Subgroups — C₃₁×#19721 #21476480

Order 7936.

798 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3227	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3207	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3184	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3152	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1721	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1720	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1716	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1715	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1709	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1705	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	—	2	2	no	—
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₃₁×#532	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃₁×#498	C₂²	1	1	yes	characteristic
17	C₃₁×#498	C₂²	1	2	yes	—
18	C₃₁×#498	C₂²	1	2	yes	—
19	C₃₁×#497	—	2	2	no	—
20	C₃₁×#494	C₂²	1	1	yes	characteristic
21	C₃₁×#494	C₂²	1	1	yes	characteristic
22	C₃₁×#493	C₂²	1	1	yes	characteristic
23	C₃₁×#493	C₂²	1	2	yes	—
24	C₃₁×#492	C₂²	1	1	yes	characteristic
25	C₃₁×#492	C₂²	1	1	yes	characteristic
26	C₃₁×#492	C₂²	1	2	yes	—
27	C₃₁×#487	C₂²	1	2	yes	—
28	C₃₁×#486	C₂²	1	2	yes	—
29	C₃₁×#486	C₂²	1	2	yes	—
30	C₃₁×#486	C₂²	1	2	yes	—
31	C₃₁×#486	C₂²	1	2	yes	—
32	C₃₁×#481	—	2	4	no	—
33	C₃₁×#476	—	2	4	no	—
34	C₃₁×#469	—	2	2	no	—
35	C₃₁×#463	—	2	4	no	—
36	C₃₁×#449	—	2	4	no	—
37	C₃₁×#442	—	2	2	no	—
38	C₃₁×#431	C₂²	1	2	yes	—
39	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#322	C₂²	1	2	yes	—
41	C₃₁×#322	C₂²	1	1	yes	characteristic
42	C₈²×C₃₁	C₂²	1	1	yes	characteristic
43	C₂×#41×C₃₁	—	2	1	no	—
44	C₂×#41×C₃₁	—	2	2	no	—
45	C₂²×Q₈×C₃₁	—	2	1	no	—
46	C₃₁×#137	—	4	8	no	—
47	C₂×Q₁₆×C₃₁	—	2	2	no	—
48	C₂×Q₁₆×C₃₁	—	2	2	no	—
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	—	2	4	no	—
52	C₂×D₈×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	2	4	no	—
54	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
55	C₃₁×C₄⋊Q₈	—	2	2	no	—
56	C₃₁×C₄⋊Q₈	—	2	1	no	—
57	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
58	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
59	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
60	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
61	C₃₁×#127	C₂³	1	2	yes	—
62	C₃₁×#127	C₂³	1	1	yes	characteristic
63	C₃₁×#124	—	2	2	no	—
64	C₃₁×#124	—	2	1	no	—
65	C₃₁×#121	—	2	2	no	—
66	C₃₁×#121	—	2	2	no	—
67	C₃₁×#121	—	2	2	no	—
68	C₄×D₄×C₃₁	—	2	2	no	—
69	C₄×D₄×C₃₁	—	2	2	no	—
70	C₄×D₄×C₃₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×#106	—	2	2	no	—
74	C₃₁×#106	—	2	4	no	—
75	C₃₁×#106	—	2	4	no	—
76	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
77	C₃₁×#103	—	2	4	no	—
78	C₃₁×#102	—	2	4	no	—
79	C₃₁×#102	—	2	4	no	—
80	C₃₁×#102	—	2	4	no	—
81	C₃₁×#102	—	2	4	no	—
82	C₃₁×C₂²⋊C₈	D₄	1	4	yes	—
83	C₃₁×#97	D₄	1	2	yes	—
84	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₃₁	C₂³	1	2	yes	—
87	C₄×C₈×C₃₁	C₂³	1	2	yes	—
