Subgroups — C₃₁×#11886 #21468645

Order 7936.

470 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	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1347	C₂	1	4	yes	maximal, complemented
5	C₂²×D₈×C₃₁	C₄	1	1	yes	characteristic, complemented
6	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×#121	—	2	1	no	—
8	C₂²×C₁₆×C₃₁	—	2	1	no	—
9	C₃₁×#466	—	2	1	no	—
10	C₃₁×#466	—	2	1	no	—
11	C₃₁×#466	C₄	1	2	yes	complemented
12	C₃₁×#466	C₂²	1	2	yes	—
13	C₂×C₃₁×#107	—	2	1	no	—
14	C₂×C₃₁×#107	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃₁×#102	—	2	1	no	—
16	C₃₁×#380	—	2	1	no	—
17	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
18	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
19	C₃₁×#340	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₃₁	—	2	1	no	—
21	C₂²×D₄×C₃₁	—	4	1	no	—
22	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
23	C₂×D₈×C₃₁	—	2	1	no	—
24	C₂×D₈×C₃₁	—	4	2	no	—
25	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₃₁	—	4	2	no	—
27	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
33	C₃₁×#121	—	2	2	no	—
34	C₃₁×#121	—	2	2	no	—
35	C₃₁×#121	—	4	1	no	—
36	C₃₁×#121	—	4	1	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
40	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
41	C₂×C₁₆×C₃₁	—	2	2	no	—
42	C₂×C₁₆×C₃₁	—	4	2	no	—
43	C₃₁×#107	—	2	1	no	—
44	C₃₁×#107	—	4	2	no	—
45	C₃₁×#107	C₂×C₄	1	2	yes	—
46	C₃₁×#102	—	2	2	no	—
47	C₃₁×#102	—	2	2	no	—
48	C₃₁×#95	—	4	1	no	—
49	C₂⁴×C₃₁	—	4	1	no	—
50	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
51	C₂×D₄×C₃₁	—	2	2	no	—
52	C₂×D₄×C₃₁	—	4	2	no	—
53	C₂×D₄×C₃₁	—	4	2	no	—
54	C₂×D₄×C₃₁	—	4	2	no	—
55	C₂×D₄×C₃₁	—	4	2	no	—
56	C₂×D₄×C₃₁	—	4	1	no	—
57	C₂×D₄×C₃₁	—	8	2	no	—
58	C₂×D₄×C₃₁	—	8	2	no	—
59	C₂²×C₄×C₃₁	—	4	1	no	—
60	C₂²×C₄×C₃₁	—	4	1	no	—
61	C₂²×C₄×C₃₁	—	4	1	no	—
62	C₂²×C₄×C₃₁	—	4	2	no	—
63	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	D₈×C₃₁	—	4	2	no	—
70	D₈×C₃₁	—	4	2	no	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	4	1	no	—
75	C₂×C₈×C₃₁	—	4	1	no	—
76	C₂×C₈×C₃₁	—	4	1	no	—
77	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₃₁	—	2	2	no	—
82	C₄⋊C₄×C₃₁	—	4	2	no	—
83	C₄⋊C₄×C₃₁	—	4	2	no	—
84	C₄⋊C₄×C₃₁	—	4	1	no	—
85	C₂²⋊C₄×C₃₁	—	4	2	no	—
86	C₂²⋊C₄×C₃₁	—	4	2	no	—
87	C₂²⋊C₄×C₃₁	—	8	2	no	—
88	C₁₆×C₃₁	—	4	2	no	—
89	#11886	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	4	1	no	—
91	C₂³×C₃₁	—	4	1	no	—
92	C₂³×C₃₁	—	4	2	no	—
93	C₂³×C₃₁	—	8	2	no	—
94	C₂³×C₃₁	—	8	2	no	—
95	C₂³×C₃₁	—	8	2	no	—
96	C₂³×C₃₁	#102	1	1	yes	characteristic
97	C₂³×C₃₁	#102	1	1	yes	characteristic
98	C₂³×C₃₁	D₁₆	1	2	yes	—
99	C₂³×C₃₁	SD₃₂	1	2	yes	—
100	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
101	D₄×C₃₁	—	4	2	no	—
102	D₄×C₃₁	—	4	2	no	—
103	D₄×C₃₁	—	8	2	no	—
104	D₄×C₃₁	—	8	4	no	—
105	D₄×C₃₁	—	8	2	no	—
106	D₄×C₃₁	—	8	2	no	—
107	C₂×C₄×C₃₁	—	2	1	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	2	1	no	—
