Subgroups — C₃₁×#19347 #21476106

Order 7936.

616 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3216	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1694	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1688	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1682	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1557	C₂	1	2	yes	maximal, complemented
9	C₃₁×#533	C₂²	1	1	yes	characteristic
10	C₃₁×#532	C₂²	1	2	yes	—
11	C₃₁×#500	C₂²	1	1	yes	characteristic
12	C₃₁×#497	—	2	2	no	—
13	C₃₁×#493	C₂²	1	1	yes	characteristic
14	C₃₁×#489	—	2	2	no	—
15	C₃₁×#486	C₂²	1	2	yes	—
16	C₃₁×#466	—	2	1	no	—
17	C₃₁×#466	—	2	1	no	—
18	C₃₁×#462	C₂²	1	4	yes	—
19	C₃₁×#460	C₂²	1	4	yes	—
20	C₃₁×#452	—	2	4	no	—
21	C₃₁×#446	—	2	1	no	—
22	C₃₁×#446	C₂²	1	2	yes	—
23	C₈×Q₈×C₃₁	—	2	1	no	—
24	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
25	C₄×D₈×C₃₁	—	2	1	no	—
26	C₄×D₈×C₃₁	—	2	1	no	—
27	C₃₁×#432	C₂²	1	2	yes	—
28	C₃₁×#424	—	2	1	no	—
29	C₃₁×#423	C₂²	1	1	yes	characteristic
30	C₃₁×#405	—	2	1	no	—
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#335	C₂²	1	2	yes	—
33	C₃₁×#334	C₂²	1	2	yes	—
34	C₃₁×#332	C₂²	1	4	yes	—
35	C₃₁×#331	C₂²	1	4	yes	—
36	C₂×#41×C₃₁	—	2	2	no	—
37	C₂×Q₁₆×C₃₁	—	2	4	no	—
38	C₂×SD₁₆×C₃₁	—	2	4	no	—
39	C₂×D₈×C₃₁	—	2	1	no	—
40	C₂×D₈×C₃₁	—	2	1	no	—
41	C₂×M₁₆×C₃₁	—	2	1	no	—
42	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
44	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
45	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
46	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₃₁×#127	C₂³	1	2	yes	—
48	C₃₁×#125	—	2	1	no	—
49	C₃₁×#124	—	2	2	no	—
50	C₃₁×C₂²⋊Q₈	—	2	1	no	—
51	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
52	C₃₁×#121	—	2	2	no	—
53	C₃₁×#121	—	2	2	no	—
54	C₄×Q₈×C₃₁	—	2	1	no	—
55	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
56	C₄×D₄×C₃₁	—	2	2	no	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₃₁×#117	—	2	1	no	—
59	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×#107	—	2	1	no	—
61	C₃₁×#107	—	2	1	no	—
62	C₃₁×C₄⋊C₈	—	2	2	no	—
63	C₃₁×C₄⋊C₈	—	2	2	no	—
64	C₃₁×C₄⋊C₈	—	2	1	no	—
65	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
66	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
67	C₃₁×#103	—	2	4	no	—
68	C₃₁×#102	—	2	2	no	—
69	C₃₁×#102	—	2	2	no	—
70	C₃₁×#102	—	2	4	no	—
71	C₃₁×C₂²⋊C₈	—	2	2	no	—
72	C₃₁×#97	—	2	2	no	—
73	C₃₁×#97	—	2	2	no	—
74	C₄×C₈×C₃₁	—	2	2	no	—
75	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
76	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
77	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	#41×C₃₁	—	8	4	no	—
80	C₂×Q₈×C₃₁	—	2	1	no	—
81	C₂×Q₈×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
83	C₂×D₄×C₃₁	—	2	2	no	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	4	2	no	—
87	C₂²×C₄×C₃₁	—	4	2	no	—
88	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
91	Q₁₆×C₃₁	—	8	4	no	—
92	SD₁₆×C₃₁	—	8	4	no	—
93	D₈×C₃₁	—	4	2	no	—
94	D₈×C₃₁	—	4	2	no	—
95	M₁₆×C₃₁	—	4	2	no	—
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	2	4	no	—
99	C₂×C₈×C₃₁	—	4	1	no	—
100	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₃₁	—	2	1	no	—
105	C₄⋊C₄×C₃₁	—	2	1	no	—
106	C₄⋊C₄×C₃₁	—	2	1	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₂²⋊C₄×C₃₁	—	2	2	no	—
113	C₂²⋊C₄×C₃₁	—	4	2	no	—
