Subgroups — C₃₁⋊C₃×#6894 #36382308

Order 22599.

932 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺²×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₉×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic
3	#21419332	C₃	1	2	yes	maximal, complemented
4	#21419331	C₃	1	2	yes	maximal, complemented
5	#21419330	C₃	1	1	yes	maximal, characteristic, complemented
6	#21419329	C₃	1	1	yes	maximal, characteristic, complemented
7	#21419328	C₃	1	6	yes	maximal, complemented
8	#21419327	C₃	1	6	yes	maximal, complemented
9	#21419326	C₃	1	2	yes	maximal, complemented
10	#21419325	C₃	1	3	yes	maximal, complemented
11	#21419324	C₃	1	3	yes	maximal, complemented
12	C₉⋊C₉×C₃₁⋊C₃	C₃	1	3	yes	maximal, complemented
13	C₉⋊C₉×C₃₁⋊C₃	C₃	1	6	yes	maximal, complemented
14	C₃²⋊C₉×C₃₁⋊C₃	C₃	1	2	yes	maximal
15	C₃₁×#6894	C₃	1	1	yes	maximal, characteristic, complemented
16	C₃³×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
17	C₃×#11719689	C₃²	1	1	yes	characteristic
18	C₃×#11719689	C₃²	1	1	yes	characteristic
19	C₃×#11719686	C₃²	1	3	yes	—
20	C₃×#11719687	C₃²	1	3	yes	—
21	3₋¹⁺²×C₃₁⋊C₃	—	3	9	no	—
22	C₃²×C₃₁⋊C₉	C₃²	1	2	yes	—
23	C₃×C₉×C₃₁⋊C₃	—	3	9	no	—
24	C₃×C₉×C₃₁⋊C₃	C₃²	1	6	yes	—
25	C₃×C₉×C₃₁⋊C₃	C₃²	1	2	yes	—
26	C₃×C₉×C₃₁⋊C₃	C₃²	1	3	yes	—
27	C₃×C₉×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
28	C₃×C₉×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
29	C₃×C₉×C₃₁⋊C₃	C₃²	1	2	yes	—
30	C₃×C₉×C₃₁⋊C₃	C₃²	1	2	yes	—
31	C₃×C₉×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
32	#14578887	C₃²	1	3	yes	complemented
33	#14578887	C₃²	1	6	yes	complemented
34	#14578886	C₃²	1	3	yes	complemented
35	#14578886	C₃²	1	6	yes	complemented
36	(C₃₁⋊C₉)⋊C₉	C₃²	1	6	yes	complemented
37	(C₃₁⋊C₉)⋊C₉	C₃²	1	6	yes	complemented
38	(C₃₁⋊C₉)⋊C₉	C₃²	1	6	yes	complemented
39	(C₃₁⋊C₉)⋊C₉	C₃²	1	6	yes	complemented
40	(C₃₁⋊C₉)⋊C₉	C₃²	1	6	yes	complemented
41	(C₃₁⋊C₉)⋊C₉	C₃²	1	6	yes	complemented
42	(C₃₁⋊C₉)⋊C₉	C₃²	1	6	yes	complemented
43	(C₃₁⋊C₉)⋊C₉	C₃²	1	6	yes	complemented
44	(C₃₁⋊C₉)⋊C₉	C₃²	1	6	yes	complemented
45	C₃²⋊(C₃₁⋊C₉)	C₃²	1	2	yes	—
46	C₃²⋊(C₃₁⋊C₉)	C₃²	1	2	yes	—
47	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	6	yes	—
48	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	6	yes	—
49	C₃×3₋¹⁺²×C₃₁	C₃²	1	1	yes	characteristic
50	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
