×C₄

yes

—

C₉×C₂²⋊C₄

—

C₄²×C₉

—

C₄²×C₉

C₂×C₄

yes

characteristic

C₄²×C₉

D₄

yes

—

C₄²×C₉

C₂³

yes

characteristic

C₂×C₉⋊C₈

—

#1544

—

Sylow

C₃×C₄⋊Q₈

C₂×D₃

yes

—

C₃×#125

Dic₃

yes

—

C₃×C₂²⋊Q₈

—

C₃×C₄×Q₈

—

C₃×#117

Dic₃

yes

—

C₂×C₃×C₄²

C₂×D₃

yes

characteristic

#830

—

C₄⋊(C₃⋊C₈)

—

C₄×C₃⋊C₈

—

C₂³×C₉

C₂²⋊C₄

yes

characteristic

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂²⋊C₄

yes

—

C₂×C₄×C₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

—

C₉⋊C₈

—

#533

D₉

yes

characteristic

#431

—

#332

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

C₂×Dic₃

yes

characteristic

C₂²×C₃×C₄

#67

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂×Dic₃

yes

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×Dic₃

yes

characteristic

C₃×C₄²

#67

yes

—

C₃×C₄²

C₂²×D₃

yes

characteristic

C₂×C₃⋊C₈

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

#101

yes

—

C₄×C₉

#135

yes

—

C₄⋊Q₈

C₂×D₉

yes

—

#125

Dic₉

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

Dic₉

yes

—

C₂×C₄²

C₂×D₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂³×C₃

#217

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#217

yes

—

C₂×C₃×C₄

#217

yes

characteristic

C₂×C₃×C₄

C₂²×Dic₃

yes

characteristic

C₂×C₃×C₄

C₂×#67

yes

—

C₃⋊C₈

—

C₂×C₉

—

C₂×C₉

C₂×#101

yes

characteristic

C₂×C₉

#416

yes

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₉

yes

characteristic

C₂²×C₄

#611

yes

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

C₂×Dic₉

yes

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

C₂×Dic₉

yes

characteristic

104

C₄²

#611

yes

—

105

C₄²

C₂²×D₉

yes

characteristic

106

C₂²×C₃

—

107

C₂²×C₃

—

108

C₂²×C₃

C₂×#217

yes

characteristic

109

C₃×C₄

—

110

C₃×C₄

—

111

C₃×C₄

—

112

C₃×C₄

#836

yes

—

113

C₃×C₄

#950

yes

—

114

C₉

#1544

yes

characteristic, complemented, Sylow

115

C₂³

#3676

yes

characteristic

116

Q₈

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

#3676

yes

—

121

C₂×C₄

#3676

yes

characteristic

122

C₂×C₄

C₂²×Dic₉

yes

characteristic

123

C₂×C₄

C₂×#611

yes

—

124

C₈

—

125

C₂×C₃

—

126

C₂×C₃

C₂×#836

yes

characteristic

127

C₂×C₃

#5233

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×#3676

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#63394

yes

—

135

C₄

#63513

yes

—

136

C₃

#71829

yes

characteristic

137

C₂

—

138

C₂

C₂×#63394

yes

characteristic

139

C₂

#10588917

yes

—

	subgroup	G/H
center	C₂×C₄	#3676
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₉×#533	C₂
socle	C₂²×C₃	C₂×#217
prime radical	C₂³×C₃	#217
cosocle	C₄²×C₉	C₂³
hypercenter	#533	D₉
hypocenter	C₉	#1544
LCS	G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

#3676

commutator

C₂×C₄×C₉

C₂²×C₄

Frattini

C₃×C₄²

C₂²×D₃

C₂

C₂²×C₃

C₂³×C₃

C₄²×C₉

C₂³

D₉

C₉

G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. #10588224	C₂	semidirect
114. C₉	#1544	semidirect

H ≅ G/N

type

3. #10588224

C₂

semidirect

114. C₉

#1544

semidirect

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

subgroup	index
C₉×#533	2
#10588605	2
#10588224	2
#71829	3

subgroup

index

Subgroups — #11776238

Conjugacy classes of subgroups

Complements

Maximal subgroups