Subgroups — C₃₁⋊C₃×#6913 #36382920

Order 22599.

992 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#737×C₃₁⋊C₃	C₃	1	3	yes	maximal, complemented
2	C₃×3₋¹⁺²×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#21419421	C₃	1	2	yes	maximal, complemented
4	#21419420	C₃	1	2	yes	maximal, complemented
5	#21419419	C₃	1	2	yes	maximal, complemented
6	#21419418	C₃	1	3	yes	maximal, complemented
7	#21419417	C₃	1	3	yes	maximal, complemented
8	#21419416	C₃	1	1	yes	maximal, characteristic, complemented
9	#21419415	C₃	1	1	yes	maximal, characteristic, complemented
10	#21419414	C₃	1	3	yes	maximal, complemented
11	#21419413	C₃	1	3	yes	maximal, complemented
12	#21419412	C₃	1	3	yes	maximal, complemented
13	#21419411	C₃	1	3	yes	maximal, complemented
14	#733×C₃₁⋊C₃	C₃	1	2	yes	maximal, complemented
15	#732×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
16	#731×C₃₁⋊C₃	C₃	1	3	yes	maximal, complemented
17	AΣL(1,𝔽₂₇)×C₃₁⋊C₃	C₃	1	3	yes	maximal, complemented
18	C₃₁×#6913	C₃	1	1	yes	maximal, characteristic, complemented
19	C₃³×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
20	#14578913	—	3	6	no	—
21	#14578913	—	3	6	no	—
22	#14578913	C₃²	1	6	yes	complemented
23	#14578913	C₃²	1	6	yes	complemented
24	#14578912	—	3	6	no	—
25	#14578912	C₃²	1	6	yes	complemented
26	#14578911	C₃²	1	3	yes	complemented
27	#14578911	C₃²	1	3	yes	complemented
28	C₃×#11719689	C₃²	1	1	yes	characteristic, complemented
29	C₃×#11719689	C₃²	1	1	yes	characteristic, complemented
30	C₃×#11719686	C₃²	1	1	yes	characteristic, complemented
31	C₃×#11719686	C₃²	1	1	yes	characteristic, complemented
32	C₃×#11719686	C₃²	1	1	yes	characteristic, complemented
33	C₃×#11719687	C₃²	1	1	yes	characteristic, complemented
34	C₃×#11719687	C₃²	1	1	yes	characteristic, complemented
35	C₃×#11719687	C₃²	1	1	yes	characteristic, complemented
36	3₋¹⁺²×C₃₁⋊C₃	—	3	3	no	—
37	3₋¹⁺²×C₃₁⋊C₃	—	3	6	no	—
38	3₋¹⁺²×C₃₁⋊C₃	—	3	6	no	—
39	3₋¹⁺²×C₃₁⋊C₃	C₃²	1	6	yes	—
40	H(3,𝔽₃)×C₃₁⋊C₃	C₃²	1	3	yes	—
41	C₃×C₉×C₃₁⋊C₃	—	3	3	no	—
42	C₃×C₉×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
43	C₃×C₉×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
44	C₃×C₉×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
45	#14578902	C₃²	1	2	yes	complemented
46	#14578902	C₃²	1	2	yes	complemented
47	#14578901	C₃²	1	6	yes	complemented
48	#14578901	C₃²	1	6	yes	complemented
49	#14578900	C₃²	1	2	yes	complemented
50	#14578899	C₃²	1	3	yes	complemented
51	#14578899	C₃²	1	3	yes	complemented
52	#14578898	C₃²	1	6	yes	complemented
53	#14578898	C₃²	1	6	yes	complemented
54	#14578897	C₃²	1	6	yes	complemented