114	C₂²⋊C₄×C₃₁	—	4	2	no	—
115	C₄²×C₃₁	—	2	2	no	—
116	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₃₁	#41	1	1	yes	characteristic
119	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
120	#19347	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₃₁	—	4	2	no	—
122	C₂³×C₃₁	#121	1	1	yes	characteristic
123	Q₈×C₃₁	—	2	4	no	—
124	Q₈×C₃₁	—	4	2	no	—
125	Q₈×C₃₁	—	4	4	no	—
126	D₄×C₃₁	—	4	2	no	—
127	D₄×C₃₁	—	4	2	no	—
128	D₄×C₃₁	—	4	4	no	—
129	D₄×C₃₁	—	8	2	no	—
130	D₄×C₃₁	—	8	4	no	—
131	C₂×C₄×C₃₁	—	2	1	no	—
132	C₂×C₄×C₃₁	—	2	1	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	2	no	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	4	1	no	—
137	C₂×C₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	8	4	no	—
139	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
140	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
141	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
142	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
145	C₈×C₃₁	—	2	1	no	—
146	C₈×C₃₁	—	2	1	no	—
147	C₈×C₃₁	—	2	1	no	—
148	C₈×C₃₁	—	4	2	no	—
149	C₈×C₃₁	—	4	4	no	—
150	C₈×C₃₁	#137	1	2	yes	—
151	#3216	C₂×C₃₁	1	2	yes	complemented
152	#3169	C₂×C₃₁	1	1	yes	characteristic
153	#2960	C₂×C₃₁	1	1	yes	characteristic, complemented
154	#1694	C₂×C₃₁	1	2	yes	complemented
155	#1688	C₂×C₃₁	1	2	yes	complemented
156	#1682	C₂×C₃₁	1	4	yes	complemented
157	#1568	C₂×C₃₁	1	1	yes	characteristic, complemented
158	#1557	C₂×C₃₁	1	2	yes	complemented
159	C₂²×C₃₁	—	2	1	no	—
160	C₂²×C₃₁	—	2	1	no	—
161	C₂²×C₃₁	—	2	1	no	—
162	C₂²×C₃₁	—	8	2	no	—
163	C₂²×C₃₁	—	8	2	no	—
164	C₂²×C₃₁	—	8	2	no	—
165	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
166	C₄×C₃₁	—	2	1	no	—
167	C₄×C₃₁	—	2	1	no	—
168	C₄×C₃₁	—	2	1	no	—
169	C₄×C₃₁	—	2	1	no	—
170	C₄×C₃₁	—	4	2	no	—
171	C₄×C₃₁	—	4	2	no	—
172	C₄×C₃₁	—	8	2	no	—
173	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
174	C₄×C₃₁	#575	1	1	yes	characteristic
175	C₄×C₃₁	#576	1	2	yes	—
176	#533	C₂²×C₃₁	1	1	yes	characteristic
177	#532	C₂²×C₃₁	1	2	yes	—
178	#500	C₂²×C₃₁	1	1	yes	characteristic
179	#497	—	2	2	no	—
180	#493	C₂²×C₃₁	1	1	yes	characteristic
181	#489	—	2	2	no	—
182	#486	C₂²×C₃₁	1	2	yes	—
183	#466	—	2	1	no	—
184	#466	—	2	1	no	—
185	#462	C₂²×C₃₁	1	4	yes	—
186	#460	C₂²×C₃₁	1	4	yes	—
187	#452	—	2	4	no	—
188	#446	—	2	1	no	—
189	#446	C₂²×C₃₁	1	2	yes	—
190	C₈×Q₈	—	2	1	no	—
191	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
192	C₄×D₈	—	2	1	no	—
193	C₄×D₈	—	2	1	no	—
194	#432	C₂²×C₃₁	1	2	yes	—
195	#424	—	2	1	no	—
196	#423	C₂²×C₃₁	1	1	yes	characteristic
197	#405	—	2	1	no	—
198	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
199	#335	C₂²×C₃₁	1	2	yes	—
200	#334	C₂²×C₃₁	1	2	yes	—
201	#332	C₂²×C₃₁	1	4	yes	—
202	#331	C₂²×C₃₁	1	4	yes	—
203	C₂×C₃₁	—	4	1	no	—
204	C₂×C₃₁	—	16	2	no	—
205	C₂×C₃₁	#3037	1	1	yes	characteristic
206	C₂×C₃₁	#3058	1	1	yes	characteristic
207	C₂×C₃₁	#3067	1	1	yes	characteristic
208	C₂×#41	—	2	2	no	—
209	C₂×Q₁₆	—	2	4	no	—
210	C₂×SD₁₆	—	2	4	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	—	2	1	no	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
216	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
217	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
218	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
219	#127	C₂³×C₃₁	1	2	yes	—
220	#125	—	2	1	no	—
