Subgroups — C₂×C₉×#870 #13152277

Order 1728.

786 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊Dic₃	C₂	1	2	yes	maximal
3	C₂×C₉×Dic₃⋊C₄	C₂	1	4	yes	maximal
4	C₉×#870	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#870	C₃	1	1	yes	maximal, characteristic
6	C₂×C₉×#125	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₉×Dic₃	C₂²	1	4	yes	—
10	C₃×C₄²×C₉	C₂²	1	4	yes	—
11	C₉×C₄⋊Dic₃	C₂²	1	8	yes	—
12	C₉×Dic₃⋊C₄	C₂²	1	16	yes	—
13	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
14	C₂×C₃×C₄⋊Dic₃	C₂×C₃	1	2	yes	—
15	C₂×C₃×Dic₃⋊C₄	C₂×C₃	1	4	yes	—
16	C₃×#870	C₂×C₃	1	8	yes	—
17	C₉×#125	—	3	8	no	—
18	C₂×C₉×C₄⋊C₄	—	3	4	no	—
19	C₂×C₉×C₄⋊C₄	—	3	2	no	—
20	C₂×C₄²×C₉	—	2	1	no	—
21	C₂×C₄²×C₉	D₃	1	1	yes	characteristic
22	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
23	C₂×C₃×C₄×C₉	—	2	4	no	—
24	C₂×C₃×C₄×C₉	Q₈	1	4	yes	—
25	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
26	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
27	C₂×C₉×Dic₃	—	2	8	no	—
28	C₂×C₉×Dic₃	C₂³	1	8	yes	—
29	C₂×C₃×#125	—	3	1	no	—
30	C₂×#870	C₉	1	1	yes	characteristic, direct product factor
31	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
32	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
33	C₂²×C₃×Dic₃	C₂²×C₃	1	4	yes	—
34	C₃²×C₄²	C₂²×C₃	1	4	yes	—
35	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
36	C₃×Dic₃⋊C₄	C₂²×C₃	1	16	yes	—
37	C₂²×C₄×C₉	—	2	2	no	—
38	C₂²×C₄×C₉	—	2	1	no	—
39	C₂²×C₄×C₉	—	3	4	no	—
40	C₂²×C₄×C₉	C₂×D₃	1	2	yes	—
41	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
42	C₉×C₄⋊C₄	—	3	16	no	—
43	C₉×C₄⋊C₄	—	3	8	no	—
44	C₄²×C₉	—	2	4	no	—
45	C₄²×C₉	C₂×D₃	1	4	yes	—
46	C₂²×C₃×C₉	C₂×Q₈	1	2	yes	—
47	C₂²×C₃×C₉	#41	1	4	yes	—
48	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
49	C₃×C₄×C₉	—	2	4	no	—
50	C₃×C₄×C₉	C₂×Q₈	1	4	yes	—
51	C₉×Dic₃	—	2	8	no	—
52	C₃×#125	—	3	8	no	—
53	C₂×C₃×C₄⋊C₄	—	3	4	no	—
54	C₂×C₃×C₄⋊C₄	—	3	2	no	—
55	C₂×C₃×C₄²	—	2	1	no	—
56	C₂×C₃×C₄²	C₂×C₉	1	1	yes	characteristic
57	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
58	C₂×C₄⋊Dic₃	C₂×C₉	1	2	yes	—
59	C₂×Dic₃⋊C₄	C₂×C₉	1	4	yes	—
60	#870	C₂×C₉	1	8	yes	direct product factor
61	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
62	C₂×C₃²×C₄	—	2	4	no	—
63	C₂×C₃²×C₄	C₃×Q₈	1	4	yes	—
64	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
65	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
66	C₂×C₃×Dic₃	—	2	8	no	—
67	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
68	C₂³×C₉	—	2	1	no	—
69	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
70	C₂×C₄×C₉	—	2	8	no	—