51	C₃₁×C₉⋊C₉	C₃²	1	3	yes	complemented
52	C₃₁×C₉⋊C₉	C₃²	1	6	yes	complemented
53	C₃₁×C₃²⋊C₉	C₃²	1	2	yes	—
54	C₃³×C₃₁	C₃³	1	1	yes	characteristic
55	C₃²×C₃₁⋊C₃	—	3	3	no	—
56	C₃²×C₃₁⋊C₃	—	3	6	no	—
57	C₃²×C₃₁⋊C₃	3₋¹⁺²	1	9	yes	—
58	C₃²×C₃₁⋊C₃	C₃³	1	1	yes	characteristic
59	C₃²×C₃₁⋊C₃	C₃³	1	1	yes	characteristic
60	C₃²×C₃₁⋊C₃	C₃³	1	1	yes	characteristic
61	3₋¹⁺²×C₃₁	—	3	3	no	—
62	C₃×C₉×C₃₁	—	3	3	no	—
63	C₃×C₉×C₃₁	C₃³	1	6	yes	—
64	C₃×C₉×C₃₁	C₃³	1	2	yes	—
65	C₃×C₉×C₃₁	C₃³	1	3	yes	—
66	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
67	#11719689	—	3	3	no	—
68	#11719689	—	3	3	no	—
69	C₃×C₃₁⋊C₉	—	3	6	no	—
70	C₃×C₃₁⋊C₉	C₃³	1	6	yes	—
71	C₃×C₃₁⋊C₉	C₃³	1	6	yes	—
72	C₃×C₃₁⋊C₉	C₃³	1	2	yes	—
73	C₃×C₃₁⋊C₉	C₃³	1	3	yes	—
74	C₃×C₃₁⋊C₉	C₃³	1	2	yes	—
75	C₃×C₃₁⋊C₉	C₃³	1	3	yes	—
76	C₃×C₃₁⋊C₉	C₃³	1	2	yes	—
77	#11719687	—	3	9	no	—
78	#11719686	—	3	9	no	—
79	C₉×C₃₁⋊C₃	—	3	6	no	—
80	C₉×C₃₁⋊C₃	—	3	9	no	—
81	C₉×C₃₁⋊C₃	—	3	18	no	—
82	C₉×C₃₁⋊C₃	—	3	3	no	—
83	C₉×C₃₁⋊C₃	—	3	3	no	—
84	C₉×C₃₁⋊C₃	—	3	6	no	—
85	C₉×C₃₁⋊C₃	—	3	6	no	—
86	C₉×C₃₁⋊C₃	3₋¹⁺²	1	9	yes	complemented
87	C₃×#6894	—	31	1	no	maximal, Sylow
88	C₃²×C₃₁	—	3	1	no	—
89	C₃²×C₃₁	—	3	2	no	—
90	C₃²×C₃₁	C₃×3₋¹⁺²	1	3	yes	—
91	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
92	C₃×C₃₁⋊C₃	—	3	1	no	—
93	C₃×C₃₁⋊C₃	—	3	1	no	—
94	C₃×C₃₁⋊C₃	—	3	2	no	—
95	C₃×C₃₁⋊C₃	—	3	2	no	—
96	C₃×C₃₁⋊C₃	—	3	27	no	—
97	C₃×C₃₁⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
98	C₃×C₃₁⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
99	C₃×C₃₁⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
100	C₃×C₃₁⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
101	C₃×C₃₁⋊C₃	#737	1	6	yes	—
102	C₉×C₃₁	—	3	2	no	—
103	C₉×C₃₁	—	3	3	no	—
104	C₉×C₃₁	—	3	6	no	—
105	C₉×C₃₁	C₃×3₋¹⁺²	1	3	yes	complemented
106	C₃₁⋊C₉	—	3	2	no	—
107	C₃₁⋊C₉	—	3	2	no	—
108	C₃₁⋊C₉	—	3	3	no	—
109	C₃₁⋊C₉	—	3	3	no	—
110	C₃₁⋊C₉	—	3	6	no	—
111	C₃₁⋊C₉	—	3	6	no	—
112	C₃₁⋊C₉	C₃×3₋¹⁺²	1	6	yes	complemented
113	C₃²×3₋¹⁺²	—	31	1	no	—
114	C₃³×C₉	—	31	1	no	—
115	#6894	—	31	2	no	—
116	#6894	—	31	1	no	—
117	#6894	—	31	1	no	—
118	#6894	—	31	3	no	—
119	#6894	—	31	3	no	—
120	#6894	—	31	6	no	—
121	#6894	—	31	6	no	—
122	#6894	—	31	2	no	—
123	#6894	—	31	2	no	—
124	#6894	C₃₁⋊C₃	1	1	yes	characteristic, direct product factor
125	C₃×C₉⋊C₉	—	31	3	no	—
126	C₃×C₉⋊C₉	—	31	6	no	—
127	C₃×C₃²⋊C₉	—	31	2	no	—