55	#14578896	C₃²	1	3	yes	complemented
56	#14578896	C₃²	1	3	yes	complemented
57	#14578895	C₃²	1	6	yes	complemented
58	#14578895	C₃²	1	6	yes	complemented
59	#14578894	C₃²	1	6	yes	complemented
60	#14578893	C₃²	1	3	yes	complemented
61	#14578893	C₃²	1	3	yes	complemented
62	C₃₁×#737	C₃²	1	3	yes	complemented
63	C₃×3₋¹⁺²×C₃₁	C₃²	1	1	yes	characteristic, complemented
64	C₃₁×#733	C₃²	1	2	yes	complemented
65	C₃₁×#732	C₃²	1	1	yes	characteristic, complemented
66	C₃₁×#731	C₃²	1	3	yes	complemented
67	C₃₁×AΣL(1,𝔽₂₇)	C₃²	1	3	yes	complemented
68	C₃³×C₃₁	C₃³	1	1	yes	characteristic
69	C₃²×C₃₁⋊C₃	—	3	3	no	—
70	C₃²×C₃₁⋊C₃	—	3	2	no	—
71	C₃²×C₃₁⋊C₃	—	3	1	no	—
72	C₃²×C₃₁⋊C₃	—	9	3	no	—
73	C₃²×C₃₁⋊C₃	C₃³	1	1	yes	characteristic
74	C₃²×C₃₁⋊C₃	C₃³	1	1	yes	characteristic
75	C₃²×C₃₁⋊C₃	C₃³	1	1	yes	characteristic
76	3₋¹⁺²×C₃₁	—	3	3	no	—
77	3₋¹⁺²×C₃₁	—	3	6	no	—
78	3₋¹⁺²×C₃₁	C₃³	1	6	yes	—
79	H(3,𝔽₃)×C₃₁	C₃³	1	3	yes	—
80	C₃×C₉×C₃₁	—	3	3	no	—
81	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
82	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
83	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
84	C₃²⋊(C₃₁⋊C₃)	—	3	6	no	—
85	C₃²⋊(C₃₁⋊C₃)	C₃³	1	6	yes	—
86	#11719689	—	3	3	no	—
87	#11719689	—	3	3	no	—
88	#11719689	—	3	6	no	—
89	#11719689	—	3	6	no	—
90	#11719689	C₃³	1	6	yes	—
91	#11719689	C₃³	1	6	yes	—
92	C₃×C₃₁⋊C₉	—	3	3	no	—
93	C₃×C₃₁⋊C₉	—	3	3	no	—
94	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
95	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
96	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
97	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
98	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
99	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
100	#11719687	—	3	6	no	—
101	#11719687	—	3	3	no	—
102	#11719687	—	3	6	no	—
103	#11719687	—	3	3	no	—
104	#11719687	—	3	3	no	—
105	#11719687	—	3	6	no	—
106	#11719687	—	3	6	no	—
107	#11719686	—	3	6	no	—
108	#11719686	—	3	3	no	—
109	#11719686	—	3	6	no	—
110	#11719686	—	3	3	no	—
111	#11719686	—	3	3	no	—
112	#11719686	—	3	6	no	—
113	#11719686	—	3	6	no	—
114	C₉×C₃₁⋊C₃	—	3	6	no	—
115	C₉×C₃₁⋊C₃	—	3	2	no	—
116	C₉×C₃₁⋊C₃	—	3	2	no	—
117	C₉×C₃₁⋊C₃	—	3	1	no	—
118	C₉×C₃₁⋊C₃	—	3	1	no	—
119	C₉×C₃₁⋊C₃	—	3	6	no	—
120	C₉×C₃₁⋊C₃	—	3	6	no	—
121	C₉×C₃₁⋊C₃	—	3	6	no	—
122	C₉×C₃₁⋊C₃	H(3,𝔽₃)	1	3	yes	—
123	C₉×C₃₁⋊C₃	H(3,𝔽₃)	1	6	yes	—
124	C₃×#6913	—	31	1	no	maximal, Sylow
125	C₃²×C₃₁	—	3	3	no	—
126	C₃²×C₃₁	—	3	1	no	—
127	C₃²×C₃₁	—	9	1	no	—
128	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
129	C₃×C₃₁⋊C₃	—	3	3	no	—
130	C₃×C₃₁⋊C₃	—	3	6	no	—