110	C₂×C₄×C₃₁	—	2	1	no	—
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	4	1	no	—
114	C₂×C₄×C₃₁	—	4	1	no	—
115	C₂×C₄×C₃₁	—	4	1	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	—	8	2	no	—
118	C₂×C₄×C₃₁	—	8	2	no	—
119	C₂×C₄×C₃₁	—	8	2	no	—
120	C₂×C₄×C₃₁	—	8	2	no	—
121	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
122	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
125	C₈×C₃₁	—	2	1	no	—
126	C₈×C₃₁	—	2	1	no	—
127	C₈×C₃₁	—	2	1	no	—
128	C₈×C₃₁	#99	1	2	yes	—
129	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
130	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
131	#1808	C₂×C₃₁	1	1	yes	characteristic, complemented
132	#1347	C₂×C₃₁	1	4	yes	complemented
133	C₂²×C₃₁	—	2	1	no	—
134	C₂²×C₃₁	—	2	1	no	—
135	C₂²×C₃₁	—	2	1	no	—
136	C₂²×C₃₁	—	2	1	no	—
137	C₂²×C₃₁	—	2	1	no	—
138	C₂²×C₃₁	—	2	1	no	—
139	C₂²×C₃₁	—	4	2	no	—
140	C₂²×C₃₁	—	4	2	no	—
141	C₂²×C₃₁	—	8	2	no	—
142	C₂²×C₃₁	—	8	2	no	—
143	C₂²×C₃₁	—	8	2	no	—
144	C₂²×C₃₁	—	16	4	no	—
145	C₂²×C₃₁	#357	1	2	yes	—
146	C₂²×C₃₁	#357	1	2	yes	—
147	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
148	C₂²×C₃₁	C₂×D₁₆	1	1	yes	characteristic
149	C₂²×C₃₁	C₂×SD₃₂	1	1	yes	characteristic
150	C₄×C₃₁	—	2	1	no	—
151	C₄×C₃₁	—	2	1	no	—
152	C₄×C₃₁	—	2	1	no	—
153	C₄×C₃₁	—	8	2	no	—
154	C₄×C₃₁	—	16	2	no	—
155	C₄×C₃₁	C₂×#99	1	1	yes	characteristic
156	C₄×C₃₁	#421	1	1	yes	characteristic
157	C₂²×D₈	C₄×C₃₁	1	1	yes	characteristic, complemented
158	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
159	C₂×#121	—	2	1	no	—
160	C₂²×C₁₆	—	2	1	no	—
161	#466	—	2	1	no	—
162	#466	—	2	1	no	—
163	#466	C₄×C₃₁	1	2	yes	complemented
164	#466	C₂²×C₃₁	1	2	yes	—
165	C₂×#107	—	2	1	no	—
166	C₂×#107	C₄×C₃₁	1	1	yes	characteristic, complemented
167	C₂×#102	—	2	1	no	—
168	#380	—	2	1	no	—
169	C₂²⋊C₁₆	—	2	1	no	—
170	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
171	#340	C₂²×C₃₁	1	2	yes	complemented
172	C₂×C₃₁	—	2	1	no	—
173	C₂×C₃₁	—	2	2	no	—
174	C₂×C₃₁	—	2	2	no	—
175	C₂×C₃₁	—	2	1	no	—
176	C₂×C₃₁	—	16	2	no	—
177	C₂×C₃₁	#1519	1	1	yes	characteristic
178	C₂×C₃₁	C₂×#357	1	1	yes	characteristic
179	C₂×C₃₁	#2145	1	1	yes	characteristic
180	C₂²×D₄	—	2	1	no	—
181	C₂²×D₄	—	4	1	no	—
182	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
183	C₂×D₈	—	2	1	no	—
184	C₂×D₈	—	4	2	no	—
185	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
186	C₂²×C₈	—	4	2	no	—
187	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
188	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
189	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
190	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
193	#121	—	2	2	no	—
194	#121	—	2	2	no	—
195	#121	—	4	1	no	—
196	#121	—	4	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₂²⋊C₄	—	4	1	no	—
200	C₂×C₂²⋊C₄	—	4	1	no	—
201	C₂×C₁₆	—	2	2	no	—
202	C₂×C₁₆	—	4	2	no	—
203	#107	—	2	1	no	—
