Subgroups — C₉×#1817 #11766538

Order 1152.

570 conjugacy classes of subgroups.

The subgroup lattice has 207 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₂²⋊C₈	C₂	1	2	yes	maximal
3	C₉×#336	C₂	1	4	yes	maximal, complemented
4	C₃×#1817	C₃	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₉	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₉	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₉	C₂²	1	4	yes	—
9	C₂×C₄²×C₉	—	2	4	no	—
10	C₂×C₄²×C₉	C₄	1	2	yes	—
11	C₂×C₄²×C₉	C₂²	1	2	yes	—
12	C₉×C₂²⋊C₈	—	2	4	no	—
13	C₂²×C₃×C₄²	C₂×C₃	1	1	yes	characteristic
14	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	2	yes	—
15	C₃×#336	C₂×C₃	1	4	yes	—
16	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₉	—	2	2	no	—
18	C₂²×C₄×C₉	—	2	2	no	—
19	C₂²×C₄×C₉	—	2	2	no	—
20	C₂²×C₄×C₉	—	2	4	no	—
21	C₂²×C₄×C₉	—	2	4	no	—
22	C₂²×C₄×C₉	C₈	1	4	yes	complemented
23	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₉	D₄	1	4	yes	—
27	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₉	—	2	4	no	—
29	C₂×C₈×C₉	—	2	8	no	—
30	C₄²×C₉	—	2	4	no	—
31	C₄²×C₉	—	2	4	no	—
32	#1817	C₉	1	1	yes	characteristic, direct product factor, Sylow
33	C₂³×C₃×C₄	C₃×C₄	1	1	yes	characteristic
34	C₂³×C₃×C₄	C₃×C₄	1	1	yes	characteristic
35	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
36	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
37	C₂×C₃×C₄²	—	2	4	no	—
38	C₂×C₃×C₄²	C₃×C₄	1	2	yes	—
39	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
40	C₃×C₂²⋊C₈	—	2	4	no	—
41	C₂³×C₉	—	2	2	no	—
42	C₂³×C₉	—	2	2	no	—
43	C₂³×C₉	C₂²⋊C₄	1	2	yes	—
44	C₂³×C₉	C₂×C₈	1	2	yes	—
45	C₂³×C₉	M₁₆	1	2	yes	—
46	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₉	—	2	8	no	—
48	C₂×C₄×C₉	—	2	8	no	—
49	C₂×C₄×C₉	—	2	2	no	—
50	C₂×C₄×C₉	—	2	2	no	—
51	C₂×C₄×C₉	—	2	2	no	—
52	C₂×C₄×C₉	—	2	2	no	—
53	C₂×C₄×C₉	—	2	2	no	—
54	C₂×C₄×C₉	—	2	8	no	—
55	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₉	C₂×C₈	1	4	yes	—
57	C₂×C₄×C₉	M₁₆	1	4	yes	—
58	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₉	#41	1	4	yes	—
60	C₈×C₉	—	4	4	no	—
61	C₂²×C₄²	C₂×C₉	1	1	yes	characteristic
62	C₂×C₂²⋊C₈	C₂×C₉	1	2	yes	—
63	#336	C₂×C₉	1	4	yes	complemented
64	C₂⁴×C₃	C₂×C₃×C₄	1	1	yes	characteristic
65	C₂²×C₃×C₄	—	2	2	no	—
66	C₂²×C₃×C₄	—	2	2	no	—
67	C₂²×C₃×C₄	—	2	2	no	—
68	C₂²×C₃×C₄	—	2	4	no	—
69	C₂²×C₃×C₄	—	2	4	no	—
70	C₂²×C₃×C₄	C₃×C₈	1	4	yes	—
71	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
73	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
74	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
75	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
76	C₂×C₃×C₈	—	2	4	no	—
77	C₂×C₃×C₈	—	2	8	no	—
78	C₃×C₄²	—	2	4	no	—
79	C₃×C₄²	—	2	4	no	—
80	C₂²×C₉	—	2	2	no	—
81	C₂²×C₉	—	2	2	no	—
82	C₂²×C₉	—	2	4	no	—
83	C₂²×C₉	—	2	4	no	—
84	C₂²×C₉	C₂²⋊C₈	1	4	yes	—
85	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₉	#117	1	2	yes	—
87	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
88	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
89	C₂²×C₉	C₂×M₁₆	1	2	yes	—
90	C₄×C₉	—	2	2	no	—
91	C₄×C₉	—	2	4	no	—
92	C₄×C₉	—	2	4	no	—
93	C₄×C₉	#123	1	4	yes	—
94	C₂³×C₄	C₄×C₉	1	1	yes	characteristic
95	C₂³×C₄	C₄×C₉	1	1	yes	characteristic
96	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
97	C₂²×C₈	C₂²×C₉	1	4	yes	—
98	C₂×C₄²	—	2	4	no	—
99	C₂×C₄²	C₄×C₉	1	2	yes	—
100	C₂×C₄²	C₂²×C₉	1	2	yes	—
101	C₂²⋊C₈	—	2	4	no	—