131	C₃×C₃₁⋊C₃	—	3	1	no	—
132	C₃×C₃₁⋊C₃	—	3	1	no	—
133	C₃×C₃₁⋊C₃	—	9	6	no	—
134	C₃×C₃₁⋊C₃	—	9	9	no	—
135	C₃×C₃₁⋊C₃	—	9	3	no	—
136	C₃×C₃₁⋊C₃	—	9	1	no	—
137	C₃×C₃₁⋊C₃	—	9	1	no	—
138	C₃×C₃₁⋊C₃	C₃×H(3,𝔽₃)	1	2	yes	—
139	C₃×C₃₁⋊C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
140	C₉×C₃₁	—	3	6	no	—
141	C₉×C₃₁	—	3	1	no	—
142	C₉×C₃₁	—	3	1	no	—
143	C₉×C₃₁	C₃×H(3,𝔽₃)	1	3	yes	—
144	C₃₁⋊C₉	—	3	6	no	—
145	C₃₁⋊C₉	—	3	6	no	—
146	C₃₁⋊C₉	—	3	1	no	—
147	C₃₁⋊C₉	—	3	1	no	—
148	C₃₁⋊C₉	—	3	1	no	—
149	C₃₁⋊C₉	—	3	1	no	—
150	C₃₁⋊C₉	C₃×H(3,𝔽₃)	1	3	yes	—
151	C₃₁⋊C₉	C₃×H(3,𝔽₃)	1	3	yes	—
152	C₃×#737	—	31	3	no	—
153	C₃²×3₋¹⁺²	—	31	1	no	—
154	#6913	—	31	3	no	—
155	#6913	—	31	3	no	—
156	#6913	—	31	2	no	—
157	#6913	—	31	2	no	—
158	#6913	—	31	2	no	—
159	#6913	—	31	1	no	—
160	#6913	—	31	1	no	—
161	#6913	—	31	3	no	—
162	#6913	—	31	3	no	—
163	#6913	—	31	3	no	—
164	#6913	—	31	3	no	—
165	#6913	C₃₁⋊C₃	1	1	yes	characteristic, direct product factor
166	C₃×#733	—	31	2	no	—
167	C₃×#732	—	31	1	no	—
168	C₃×#731	—	31	3	no	—
169	C₃×AΣL(1,𝔽₂₇)	—	31	3	no	—
170	C₃×C₃₁	—	3	1	no	—
171	C₃×C₃₁	—	9	1	no	—
172	C₃×C₃₁	—	9	3	no	—
173	C₃×C₃₁	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
174	C₃₁⋊C₃	—	3	2	no	—
175	C₃₁⋊C₃	—	9	6	no	—
176	C₃₁⋊C₃	—	9	1	no	—
177	C₃₁⋊C₃	—	9	1	no	—
178	C₃₁⋊C₃	#6913	1	3	yes	direct product factor
179	C₃⁴	—	31	1	no	—
180	#737	—	31	6	no	—
181	#737	—	31	6	no	—
182	#737	—	31	3	no	—
183	#737	—	31	3	no	—
184	#737	—	31	6	no	—
185	#737	—	93	6	no	—
186	#737	—	93	6	no	—
187	#737	—	93	6	no	—
188	#737	C₃×C₃₁⋊C₃	1	3	yes	complemented
189	C₃×3₋¹⁺²	—	31	1	no	—
190	C₃×3₋¹⁺²	—	31	1	no	—
191	C₃×3₋¹⁺²	—	31	1	no	—
192	C₃×3₋¹⁺²	—	31	1	no	—
193	C₃×3₋¹⁺²	—	31	1	no	—
194	C₃×3₋¹⁺²	—	31	1	no	—
195	C₃×3₋¹⁺²	—	31	1	no	—
196	C₃×3₋¹⁺²	—	31	1	no	—
197	C₃×3₋¹⁺²	—	31	6	no	—
198	C₃×3₋¹⁺²	—	93	3	no	—
199	C₃×3₋¹⁺²	—	93	6	no	—
200	C₃×3₋¹⁺²	—	93	6	no	—
201	C₃×3₋¹⁺²	C₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented
202	C₃×H(3,𝔽₃)	—	31	3	no	—
203	C₃²×C₉	—	31	1	no	—
204	C₃²×C₉	—	31	1	no	—
205	C₃²×C₉	—	31	1	no	—
206	C₃²×C₉	—	93	3	no	—
207	#733	—	31	6	no	—
208	#733	—	31	6	no	—
209	#733	—	31	2	no	—
210	#733	—	31	2	no	—
211	#733	C₃×C₃₁⋊C₃	1	2	yes	complemented
212	#732	—	31	3	no	—
213	#732	—	31	3	no	—
214	#732	—	31	2	no	—
215	#732	C₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented
216	#731	—	31	6	no	—
217	#731	—	31	6	no	—
218	#731	—	31	3	no	—
219	#731	—	31	3	no	—
220	#731	—	31	6	no	—
221	#731	C₃×C₃₁⋊C₃	1	3	yes	complemented
222	AΣL(1,𝔽₂₇)	—	31	6	no	—
223	AΣL(1,𝔽₂₇)	—	31	6	no	—
224	AΣL(1,𝔽₂₇)	—	31	6	no	—
225	AΣL(1,𝔽₂₇)	—	31	3	no	—
226	AΣL(1,𝔽₂₇)	—	31	3	no	—