204	#107	—	4	2	no	—
205	#107	C₂×C₄×C₃₁	1	2	yes	—
206	#102	—	2	2	no	—
207	#102	—	2	2	no	—
208	#95	—	4	1	no	—
209	C₃₁	#11886	1	1	yes	characteristic, direct product factor, Sylow
210	C₂⁴	—	4	1	no	—
211	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂×D₄	—	8	2	no	—
219	C₂×D₄	—	8	2	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
225	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
226	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
227	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
228	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
230	D₈	—	4	2	no	—
231	D₈	—	4	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
239	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
240	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
241	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	1	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	8	2	no	—
249	C₁₆	—	4	2	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	8	2	no	—
254	C₂³	—	8	2	no	—
255	C₂³	—	8	2	no	—
256	C₂³	C₃₁×#102	1	1	yes	characteristic
257	C₂³	C₃₁×#102	1	1	yes	characteristic
258	C₂³	C₃₁×D₁₆	1	2	yes	—
259	C₂³	C₃₁×SD₃₂	1	2	yes	—
260	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	8	2	no	—
264	D₄	—	8	2	no	—
265	D₄	—	8	2	no	—
266	D₄	—	8	4	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
282	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
283	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
284	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	C₃₁×#99	1	2	yes	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	16	4	no	—
301	C₂²	C₃₁×#357	1	2	yes	—
302	C₂²	C₃₁×#357	1	2	yes	—
303	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
304	C₂²	C₂×C₃₁×D₁₆	1	1	yes	characteristic
305	C₂²	C₂×C₃₁×SD₃₂	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	16	2	no	—
311	C₄	C₂×C₃₁×#99	1	1	yes	characteristic
312	C₄	C₃₁×#421	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	16	2	no	—
318	C₂	C₃₁×#1519	1	1	yes	characteristic
319	C₂	C₂×C₃₁×#357	1	1	yes	characteristic
320	C₂	C₃₁×#2145	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×C₂²⋊C₁₆	C₂	semidirect
3. C₃₁×#1808	C₂	semidirect
4. C₃₁×#1347	C₂	semidirect
5. C₂²×D₈×C₃₁	C₄	semidirect
11. C₃₁×#466	C₄	semidirect
14. C₂×C₃₁×#107	C₄	semidirect
18. C₃₁×C₂²⋊C₁₆	C₂²	semidirect
19. C₃₁×#340	C₂²	semidirect
89. #11886	C₃₁	direct (N × H)
130. C₂×C₂²⋊C₁₆	C₂×C₃₁	semidirect
131. #1808	C₂×C₃₁	semidirect
132. #1347	C₂×C₃₁	semidirect
157. C₂²×D₈	C₄×C₃₁	semidirect
163. #466	C₄×C₃₁	semidirect
166. C₂×#107	C₄×C₃₁	semidirect
170. C₂²⋊C₁₆	C₂²×C₃₁	semidirect
171. #340	C₂²×C₃₁	semidirect
209. C₃₁	#11886	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×C₃₁×#466	2
C₂×C₃₁×C₂²⋊C₁₆	2
C₃₁×#1808	2
C₃₁×#1347	2
#11886	31

Subgroups — C31×#11886 #21468645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11886 #21468645