Subgroups — C₃₁×#19961 #21476720

Order 7936.

638 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3397	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3304	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1686	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1681	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1650	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1621	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1591	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#556	C₂²	1	2	yes	—
11	C₃₁×#550	C₂²	1	2	yes	—
12	C₃₁×#500	C₂²	1	1	yes	characteristic
13	C₃₁×#496	—	2	2	no	—
14	C₃₁×#493	C₂²	1	1	yes	characteristic
15	C₃₁×#492	C₂²	1	2	yes	—
16	C₃₁×#486	C₂²	1	2	yes	—
17	C₃₁×#482	—	2	2	no	—
18	C₃₁×#475	C₂²	1	2	yes	—
19	C₃₁×#465	—	2	2	no	—
20	C₃₁×#463	—	2	2	no	—
21	C₃₁×#461	C₂²	1	2	yes	—
22	C₃₁×#460	—	2	2	no	—
23	C₃₁×#460	C₂²	1	2	yes	—
24	C₃₁×#459	C₂²	1	2	yes	—
25	C₃₁×#452	—	2	2	no	—
26	C₃₁×#446	C₂²	1	2	yes	—
27	C₃₁×#440	—	2	2	no	—
28	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
29	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₈×D₄×C₃₁	C₂²	1	2	yes	—
31	C₃₁×#416	—	2	2	no	—
32	C₃₁×#335	C₂²	1	1	yes	characteristic
33	C₃₁×#335	C₂²	1	1	yes	characteristic
34	C₃₁×#332	C₂²	1	2	yes	—
35	C₃₁×#331	C₂²	1	2	yes	—
36	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×#41×C₃₁	—	2	2	no	—
40	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
41	C₂×SD₁₆×C₃₁	—	2	2	no	—
42	C₂×SD₁₆×C₃₁	—	2	2	no	—
43	C₂×SD₁₆×C₃₁	—	2	2	no	—
44	C₂×SD₁₆×C₃₁	—	2	2	no	—
45	C₂×D₈×C₃₁	—	2	2	no	—
46	C₂×D₈×C₃₁	—	4	1	no	—
47	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	—	2	2	no	—
49	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₃₁×#127	—	2	2	no	—
51	C₃₁×#127	C₂³	1	2	yes	—
52	C₃₁×#126	—	4	2	no	—
53	C₃₁×#124	—	2	2	no	—
54	C₃₁×C₂²⋊Q₈	—	2	2	no	—
55	C₃₁×#121	—	2	2	no	—
56	C₃₁×#121	—	4	2	no	—
57	C₄×Q₈×C₃₁	—	2	2	no	—
58	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
59	C₄×D₄×C₃₁	—	2	2	no	—
60	C₄×D₄×C₃₁	C₂³	1	2	yes	—
61	C₃₁×#117	—	2	2	no	—
62	C₃₁×#107	—	4	1	no	—
63	C₃₁×#107	D₄	1	1	yes	characteristic
64	C₃₁×#107	D₄	1	1	yes	characteristic
65	C₃₁×#106	—	2	2	no	—
66	C₃₁×C₄⋊C₈	—	2	2	no	—
67	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
68	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
69	C₃₁×#103	—	2	2	no	—
70	C₃₁×#103	—	2	2	no	—
71	C₃₁×#103	D₄	1	2	yes	—
72	C₃₁×#102	—	2	2	no	—
73	C₃₁×#102	—	2	2	no	—
74	C₃₁×#102	—	2	2	no	—
75	C₃₁×#102	—	2	2	no	—
76	C₃₁×#102	D₄	1	2	yes	—
77	C₃₁×C₂²⋊C₈	—	2	2	no	—
78	C₃₁×#97	—	2	2	no	—
79	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	#41×C₃₁	—	4	4	no	—
83	#41×C₃₁	—	8	2	no	—
84	C₂×Q₈×C₃₁	—	2	2	no	—
85	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	4	2	no	—
90	C₂×D₄×C₃₁	—	4	2	no	—
91	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	4	2	no	—
94	Q₁₆×C₃₁	—	4	1	no	—
95	SD₁₆×C₃₁	—	4	4	no	—
96	SD₁₆×C₃₁	—	8	2	no	—
97	SD₁₆×C₃₁	—	8	2	no	—
98	SD₁₆×C₃₁	—	8	2	no	—
99	D₈×C₃₁	—	2	2	no	—
100	D₈×C₃₁	—	8	2	no	—
101	D₈×C₃₁	—	8	2	no	—
102	C₂×C₈×C₃₁	—	2	1	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	8	2	no	—
106	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₄⋊C₄×C₃₁	—	2	2	no	—
113	C₄⋊C₄×C₃₁	—	4	2	no	—
114	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
115	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
116	C₂²⋊C₄×C₃₁	—	2	2	no	—
117	C₂²⋊C₄×C₃₁	—	4	2	no	—
118	C₂²⋊C₄×C₃₁	—	4	2	no	—