102	C₂³×C₃	—	2	2	no	—
103	C₂³×C₃	—	2	2	no	—
104	C₂³×C₃	C₃×C₂²⋊C₄	1	2	yes	—
105	C₂³×C₃	C₂×C₃×C₈	1	2	yes	—
106	C₂³×C₃	C₃×M₁₆	1	2	yes	—
107	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
108	C₂×C₃×C₄	—	2	8	no	—
109	C₂×C₃×C₄	—	2	8	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	8	no	—
116	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
117	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
118	C₂×C₃×C₄	C₃×M₁₆	1	4	yes	—
119	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
120	C₂×C₃×C₄	C₃×#41	1	4	yes	—
121	C₃×C₈	—	4	4	no	—
122	C₂×C₉	—	2	4	no	—
123	C₂×C₉	#381	1	1	yes	characteristic
124	C₂×C₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
125	C₂×C₉	#407	1	1	yes	characteristic
126	C₂×C₉	#431	1	2	yes	—
127	C₂×C₉	#432	1	2	yes	—
128	C₂⁴	C₂×C₄×C₉	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	C₈×C₉	1	4	yes	complemented
135	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
137	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
138	C₂²×C₄	D₄×C₉	1	4	yes	—
139	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	8	no	—
142	C₄²	—	2	4	no	—
143	C₄²	—	2	4	no	—
144	C₂²×C₃	—	2	2	no	—
145	C₂²×C₃	—	2	2	no	—
146	C₂²×C₃	—	2	4	no	—
147	C₂²×C₃	—	2	4	no	—
148	C₂²×C₃	C₃×C₂²⋊C₈	1	4	yes	—
149	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₃	C₃×#117	1	2	yes	—
151	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
152	C₂²×C₃	C₂×C₃×M₁₆	1	1	yes	characteristic
153	C₂²×C₃	C₂×C₃×M₁₆	1	2	yes	—
154	C₃×C₄	—	2	2	no	—
155	C₃×C₄	—	2	4	no	—
156	C₃×C₄	—	2	4	no	—
157	C₃×C₄	C₃×#123	1	4	yes	—
158	C₉	#1817	1	1	yes	characteristic, direct product factor, Sylow
159	C₂³	—	2	2	no	—
160	C₂³	—	2	2	no	—
161	C₂³	C₉×C₂²⋊C₄	1	2	yes	—
162	C₂³	C₂×C₈×C₉	1	2	yes	—
163	C₂³	C₉×M₁₆	1	2	yes	—
164	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
174	C₂×C₄	C₂×C₈×C₉	1	4	yes	—
175	C₂×C₄	C₉×M₁₆	1	4	yes	—
176	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
177	C₂×C₄	C₉×#41	1	4	yes	—
178	C₈	—	4	4	no	—
179	C₂×C₃	—	2	4	no	—
180	C₂×C₃	C₃×#381	1	1	yes	characteristic
181	C₂×C₃	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
182	C₂×C₃	C₃×#407	1	1	yes	characteristic
183	C₂×C₃	C₃×#431	1	2	yes	—
184	C₂×C₃	C₃×#432	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	2	4	no	—
188	C₂²	—	2	4	no	—
189	C₂²	C₉×C₂²⋊C₈	1	4	yes	—
190	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
191	C₂²	C₉×#117	1	2	yes	—
192	C₂²	C₂²×C₈×C₉	1	1	yes	characteristic
193	C₂²	C₂×C₉×M₁₆	1	1	yes	characteristic
194	C₂²	C₂×C₉×M₁₆	1	2	yes	—
195	C₄	—	2	2	no	—
196	C₄	—	2	4	no	—
197	C₄	—	2	4	no	—
198	C₄	C₉×#123	1	4	yes	—
199	C₃	C₃×#1817	1	1	yes	characteristic
200	C₂	—	2	4	no	—
201	C₂	C₉×#381	1	1	yes	characteristic
202	C₂	C₂×C₉×C₂²⋊C₈	1	1	yes	characteristic
203	C₂	C₉×#407	1	1	yes	characteristic
204	C₂	C₉×#431	1	2	yes	—
205	C₂	C₉×#432	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₉	C₂³
commutator	C₂²	C₂²×C₈×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂²×C₃×C₄
prime radical	C₂⁴×C₃	C₂×C₃×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₉×#336	C₂	semidirect
22. C₂²×C₄×C₉	C₈	semidirect
32. #1817	C₉	direct (N × H)
63. #336	C₂×C₉	semidirect
134. C₂²×C₄	C₈×C₉	semidirect
158. C₉	#1817	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄²×C₉	2
C₂×C₉×C₂²⋊C₈	2
C₉×#336	2
C₃×#1817	3

Subgroups — C9×#1817 #11766538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1817 #11766538