Subgroups — C₃₁×#19888 #21476647

Order 7936.

710 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3358	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3278	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1685	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1682	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1649	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1621	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1589	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#137	—	2	2	no	—
11	C₃₁×#550	C₂²	1	2	yes	—
12	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#500	C₂²	1	1	yes	characteristic
14	C₃₁×#493	C₂²	1	1	yes	characteristic
15	C₃₁×#492	C₂²	1	2	yes	—
16	C₃₁×#486	—	2	2	no	—
17	C₃₁×#486	C₂²	1	2	yes	—
18	C₃₁×#474	—	2	2	no	—
19	C₃₁×#474	C₂²	1	2	yes	—
20	C₃₁×#468	—	2	2	no	—
21	C₃₁×#462	C₂²	1	2	yes	—
22	C₃₁×#460	C₂²	1	2	yes	—
23	C₃₁×#459	—	2	2	no	—
24	C₃₁×#459	C₂²	1	2	yes	—
25	C₃₁×#452	—	2	2	no	—
26	C₃₁×#448	—	2	2	no	—
27	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
28	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
29	C₄×D₈×C₃₁	—	2	2	no	—
30	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#436	C₂²	1	2	yes	—
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	C₂²×Q₈×C₃₁	—	2	2	no	—
41	C₃₁×#137	—	4	8	no	—
42	C₂×Q₁₆×C₃₁	—	2	2	no	—
43	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
44	C₂×SD₁₆×C₃₁	—	2	2	no	—
45	C₂×SD₁₆×C₃₁	—	2	2	no	—
46	C₂×SD₁₆×C₃₁	—	2	2	no	—
47	C₂×D₈×C₃₁	—	2	2	no	—
48	C₂×D₈×C₃₁	—	4	1	no	—
49	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₃₁×C₄⋊Q₈	—	2	2	no	—
52	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
53	C₃₁×#127	—	2	2	no	—
54	C₃₁×#127	C₂³	1	2	yes	—
55	C₃₁×C₂²⋊Q₈	—	2	2	no	—
56	C₃₁×C₂²⋊Q₈	—	4	2	no	—
57	C₃₁×#121	—	2	2	no	—
58	C₃₁×#121	—	4	2	no	—
59	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
60	C₄×D₄×C₃₁	—	2	2	no	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	C₂³	1	2	yes	—
63	C₃₁×#107	—	2	2	no	—
64	C₃₁×#107	—	4	1	no	—
65	C₃₁×#107	D₄	1	1	yes	characteristic
66	C₃₁×#107	D₄	1	1	yes	characteristic
67	C₃₁×C₄⋊C₈	—	2	2	no	—
68	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
69	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
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	—	2	2	no	—
77	C₃₁×#102	D₄	1	2	yes	—
78	C₃₁×C₂²⋊C₈	—	2	2	no	—
79	C₄×C₈×C₃₁	—	2	2	no	—
80	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
83	#41×C₃₁	—	4	4	no	—
84	#41×C₃₁	—	8	2	no	—
85	C₂×Q₈×C₃₁	—	2	2	no	—
86	C₂×Q₈×C₃₁	—	4	2	no	—
87	C₂×Q₈×C₃₁	—	4	4	no	—
88	C₂×Q₈×C₃₁	—	8	2	no	—
89	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
90	C₂×D₄×C₃₁	—	2	2	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	4	2	no	—
94	C₂×D₄×C₃₁	—	4	2	no	—
95	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	4	2	no	—
98	C₂²×C₄×C₃₁	—	4	2	no	—
99	Q₁₆×C₃₁	—	4	4	no	—
100	Q₁₆×C₃₁	D₈	1	4	yes	complemented
101	SD₁₆×C₃₁	—	4	4	no	—
102	SD₁₆×C₃₁	—	4	4	no	—
103	SD₁₆×C₃₁	—	8	2	no	—
104	D₈×C₃₁	—	4	1	no	—
105	D₈×C₃₁	—	4	4	no	—
106	D₈×C₃₁	—	8	2	no	—
107	M₁₆×C₃₁	—	4	4	no	—
108	C₂×C₈×C₃₁	—	2	1	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	4	2	no	—
119	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
120	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
121	C₂²⋊C₄×C₃₁	—	2	2	no	—
122	C₂²⋊C₄×C₃₁	—	4	2	no	—
123	C₂²⋊C₄×C₃₁	—	4	2	no	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
126	#19888	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂³×C₃₁	—	2	2	no	—
128	C₂³×C₃₁	—	4	2	no	—
129	Q₈×C₃₁	—	2	2	no	—
130	Q₈×C₃₁	—	4	2	no	—