88	#41×C₃₁	—	4	4	no	—
89	#41×C₃₁	—	8	4	no	—
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	—	2	1	no	—
92	C₂×Q₈×C₃₁	—	2	1	no	—
93	C₂×Q₈×C₃₁	—	2	1	no	—
94	C₂×Q₈×C₃₁	—	4	4	no	—
95	C₂×Q₈×C₃₁	—	4	1	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	2	1	no	—
101	C₂×D₄×C₃₁	—	4	1	no	—
102	C₂×D₄×C₃₁	—	4	2	no	—
103	C₂²×C₄×C₃₁	—	4	1	no	—
104	C₂²×C₄×C₃₁	—	4	1	no	—
105	C₂²×C₄×C₃₁	—	4	2	no	—
106	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	Q₁₆×C₃₁	—	4	4	no	—
109	Q₁₆×C₃₁	—	4	4	no	—
110	SD₁₆×C₃₁	—	4	4	no	—
111	SD₁₆×C₃₁	—	4	4	no	—
112	SD₁₆×C₃₁	—	8	4	no	—
113	D₈×C₃₁	—	4	4	no	—
114	D₈×C₃₁	—	8	4	no	—
115	M₁₆×C₃₁	—	2	4	no	—
116	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
118	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
119	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
120	C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	4	2	no	—
126	C₂²⋊C₄×C₃₁	—	4	2	no	—
127	C₂²⋊C₄×C₃₁	—	4	2	no	—
128	C₂²⋊C₄×C₃₁	—	4	2	no	—
129	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₃₁	C₂×D₄	1	2	yes	—
131	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
132	#19721	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂³×C₃₁	—	4	1	no	—
134	C₂³×C₃₁	—	4	2	no	—
135	C₂³×C₃₁	#127	1	1	yes	characteristic
136	Q₈×C₃₁	—	4	2	no	—
137	Q₈×C₃₁	—	4	2	no	—
138	Q₈×C₃₁	—	4	2	no	—
139	Q₈×C₃₁	—	4	4	no	—
140	Q₈×C₃₁	—	8	2	no	—
141	D₄×C₃₁	—	4	4	no	—
142	D₄×C₃₁	—	4	2	no	—
143	D₄×C₃₁	—	4	4	no	—
144	D₄×C₃₁	—	4	4	no	—
145	D₄×C₃₁	—	4	4	no	—
146	D₄×C₃₁	—	8	2	no	—
147	D₄×C₃₁	—	8	4	no	—
148	C₂×C₄×C₃₁	—	2	2	no	—
149	C₂×C₄×C₃₁	—	2	4	no	—
150	C₂×C₄×C₃₁	—	4	1	no	—
151	C₂×C₄×C₃₁	—	4	2	no	—
152	C₂×C₄×C₃₁	—	4	1	no	—
153	C₂×C₄×C₃₁	—	4	1	no	—
154	C₂×C₄×C₃₁	—	8	2	no	—
155	C₂×C₄×C₃₁	—	8	4	no	—
156	C₂×C₄×C₃₁	—	8	2	no	—
157	C₂×C₄×C₃₁	#127	1	2	yes	—
158	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
159	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
161	C₈×C₃₁	—	2	2	no	—
162	C₈×C₃₁	—	2	4	no	—
163	C₈×C₃₁	—	2	4	no	—
164	C₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
165	#3227	C₂×C₃₁	1	2	yes	complemented
166	#3207	C₂×C₃₁	1	2	yes	complemented
167	#3184	C₂×C₃₁	1	1	yes	characteristic, complemented
168	#3152	C₂×C₃₁	1	1	yes	characteristic, complemented
169	#1721	C₂×C₃₁	1	1	yes	characteristic, complemented
170	#1720	C₂×C₃₁	1	2	yes	complemented
171	#1716	C₂×C₃₁	1	1	yes	characteristic, complemented
172	#1715	C₂×C₃₁	1	2	yes	complemented
173	#1709	C₂×C₃₁	1	1	yes	characteristic, complemented
174	#1705	C₂×C₃₁	1	1	yes	characteristic, complemented
175	#1453	C₂×C₃₁	1	1	yes	characteristic, complemented
176	C₂²×C₃₁	—	2	1	no	—
177	C₂²×C₃₁	—	2	2	no	—
178	C₂²×C₃₁	—	8	1	no	—
179	C₂²×C₃₁	—	8	2	no	—
180	C₂²×C₃₁	—	8	2	no	—
181	C₂²×C₃₁	—	8	2	no	—
182	C₂²×C₃₁	—	8	2	no	—
183	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
184	C₄×C₃₁	—	2	1	no	—
185	C₄×C₃₁	—	2	2	no	—
186	C₄×C₃₁	—	8	1	no	—
187	C₄×C₃₁	—	8	1	no	—
188	C₄×C₃₁	—	8	1	no	—
189	C₄×C₃₁	—	8	2	no	—
190	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
