Subgroups — C₃₁×#10160 #21466919

Order 7936.

578 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3141	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1965	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1492	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#532	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
11	C₃₁×#493	—	2	1	no	—
12	C₃₁×#489	—	2	2	no	—
13	C₃₁×#487	—	2	1	no	—
14	C₃₁×#486	—	2	2	no	—
15	C₃₁×#486	—	2	1	no	—
16	C₃₁×#486	—	2	1	no	—
17	C₃₁×#486	—	2	2	no	—
18	C₃₁×#459	—	2	2	no	—
19	C₃₁×#447	—	2	2	no	—
20	C₃₁×#431	C₂²	1	1	yes	characteristic
21	C₃₁×#423	C₂²	1	1	yes	characteristic
22	C₃₁×#417	—	2	2	no	—
23	C₂×C₃₁×#102	—	2	2	no	—
24	C₂×C₃₁×#100	—	2	2	no	—
25	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#333	—	2	2	no	—
28	C₃₁×D₄⋊C₈	—	2	2	no	—
29	C₂×#41×C₃₁	—	2	1	no	—
30	C₂²×Q₈×C₃₁	—	2	1	no	—
31	C₂²×D₄×C₃₁	—	2	1	no	—
32	C₂²×D₄×C₃₁	—	2	1	no	—
33	C₂²×D₄×C₃₁	D₄	1	2	yes	—
34	C₂×D₈×C₃₁	—	2	4	no	—
35	C₂×M₁₆×C₃₁	—	2	1	no	—
36	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
37	C₂²×C₈×C₃₁	—	2	1	no	—
38	C₃₁×C₄⋊Q₈	—	2	1	no	—
39	C₃₁×C₄⋊Q₈	—	2	1	no	—
40	C₃₁×C₄⋊Q₈	—	2	1	no	—
41	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
42	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
43	C₃₁×#127	—	2	1	no	—
44	C₃₁×#127	—	2	1	no	—
45	C₃₁×#127	—	2	1	no	—
46	C₃₁×#127	—	2	1	no	—
47	C₃₁×#127	D₄	1	1	yes	characteristic
48	C₃₁×#124	—	2	1	no	—
49	C₃₁×#121	—	2	1	no	—
50	C₃₁×#121	—	2	1	no	—
51	C₄×D₄×C₃₁	—	2	1	no	—
52	C₄×D₄×C₃₁	—	2	1	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×C₄⋊C₈	—	2	1	no	—
57	C₃₁×C₄⋊C₈	—	2	1	no	—
58	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
59	C₃₁×#103	—	2	4	no	—
60	C₃₁×#102	—	2	8	no	—
61	C₃₁×#102	—	2	4	no	—
62	C₃₁×#102	—	2	4	no	—
63	C₃₁×#100	—	4	4	no	—
64	C₃₁×C₂²⋊C₈	—	2	1	no	—
65	C₃₁×#97	—	2	1	no	—
66	C₄×C₈×C₃₁	—	2	1	no	—
67	C₄×C₈×C₃₁	—	2	1	no	—
68	C₄×C₈×C₃₁	—	2	1	no	—
69	C₄×C₈×C₃₁	D₄	1	2	yes	—
70	C₂⁴×C₃₁	—	2	2	no	—
71	#41×C₃₁	—	4	2	no	—
72	#41×C₃₁	—	4	2	no	—
73	C₂×Q₈×C₃₁	—	2	1	no	—
74	C₂×Q₈×C₃₁	—	2	2	no	—
75	C₂×Q₈×C₃₁	—	4	1	no	—
76	C₂×Q₈×C₃₁	—	8	2	no	—
77	C₂×D₄×C₃₁	—	2	1	no	—
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	—	2	2	no	—
80	C₂×D₄×C₃₁	—	2	1	no	—
81	C₂×D₄×C₃₁	—	2	1	no	—
82	C₂×D₄×C₃₁	—	4	4	no	—
83	C₂×D₄×C₃₁	—	4	1	no	—
84	C₂×D₄×C₃₁	—	4	1	no	—
85	C₂×D₄×C₃₁	—	4	1	no	—
86	C₂×D₄×C₃₁	—	4	1	no	—
87	C₂×D₄×C₃₁	—	4	1	no	—
88	C₂×D₄×C₃₁	—	4	2	no	—
89	C₂×D₄×C₃₁	—	4	2	no	—
90	C₂×D₄×C₃₁	—	8	2	no	—
91	C₂²×C₄×C₃₁	—	4	1	no	—
92	C₂²×C₄×C₃₁	—	4	1	no	—
93	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
96	D₈×C₃₁	—	8	4	no	—
97	M₁₆×C₃₁	—	2	2	no	—
98	M₁₆×C₃₁	—	2	2	no	—
99	M₁₆×C₃₁	—	4	2	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	4	2	no	—
104	C₂×C₈×C₃₁	—	4	1	no	—
105	C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₄⋊C₄×C₃₁	—	2	1	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	2	1	no	—
109	C₄⋊C₄×C₃₁	—	4	1	no	—
110	C₄⋊C₄×C₃₁	—	4	1	no	—
111	C₂²⋊C₄×C₃₁	—	4	1	no	—
112	C₂²⋊C₄×C₃₁	—	4	1	no	—
113	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
117	#10160	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂³×C₃₁	—	4	1	no	—
119	C₂³×C₃₁	—	4	2	no	—
120	C₂³×C₃₁	—	4	4	no	—
121	C₂³×C₃₁	—	8	2	no	—
122	C₂³×C₃₁	—	8	2	no	—
123	C₂³×C₃₁	#121	1	1	yes	characteristic
124	Q₈×C₃₁	—	4	2	no	—
125	Q₈×C₃₁	—	8	2	no	—
126	Q₈×C₃₁	—	8	2	no	—
127	D₄×C₃₁	—	4	4	no	—
128	D₄×C₃₁	—	4	2	no	—
129	D₄×C₃₁	—	4	2	no	—
130	D₄×C₃₁	—	4	4	no	—
131	D₄×C₃₁	—	4	2	no	—
