Subgroups — C₂×C₉×#235 #11722598

Order 864.

546 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#67	C₂	1	2	yes	maximal, complemented
3	C₉×#235	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₉×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×D₃×C₉	C₂	1	2	yes	maximal, complemented
6	C₂×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#235	C₃	1	1	yes	maximal, characteristic
8	C₂×C₉×#41	—	3	1	no	maximal
9	C₂³×C₃×C₉	C₂²	1	1	yes	characteristic
10	C₂²×D₃×C₉	C₂²	1	2	yes	—
11	C₂×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄×C₉	C₂²	1	4	yes	complemented
14	C₉×#67	C₂²	1	8	yes	complemented
15	C₂×C₉×Dic₃	C₂²	1	2	yes	—
16	C₉×D₁₂	C₂²	1	4	yes	complemented
17	C₄×D₃×C₉	C₂²	1	8	yes	complemented
18	C₉×Dic₆	C₂²	1	4	yes	complemented
19	C₂²×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×#67	C₂×C₃	1	2	yes	—
21	C₃×#235	C₂×C₃	1	8	yes	—
22	C₂×C₃×D₁₂	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×C₄×D₃	C₂×C₃	1	2	yes	—
24	C₂×C₃×Dic₆	C₂×C₃	1	1	yes	characteristic
25	C₉×#41	—	3	8	no	—
26	C₂×Q₈×C₉	—	3	1	no	—
27	C₂×D₄×C₉	—	3	2	no	—
28	C₂×D₄×C₉	—	3	1	no	—
29	C₂²×C₄×C₉	—	2	1	no	—
30	C₂²×C₄×C₉	—	3	2	no	—
31	C₂²×C₄×C₉	D₃	1	1	yes	characteristic, complemented
32	C₂²×C₃×C₉	—	2	2	no	—
33	C₂²×C₃×C₉	C₂³	1	2	yes	—
34	C₂²×C₃×C₉	C₂³	1	1	yes	characteristic
35	C₂×D₃×C₉	—	2	4	no	—
36	C₂×D₃×C₉	C₂³	1	4	yes	—
37	C₃×C₄×C₉	C₂³	1	2	yes	—
38	C₃×C₄×C₉	C₂³	1	2	yes	—
39	C₉×Dic₃	C₂³	1	4	yes	—
40	C₂×C₃×#41	—	3	1	no	—
41	C₂×#235	C₉	1	1	yes	characteristic, direct product factor
42	C₂³×C₃²	C₂²×C₃	1	1	yes	characteristic
43	C₂²×C₃×D₃	C₂²×C₃	1	2	yes	—
44	C₂×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
46	C₂×C₃²×C₄	C₂²×C₃	1	4	yes	—
47	C₃×#67	C₂²×C₃	1	8	yes	—
48	C₂×C₃×Dic₃	C₂²×C₃	1	2	yes	—
49	C₃×D₁₂	C₂²×C₃	1	4	yes	—
50	C₃×C₄×D₃	C₂²×C₃	1	8	yes	—
51	C₃×Dic₆	C₂²×C₃	1	4	yes	—
52	C₂³×C₉	—	2	1	no	—
53	C₂³×C₉	—	3	2	no	—
54	C₂³×C₉	C₂×D₃	1	1	yes	characteristic
55	Q₈×C₉	—	3	4	no	—
56	D₄×C₉	—	3	8	no	—
57	D₄×C₉	—	3	4	no	—
58	C₂×C₄×C₉	—	2	1	no	—
59	C₂×C₄×C₉	—	2	1	no	—
60	C₂×C₄×C₉	—	2	4	no	—
61	C₂×C₄×C₉	—	3	2	no	—
62	C₂×C₄×C₉	—	3	8	no	—
63	C₂×C₄×C₉	C₂×D₃	1	1	yes	characteristic
64	C₂×C₄×C₉	C₂×D₃	1	1	yes	characteristic
65	C₂×C₄×C₉	C₂×D₃	1	4	yes	complemented
66	C₂×C₃×C₉	—	2	2	no	—
67	C₂×C₃×C₉	#41	1	2	yes	complemented
68	C₂×C₃×C₉	C₂⁴	1	1	yes	characteristic
69	D₃×C₉	—	2	4	no	—
70	C₃×#41	—	3	8	no	—
71	C₂×C₃×Q₈	—	3	1	no	—
72	C₂×C₃×D₄	—	3	2	no	—
73	C₂×C₃×D₄	—	3	1	no	—
74	C₂²×C₃×C₄	—	2	1	no	—
75	C₂²×C₃×C₄	—	3	2	no	—
76	C₂²×C₃×C₄	C₂×C₉	1	1	yes	characteristic, complemented
77	C₂²×C₃×C₄	C₃×D₃	1	1	yes	characteristic
78	C₂×#67	C₂×C₉	1	2	yes	complemented
79	#235	C₂×C₉	1	8	yes	direct product factor
80	C₂×D₁₂	C₂×C₉	1	1	yes	characteristic, complemented
81	C₂×C₄×D₃	C₂×C₉	1	2	yes	complemented
82	C₂×Dic₆	C₂×C₉	1	1	yes	characteristic, complemented
83	C₂²×C₃²	—	2	2	no	—
84	C₂²×C₃²	C₂³×C₃	1	2	yes	—
85	C₂²×C₃²	C₂³×C₃	1	1	yes	characteristic
86	C₂×C₃×D₃	—	2	4	no	—
87	C₂×C₃×D₃	C₂³×C₃	1	4	yes	—
88	C₃²×C₄	C₂³×C₃	1	2	yes	—
89	C₃²×C₄	C₂³×C₃	1	2	yes	—
90	C₃×Dic₃	C₂³×C₃	1	4	yes	—
91	C₂²×C₉	—	2	4	no	—
92	C₂²×C₉	—	2	2	no	—
93	C₂²×C₉	—	2	2	no	—
94	C₂²×C₉	—	2	1	no	—
95	C₂²×C₉	—	3	4	no	—
96	C₂²×C₉	—	6	4	no	—
97	C₂²×C₉	C₂²×D₃	1	2	yes	—
98	C₂²×C₉	C₂²×D₃	1	1	yes	characteristic
99	C₄×C₉	—	2	2	no	—
100	C₄×C₉	—	2	2	no	—
101	C₄×C₉	—	3	4	no	—
102	C₄×C₉	C₂²×D₃	1	2	yes	—
103	C₄×C₉	C₂²×D₃	1	2	yes	—