71	C₂×C₄×C₉	—	2	4	no	—
72	C₂×C₄×C₉	—	2	4	no	—
73	C₂×C₄×C₉	—	2	4	no	—
74	C₂×C₄×C₉	—	2	2	no	—
75	C₂×C₄×C₉	—	3	8	no	—
76	C₂×C₄×C₉	—	6	8	no	—
77	C₂×C₄×C₉	Dic₆	1	4	yes	—
78	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
79	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
80	C₂×#125	—	3	1	no	Sylow
81	C₂×C₃×C₉	#125	1	4	yes	complemented
82	C₂×C₃×C₉	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₃×C₉	C₂×#41	1	2	yes	—
84	C₂²×C₃×C₄	—	2	2	no	—
85	C₂²×C₃×C₄	—	2	1	no	—
86	C₂²×C₃×C₄	—	3	4	no	—
87	C₂²×C₃×C₄	C₂²×C₉	1	2	yes	—
88	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
90	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
91	C₂²×Dic₃	C₂²×C₉	1	4	yes	—
92	C₃×C₄⋊C₄	—	3	16	no	—
93	C₃×C₄⋊C₄	—	3	8	no	—
94	C₃×C₄²	—	2	4	no	—
95	C₃×C₄²	C₂²×C₉	1	4	yes	—
96	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
97	C₄⋊Dic₃	C₂²×C₉	1	8	yes	—
98	Dic₃⋊C₄	C₂²×C₉	1	16	yes	—
99	C₂²×C₃²	C₂×C₃×Q₈	1	2	yes	—
100	C₂²×C₃²	C₃×#41	1	4	yes	—
101	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
102	C₃²×C₄	—	2	4	no	—
103	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
104	C₃×Dic₃	—	2	8	no	—
105	C₂²×C₉	—	2	1	no	—
106	C₂²×C₉	—	2	2	no	—
107	C₂²×C₉	—	2	4	no	—
108	C₂²×C₉	C₂×Dic₆	1	2	yes	—
109	C₂²×C₉	#235	1	4	yes	—
110	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
111	C₄×C₉	—	2	8	no	—
112	C₄×C₉	—	2	4	no	—
113	C₄×C₉	—	2	4	no	—
114	C₄×C₉	—	6	8	no	—
115	C₄×C₉	C₂×Dic₆	1	4	yes	—
116	#125	—	3	8	no	—
117	C₂×C₄⋊C₄	—	3	4	no	—
118	C₂×C₄⋊C₄	—	3	2	no	—
119	C₂×C₄²	D₃×C₉	1	1	yes	characteristic
120	C₃×C₉	C₂×#125	1	1	yes	characteristic, complemented, Sylow
121	C₂³×C₃	—	2	1	no	—
122	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
123	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
124	C₂×C₃×C₄	—	2	8	no	—
125	C₂×C₃×C₄	—	2	4	no	—
126	C₂×C₃×C₄	—	2	4	no	—
127	C₂×C₃×C₄	—	2	4	no	—
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	3	8	no	—
131	C₂×C₃×C₄	—	6	8	no	—
132	C₂×C₃×C₄	Q₈×C₉	1	4	yes	—
133	C₂×C₃×C₄	C₂³×C₉	1	4	yes	—
134	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
135	C₂×C₃×C₄	C₃×Dic₆	1	4	yes	—
136	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
137	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
138	C₂×Dic₃	—	2	8	no	—
139	C₂×Dic₃	C₂³×C₉	1	8	yes	—
140	C₂×C₃²	C₃×#125	1	4	yes	—
141	C₂×C₃²	C₂²×C₃×Q₈	1	1	yes	characteristic
142	C₂×C₃²	C₂×C₃×#41	1	2	yes	—
143	C₂×C₉	—	2	1	no	—
144	C₂×C₉	—	2	2	no	—
145	C₂×C₉	—	2	4	no	—
146	C₂×C₉	#870	1	4	yes	direct product factor
147	C₂×C₉	C₂²×Dic₆	1	1	yes	characteristic
148	C₂×C₉	C₂×#235	1	2	yes	—