132	D₄×C₃₁	—	8	2	no	—
133	D₄×C₃₁	—	8	2	no	—
134	D₄×C₃₁	—	8	2	no	—
135	D₄×C₃₁	—	8	2	no	—
136	D₄×C₃₁	—	8	1	no	—
137	D₄×C₃₁	—	8	1	no	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	4	1	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	—	8	2	no	—
143	C₂×C₄×C₃₁	—	8	1	no	—
144	C₂×C₄×C₃₁	—	8	1	no	—
145	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
146	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
147	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
148	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
149	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
150	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
151	C₂×C₄×C₃₁	#135	1	2	yes	—
152	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
153	C₈×C₃₁	—	4	2	no	—
154	C₈×C₃₁	—	4	2	no	—
155	C₈×C₃₁	—	4	2	no	—
156	C₈×C₃₁	—	8	1	no	—
157	#3205	C₂×C₃₁	1	1	yes	characteristic
158	#3141	C₂×C₃₁	1	1	yes	characteristic, complemented
159	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
160	#1965	C₂×C₃₁	1	1	yes	characteristic, complemented
161	#1548	C₂×C₃₁	1	1	yes	characteristic, complemented
162	#1492	C₂×C₃₁	1	1	yes	characteristic, complemented
163	#1278	C₂×C₃₁	1	1	yes	characteristic, complemented
164	C₂²×C₃₁	—	2	1	no	—
165	C₂²×C₃₁	—	2	1	no	—
166	C₂²×C₃₁	—	8	2	no	—
167	C₂²×C₃₁	—	8	1	no	—
168	C₂²×C₃₁	—	8	1	no	—
169	C₂²×C₃₁	—	8	2	no	—
170	C₂²×C₃₁	—	8	1	no	—
171	C₂²×C₃₁	—	8	2	no	—
172	C₂²×C₃₁	—	16	4	no	—
173	C₂²×C₃₁	#392	1	1	yes	characteristic
174	C₂²×C₃₁	#463	1	2	yes	—
175	C₄×C₃₁	—	2	1	no	—
176	C₄×C₃₁	—	2	1	no	—
177	C₄×C₃₁	—	8	1	no	—
178	C₄×C₃₁	—	8	2	no	—
179	C₄×C₃₁	#449	1	2	yes	—
180	C₄×C₃₁	#455	1	2	yes	—
181	C₄×C₃₁	#472	1	2	yes	—
182	C₄×C₃₁	#495	1	1	yes	characteristic
183	C₄×C₃₁	#496	1	1	yes	characteristic
184	#532	C₂²×C₃₁	1	1	yes	characteristic
185	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
186	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
187	#493	—	2	1	no	—
188	#489	—	2	2	no	—
189	#487	—	2	1	no	—
190	#486	—	2	2	no	—
191	#486	—	2	1	no	—
192	#486	—	2	1	no	—
193	#486	—	2	2	no	—
194	#459	—	2	2	no	—
195	#447	—	2	2	no	—
196	#431	C₂²×C₃₁	1	1	yes	characteristic
197	#423	C₂²×C₃₁	1	1	yes	characteristic
198	#417	—	2	2	no	—
199	C₂×#102	—	2	2	no	—
200	C₂×#100	—	2	2	no	—
201	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
202	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
203	#333	—	2	2	no	—
204	D₄⋊C₈	—	2	2	no	—
205	C₂×C₃₁	—	2	2	no	—
206	C₂×C₃₁	—	16	1	no	—
207	C₂×C₃₁	—	16	2	no	—
208	C₂×C₃₁	#2004	1	1	yes	characteristic
209	C₂×C₃₁	#2012	1	1	yes	characteristic
210	C₂×C₃₁	#2018	1	1	yes	characteristic
211	C₂×#41	—	2	1	no	—
212	C₂²×Q₈	—	2	1	no	—
213	C₂²×D₄	—	2	1	no	—
214	C₂²×D₄	—	2	1	no	—
215	C₂²×D₄	D₄×C₃₁	1	2	yes	—
216	C₂×D₈	—	2	4	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
219	C₂²×C₈	—	2	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
224	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
225	#127	—	2	1	no	—
226	#127	—	2	1	no	—
227	#127	—	2	1	no	—
228	#127	—	2	1	no	—
229	#127	D₄×C₃₁	1	1	yes	characteristic
230	#124	—	2	1	no	—
231	#121	—	2	1	no	—
232	#121	—	2	1	no	—
233	C₄×D₄	—	2	1	no	—
234	C₄×D₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
241	#103	—	2	4	no	—
242	#102	—	2	8	no	—
243	#102	—	2	4	no	—
244	#102	—	2	4	no	—
245	#100	—	4	4	no	—
246	C₂²⋊C₈	—	2	1	no	—
247	#97	—	2	1	no	—
248	C₄×C₈	—	2	1	no	—
249	C₄×C₈	—	2	1	no	—
250	C₄×C₈	—	2	1	no	—
251	C₄×C₈	D₄×C₃₁	1	2	yes	—
