Group #11777837 — subgroups

Subgroups — #11777837

Order 1152.

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

normal subgroups only

This group has 228 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	#10588509	C₂	1	2	yes	maximal
2	C₄×Dic₃₆	C₂	1	1	yes	maximal, characteristic
3	C₉×#335	C₂	1	1	yes	maximal, characteristic
4	#10588160	C₂	1	2	yes	maximal
5	C₈×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#68564	—	3	1	no	maximal
7	C₂×Dic₃₆	C₄	1	1	yes	characteristic
8	C₄×Dic₁₈	C₂²	1	2	yes	—
9	C₉×C₄⋊C₈	C₂²	1	2	yes	—
10	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
11	#63378	C₄	1	1	yes	characteristic
12	#63376	C₄	1	2	yes	—
13	#63375	—	2	2	no	—
14	Dic₉⋊C₈	—	2	2	no	—
15	C₄×C₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₉⋊C₈	C₂²	1	1	yes	characteristic
17	#4831	—	3	2	no	—
18	C₄×Dic₁₂	—	3	1	no	—
19	C₃×#335	D₃	1	1	yes	characteristic
20	#4484	—	3	2	no	—
21	C₈×C₃⋊C₈	—	3	1	no	—
22	C₂×Dic₁₈	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₉	—	2	2	no	—
24	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₉	C₂³	1	1	yes	characteristic
27	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
28	Dic₉⋊C₄	—	2	2	no	—
29	C₄×Dic₉	—	2	2	no	—
30	C₂×C₉⋊C₈	—	2	1	no	—
31	C₂×C₉⋊C₈	D₄	1	1	yes	characteristic
32	C₂×C₉⋊C₈	D₄	1	1	yes	characteristic
33	Dic₃₆	—	2	2	no	—
34	#1591	—	9	1	no	Sylow
35	C₂×Dic₁₂	—	3	1	no	—
36	C₄×Dic₆	—	3	2	no	—
37	C₃×C₄⋊C₈	C₂×D₃	1	2	yes	—
38	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
39	#818	—	3	1	no	—
40	#816	—	3	2	no	—
41	#815	—	6	2	no	—
42	Dic₃⋊C₈	—	6	2	no	—
43	C₄×C₃⋊C₈	—	3	1	no	—
44	C₄×C₃⋊C₈	—	3	1	no	—
45	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₉	#41	1	1	yes	characteristic
48	C₂×Dic₉	—	2	2	no	—
49	C₂×Dic₉	—	2	2	no	—
50	Dic₁₈	—	2	2	no	—
51	Dic₁₈	—	2	2	no	—
52	C₈×C₉	—	2	1	no	—
53	C₈×C₉	—	4	2	no	—
54	C₈×C₉	M₁₆	1	2	yes	—
55	C₉⋊C₈	—	2	2	no	—
56	C₉⋊C₈	—	2	1	no	—
57	C₉⋊C₈	Q₁₆	1	2	yes	complemented
58	#446	—	9	2	no	—
59	C₄×Q₁₆	—	9	1	no	—
60	#335	D₉	1	1	yes	characteristic
61	Q₈⋊C₈	—	9	2	no	—
62	C₈²	—	9	1	no	—
63	C₂×Dic₆	—	3	2	no	—
64	C₂×C₃×C₈	—	2	2	no	—
65	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
66	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
67	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
68	C₄⋊Dic₃	—	3	2	no	—
69	Dic₃⋊C₄	—	6	2	no	—
70	C₄×Dic₃	—	6	2	no	—
71	C₂×C₃⋊C₈	—	3	1	no	—
72	C₂×C₃⋊C₈	—	3	1	no	—
73	C₂×C₃⋊C₈	—	6	1	no	—
74	Dic₁₂	—	6	2	no	—
75	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
76	C₄×C₉	—	2	1	no	—
77	C₄×C₉	C₂×M₁₆	1	1	yes	characteristic
78	C₄×C₉	#131	1	1	yes	characteristic
79	C₄×C₉	C₂×Q₁₆	1	1	yes	characteristic
80	C₄×C₉	#135	1	1	yes	characteristic
81	Dic₉	—	4	2	no	—
82	Dic₉	—	4	2	no	—
83	C₂×Q₁₆	—	9	1	no	—
84	C₄×Q₈	—	9	2	no	—
85	#107	—	9	1	no	—
86	C₄⋊C₈	—	18	2	no	—
87	C₄⋊C₈	C₂×D₉	1	2	yes	—
88	#103	—	9	2	no	—
89	#97	—	18	2	no	—
90	C₄×C₈	—	9	1	no	—
91	C₄×C₈	—	9	1	no	—
92	C₄×C₈	C₂×D₉	1	1	yes	characteristic
93	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
94	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
95	C₂×C₃×C₄	#239	1	1	yes	characteristic
96	C₂×Dic₃	—	6	2	no	—
97	C₂×Dic₃	—	6	2	no	—
98	Dic₆	—	6	2	no	—
99	Dic₆	—	6	2	no	—
100	C₃×C₈	—	2	1	no	—
101	C₃×C₈	—	4	2	no	—
102	C₃×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
103	C₃⋊C₈	—	3	2	no	—
104	C₃⋊C₈	—	6	2	no	—
105	C₃⋊C₈	—	6	1	no	—
106	C₂×C₉	#436	1	1	yes	characteristic
107	C₂×C₉	C₄×Q₁₆	1	1	yes	characteristic
108	C₂×C₉	#443	1	1	yes	characteristic
109	C₂×Q₈	—	9	2	no	—
110	Q₁₆	—	18	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	9	1	no	—
113	C₂×C₈	—	9	1	no	—
114	C₂×C₈	—	18	1	no	—
115	C₂×C₈	C₄×D₉	1	1	yes	characteristic
116	C₂×C₈	C₄×D₉	1	1	yes	characteristic
117	C₄⋊C₄	—	9	2	no	—
118	C₄⋊C₄	—	18	2	no	—
119	C₄²	—	18	2	no	—
120	C₄²	C₂²×D₉	1	1	yes	characteristic
121	C₂²×C₃	#893	1	1	yes	characteristic
122	C₃×C₄	—	2	1	no	—
123	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
124	C₃×C₄	#907	1	1	yes	characteristic
125	C₃×C₄	#912	1	1	yes	characteristic
126	C₃×C₄	D₃×Q₁₆	1	1	yes	characteristic
127	Dic₃	—	12	2	no	—
128	Dic₃	—	12	2	no	—
129	C₉	#1591	1	1	yes	characteristic, complemented, Sylow
130	Q₈	—	18	2	no	—
131	Q₈	—	18	2	no	—
132	C₂×C₄	—	18	2	no	—
133	C₂×C₄	—	18	2	no	—
134	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
135	C₂×C₄	D₄×D₉	1	1	yes	characteristic
136	C₂×C₄	#3701	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	4	2	no	—
139	C₈	—	9	2	no	—
140	C₈	—	18	2	no	—
141	C₈	—	18	1	no	—
142	C₈	#3663	1	2	yes	—
143	C₂×C₃	#4838	1	1	yes	characteristic
144	C₂×C₃	#4873	1	1	yes	characteristic
145	C₂×C₃	#4895	1	1	yes	characteristic
146	C₂²	#63455	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	36	2	no	—
149	C₄	—	36	2	no	—
150	C₄	C₂×#3663	1	1	yes	characteristic
151	C₄	#63469	1	1	yes	characteristic
152	C₄	#63474	1	1	yes	characteristic
153	C₄	Q₁₆×D₉	1	1	yes	characteristic
154	C₃	#68564	1	1	yes	characteristic
155	C₂	#10588522	1	1	yes	characteristic
156	C₂	#10588553	1	1	yes	characteristic
157	C₂	#10588575	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10588509

