Subgroups — C₉×#1303 #11766024

Order 1152.

264 conjugacy classes of subgroups.

The subgroup lattice has 201 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#431	C₂	1	1	yes	maximal, characteristic
2	C₉×#423	C₂	1	1	yes	maximal, characteristic
3	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×#1303	C₃	1	1	yes	maximal, characteristic
5	C₂×C₉×M₁₆	—	2	1	no	—
6	C₂×C₉×C₄⋊C₄	—	2	1	no	—
7	C₂×C₄²×C₉	—	2	1	no	—
8	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
9	C₉×C₄⋊C₈	—	2	1	no	—
10	C₉×C₄⋊C₈	C₄	1	2	yes	complemented
11	C₉×C₄⋊C₈	C₄	1	2	yes	complemented
12	C₉×C₂²⋊C₈	—	2	1	no	—
13	C₄×C₈×C₉	—	2	1	no	—
14	C₉×#95	C₄	1	2	yes	—
15	C₃×#431	C₂×C₃	1	1	yes	characteristic
16	C₃×#423	C₂×C₃	1	1	yes	characteristic
17	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
18	C₂²×C₄×C₉	—	2	2	no	—
19	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
22	C₉×M₁₆	—	4	1	no	—
23	C₉×M₁₆	—	4	1	no	—
24	C₂×C₈×C₉	—	2	2	no	—
25	C₂×C₈×C₉	—	2	2	no	—
26	C₉×C₄⋊C₄	—	2	4	no	—
27	C₄²×C₉	—	4	1	no	—
28	C₄²×C₉	—	4	1	no	—
29	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₉	D₄	1	1	yes	characteristic
32	C₄²×C₉	Q₈	1	1	yes	characteristic
33	#1303	C₉	1	1	yes	characteristic, direct product factor, Sylow
34	C₂×C₃×M₁₆	—	2	1	no	—
35	C₂×C₃×C₄⋊C₄	—	2	1	no	—
36	C₂×C₃×C₄²	—	2	1	no	—
37	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
38	C₃×C₄⋊C₈	—	2	1	no	—
39	C₃×C₄⋊C₈	C₃×C₄	1	2	yes	—
40	C₃×C₄⋊C₈	C₃×C₄	1	2	yes	—
41	C₃×C₂²⋊C₈	—	2	1	no	—
42	C₃×C₄×C₈	—	2	1	no	—
43	C₃×#95	C₃×C₄	1	2	yes	—
44	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₉	—	2	2	no	—
46	C₂×C₄×C₉	—	2	4	no	—
47	C₂×C₄×C₉	—	4	1	no	—
48	C₂×C₄×C₉	—	4	2	no	—
49	C₂×C₄×C₉	—	4	2	no	—
50	C₂×C₄×C₉	C₄²	1	1	yes	characteristic
51	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₉	D₈	1	1	yes	characteristic
57	C₂×C₄×C₉	SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₉	SD₁₆	1	1	yes	characteristic
59	C₂×C₄×C₉	Q₁₆	1	1	yes	characteristic
60	C₈×C₉	—	4	2	no	—
61	C₈×C₉	—	4	2	no	—
62	#431	C₂×C₉	1	1	yes	characteristic
63	#423	C₂×C₉	1	1	yes	characteristic
64	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
65	C₂²×C₃×C₄	—	2	2	no	—
66	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
69	C₃×M₁₆	—	4	1	no	—
70	C₃×M₁₆	—	4	1	no	—
71	C₂×C₃×C₈	—	2	2	no	—
72	C₂×C₃×C₈	—	2	2	no	—
73	C₃×C₄⋊C₄	—	2	4	no	—
74	C₃×C₄²	—	4	1	no	—
75	C₃×C₄²	—	4	1	no	—
76	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
77	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
78	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
79	C₃×C₄²	C₃×Q₈	1	1	yes	characteristic
80	C₂²×C₉	—	2	1	no	—
81	C₂²×C₉	—	2	1	no	—
82	C₂²×C₉	#95	1	1	yes	characteristic
83	C₂²×C₉	#102	1	1	yes	characteristic
84	C₂²×C₉	#103	1	1	yes	characteristic
85	C₄×C₉	—	2	1	no	—
86	C₄×C₉	—	2	1	no	—
87	C₄×C₉	—	2	1	no	—
88	C₄×C₉	—	4	4	no	—
89	C₄×C₉	#102	1	1	yes	characteristic
90	C₄×C₉	#103	1	1	yes	characteristic
91	C₄×C₉	#104	1	2	yes	—
92	C₄×C₉	#106	1	1	yes	characteristic
93	C₄×C₉	#107	1	1	yes	characteristic
94	C₂×M₁₆	—	2	1	no	—
95	C₂×C₄⋊C₄	—	2	1	no	—
96	C₂×C₄²	—	2	1	no	—
97	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
98	C₄⋊C₈	—	2	1	no	—
99	C₄⋊C₈	C₄×C₉	1	2	yes	complemented
100	C₄⋊C₈	C₄×C₉	1	2	yes	complemented
101	C₂²⋊C₈	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	#95	C₄×C₉	1	2	yes	—
104	C₂³×C₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	4	no	—
107	C₂×C₃×C₄	—	4	1	no	—