119	C₂²⋊C₄×C₃₁	—	4	2	no	—
120	C₄²×C₃₁	—	2	2	no	—
121	C₄²×C₃₁	—	4	2	no	—
122	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
123	#19961	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂³×C₃₁	—	2	2	no	—
125	C₂³×C₃₁	—	4	2	no	—
126	Q₈×C₃₁	—	2	2	no	—
127	Q₈×C₃₁	—	2	2	no	—
128	Q₈×C₃₁	—	4	2	no	—
129	Q₈×C₃₁	—	4	2	no	—
130	D₄×C₃₁	—	2	2	no	—
131	D₄×C₃₁	—	2	2	no	—
132	D₄×C₃₁	—	4	2	no	—
133	D₄×C₃₁	—	4	2	no	—
134	D₄×C₃₁	—	4	2	no	—
135	D₄×C₃₁	—	4	2	no	—
136	D₄×C₃₁	—	8	2	no	—
137	D₄×C₃₁	—	8	4	no	—
138	D₄×C₃₁	—	16	2	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	4	2	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	8	4	no	—
146	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
149	C₈×C₃₁	—	2	2	no	—
150	C₈×C₃₁	—	2	1	no	—
151	C₈×C₃₁	—	2	1	no	—
152	C₈×C₃₁	—	2	1	no	—
153	C₈×C₃₁	—	4	2	no	—
154	C₈×C₃₁	—	4	2	no	—
155	C₈×C₃₁	#135	1	2	yes	—
156	#3397	C₂×C₃₁	1	2	yes	complemented
157	#3304	C₂×C₃₁	1	2	yes	complemented
158	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
159	#1686	C₂×C₃₁	1	2	yes	complemented
160	#1681	C₂×C₃₁	1	2	yes	complemented
161	#1650	C₂×C₃₁	1	2	yes	complemented
162	#1621	C₂×C₃₁	1	2	yes	complemented
163	#1591	C₂×C₃₁	1	1	yes	characteristic, complemented
164	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
165	C₂²×C₃₁	—	4	2	no	—
166	C₂²×C₃₁	—	4	2	no	—
167	C₂²×C₃₁	—	4	2	no	—
168	C₂²×C₃₁	—	8	2	no	—
169	C₂²×C₃₁	—	8	2	no	—
170	C₂²×C₃₁	—	8	2	no	—
171	C₂²×C₃₁	D₄²	1	1	yes	characteristic
172	C₄×C₃₁	—	2	1	no	—
173	C₄×C₃₁	—	4	2	no	—
174	C₄×C₃₁	—	4	2	no	—
175	C₄×C₃₁	—	4	2	no	—
176	C₄×C₃₁	—	8	2	no	—
177	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
178	C₄×C₃₁	#575	1	1	yes	characteristic
179	C₄×C₃₁	#576	1	1	yes	characteristic
180	C₄×C₃₁	#576	1	1	yes	characteristic
181	#556	C₂²×C₃₁	1	2	yes	—
182	#550	C₂²×C₃₁	1	2	yes	—
183	#500	C₂²×C₃₁	1	1	yes	characteristic
184	#496	—	2	2	no	—
185	#493	C₂²×C₃₁	1	1	yes	characteristic
186	#492	C₂²×C₃₁	1	2	yes	—
187	#486	C₂²×C₃₁	1	2	yes	—
188	#482	—	2	2	no	—
189	#475	C₂²×C₃₁	1	2	yes	—
190	#465	—	2	2	no	—
191	#463	—	2	2	no	—
192	#461	C₂²×C₃₁	1	2	yes	—
193	#460	—	2	2	no	—
194	#460	C₂²×C₃₁	1	2	yes	—
195	#459	C₂²×C₃₁	1	2	yes	—
196	#452	—	2	2	no	—
197	#446	C₂²×C₃₁	1	2	yes	—
198	#440	—	2	2	no	—
199	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
200	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
201	C₈×D₄	C₂²×C₃₁	1	2	yes	—
202	#416	—	2	2	no	—
203	#335	C₂²×C₃₁	1	1	yes	characteristic
204	#335	C₂²×C₃₁	1	1	yes	characteristic
205	#332	C₂²×C₃₁	1	2	yes	—
206	#331	C₂²×C₃₁	1	2	yes	—
207	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
208	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
209	C₈²	C₂²×C₃₁	1	1	yes	characteristic
210	C₂×C₃₁	—	8	2	no	—
211	C₂×C₃₁	—	16	2	no	—
212	C₂×C₃₁	#3273	1	1	yes	characteristic
213	C₂×C₃₁	#3287	1	1	yes	characteristic
214	C₂×C₃₁	#3291	1	1	yes	characteristic
215	C₂×#41	—	2	2	no	—
216	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
217	C₂×SD₁₆	—	2	2	no	—
218	C₂×SD₁₆	—	2	2	no	—
219	C₂×SD₁₆	—	2	2	no	—
220	C₂×SD₁₆	—	2	2	no	—
221	C₂×D₈	—	2	2	no	—
222	C₂×D₈	—	4	1	no	—
223	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
224	C₂²×C₈	—	2	2	no	—
225	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
226	#127	—	2	2	no	—
227	#127	C₂³×C₃₁	1	2	yes	—
228	#126	—	4	2	no	—
229	#124	—	2	2	no	—
230	C₂²⋊Q₈	—	2	2	no	—
231	#121	—	2	2	no	—
232	#121	—	4	2	no	—
233	C₄×Q₈	—	2	2	no	—