131	Q₈×C₃₁	—	4	2	no	—
132	Q₈×C₃₁	—	8	4	no	—
133	Q₈×C₃₁	C₂×D₈	1	4	yes	—
134	D₄×C₃₁	—	2	2	no	—
135	D₄×C₃₁	—	2	2	no	—
136	D₄×C₃₁	—	4	2	no	—
137	D₄×C₃₁	—	4	2	no	—
138	D₄×C₃₁	—	4	2	no	—
139	D₄×C₃₁	—	4	2	no	—
140	D₄×C₃₁	—	8	4	no	—
141	D₄×C₃₁	—	8	2	no	—
142	D₄×C₃₁	—	16	2	no	—
143	C₂×C₄×C₃₁	—	2	2	no	—
144	C₂×C₄×C₃₁	—	2	2	no	—
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	4	2	no	—
147	C₂×C₄×C₃₁	—	4	2	no	—
148	C₂×C₄×C₃₁	—	4	2	no	—
149	C₂×C₄×C₃₁	—	8	4	no	—
150	C₂×C₄×C₃₁	—	8	4	no	—
151	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
154	C₈×C₃₁	—	2	2	no	—
155	C₈×C₃₁	—	2	1	no	—
156	C₈×C₃₁	—	2	1	no	—
157	C₈×C₃₁	—	2	1	no	—
158	C₈×C₃₁	—	4	2	no	—
159	C₈×C₃₁	—	4	2	no	—
160	C₈×C₃₁	C₂×D₈	1	2	yes	—
161	#3358	C₂×C₃₁	1	2	yes	complemented
162	#3278	C₂×C₃₁	1	2	yes	complemented
163	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
164	#1685	C₂×C₃₁	1	2	yes	complemented
165	#1682	C₂×C₃₁	1	2	yes	complemented
166	#1649	C₂×C₃₁	1	2	yes	complemented
167	#1621	C₂×C₃₁	1	2	yes	complemented
168	#1589	C₂×C₃₁	1	1	yes	characteristic, complemented
169	C₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
170	C₂²×C₃₁	—	4	2	no	—
171	C₂²×C₃₁	—	4	2	no	—
172	C₂²×C₃₁	—	4	2	no	—
173	C₂²×C₃₁	—	8	2	no	—
174	C₂²×C₃₁	—	8	2	no	—
175	C₂²×C₃₁	—	8	2	no	—
176	C₂²×C₃₁	D₄²	1	1	yes	characteristic
177	C₄×C₃₁	—	2	4	no	—
178	C₄×C₃₁	—	2	1	no	—
179	C₄×C₃₁	—	4	2	no	—
180	C₄×C₃₁	—	4	2	no	—
181	C₄×C₃₁	—	8	2	no	—
182	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
183	C₄×C₃₁	#575	1	1	yes	characteristic
184	C₄×C₃₁	#576	1	1	yes	characteristic
185	C₄×C₃₁	#578	1	1	yes	characteristic
186	C₂×#137	—	2	2	no	—
187	#550	C₂²×C₃₁	1	2	yes	—
188	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
189	#500	C₂²×C₃₁	1	1	yes	characteristic
190	#493	C₂²×C₃₁	1	1	yes	characteristic
191	#492	C₂²×C₃₁	1	2	yes	—
192	#486	—	2	2	no	—
193	#486	C₂²×C₃₁	1	2	yes	—
194	#474	—	2	2	no	—
195	#474	C₂²×C₃₁	1	2	yes	—
196	#468	—	2	2	no	—
197	#462	C₂²×C₃₁	1	2	yes	—
198	#460	C₂²×C₃₁	1	2	yes	—
199	#459	—	2	2	no	—
200	#459	C₂²×C₃₁	1	2	yes	—
201	#452	—	2	2	no	—
202	#448	—	2	2	no	—
203	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
204	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
205	C₄×D₈	—	2	2	no	—
206	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
207	#436	C₂²×C₃₁	1	2	yes	—
208	#335	C₂²×C₃₁	1	1	yes	characteristic
209	#335	C₂²×C₃₁	1	1	yes	characteristic
210	#332	C₂²×C₃₁	1	2	yes	—
211	#331	C₂²×C₃₁	1	2	yes	—
212	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
213	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
214	C₈²	C₂²×C₃₁	1	1	yes	characteristic
215	C₂×C₃₁	—	8	2	no	—
216	C₂×C₃₁	—	16	2	no	—
217	C₂×C₃₁	#3277	1	1	yes	characteristic
218	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
219	C₂×C₃₁	#3291	1	1	yes	characteristic
220	C₂×#41	—	2	2	no	—
221	C₂²×Q₈	—	2	2	no	—
222	#137	—	4	8	no	—
223	C₂×Q₁₆	—	2	2	no	—
224	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×SD₁₆	—	2	2	no	—
228	C₂×D₈	—	2	2	no	—
229	C₂×D₈	—	4	1	no	—
230	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
231	C₂×M₁₆	—	2	2	no	—
232	C₄⋊Q₈	—	2	2	no	—
233	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
234	#127	—	2	2	no	—
235	#127	C₂³×C₃₁	1	2	yes	—
236	C₂²⋊Q₈	—	2	2	no	—
237	C₂²⋊Q₈	—	4	2	no	—
238	#121	—	2	2	no	—
239	#121	—	4	2	no	—
240	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
241	C₄×D₄	—	2	2	no	—
242	C₄×D₄	—	2	2	no	—
243	C₄×D₄	C₂³×C₃₁	1	2	yes	—
244	#107	—	2	2	no	—
245	#107	—	4	1	no	—
246	#107	D₄×C₃₁	1	1	yes	characteristic
247	#107	D₄×C₃₁	1	1	yes	characteristic
