Subgroups — C₃₁×#19747 #21476506

Order 7936.

908 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×#3184	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3182	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3151	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3138	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1721	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1717	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1714	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1706	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1455	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
11	C₃₁×#533	C₂²	1	1	yes	characteristic
12	C₃₁×#532	C₂²	1	2	yes	—
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₃₁×#500	C₂²	1	2	yes	complemented
15	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
16	C₃₁×#496	—	2	4	no	—
17	C₃₁×#493	C₂²	1	2	yes	—
18	C₃₁×#493	C₂²	1	2	yes	—
19	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
20	C₃₁×#492	C₂²	1	2	yes	—
21	C₃₁×#492	C₂²	1	2	yes	—
22	C₃₁×#492	C₂²	1	2	yes	complemented
23	C₃₁×#490	—	2	2	no	—
24	C₃₁×#487	C₂²	1	2	yes	complemented
25	C₃₁×#486	C₂²	1	2	yes	—
26	C₃₁×#486	C₂²	1	2	yes	—
27	C₃₁×#486	C₂²	1	2	yes	complemented
28	C₃₁×#486	C₂²	1	2	yes	complemented
29	C₃₁×#469	—	2	2	no	—
30	C₃₁×#468	—	2	4	no	—
31	C₃₁×#442	—	2	2	no	—
32	C₃₁×#440	—	2	4	no	—
33	C₃₁×#418	—	2	2	no	—
34	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
36	C₃₁×#322	C₂²	1	2	yes	complemented
37	C₃₁×#322	C₂²	1	1	yes	characteristic, complemented
38	C₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
39	C₂×#41×C₃₁	—	2	2	no	—
40	C₂²×D₄×C₃₁	—	2	1	no	—
41	C₂²×D₄×C₃₁	—	2	2	no	—
42	C₃₁×AGL(1,ℤ/8)	—	4	16	no	—
43	C₂×SD₁₆×C₃₁	—	2	4	no	—
44	C₂×SD₁₆×C₃₁	—	2	4	no	—
45	C₂×SD₁₆×C₃₁	—	2	4	no	—
46	C₂×D₈×C₃₁	—	2	2	no	—
47	C₂×D₈×C₃₁	—	2	4	no	—
48	C₂×D₈×C₃₁	—	2	4	no	—
49	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
50	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
51	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
52	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
53	C₃₁×#127	—	2	1	no	—
54	C₃₁×#127	—	2	2	no	—
55	C₃₁×#127	C₂³	1	2	yes	—
56	C₃₁×#127	C₂³	1	2	yes	—
57	C₃₁×#125	—	2	1	no	—
58	C₃₁×#124	—	2	2	no	—
59	C₃₁×C₂²⋊Q₈	—	2	2	no	—
60	C₃₁×#121	—	2	2	no	—
61	C₃₁×#121	—	2	2	no	—
62	C₄×Q₈×C₃₁	—	2	2	no	—
63	C₄×D₄×C₃₁	—	2	2	no	—
64	C₄×D₄×C₃₁	—	2	2	no	—
65	C₃₁×#117	—	2	1	no	—
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#107	—	2	4	no	—
68	C₃₁×#106	—	2	2	no	—
69	C₃₁×#103	—	2	4	no	—
70	C₃₁×#102	—	2	4	no	—
71	C₃₁×#102	—	2	4	no	—
72	C₃₁×#102	—	2	4	no	—
73	C₃₁×#102	—	2	4	no	—
74	C₃₁×#97	D₄	1	4	yes	—
75	C₃₁×#97	D₄	1	2	yes	—
76	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₃₁	C₂³	1	2	yes	—
79	C₄×C₈×C₃₁	C₂³	1	2	yes	—
80	C₂⁴×C₃₁	—	4	1	no	—
81	#41×C₃₁	—	4	8	no	—
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
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	—	2	2	no	—
90	C₂×D₄×C₃₁	—	4	8	no	—
91	C₂×D₄×C₃₁	—	4	4	no	—
92	C₂×D₄×C₃₁	—	4	2	no	—
93	C₂×D₄×C₃₁	—	4	1	no	—
94	C₂×D₄×C₃₁	—	4	2	no	—
95	C₂²×C₄×C₃₁	—	4	2	no	—
96	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
98	SD₁₆×C₃₁	—	4	8	no	—
99	SD₁₆×C₃₁	—	4	8	no	—
100	SD₁₆×C₃₁	—	4	8	no	—