227	AΣL(1,𝔽₂₇)	C₃×C₃₁⋊C₃	1	3	yes	complemented
228	C₃₁	C₃×#6913	1	1	yes	characteristic, complemented, Sylow
229	C₃³	—	279	3	no	—
230	C₃³	—	31	1	no	—
231	C₃³	—	31	1	no	—
232	C₃³	—	31	1	no	—
233	C₃³	—	93	3	no	—
234	C₃³	—	93	1	no	—
235	C₃³	—	93	2	no	—
236	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
237	3₋¹⁺²	—	3	3	no	—
238	3₋¹⁺²	—	3	6	no	—
239	3₋¹⁺²	—	31	6	no	—
240	3₋¹⁺²	—	31	6	no	—
241	3₋¹⁺²	—	93	3	no	—
242	3₋¹⁺²	—	93	3	no	—
243	3₋¹⁺²	—	93	6	no	—
244	3₋¹⁺²	—	93	6	no	—
245	3₋¹⁺²	—	93	3	no	—
246	3₋¹⁺²	—	93	3	no	—
247	3₋¹⁺²	—	93	6	no	—
248	3₋¹⁺²	—	93	6	no	—
249	3₋¹⁺²	—	93	6	no	—
250	3₋¹⁺²	—	93	6	no	—
251	3₋¹⁺²	—	93	3	no	—
252	3₋¹⁺²	—	93	3	no	—
253	3₋¹⁺²	—	93	6	no	—
254	3₋¹⁺²	—	93	6	no	—
255	3₋¹⁺²	—	93	3	no	—
256	3₋¹⁺²	—	93	6	no	—
257	3₋¹⁺²	—	93	3	no	—
258	3₋¹⁺²	—	93	6	no	—
259	3₋¹⁺²	C₃²×C₃₁⋊C₃	1	6	yes	—
260	H(3,𝔽₃)	—	31	6	no	—
261	H(3,𝔽₃)	—	93	6	no	—
262	H(3,𝔽₃)	C₃²×C₃₁⋊C₃	1	3	yes	—
263	C₃×C₉	—	3	3	no	—
264	C₃×C₉	—	31	1	no	—
265	C₃×C₉	—	31	1	no	—
266	C₃×C₉	—	31	3	no	—
267	C₃×C₉	—	31	6	no	—
268	C₃×C₉	—	31	1	no	—
269	C₃×C₉	—	31	1	no	—
270	C₃×C₉	—	31	1	no	—
271	C₃×C₉	—	31	1	no	—
272	C₃×C₉	—	93	6	no	—
273	C₃×C₉	—	93	6	no	—
274	C₃×C₉	—	93	1	no	—
275	C₃×C₉	—	93	1	no	—
276	C₃×C₉	—	93	2	no	—
277	C₃×C₉	—	93	2	no	—
278	C₃×C₉	—	93	6	no	—
279	C₃×C₉	—	93	3	no	—
280	C₃×C₉	—	93	3	no	—
281	C₃×C₉	—	93	6	no	—
282	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
283	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
284	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
285	C₃²	—	3	3	no	—
286	C₃²	—	3	1	no	—
287	C₃²	—	9	1	no	—
288	C₃²	—	279	1	no	—
289	C₃²	—	279	1	no	—
290	C₃²	—	279	3	no	—
291	C₃²	—	279	9	no	—
292	C₃²	—	279	6	no	—
293	C₃²	—	31	1	no	—
294	C₃²	—	31	2	no	—
295	C₃²	—	93	3	no	—
296	C₃²	—	93	6	no	—
297	C₃²	—	93	1	no	—
298	C₃²	—	93	1	no	—
299	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
300	C₉	—	3	6	no	—
301	C₉	—	3	1	no	—
302	C₉	—	3	1	no	—
303	C₉	—	31	3	no	—
304	C₉	—	31	3	no	—
305	C₉	—	93	1	no	—
306	C₉	—	93	1	no	—
307	C₉	—	93	1	no	—
308	C₉	—	93	1	no	—
309	C₉	—	93	6	no	—
310	C₉	—	93	6	no	—
311	C₉	H(3,𝔽₃)×C₃₁⋊C₃	1	3	yes	—
312	C₃	—	3	1	no	—
313	C₃	—	9	1	no	—
314	C₃	—	9	3	no	—
315	C₃	—	279	1	no	—
316	C₃	—	279	1	no	—
317	C₃	—	279	6	no	—
318	C₃	—	31	3	no	—
319	C₃	—	93	2	no	—
320	C₃	C₃×H(3,𝔽₃)×C₃₁⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×H(3,𝔽₃)×C₃₁⋊C₃
commutator, cosocle	C₃²×C₃₁	C₃⁴
Frattini	C₃²	C₃³×C₃₁⋊C₃
Fitting	C₃₁×#6913	C₃
socle	C₃×C₃₁	C₃²×H(3,𝔽₃)
hypercenter	#6913	C₃₁⋊C₃
hypocenter	C₃₁	C₃×#6913
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ #6913