C₂

yes

maximal

C₄×Dic₃₆

C₂

yes

maximal, characteristic

C₉×#335

C₂

yes

maximal, characteristic

#10588160

C₂

yes

maximal

C₈×C₉⋊C₈

C₂

yes

maximal, characteristic

#68564

—

maximal

C₂×Dic₃₆

C₄

yes

characteristic

C₄×Dic₁₈

C₂²

yes

—

C₉×C₄⋊C₈

C₂²

yes

—

C₄×C₈×C₉

C₂²

yes

characteristic

#63378

C₄

yes

characteristic

#63376

C₄

yes

—

#63375

—

Dic₉⋊C₈

—

C₄×C₉⋊C₈

C₂²

yes

characteristic

C₄×C₉⋊C₈

C₂²

yes

characteristic

#4831

—

C₄×Dic₁₂

—

C₃×#335

D₃

yes

characteristic

#4484

—

C₈×C₃⋊C₈

—

C₂×Dic₁₈

C₂×C₄

yes

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂×C₄

yes

characteristic

C₂×C₈×C₉

C₂×C₄

yes

characteristic

C₄²×C₉

C₂³

yes

characteristic

C₄⋊Dic₉

C₂×C₄

yes

—

Dic₉⋊C₄

—

C₄×Dic₉

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

D₄

yes

characteristic

C₂×C₉⋊C₈

D₄

yes

characteristic

Dic₃₆

—

#1591

—

Sylow

C₂×Dic₁₂

—

C₄×Dic₆

—

C₃×C₄⋊C₈

C₂×D₃

yes

—

C₃×C₄×C₈

C₂×D₃

yes

—

—

—

Dic₃⋊C₈

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂×C₄×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