104	C₂×#41	—	3	1	no	Sylow
105	C₃×C₉	C₂×#41	1	1	yes	characteristic, complemented, Sylow
106	C₂³×C₃	—	2	1	no	—
107	C₂³×C₃	—	3	2	no	—
108	C₂³×C₃	C₂²×C₉	1	1	yes	characteristic
109	C₂³×C₃	C₂×C₃×D₃	1	1	yes	characteristic
110	C₂²×D₃	C₂²×C₉	1	2	yes	—
111	C₃×Q₈	—	3	4	no	—
112	C₃×D₄	—	3	8	no	—
113	C₃×D₄	—	3	4	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	—	2	4	no	—
117	C₂×C₃×C₄	—	3	2	no	—
118	C₂×C₃×C₄	—	3	8	no	—
119	C₂×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂²×C₉	1	4	yes	complemented
122	C₂×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
125	#67	C₂²×C₉	1	8	yes	complemented
126	C₂×Dic₃	C₂²×C₉	1	2	yes	—
127	D₁₂	C₂²×C₉	1	4	yes	complemented
128	C₄×D₃	C₂²×C₉	1	8	yes	complemented
129	Dic₆	C₂²×C₉	1	4	yes	complemented
130	C₂×C₃²	—	2	2	no	—
131	C₂×C₃²	C₃×#41	1	2	yes	—
132	C₂×C₃²	C₂⁴×C₃	1	1	yes	characteristic
133	C₃×D₃	—	2	4	no	—
134	C₂×C₉	—	2	4	no	—
135	C₂×C₉	—	2	2	no	—
136	C₂×C₉	—	2	1	no	—
137	C₂×C₉	—	2	2	no	—
138	C₂×C₉	—	6	4	no	—
139	C₂×C₉	#235	1	2	yes	direct product factor
140	C₂×C₉	C₂³×D₃	1	1	yes	characteristic
141	#41	—	3	8	no	—
142	C₂×Q₈	—	3	1	no	—
143	C₂×D₄	—	3	2	no	—
144	C₂×D₄	—	3	1	no	—
145	C₂²×C₄	—	3	2	no	—
146	C₂²×C₄	D₃×C₉	1	1	yes	characteristic, complemented
147	C₂²×C₃	—	2	4	no	—
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	—	2	2	no	—
150	C₂²×C₃	—	2	2	no	—
151	C₂²×C₃	—	2	1	no	—
152	C₂²×C₃	—	3	4	no	—
153	C₂²×C₃	—	6	4	no	—
154	C₂²×C₃	C₂³×C₉	1	2	yes	—
155	C₂²×C₃	C₂³×C₉	1	1	yes	characteristic
156	C₂²×C₃	C₂²×C₃×D₃	1	2	yes	—
157	C₂²×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
158	C₂×D₃	—	2	4	no	—
159	C₂×D₃	C₂³×C₉	1	4	yes	—
160	C₃×C₄	—	2	2	no	—
161	C₃×C₄	—	2	2	no	—
162	C₃×C₄	—	3	4	no	—
163	C₃×C₄	C₂³×C₉	1	2	yes	—
164	C₃×C₄	C₂³×C₉	1	2	yes	—
165	C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
166	C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
167	Dic₃	C₂³×C₉	1	4	yes	—
168	C₃²	C₂×C₃×#41	1	1	yes	characteristic
169	C₉	—	2	1	no	—
170	C₉	C₂×#235	1	1	yes	characteristic, direct product factor
171	C₂³	—	3	2	no	—
172	C₂³	C₂×D₃×C₉	1	1	yes	characteristic
173	Q₈	—	3	4	no	—
174	D₄	—	3	8	no	—
175	D₄	—	3	4	no	—
176	C₂×C₄	—	3	2	no	—
177	C₂×C₄	—	3	8	no	—
178	C₂×C₄	C₂×D₃×C₉	1	1	yes	characteristic
179	C₂×C₄	C₂×D₃×C₉	1	1	yes	characteristic
180	C₂×C₄	C₂×D₃×C₉	1	4	yes	complemented
181	C₂×C₃	—	2	4	no	—
182	C₂×C₃	—	2	2	no	—
183	C₂×C₃	—	2	2	no	—
184	C₂×C₃	—	2	1	no	—
185	C₂×C₃	—	2	2	no	—
186	C₂×C₃	—	6	4	no	—
187	C₂×C₃	C₉×#41	1	2	yes	complemented
188	C₂×C₃	C₂⁴×C₉	1	1	yes	characteristic
189	C₂×C₃	C₃×#235	1	2	yes	—
190	C₂×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
191	D₃	—	2	4	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	3	4	no	—
194	C₂²	—	6	4	no	—
195	C₂²	C₂²×D₃×C₉	1	2	yes	—
196	C₂²	C₂²×D₃×C₉	1	1	yes	characteristic
197	C₄	—	3	4	no	—
198	C₄	C₂²×D₃×C₉	1	2	yes	—
199	C₄	C₂²×D₃×C₉	1	2	yes	—
200	C₃	—	2	1	no	—
201	C₃	C₂×C₉×#41	1	1	yes	characteristic, complemented
202	C₃	C₂×C₃×#235	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	—	6	4	no	—
205	C₂	C₉×#235	1	2	yes	direct product factor
206	C₂	C₂³×D₃×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂×D₃
commutator	C₂×C₃	C₂⁴×C₉
Frattini	C₂×C₃	C₂³×C₃×D₃
Fitting	C₂²×C₃×C₄×C₉	C₂
socle	C₂²×C₃²	C₂³×C₃
prime radical	C₂×C₃×#235	C₃
cosocle	C₂×C₃²	C₂⁴×C₃
hypercenter	C₂²×C₄×C₉	D₃
hypocenter	C₃	C₂×C₉×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂²×C₄×C₉

