Subgroups — C₉×#5700 #13153174

Order 1728.

716 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#896	C₂	1	4	yes	maximal, complemented
3	C₉×#895	C₂	1	4	yes	maximal, complemented
4	C₉×#894	C₂	1	4	yes	maximal, complemented
5	C₉×#875	C₂	1	2	yes	maximal, complemented
6	C₃×#5700	C₃	1	1	yes	maximal, characteristic
7	C₉×#562	—	3	1	no	maximal
8	C₂×C₉×D₁₂	C₂²	1	2	yes	—
9	C₂×C₉×D₁₂	C₂²	1	4	yes	—
10	C₂×C₄×D₃×C₉	C₂²	1	4	yes	—
11	C₂×C₉×Dic₆	C₂²	1	2	yes	—
12	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
13	C₃×C₉×C₄⋊C₄	C₂²	1	4	yes	—
14	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₉×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
16	C₉×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
17	C₉×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
18	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
19	C₃²×#125	C₂×C₃	1	1	yes	characteristic
20	C₃×#896	C₂×C₃	1	4	yes	—
21	C₃×#895	C₂×C₃	1	4	yes	—
22	C₃×#894	C₂×C₃	1	4	yes	—
23	C₃×#875	C₂×C₃	1	2	yes	—
24	C₉×#125	—	2	1	no	—
25	C₉×#125	D₃	1	1	yes	characteristic, complemented
26	C₉×#124	—	3	2	no	—
27	C₉×#123	—	3	4	no	—
28	C₉×C₂²⋊Q₈	—	3	4	no	—
29	C₉×#121	—	3	4	no	—
30	C₂²×D₃×C₉	C₂³	1	4	yes	—
31	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
33	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₂×C₉×Dic₃	C₂³	1	4	yes	—
35	C₉×D₁₂	—	4	2	no	—
36	C₉×D₁₂	—	4	4	no	—
37	C₄×D₃×C₉	—	4	4	no	—
38	C₉×Dic₆	—	4	2	no	—
39	C₃×#562	—	3	1	no	—
40	#5700	C₉	1	1	yes	characteristic, direct product factor
41	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
42	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
43	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
44	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
45	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
46	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
47	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
48	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
49	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
50	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
51	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
52	C₂×Q₈×C₉	—	3	2	no	—
53	C₂×D₄×C₉	—	3	4	no	—
54	C₂×D₄×C₉	—	3	2	no	—
55	C₂²×C₄×C₉	—	3	4	no	—
56	C₉×C₄⋊C₄	—	2	2	no	—
57	C₉×C₄⋊C₄	—	2	4	no	—
58	C₉×C₄⋊C₄	—	3	4	no	—
59	C₉×C₄⋊C₄	C₂×D₃	1	4	yes	—
60	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
61	C₉×C₂²⋊C₄	—	3	4	no	—
62	C₉×C₂²⋊C₄	—	3	4	no	—
63	C₉×C₂²⋊C₄	—	3	4	no	—
64	C₄²×C₉	—	2	1	no	—
65	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
66	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
67	C₂×D₃×C₉	—	2	4	no	—
68	C₂×D₃×C₉	—	2	4	no	—
69	C₂×D₃×C₉	—	2	4	no	—
70	C₃×C₄×C₉	—	2	2	no	—
71	C₃×C₄×C₉	—	2	4	no	—
72	C₃×C₄×C₉	—	2	1	no	—
73	C₉×Dic₃	—	2	4	no	—
74	C₃×#125	—	2	1	no	—
75	C₃×#125	C₂×C₉	1	1	yes	characteristic, complemented
76	C₃×#125	C₃×D₃	1	1	yes	characteristic
77	C₃×#124	—	3	2	no	—
78	C₃×#123	—	3	4	no	—
79	C₃×C₂²⋊Q₈	—	3	4	no	—
80	C₃×#121	—	3	4	no	—
81	#896	C₂×C₉	1	4	yes	complemented
82	#895	C₂×C₉	1	4	yes	complemented