#41

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

M₁₆

yes

—

C₉⋊C₈

—

C₉⋊C₈

—

C₉⋊C₈

Q₁₆

yes

complemented

#446

—

C₄×Q₁₆

—

#335

D₉

yes

characteristic

Q₈⋊C₈

—

C₈²

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄×D₃

yes

characteristic

C₂×C₃×C₈

C₄×D₃

yes

characteristic

C₃×C₄²

C₂²×D₃

yes

characteristic

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

Dic₁₂

—

C₂²×C₉

C₄×D₄

yes

characteristic

C₄×C₉

—

C₄×C₉

C₂×M₁₆

yes

characteristic

C₄×C₉

#131

yes

characteristic

C₄×C₉

C₂×Q₁₆

yes

characteristic

C₄×C₉

#135

yes

characteristic

Dic₉

—

Dic₉

—

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₉

yes

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₉

yes

characteristic

C₂×C₃×C₄

C₂×C₄×D₃

yes

characteristic

C₂×C₃×C₄

D₃×D₄

yes

characteristic

C₂×C₃×C₄

#239

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

100

C₃×C₈

—

101

C₃×C₈

—

102

C₃×C₈

ΓL(1,𝔽₂₅)

yes

—

103

C₃⋊C₈

—

104

C₃⋊C₈

—

105

C₃⋊C₈

—

106

C₂×C₉

#436

yes

characteristic

107

C₂×C₉

C₄×Q₁₆

yes

characteristic

108

C₂×C₉

#443

yes

characteristic

109

C₂×Q₈

—

110

Q₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₄×D₉

yes

characteristic

116

C₂×C₈

C₄×D₉

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

—

120

C₄²

C₂²×D₉

yes

characteristic

121

C₂²×C₃

#893

yes

characteristic

122

C₃×C₄

—

123

C₃×C₄

C₂×ΓL(1,𝔽₂₅)

yes

characteristic

124

C₃×C₄

#907

yes

characteristic

125

C₃×C₄

#912

yes

characteristic

126

C₃×C₄

D₃×Q₁₆

yes

characteristic

127

Dic₃

—

128

Dic₃

—

129

C₉

#1591

yes

characteristic, complemented, Sylow

130

Q₈

—

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₂×C₄×D₉

yes

characteristic

135

C₂×C₄

D₄×D₉

yes

characteristic

136

C₂×C₄

#3701

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

#3663

yes

—

143

C₂×C₃

#4838

yes

characteristic

144

C₂×C₃

#4873

yes

characteristic

145

C₂×C₃

#4895

yes

characteristic

146

C₂²

#63455

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

C₂×#3663

yes

characteristic

151

C₄

#63469

yes

characteristic

152

C₄

#63474

yes

characteristic

153

C₄

Q₁₆×D₉

yes

characteristic

154

C₃

#68564

yes

characteristic

155

C₂

#10588522

yes

characteristic

156

C₂

#10588553

yes

characteristic

157

C₂

#10588575

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₉
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₉×#335	C₂
socle, prime radical	C₂²×C₃	#893
cosocle	C₄²×C₉	C₂³
hypercenter	#335	D₉
hypocenter	C₉	#1591
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×D₉

commutator

C₂×C₄×C₉

C₂²×C₄

Frattini

C₃×C₄²

C₂²×D₃

C₂

C₂²×C₃

C₄²×C₉

C₂³

D₉

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
57. C₉⋊C₈	Q₁₆	semidirect
129. C₉	#1591	semidirect

H ≅ G/N

type

57. C₉⋊C₈

Q₁₆

semidirect

129. C₉

#1591

semidirect

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

subgroup	index
#10588509	2
C₄×Dic₃₆	2
C₉×#335	2
#10588160	2
C₈×C₉⋊C₈	2
#68564	3

subgroup

index

C₄×Dic₃₆

C₈×C₉⋊C₈

Subgroups — #11777837

Conjugacy classes of subgroups

Complements

Maximal subgroups