252	C₃₁	#10160	1	1	yes	characteristic, direct product factor, Sylow
253	C₂⁴	—	2	2	no	—
254	#41	—	4	2	no	—
255	#41	—	4	2	no	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	8	2	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	—	4	4	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	—	4	2	no	—
272	C₂×D₄	—	4	2	no	—
273	C₂×D₄	—	8	2	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	—	4	1	no	—
276	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
277	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
278	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
279	D₈	—	8	4	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	2	2	no	—
282	M₁₆	—	4	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	—	4	1	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	4	1	no	—
296	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
297	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
298	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
299	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
300	C₂³	—	4	1	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	4	no	—
303	C₂³	—	8	2	no	—
304	C₂³	—	8	2	no	—
305	C₂³	C₃₁×#121	1	1	yes	characteristic
306	Q₈	—	4	2	no	—
307	Q₈	—	8	2	no	—
308	Q₈	—	8	2	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	4	no	—
313	D₄	—	4	2	no	—
314	D₄	—	8	2	no	—
315	D₄	—	8	2	no	—
316	D₄	—	8	2	no	—
317	D₄	—	8	2	no	—
318	D₄	—	8	1	no	—
319	D₄	—	8	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	8	1	no	—
326	C₂×C₄	—	8	1	no	—
327	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
328	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
329	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
330	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
331	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
332	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
333	C₂×C₄	C₃₁×#135	1	2	yes	—
334	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
335	C₈	—	4	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₈	—	8	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	1	no	—
343	C₂²	—	8	1	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	1	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	16	4	no	—
348	C₂²	C₃₁×#392	1	1	yes	characteristic
349	C₂²	C₃₁×#463	1	2	yes	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	8	1	no	—
353	C₄	—	8	2	no	—
354	C₄	C₃₁×#449	1	2	yes	—
355	C₄	C₃₁×#455	1	2	yes	—
356	C₄	C₃₁×#472	1	2	yes	—
357	C₄	C₃₁×#495	1	1	yes	characteristic
358	C₄	C₃₁×#496	1	1	yes	characteristic
359	C₂	—	2	2	no	—
360	C₂	—	16	2	no	—
361	C₂	—	16	1	no	—
362	C₂	C₃₁×#2004	1	1	yes	characteristic
363	C₂	C₃₁×#2012	1	1	yes	characteristic
364	C₂	C₃₁×#2018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₃₁×#3205	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3141	C₂	semidirect
3. C₂×C₃₁×#486	C₂	semidirect
4. C₃₁×#1965	C₂	semidirect
5. C₃₁×#1548	C₂	semidirect
6. C₃₁×#1492	C₂	semidirect
7. C₃₁×#1278	C₂	semidirect
117. #10160	C₃₁	direct (N × H)
158. #3141	C₂×C₃₁	semidirect
159. C₂×#486	C₂×C₃₁	semidirect
160. #1965	C₂×C₃₁	semidirect
161. #1548	C₂×C₃₁	semidirect
162. #1492	C₂×C₃₁	semidirect
163. #1278	C₂×C₃₁	semidirect
252. C₃₁	#10160	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3205	2
C₃₁×#3141	2
C₂×C₃₁×#486	2
C₃₁×#1965	2
C₃₁×#1548	2
C₃₁×#1492	2
C₃₁×#1278	2
#10160	31

Subgroups — C31×#10160 #21466919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10160 #21466919