83	#894	C₂×C₉	1	4	yes	complemented
84	#875	C₂×C₉	1	2	yes	complemented
85	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	—
86	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
87	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
89	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
90	C₃×D₁₂	—	4	2	no	—
91	C₃×D₁₂	—	4	4	no	—
92	C₃×C₄×D₃	—	4	4	no	—
93	C₃×Dic₆	—	4	2	no	—
94	C₂³×C₉	—	3	4	no	—
95	Q₈×C₉	—	12	2	no	—
96	D₄×C₉	—	12	4	no	—
97	D₄×C₉	—	12	2	no	—
98	C₂×C₄×C₉	—	2	1	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	4	no	—
101	C₂×C₄×C₉	—	3	4	no	—
102	C₂×C₄×C₉	—	12	4	no	—
103	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
104	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
105	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
106	#562	—	3	1	no	Sylow
107	C₂×C₃×C₉	H(5,𝔽₂)	1	2	yes	—
108	C₂×C₃×C₉	2₋¹⁺⁴	1	1	yes	characteristic
109	D₃×C₉	—	4	4	no	—
110	C₂×C₃×Q₈	—	3	2	no	—
111	C₂×C₃×D₄	—	3	4	no	—
112	C₂×C₃×D₄	—	3	2	no	—
113	C₂²×C₃×C₄	—	3	4	no	—
114	C₂×D₁₂	C₂²×C₉	1	2	yes	—
115	C₂×D₁₂	C₂²×C₉	1	4	yes	—
116	C₂×C₄×D₃	C₂²×C₉	1	4	yes	—
117	C₂×Dic₆	C₂²×C₉	1	2	yes	—
118	C₃×C₄⋊C₄	—	2	4	no	—
119	C₃×C₄⋊C₄	—	2	2	no	—
120	C₃×C₄⋊C₄	—	3	4	no	—
121	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
122	C₃×C₄⋊C₄	C₂²×C₉	1	4	yes	—
123	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
124	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
125	C₃×C₂²⋊C₄	—	3	4	no	—
126	C₃×C₂²⋊C₄	—	3	4	no	—
127	C₃×C₂²⋊C₄	—	3	4	no	—
128	C₃×C₄²	—	2	1	no	—
129	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
130	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
131	(C₂×D₃)⋊C₄	C₂²×C₉	1	4	yes	—
132	(C₂×D₃)⋊C₄	C₂²×C₉	1	4	yes	—
133	(C₂×D₃)⋊C₄	C₂²×C₉	1	4	yes	—
134	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
135	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
136	C₂×C₃×D₃	—	2	4	no	—
137	C₂×C₃×D₃	—	2	4	no	—
138	C₂×C₃×D₃	—	2	4	no	—
139	C₃²×C₄	—	2	1	no	—
140	C₃²×C₄	—	2	2	no	—
141	C₃²×C₄	—	2	4	no	—
142	C₃×Dic₃	—	2	4	no	—
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	—	6	4	no	—
145	C₂²×C₉	—	6	4	no	—
146	C₂²×C₉	—	6	4	no	—
147	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
148	C₄×C₉	—	2	2	no	—
149	C₄×C₉	—	2	1	no	—
150	C₄×C₉	—	2	4	no	—
151	C₄×C₉	—	2	2	no	—
152	C₄×C₉	—	4	4	no	—
153	C₄×C₉	—	4	2	no	—
154	C₄×C₉	—	6	4	no	—
155	#125	D₃×C₉	1	1	yes	characteristic, complemented
156	#124	—	3	2	no	—
157	#123	—	3	4	no	—
158	C₂²⋊Q₈	—	3	4	no	—
159	#121	—	3	4	no	—
160	C₃×C₉	#562	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₃	—	3	4	no	—
162	C₂²×D₃	C₂³×C₉	1	4	yes	—
163	C₃×Q₈	—	12	2	no	—
164	C₃×D₄	—	12	4	no	—
165	C₃×D₄	—	12	2	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	3	4	no	—
170	C₂×C₃×C₄	—	12	4	no	—
171	C₂×C₃×C₄	C₂³×C₉	1	4	yes	—
172	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
174	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
176	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
177	C₂×Dic₃	C₂³×C₉	1	4	yes	—
178	D₁₂	—	4	2	no	—
179	D₁₂	—	4	4	no	—
180	C₄×D₃	—	4	4	no	—
181	Dic₆	—	4	2	no	—