108	C₂×C₃×C₄	—	4	2	no	—
109	C₂×C₃×C₄	—	4	2	no	—
110	C₂×C₃×C₄	C₃×C₄²	1	1	yes	characteristic
111	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₃×C₄	C₃×D₈	1	1	yes	characteristic
117	C₂×C₃×C₄	C₃×SD₁₆	1	1	yes	characteristic
118	C₂×C₃×C₄	C₃×SD₁₆	1	1	yes	characteristic
119	C₂×C₃×C₄	C₃×Q₁₆	1	1	yes	characteristic
120	C₃×C₈	—	4	2	no	—
121	C₃×C₈	—	4	2	no	—
122	C₂×C₉	—	2	1	no	—
123	C₂×C₉	—	2	1	no	—
124	C₂×C₉	#339	1	1	yes	characteristic
125	C₂×C₉	#340	1	1	yes	characteristic
126	C₂×C₉	#344	1	1	yes	characteristic
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
129	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
130	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
131	M₁₆	—	4	1	no	—
132	M₁₆	—	4	1	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₄⋊C₄	—	2	4	no	—
136	C₄²	—	4	1	no	—
137	C₄²	—	4	1	no	—
138	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
139	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
140	C₄²	D₄×C₉	1	1	yes	characteristic
141	C₄²	Q₈×C₉	1	1	yes	characteristic
142	C₂²×C₃	—	2	1	no	—
143	C₂²×C₃	—	2	1	no	—
144	C₂²×C₃	C₃×#95	1	1	yes	characteristic
145	C₂²×C₃	C₃×#102	1	1	yes	characteristic
146	C₂²×C₃	C₃×#103	1	1	yes	characteristic
147	C₃×C₄	—	2	1	no	—
148	C₃×C₄	—	2	1	no	—
149	C₃×C₄	—	2	1	no	—
150	C₃×C₄	—	4	4	no	—
151	C₃×C₄	C₃×#102	1	1	yes	characteristic
152	C₃×C₄	C₃×#103	1	1	yes	characteristic
153	C₃×C₄	C₃×#104	1	2	yes	—
154	C₃×C₄	C₃×#106	1	1	yes	characteristic
155	C₃×C₄	C₃×#107	1	1	yes	characteristic
156	C₉	#1303	1	1	yes	characteristic, direct product factor, Sylow
157	C₂³	C₉×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	C₄²×C₉	1	1	yes	characteristic
164	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₄	C₉×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₄	C₉×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₄	C₉×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₄	C₉×D₈	1	1	yes	characteristic
170	C₂×C₄	C₉×SD₁₆	1	1	yes	characteristic
171	C₂×C₄	C₉×SD₁₆	1	1	yes	characteristic
172	C₂×C₄	C₉×Q₁₆	1	1	yes	characteristic
173	C₈	—	4	2	no	—
174	C₈	—	4	2	no	—
175	C₂×C₃	—	2	1	no	—
176	C₂×C₃	—	2	1	no	—
177	C₂×C₃	C₃×#339	1	1	yes	characteristic
178	C₂×C₃	C₃×#340	1	1	yes	characteristic
179	C₂×C₃	C₃×#344	1	1	yes	characteristic
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₉×#95	1	1	yes	characteristic
183	C₂²	C₉×#102	1	1	yes	characteristic
184	C₂²	C₉×#103	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	4	4	no	—
189	C₄	C₉×#102	1	1	yes	characteristic
190	C₄	C₉×#103	1	1	yes	characteristic
191	C₄	C₉×#104	1	2	yes	—
192	C₄	C₉×#106	1	1	yes	characteristic
193	C₄	C₉×#107	1	1	yes	characteristic
194	C₃	C₃×#1303	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	C₉×#339	1	1	yes	characteristic
198	C₂	C₉×#340	1	1	yes	characteristic
199	C₂	C₉×#344	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂²⋊C₄
commutator	C₂×C₄	C₄²×C₉
Frattini, cosocle	C₂×C₃×C₄²	C₂²×C₃
socle	C₂²×C₃	C₃×#95
prime radical	C₂³×C₃	C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
10. C₉×C₄⋊C₈	C₄	semidirect
11. C₉×C₄⋊C₈	C₄	semidirect
33. #1303	C₉	direct (N × H)
99. C₄⋊C₈	C₄×C₉	semidirect
100. C₄⋊C₈	C₄×C₉	semidirect
156. C₉	#1303	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#431	2
C₉×#423	2
C₄×C₉×C₄⋊C₄	2
C₃×#1303	3

Subgroups — C9×#1303 #11766024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1303 #11766024