Complements

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

N	H ≅ G/N	type
1. #737×C₃₁⋊C₃	C₃	semidirect
2. C₃×3₋¹⁺²×C₃₁⋊C₃	C₃	semidirect
3. #21419421	C₃	semidirect
4. #21419420	C₃	semidirect
5. #21419419	C₃	semidirect
6. #21419418	C₃	semidirect
7. #21419417	C₃	semidirect
8. #21419416	C₃	semidirect
9. #21419415	C₃	semidirect
10. #21419414	C₃	semidirect
11. #21419413	C₃	semidirect
12. #21419412	C₃	semidirect
13. #21419411	C₃	semidirect
14. #733×C₃₁⋊C₃	C₃	semidirect
15. #732×C₃₁⋊C₃	C₃	semidirect
16. #731×C₃₁⋊C₃	C₃	semidirect
17. AΣL(1,𝔽₂₇)×C₃₁⋊C₃	C₃	semidirect
18. C₃₁×#6913	C₃	semidirect
22. #14578913	C₃²	semidirect
23. #14578913	C₃²	semidirect
25. #14578912	C₃²	semidirect
26. #14578911	C₃²	semidirect
27. #14578911	C₃²	semidirect
28. C₃×#11719689	C₃²	semidirect
29. C₃×#11719689	C₃²	semidirect
30. C₃×#11719686	C₃²	semidirect
31. C₃×#11719686	C₃²	semidirect
32. C₃×#11719686	C₃²	semidirect
33. C₃×#11719687	C₃²	semidirect
34. C₃×#11719687	C₃²	semidirect
35. C₃×#11719687	C₃²	semidirect
45. #14578902	C₃²	semidirect
46. #14578902	C₃²	semidirect
47. #14578901	C₃²	semidirect
48. #14578901	C₃²	semidirect
49. #14578900	C₃²	semidirect
50. #14578899	C₃²	semidirect
51. #14578899	C₃²	semidirect
52. #14578898	C₃²	semidirect
53. #14578898	C₃²	semidirect
54. #14578897	C₃²	semidirect
55. #14578896	C₃²	semidirect
56. #14578896	C₃²	semidirect
57. #14578895	C₃²	semidirect
58. #14578895	C₃²	semidirect
59. #14578894	C₃²	semidirect
60. #14578893	C₃²	semidirect
61. #14578893	C₃²	semidirect
62. C₃₁×#737	C₃²	semidirect
63. C₃×3₋¹⁺²×C₃₁	C₃²	semidirect
64. C₃₁×#733	C₃²	semidirect
65. C₃₁×#732	C₃²	semidirect
66. C₃₁×#731	C₃²	semidirect
67. C₃₁×AΣL(1,𝔽₂₇)	C₃²	semidirect
165. #6913	C₃₁⋊C₃	direct (N × H)
178. C₃₁⋊C₃	#6913	direct (N × H)
188. #737	C₃×C₃₁⋊C₃	semidirect
201. C₃×3₋¹⁺²	C₃×C₃₁⋊C₃	semidirect
211. #733	C₃×C₃₁⋊C₃	semidirect
215. #732	C₃×C₃₁⋊C₃	semidirect
221. #731	C₃×C₃₁⋊C₃	semidirect
227. AΣL(1,𝔽₂₇)	C₃×C₃₁⋊C₃	semidirect
228. C₃₁	C₃×#6913	semidirect

Maximal subgroups

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

subgroup	index
#737×C₃₁⋊C₃	3
C₃×3₋¹⁺²×C₃₁⋊C₃	3
#21419421	3
#21419420	3
#21419419	3
#21419418	3
#21419417	3
#21419416	3
#21419415	3
#21419414	3
#21419413	3
#21419412	3
#21419411	3
#733×C₃₁⋊C₃	3
#732×C₃₁⋊C₃	3
#731×C₃₁⋊C₃	3
AΣL(1,𝔽₂₇)×C₃₁⋊C₃	3
C₃₁×#6913	3
C₃×#6913	31

Subgroups — C31⋊C3×#6913 #36382920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁⋊C₃×#6913 #36382920