128	C₃×C₃₁	—	3	3	no	—
129	C₃×C₃₁	C₃²×3₋¹⁺²	1	1	yes	characteristic
130	C₃×C₃₁	C₃²×3₋¹⁺²	1	1	yes	characteristic
131	C₃×C₃₁	C₃×#737	1	2	yes	—
132	C₃₁⋊C₃	—	3	3	no	—
133	C₃₁⋊C₃	—	3	3	no	—
134	C₃₁⋊C₃	#6894	1	9	yes	direct product factor
135	C₃⁴	—	31	1	no	—
136	C₃×3₋¹⁺²	—	31	1	no	—
137	C₃×3₋¹⁺²	—	31	1	no	—
138	C₃×3₋¹⁺²	—	31	3	no	—
139	C₃×3₋¹⁺²	—	31	3	no	—
140	C₃×3₋¹⁺²	—	93	9	no	—
141	C₃×3₋¹⁺²	C₃×C₃₁⋊C₃	1	1	yes	characteristic
142	C₃²×C₉	—	31	1	no	—
143	C₃²×C₉	—	31	2	no	—
144	C₃²×C₉	—	31	2	no	—
145	C₃²×C₉	—	31	2	no	—
146	C₃²×C₉	—	31	2	no	—
147	C₃²×C₉	—	31	1	no	—
148	C₃²×C₉	—	31	1	no	—
149	C₃²×C₉	—	31	3	no	—
150	C₃²×C₉	—	31	6	no	—
151	C₃²×C₉	—	93	9	no	—
152	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
153	C₉⋊C₉	—	31	6	no	—
154	C₉⋊C₉	—	31	6	no	—
155	C₉⋊C₉	—	31	6	no	—
156	C₉⋊C₉	—	31	6	no	—
157	C₉⋊C₉	—	31	6	no	—
158	C₉⋊C₉	—	31	6	no	—
159	C₉⋊C₉	—	31	6	no	—
160	C₉⋊C₉	—	31	6	no	—
161	C₉⋊C₉	—	31	6	no	—
162	C₉⋊C₉	—	31	3	no	—
163	C₉⋊C₉	—	31	6	no	—
164	C₉⋊C₉	—	31	3	no	—
165	C₉⋊C₉	—	31	6	no	—
166	C₉⋊C₉	C₃×C₃₁⋊C₃	1	6	yes	complemented
167	C₉⋊C₉	C₃×C₃₁⋊C₃	1	3	yes	complemented
168	C₃²⋊C₉	—	31	2	no	—
169	C₃²⋊C₉	—	31	2	no	—
170	C₃²⋊C₉	—	31	6	no	—
171	C₃²⋊C₉	—	31	6	no	—
172	C₃²⋊C₉	C₃×C₃₁⋊C₃	1	2	yes	—
173	C₃₁	C₃×#6894	1	1	yes	characteristic, complemented, Sylow
174	C₃³	—	31	1	no	—
175	C₃³	—	31	9	no	—
176	C₃³	—	31	1	no	—
177	C₃³	—	31	1	no	—
178	C₃³	—	93	3	no	—
179	C₃³	—	93	6	no	—
180	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
181	3₋¹⁺²	—	3	3	no	—
182	3₋¹⁺²	—	93	9	no	—
183	3₋¹⁺²	—	93	9	no	—
184	3₋¹⁺²	—	93	3	no	—
185	3₋¹⁺²	—	93	3	no	—
186	C₃×C₉	—	3	3	no	—
187	C₃×C₉	—	31	2	no	—
188	C₃×C₉	—	31	9	no	—
189	C₃×C₉	—	31	6	no	—
190	C₃×C₉	—	31	6	no	—
191	C₃×C₉	—	31	2	no	—
192	C₃×C₉	—	31	2	no	—
193	C₃×C₉	—	31	3	no	—
194	C₃×C₉	—	31	3	no	—
195	C₃×C₉	—	93	6	no	—
196	C₃×C₉	—	93	6	no	—
197	C₃×C₉	—	93	18	no	—
198	C₃×C₉	—	93	6	no	—
199	C₃×C₉	—	93	9	no	—
200	C₃×C₉	—	93	6	no	—
201	C₃×C₉	—	93	3	no	—
202	C₃×C₉	—	93	3	no	—
203	C₃×C₉	C₃²×C₃₁⋊C₃	1	6	yes	—
204	C₃×C₉	C₃²×C₃₁⋊C₃	1	2	yes	—
205	C₃×C₉	C₃²×C₃₁⋊C₃	1	3	yes	—
206	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
207	C₃²	—	3	1	no	—
208	C₃²	—	3	2	no	—
209	C₃²	—	31	3	no	—
210	C₃²	—	31	3	no	—
211	C₃²	—	31	6	no	—
212	C₃²	—	31	3	no	—
213	C₃²	—	31	3	no	—
214	C₃²	—	93	27	no	—
215	C₃²	—	93	1	no	—
216	C₃²	—	93	1	no	—
217	C₃²	—	93	2	no	—