182	C₂×C₃²	C₃×H(5,𝔽₂)	1	2	yes	—
183	C₂×C₃²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
184	C₃×D₃	—	4	4	no	—
185	C₂×C₉	—	2	1	no	—
186	C₂×C₉	—	2	2	no	—
187	C₂×C₉	—	12	4	no	—
188	C₂×C₉	#1007	1	1	yes	characteristic
189	C₂×C₉	#1009	1	2	yes	—
190	C₂×Q₈	—	3	2	no	—
191	C₂×D₄	—	3	4	no	—
192	C₂×D₄	—	3	2	no	—
193	C₂²×C₄	—	3	4	no	—
194	C₄⋊C₄	—	3	4	no	—
195	C₄⋊C₄	C₂×D₃×C₉	1	4	yes	—
196	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
197	C₂²⋊C₄	—	3	4	no	—
198	C₂²⋊C₄	—	3	4	no	—
199	C₂²⋊C₄	—	3	4	no	—
200	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	—	6	4	no	—
203	C₂²×C₃	—	6	4	no	—
204	C₂²×C₃	—	6	4	no	—
205	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
206	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
207	C₂×D₃	—	2	4	no	—
208	C₂×D₃	—	2	4	no	—
209	C₂×D₃	—	2	4	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	4	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	2	4	no	—
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	4	4	no	—
218	C₃×C₄	—	4	2	no	—
219	C₃×C₄	—	6	4	no	—
220	Dic₃	—	2	4	no	—
221	C₃²	C₃×#562	1	1	yes	characteristic
222	C₉	—	2	1	no	—
223	C₉	#5700	1	1	yes	characteristic, direct product factor
224	C₂³	—	3	4	no	—
225	Q₈	—	12	2	no	—
226	D₄	—	12	4	no	—
227	D₄	—	12	2	no	—
228	C₂×C₄	—	3	4	no	—
229	C₂×C₄	—	12	4	no	—
230	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
231	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
232	C₂×C₄	C₂²×D₃×C₉	1	4	yes	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	2	2	no	—
235	C₂×C₃	—	12	4	no	—
236	C₂×C₃	C₉×H(5,𝔽₂)	1	2	yes	—
237	C₂×C₃	C₉×2₋¹⁺⁴	1	1	yes	characteristic
238	C₂×C₃	C₃×#1007	1	1	yes	characteristic
239	C₂×C₃	C₃×#1009	1	2	yes	—
240	D₃	—	4	4	no	—
241	C₂²	—	6	4	no	—
242	C₂²	—	6	4	no	—
243	C₂²	—	6	4	no	—
244	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
245	C₄	—	2	1	no	—
246	C₄	—	2	4	no	—
247	C₄	—	2	2	no	—
248	C₄	—	6	4	no	—
249	C₃	—	2	1	no	—
250	C₃	C₉×#562	1	1	yes	characteristic, complemented
251	C₃	C₃×#5700	1	1	yes	characteristic
252	C₂	—	12	4	no	—
253	C₂	C₉×#1007	1	1	yes	characteristic
254	C₂	C₉×#1009	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂³×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₉×#125	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₃×#5700	C₃
hypercenter	C₉×#125	D₃
hypocenter	C₃	C₉×#562
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#125

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×#125	C₂	semidirect
2. C₉×#896	C₂	semidirect
3. C₉×#895	C₂	semidirect
4. C₉×#894	C₂	semidirect
5. C₉×#875	C₂	semidirect
25. C₉×#125	D₃	semidirect
40. #5700	C₉	direct (N × H)
75. C₃×#125	C₂×C₉	semidirect
81. #896	C₂×C₉	semidirect
82. #895	C₂×C₉	semidirect
83. #894	C₂×C₉	semidirect
84. #875	C₂×C₉	semidirect
155. #125	D₃×C₉	semidirect
160. C₃×C₉	#562	semidirect
223. C₉	#5700	direct (N × H)
250. C₃	C₉×#562	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#125	2
C₉×#896	2
C₉×#895	2
C₉×#894	2
C₉×#875	2
C₃×#5700	3
C₉×#562	3

Subgroups — C9×#5700 #13153174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5700 #13153174