248	C₄⋊C₈	—	2	2	no	—
249	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
250	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
251	#103	—	2	2	no	—
252	#103	D₄×C₃₁	1	2	yes	—
253	#102	—	2	2	no	—
254	#102	—	2	2	no	—
255	#102	—	2	2	no	—
256	#102	—	2	2	no	—
257	#102	—	2	2	no	—
258	#102	D₄×C₃₁	1	2	yes	—
259	C₂²⋊C₈	—	2	2	no	—
260	C₄×C₈	—	2	2	no	—
261	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
262	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
263	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
264	C₃₁	#19888	1	1	yes	characteristic, direct product factor, Sylow
265	#41	—	4	4	no	—
266	#41	—	8	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×Q₈	—	4	4	no	—
270	C₂×Q₈	—	8	2	no	—
271	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	4	2	no	—
276	C₂×D₄	—	4	2	no	—
277	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	4	2	no	—
280	C₂²×C₄	—	4	2	no	—
281	Q₁₆	—	4	4	no	—
282	Q₁₆	D₈×C₃₁	1	4	yes	complemented
283	SD₁₆	—	4	4	no	—
284	SD₁₆	—	4	4	no	—
285	SD₁₆	—	8	2	no	—
286	D₈	—	4	1	no	—
287	D₈	—	4	4	no	—
288	D₈	—	8	2	no	—
289	M₁₆	—	4	4	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
294	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
295	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
296	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	4	2	no	—
301	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
302	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	4	2	no	—
305	C₂²⋊C₄	—	4	2	no	—
306	C₄²	—	2	2	no	—
307	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
308	C₂³	—	2	2	no	—
309	C₂³	—	4	2	no	—
310	Q₈	—	2	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	8	4	no	—
314	Q₈	C₂×D₈×C₃₁	1	4	yes	—
315	D₄	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	4	2	no	—
321	D₄	—	8	4	no	—
322	D₄	—	8	2	no	—
323	D₄	—	16	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	8	4	no	—
331	C₂×C₄	—	8	4	no	—
332	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
334	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	4	2	no	—
341	C₈	C₂×D₈×C₃₁	1	2	yes	—
342	C₂²	—	4	2	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	D₄²×C₃₁	1	1	yes	characteristic
349	C₄	—	2	4	no	—
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	8	2	no	—
354	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
355	C₄	C₃₁×#575	1	1	yes	characteristic
356	C₄	C₃₁×#576	1	1	yes	characteristic
357	C₄	C₃₁×#578	1	1	yes	characteristic
358	C₂	—	8	2	no	—
359	C₂	—	16	2	no	—
360	C₂	C₃₁×#3277	1	1	yes	characteristic
361	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
362	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×#3358	C₂	semidirect
2. C₃₁×#3278	C₂	semidirect
3. C₃₁×#1706	C₂	semidirect
4. C₃₁×#1685	C₂	semidirect
5. C₃₁×#1682	C₂	semidirect
6. C₃₁×#1649	C₂	semidirect
7. C₃₁×#1621	C₂	semidirect
8. C₃₁×#1589	C₂	semidirect
9. C₈×Q₁₆×C₃₁	C₂	semidirect
100. Q₁₆×C₃₁	D₈	semidirect
126. #19888	C₃₁	direct (N × H)
161. #3358	C₂×C₃₁	semidirect
162. #3278	C₂×C₃₁	semidirect
163. #1706	C₂×C₃₁	semidirect
164. #1685	C₂×C₃₁	semidirect
165. #1682	C₂×C₃₁	semidirect
166. #1649	C₂×C₃₁	semidirect
167. #1621	C₂×C₃₁	semidirect
168. #1589	C₂×C₃₁	semidirect
169. C₈×Q₁₆	C₂×C₃₁	semidirect
264. C₃₁	#19888	direct (N × H)
282. Q₁₆	D₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3358	2
C₃₁×#3278	2
C₃₁×#1706	2
C₃₁×#1685	2
C₃₁×#1682	2
C₃₁×#1649	2
C₃₁×#1621	2
C₃₁×#1589	2
C₈×Q₁₆×C₃₁	2
#19888	31

Subgroups — C31×#19888 #21476647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19888 #21476647