234	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
235	C₄×D₄	—	2	2	no	—
236	C₄×D₄	C₂³×C₃₁	1	2	yes	—
237	#117	—	2	2	no	—
238	#107	—	4	1	no	—
239	#107	D₄×C₃₁	1	1	yes	characteristic
240	#107	D₄×C₃₁	1	1	yes	characteristic
241	#106	—	2	2	no	—
242	C₄⋊C₈	—	2	2	no	—
243	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
244	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
245	#103	—	2	2	no	—
246	#103	—	2	2	no	—
247	#103	D₄×C₃₁	1	2	yes	—
248	#102	—	2	2	no	—
249	#102	—	2	2	no	—
250	#102	—	2	2	no	—
251	#102	—	2	2	no	—
252	#102	D₄×C₃₁	1	2	yes	—
253	C₂²⋊C₈	—	2	2	no	—
254	#97	—	2	2	no	—
255	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
257	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
258	C₃₁	#19961	1	1	yes	characteristic, direct product factor, Sylow
259	#41	—	4	4	no	—
260	#41	—	8	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	4	2	no	—
271	Q₁₆	—	4	1	no	—
272	SD₁₆	—	4	4	no	—
273	SD₁₆	—	8	2	no	—
274	SD₁₆	—	8	2	no	—
275	SD₁₆	—	8	2	no	—
276	D₈	—	2	2	no	—
277	D₈	—	8	2	no	—
278	D₈	—	8	2	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	8	2	no	—
283	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
284	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
285	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
286	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
292	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
293	C₂²⋊C₄	—	2	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	4	2	no	—
299	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
300	C₂³	—	2	2	no	—
301	C₂³	—	4	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	2	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	2	no	—
306	D₄	—	2	2	no	—
307	D₄	—	2	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	8	2	no	—
313	D₄	—	8	4	no	—
314	D₄	—	16	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	8	4	no	—
322	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
324	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	4	2	no	—
330	C₈	—	4	2	no	—
331	C₈	C₃₁×#135	1	2	yes	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	D₄²×C₃₁	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	4	2	no	—
343	C₄	—	8	2	no	—
344	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
345	C₄	C₃₁×#575	1	1	yes	characteristic
346	C₄	C₃₁×#576	1	1	yes	characteristic
347	C₄	C₃₁×#576	1	1	yes	characteristic
348	C₂	—	8	2	no	—
349	C₂	—	16	2	no	—
350	C₂	C₃₁×#3273	1	1	yes	characteristic
351	C₂	C₃₁×#3287	1	1	yes	characteristic
352	C₂	C₃₁×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3397	C₂	semidirect
2. C₃₁×#3304	C₂	semidirect
3. C₃₁×#1706	C₂	semidirect
4. C₃₁×#1686	C₂	semidirect
5. C₃₁×#1681	C₂	semidirect
6. C₃₁×#1650	C₂	semidirect
7. C₃₁×#1621	C₂	semidirect
8. C₃₁×#1591	C₂	semidirect
9. C₈×D₈×C₃₁	C₂	semidirect
123. #19961	C₃₁	direct (N × H)
156. #3397	C₂×C₃₁	semidirect
157. #3304	C₂×C₃₁	semidirect
158. #1706	C₂×C₃₁	semidirect
159. #1686	C₂×C₃₁	semidirect
160. #1681	C₂×C₃₁	semidirect
161. #1650	C₂×C₃₁	semidirect
162. #1621	C₂×C₃₁	semidirect
163. #1591	C₂×C₃₁	semidirect
164. C₈×D₈	C₂×C₃₁	semidirect
258. C₃₁	#19961	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×#3397	2
C₃₁×#3304	2
C₃₁×#1706	2
C₃₁×#1686	2
C₃₁×#1681	2
C₃₁×#1650	2
C₃₁×#1621	2
C₃₁×#1591	2
C₈×D₈×C₃₁	2
#19961	31

Subgroups — C31×#19961 #21476720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19961 #21476720