191	C₄×C₃₁	#577	1	4	yes	—
192	C₂×#137	—	2	2	no	—
193	#532	C₂²×C₃₁	1	1	yes	characteristic
194	#532	C₂²×C₃₁	1	2	yes	—
195	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
196	#498	C₂²×C₃₁	1	1	yes	characteristic
197	#498	C₂²×C₃₁	1	2	yes	—
198	#498	C₂²×C₃₁	1	2	yes	—
199	#497	—	2	2	no	—
200	#494	C₂²×C₃₁	1	1	yes	characteristic
201	#494	C₂²×C₃₁	1	1	yes	characteristic
202	#493	C₂²×C₃₁	1	1	yes	characteristic
203	#493	C₂²×C₃₁	1	2	yes	—
204	#492	C₂²×C₃₁	1	1	yes	characteristic
205	#492	C₂²×C₃₁	1	1	yes	characteristic
206	#492	C₂²×C₃₁	1	2	yes	—
207	#487	C₂²×C₃₁	1	2	yes	—
208	#486	C₂²×C₃₁	1	2	yes	—
209	#486	C₂²×C₃₁	1	2	yes	—
210	#486	C₂²×C₃₁	1	2	yes	—
211	#486	C₂²×C₃₁	1	2	yes	—
212	#481	—	2	4	no	—
213	#476	—	2	4	no	—
214	#469	—	2	2	no	—
215	#463	—	2	4	no	—
216	#449	—	2	4	no	—
217	#442	—	2	2	no	—
218	#431	C₂²×C₃₁	1	2	yes	—
219	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
220	#322	C₂²×C₃₁	1	2	yes	—
221	#322	C₂²×C₃₁	1	1	yes	characteristic
222	C₈²	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×C₃₁	—	4	1	no	—
224	C₂×C₃₁	—	16	1	no	—
225	C₂×C₃₁	—	16	2	no	—
226	C₂×C₃₁	#3151	1	1	yes	characteristic
227	C₂×C₃₁	#3154	1	2	yes	—
228	C₂×#41	—	2	1	no	—
229	C₂×#41	—	2	2	no	—
230	C₂²×Q₈	—	2	1	no	—
231	#137	—	4	8	no	—
232	C₂×Q₁₆	—	2	2	no	—
233	C₂×Q₁₆	—	2	2	no	—
234	C₂×SD₁₆	—	2	2	no	—
235	C₂×SD₁₆	—	2	2	no	—
236	C₂×SD₁₆	—	2	4	no	—
237	C₂×D₈	—	2	2	no	—
238	C₂×D₈	—	2	4	no	—
239	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
240	C₄⋊Q₈	—	2	2	no	—
241	C₄⋊Q₈	—	2	1	no	—
242	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
243	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
244	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
245	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
246	#127	C₂³×C₃₁	1	2	yes	—
247	#127	C₂³×C₃₁	1	1	yes	characteristic
248	#124	—	2	2	no	—
249	#124	—	2	1	no	—
250	#121	—	2	2	no	—
251	#121	—	2	2	no	—
252	#121	—	2	2	no	—
253	C₄×D₄	—	2	2	no	—
254	C₄×D₄	—	2	2	no	—
255	C₄×D₄	—	2	2	no	—
256	C₂×C₄⋊C₄	—	2	2	no	—
257	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
258	#106	—	2	2	no	—
259	#106	—	2	4	no	—
260	#106	—	2	4	no	—
261	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
262	#103	—	2	4	no	—
263	#102	—	2	4	no	—
264	#102	—	2	4	no	—
265	#102	—	2	4	no	—
266	#102	—	2	4	no	—
267	C₂²⋊C₈	D₄×C₃₁	1	4	yes	—
268	#97	D₄×C₃₁	1	2	yes	—
269	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
270	C₄×C₈	C₂³×C₃₁	1	2	yes	—
271	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
272	C₄×C₈	C₂³×C₃₁	1	2	yes	—
273	C₃₁	#19721	1	1	yes	characteristic, direct product factor, Sylow
274	#41	—	4	4	no	—
275	#41	—	8	4	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	2	1	no	—
279	C₂×Q₈	—	2	1	no	—
280	C₂×Q₈	—	4	4	no	—
281	C₂×Q₈	—	4	1	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	—	4	1	no	—
288	C₂×D₄	—	4	2	no	—
289	C₂²×C₄	—	4	1	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	4	2	no	—