149	C₂²×C₄	—	3	4	no	—
150	C₂²×C₄	C₂×D₃×C₉	1	2	yes	—
151	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
152	C₄⋊C₄	—	3	16	no	—
153	C₄⋊C₄	—	3	8	no	—
154	C₄²	C₂×D₃×C₉	1	4	yes	—
155	C₂²×C₃	—	2	1	no	—
156	C₂²×C₃	—	2	2	no	—
157	C₂²×C₃	—	2	4	no	—
158	C₂²×C₃	C₂×Q₈×C₉	1	2	yes	—
159	C₂²×C₃	C₉×#41	1	4	yes	—
160	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
161	C₂²×C₃	C₂×C₃×Dic₆	1	2	yes	—
162	C₂²×C₃	C₃×#235	1	4	yes	—
163	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
164	C₃×C₄	—	2	8	no	—
165	C₃×C₄	—	2	4	no	—
166	C₃×C₄	—	2	4	no	—
167	C₃×C₄	—	2	4	no	—
168	C₃×C₄	—	6	8	no	—
169	C₃×C₄	C₂×Q₈×C₉	1	4	yes	—
170	C₃×C₄	C₂×C₃×Dic₆	1	4	yes	—
171	Dic₃	—	2	8	no	—
172	C₃²	C₂×C₃×#125	1	1	yes	characteristic
173	C₉	—	2	1	no	—
174	C₉	C₂×#870	1	1	yes	characteristic, direct product factor
175	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	3	8	no	—
178	C₂×C₄	—	6	8	no	—
179	C₂×C₄	C₉×Dic₆	1	4	yes	—
180	C₂×C₄	C₂²×D₃×C₉	1	4	yes	—
181	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
182	C₂×C₃	—	2	1	no	—
183	C₂×C₃	—	2	2	no	—
184	C₂×C₃	—	2	4	no	—
185	C₂×C₃	C₉×#125	1	4	yes	complemented
186	C₂×C₃	C₂²×Q₈×C₉	1	1	yes	characteristic
187	C₂×C₃	C₂×C₉×#41	1	2	yes	—
188	C₂×C₃	C₃×#870	1	4	yes	—
189	C₂×C₃	C₂²×C₃×Dic₆	1	1	yes	characteristic
190	C₂×C₃	C₂×C₃×#235	1	2	yes	—
191	C₂²	C₂×C₉×Dic₆	1	2	yes	—
192	C₂²	C₉×#235	1	4	yes	—
193	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	6	8	no	—
196	C₄	C₂×C₉×Dic₆	1	4	yes	—
197	C₃	—	2	1	no	—
198	C₃	C₂×C₉×#125	1	1	yes	characteristic, complemented
199	C₃	C₂×C₃×#870	1	1	yes	characteristic
200	C₂	C₉×#870	1	4	yes	direct product factor
201	C₂	C₂²×C₉×Dic₆	1	1	yes	characteristic
202	C₂	C₂×C₉×#235	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂²×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle, prime radical	C₂³×C₃²	C₂³×C₃
cosocle	C₂²×C₃²	C₂⁴×C₃
hypercenter	C₂×C₄²×C₉	D₃
hypocenter	C₃	C₂×C₉×#125
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉

Complements

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

N	H ≅ G/N	type
4. C₉×#870	C₂	direct (N × H)
30. C₂×#870	C₉	direct (N × H)
60. #870	C₂×C₉	direct (N × H)
81. C₂×C₃×C₉	#125	semidirect
120. C₃×C₉	C₂×#125	semidirect
146. C₂×C₉	#870	direct (N × H)
174. C₉	C₂×#870	direct (N × H)
185. C₂×C₃	C₉×#125	semidirect
198. C₃	C₂×C₉×#125	semidirect
200. C₂	C₉×#870	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²×C₉	2
C₂×C₉×C₄⋊Dic₃	2
C₂×C₉×Dic₃⋊C₄	2
C₉×#870	2
C₂×C₃×#870	3
C₂×C₉×#125	3

Subgroups — C2×C9×#870 #13152277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#870 #13152277