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₄×C₉	C₂	semidirect
2. C₂×C₉×#67	C₂	semidirect
3. C₉×#235	C₂	direct (N × H)
4. C₂×C₉×D₁₂	C₂	semidirect
5. C₂×C₄×D₃×C₉	C₂	semidirect
6. C₂×C₉×Dic₆	C₂	semidirect
13. C₂×C₃×C₄×C₉	C₂²	semidirect
14. C₉×#67	C₂²	semidirect
16. C₉×D₁₂	C₂²	semidirect
17. C₄×D₃×C₉	C₂²	semidirect
18. C₉×Dic₆	C₂²	semidirect
31. C₂²×C₄×C₉	D₃	semidirect
41. C₂×#235	C₉	direct (N × H)
65. C₂×C₄×C₉	C₂×D₃	semidirect
67. C₂×C₃×C₉	#41	semidirect
76. C₂²×C₃×C₄	C₂×C₉	semidirect
78. C₂×#67	C₂×C₉	semidirect
79. #235	C₂×C₉	direct (N × H)
80. C₂×D₁₂	C₂×C₉	semidirect
81. C₂×C₄×D₃	C₂×C₉	semidirect
82. C₂×Dic₆	C₂×C₉	semidirect
105. C₃×C₉	C₂×#41	semidirect
121. C₂×C₃×C₄	C₂²×C₉	semidirect
125. #67	C₂²×C₉	semidirect
127. D₁₂	C₂²×C₉	semidirect
128. C₄×D₃	C₂²×C₉	semidirect
129. Dic₆	C₂²×C₉	semidirect
139. C₂×C₉	#235	direct (N × H)
146. C₂²×C₄	D₃×C₉	semidirect
170. C₉	C₂×#235	direct (N × H)
180. C₂×C₄	C₂×D₃×C₉	semidirect
187. C₂×C₃	C₉×#41	semidirect
201. C₃	C₂×C₉×#41	semidirect
205. C₂	C₉×#235	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×C₃×C₄×C₉	2
C₂×C₉×#67	2
C₉×#235	2
C₂×C₉×D₁₂	2
C₂×C₄×D₃×C₉	2
C₂×C₉×Dic₆	2
C₂×C₃×#235	3
C₂×C₉×#41	3

Subgroups — C2×C9×#235 #11722598

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#235 #11722598