218	C₃²	—	93	2	no	—
219	C₃²	3₋¹⁺²×C₃₁⋊C₃	1	3	yes	—
220	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
221	C₉	—	3	2	no	—
222	C₉	—	3	3	no	—
223	C₉	—	3	6	no	—
224	C₉	—	31	6	no	—
225	C₉	—	93	2	no	—
226	C₉	—	93	2	no	—
227	C₉	—	93	3	no	—
228	C₉	—	93	3	no	—
229	C₉	—	93	6	no	—
230	C₉	—	93	6	no	—
231	C₉	3₋¹⁺²×C₃₁⋊C₃	1	3	yes	complemented
232	C₃	—	3	3	no	—
233	C₃	—	31	9	no	—
234	C₃	—	93	3	no	—
235	C₃	—	93	3	no	—
236	C₃	C₃×3₋¹⁺²×C₃₁⋊C₃	1	1	yes	characteristic
237	C₃	C₃×3₋¹⁺²×C₃₁⋊C₃	1	1	yes	characteristic
238	C₃	#737×C₃₁⋊C₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₁⋊C₃
commutator, socle, cosocle	C₃²×C₃₁	C₃⁴
Fitting	C₃₁×#6894	C₃
prime radical	C₃³×C₃₁⋊C₃	C₃²
hypercenter	#6894	C₃₁⋊C₃
hypocenter	C₃₁	C₃×#6894
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃² ◁ #6894

Complements

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

N	H ≅ G/N	type
3. #21419332	C₃	semidirect
4. #21419331	C₃	semidirect
5. #21419330	C₃	semidirect
6. #21419329	C₃	semidirect
7. #21419328	C₃	semidirect
8. #21419327	C₃	semidirect
9. #21419326	C₃	semidirect
10. #21419325	C₃	semidirect
11. #21419324	C₃	semidirect
12. C₉⋊C₉×C₃₁⋊C₃	C₃	semidirect
13. C₉⋊C₉×C₃₁⋊C₃	C₃	semidirect
15. C₃₁×#6894	C₃	semidirect
32. #14578887	C₃²	semidirect
33. #14578887	C₃²	semidirect
34. #14578886	C₃²	semidirect
35. #14578886	C₃²	semidirect
36. (C₃₁⋊C₉)⋊C₉	C₃²	semidirect
37. (C₃₁⋊C₉)⋊C₉	C₃²	semidirect
38. (C₃₁⋊C₉)⋊C₉	C₃²	semidirect
39. (C₃₁⋊C₉)⋊C₉	C₃²	semidirect
40. (C₃₁⋊C₉)⋊C₉	C₃²	semidirect
41. (C₃₁⋊C₉)⋊C₉	C₃²	semidirect
42. (C₃₁⋊C₉)⋊C₉	C₃²	semidirect
43. (C₃₁⋊C₉)⋊C₉	C₃²	semidirect
44. (C₃₁⋊C₉)⋊C₉	C₃²	semidirect
51. C₃₁×C₉⋊C₉	C₃²	semidirect
52. C₃₁×C₉⋊C₉	C₃²	semidirect
86. C₉×C₃₁⋊C₃	3₋¹⁺²	semidirect
105. C₉×C₃₁	C₃×3₋¹⁺²	semidirect
112. C₃₁⋊C₉	C₃×3₋¹⁺²	semidirect
124. #6894	C₃₁⋊C₃	direct (N × H)
134. C₃₁⋊C₃	#6894	direct (N × H)
166. C₉⋊C₉	C₃×C₃₁⋊C₃	semidirect
167. C₉⋊C₉	C₃×C₃₁⋊C₃	semidirect
173. C₃₁	C₃×#6894	semidirect
231. C₉	3₋¹⁺²×C₃₁⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×3₋¹⁺²×C₃₁⋊C₃	3
C₃²×C₉×C₃₁⋊C₃	3
#21419332	3
#21419331	3
#21419330	3
#21419329	3
#21419328	3
#21419327	3
#21419326	3
#21419325	3
#21419324	3
C₉⋊C₉×C₃₁⋊C₃	3
C₃²⋊C₉×C₃₁⋊C₃	3
C₃₁×#6894	3
C₃×#6894	31

Subgroups — C31⋊C3×#6894 #36382308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁⋊C₃×#6894 #36382308