292	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
293	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
294	Q₁₆	—	4	4	no	—
295	Q₁₆	—	4	4	no	—
296	SD₁₆	—	4	4	no	—
297	SD₁₆	—	4	4	no	—
298	SD₁₆	—	8	4	no	—
299	D₈	—	4	4	no	—
300	D₈	—	8	4	no	—
301	M₁₆	—	2	4	no	—
302	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
303	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
304	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
305	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₂²⋊C₄	—	4	2	no	—
313	C₂²⋊C₄	—	4	2	no	—
314	C₂²⋊C₄	—	4	2	no	—
315	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
316	C₄²	C₂×D₄×C₃₁	1	2	yes	—
317	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
318	C₂³	—	4	1	no	—
319	C₂³	—	4	2	no	—
320	C₂³	C₃₁×#127	1	1	yes	characteristic
321	Q₈	—	4	2	no	—
322	Q₈	—	4	2	no	—
323	Q₈	—	4	2	no	—
324	Q₈	—	4	4	no	—
325	Q₈	—	8	2	no	—
326	D₄	—	4	4	no	—
327	D₄	—	4	2	no	—
328	D₄	—	4	4	no	—
329	D₄	—	4	4	no	—
330	D₄	—	4	4	no	—
331	D₄	—	8	2	no	—
332	D₄	—	8	4	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	4	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	8	2	no	—
340	C₂×C₄	—	8	4	no	—
341	C₂×C₄	—	8	2	no	—
342	C₂×C₄	C₃₁×#127	1	2	yes	—
343	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
344	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
345	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
346	C₈	—	2	2	no	—
347	C₈	—	2	4	no	—
348	C₈	—	2	4	no	—
349	C₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	8	1	no	—
353	C₂²	—	8	2	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
358	C₄	—	2	1	no	—
359	C₄	—	2	2	no	—
360	C₄	—	8	1	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	1	no	—
363	C₄	—	8	2	no	—
364	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
365	C₄	C₃₁×#577	1	4	yes	—
366	C₂	—	4	1	no	—
367	C₂	—	16	1	no	—
368	C₂	—	16	2	no	—
369	C₂	C₃₁×#3151	1	1	yes	characteristic
370	C₂	C₃₁×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ 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
1. C₃₁×#3227	C₂	semidirect
2. C₃₁×#3207	C₂	semidirect
3. C₃₁×#3184	C₂	semidirect
4. C₃₁×#3152	C₂	semidirect
5. C₃₁×#1721	C₂	semidirect
6. C₃₁×#1720	C₂	semidirect
7. C₃₁×#1716	C₂	semidirect
8. C₃₁×#1715	C₂	semidirect
9. C₃₁×#1709	C₂	semidirect
10. C₃₁×#1705	C₂	semidirect
11. C₃₁×#1453	C₂	semidirect
132. #19721	C₃₁	direct (N × H)
165. #3227	C₂×C₃₁	semidirect
166. #3207	C₂×C₃₁	semidirect
167. #3184	C₂×C₃₁	semidirect
168. #3152	C₂×C₃₁	semidirect
169. #1721	C₂×C₃₁	semidirect
170. #1720	C₂×C₃₁	semidirect
171. #1716	C₂×C₃₁	semidirect
172. #1715	C₂×C₃₁	semidirect
173. #1709	C₂×C₃₁	semidirect
174. #1705	C₂×C₃₁	semidirect
175. #1453	C₂×C₃₁	semidirect
273. C₃₁	#19721	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×#3227	2
C₃₁×#3207	2
C₃₁×#3184	2
C₃₁×#3152	2
C₃₁×#1721	2
C₃₁×#1720	2
C₃₁×#1716	2
C₃₁×#1715	2
C₃₁×#1709	2
C₃₁×#1705	2
C₃₁×#1453	2
#19721	31

Subgroups — C31×#19721 #21476480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19721 #21476480