221	#124	—	2	2	no	—
222	C₂²⋊Q₈	—	2	1	no	—
223	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
224	#121	—	2	2	no	—
225	#121	—	2	2	no	—
226	C₄×Q₈	—	2	1	no	—
227	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
228	C₄×D₄	—	2	2	no	—
229	C₄×D₄	—	2	2	no	—
230	#117	—	2	1	no	—
231	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
232	#107	—	2	1	no	—
233	#107	—	2	1	no	—
234	C₄⋊C₈	—	2	2	no	—
235	C₄⋊C₈	—	2	2	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
238	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
239	#103	—	2	4	no	—
240	#102	—	2	2	no	—
241	#102	—	2	2	no	—
242	#102	—	2	4	no	—
243	C₂²⋊C₈	—	2	2	no	—
244	#97	—	2	2	no	—
245	#97	—	2	2	no	—
246	C₄×C₈	—	2	2	no	—
247	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
248	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
250	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
251	C₃₁	#19347	1	1	yes	characteristic, direct product factor, Sylow
252	#41	—	8	4	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
263	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
264	Q₁₆	—	8	4	no	—
265	SD₁₆	—	8	4	no	—
266	D₈	—	4	2	no	—
267	D₈	—	4	2	no	—
268	M₁₆	—	4	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
274	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
285	C₂²⋊C₄	—	2	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
290	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
291	C₄²	#41×C₃₁	1	1	yes	characteristic
292	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
293	C₂³	—	4	2	no	—
294	C₂³	C₃₁×#121	1	1	yes	characteristic
295	Q₈	—	2	4	no	—
296	Q₈	—	4	2	no	—
297	Q₈	—	4	4	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	4	no	—
301	D₄	—	8	2	no	—
302	D₄	—	8	4	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	8	4	no	—
311	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
312	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
313	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
315	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
316	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	4	2	no	—
321	C₈	—	4	4	no	—
322	C₈	C₃₁×#137	1	2	yes	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	2	no	—
335	C₄	—	4	2	no	—
336	C₄	—	8	2	no	—
337	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
338	C₄	C₃₁×#575	1	1	yes	characteristic
339	C₄	C₃₁×#576	1	2	yes	—
340	C₂	—	4	1	no	—
341	C₂	—	16	2	no	—
342	C₂	C₃₁×#3037	1	1	yes	characteristic
343	C₂	C₃₁×#3058	1	1	yes	characteristic
344	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3169	C₂
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×#3216	C₂	semidirect
3. C₃₁×#2960	C₂	semidirect
4. C₃₁×#1694	C₂	semidirect
5. C₃₁×#1688	C₂	semidirect
6. C₃₁×#1682	C₂	semidirect
7. C₃₁×#1568	C₂	semidirect
8. C₃₁×#1557	C₂	semidirect
120. #19347	C₃₁	direct (N × H)
151. #3216	C₂×C₃₁	semidirect
153. #2960	C₂×C₃₁	semidirect
154. #1694	C₂×C₃₁	semidirect
155. #1688	C₂×C₃₁	semidirect
156. #1682	C₂×C₃₁	semidirect
157. #1568	C₂×C₃₁	semidirect
158. #1557	C₂×C₃₁	semidirect
251. C₃₁	#19347	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×#3216	2
C₃₁×#3169	2
C₃₁×#2960	2
C₃₁×#1694	2
C₃₁×#1688	2
C₃₁×#1682	2
C₃₁×#1568	2
C₃₁×#1557	2
#19347	31

Subgroups — C31×#19347 #21476106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19347 #21476106