101	D₈×C₃₁	—	4	4	no	—
102	D₈×C₃₁	—	4	8	no	—
103	D₈×C₃₁	—	4	8	no	—
104	M₁₆×C₃₁	—	2	8	no	—
105	M₁₆×C₃₁	—	4	2	no	—
106	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
109	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
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	—	2	2	no	—
114	C₄⋊C₄×C₃₁	—	4	2	no	—
115	C₂²⋊C₄×C₃₁	—	4	1	no	—
116	C₂²⋊C₄×C₃₁	—	4	2	no	—
117	C₂²⋊C₄×C₃₁	—	4	2	no	—
118	C₄²×C₃₁	—	4	1	no	—
119	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₃₁	C₂×D₄	1	2	yes	—
121	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
122	#19747	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂³×C₃₁	—	4	2	no	—
124	C₂³×C₃₁	—	4	2	no	—
125	C₂³×C₃₁	—	8	4	no	—
126	C₂³×C₃₁	—	8	2	no	—
127	C₂³×C₃₁	#127	1	1	yes	characteristic
128	Q₈×C₃₁	—	4	4	no	—
129	Q₈×C₃₁	—	4	4	no	—
130	D₄×C₃₁	—	4	4	no	—
131	D₄×C₃₁	—	4	4	no	—
132	D₄×C₃₁	—	4	4	no	—
133	D₄×C₃₁	—	4	4	no	—
134	D₄×C₃₁	—	4	4	no	—
135	D₄×C₃₁	—	4	4	no	—
136	D₄×C₃₁	—	8	4	no	—
137	D₄×C₃₁	—	8	2	no	—
138	D₄×C₃₁	—	8	4	no	—
139	C₂×C₄×C₃₁	—	2	4	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	—	4	2	no	—
143	C₂×C₄×C₃₁	—	8	4	no	—
144	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
145	C₂×C₄×C₃₁	#127	1	2	yes	—
146	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
147	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₃₁	—	2	4	no	—
149	C₈×C₃₁	—	2	2	no	—
150	C₈×C₃₁	—	2	2	no	—
151	C₈×C₃₁	AGL(1,ℤ/8)	1	8	yes	complemented
152	#3184	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#3182	C₂×C₃₁	1	2	yes	complemented
154	#3151	C₂×C₃₁	1	2	yes	complemented
155	#3138	C₂×C₃₁	1	1	yes	characteristic, complemented
156	#1721	C₂×C₃₁	1	2	yes	complemented
157	#1717	C₂×C₃₁	1	2	yes	complemented
158	#1714	C₂×C₃₁	1	2	yes	complemented
159	#1706	C₂×C₃₁	1	2	yes	complemented
160	#1455	C₂×C₃₁	1	1	yes	characteristic, complemented
161	C₂²×C₃₁	—	2	2	no	—
162	C₂²×C₃₁	—	2	1	no	—
163	C₂²×C₃₁	—	8	2	no	—
164	C₂²×C₃₁	—	8	2	no	—
165	C₂²×C₃₁	—	8	2	no	—
166	C₂²×C₃₁	—	8	2	no	—
167	C₂²×C₃₁	—	8	2	no	—
168	C₂²×C₃₁	—	8	2	no	—
169	C₂²×C₃₁	—	16	4	no	—
170	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
171	C₄×C₃₁	—	2	2	no	—
172	C₄×C₃₁	—	2	1	no	—
173	C₄×C₃₁	—	8	2	no	—
174	C₄×C₃₁	—	8	2	no	—
175	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
176	C₄×C₃₁	#575	1	2	yes	—
177	C₂×AGL(1,ℤ/8)	—	2	4	no	—
178	#533	C₂²×C₃₁	1	1	yes	characteristic
179	#532	C₂²×C₃₁	1	2	yes	—
180	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
181	#500	C₂²×C₃₁	1	2	yes	complemented
182	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
183	#496	—	2	4	no	—
184	#493	C₂²×C₃₁	1	2	yes	—
185	#493	C₂²×C₃₁	1	2	yes	—
186	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
187	#492	C₂²×C₃₁	1	2	yes	—
188	#492	C₂²×C₃₁	1	2	yes	—
189	#492	C₂²×C₃₁	1	2	yes	complemented
190	#490	—	2	2	no	—
191	#487	C₂²×C₃₁	1	2	yes	complemented
192	#486	C₂²×C₃₁	1	2	yes	—
193	#486	C₂²×C₃₁	1	2	yes	—
194	#486	C₂²×C₃₁	1	2	yes	complemented
195	#486	C₂²×C₃₁	1	2	yes	complemented
196	#469	—	2	2	no	—
197	#468	—	2	4	no	—
198	#442	—	2	2	no	—
199	#440	—	2	4	no	—
200	#418	—	2	2	no	—
201	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
202	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
203	#322	C₂²×C₃₁	1	2	yes	complemented
204	#322	C₂²×C₃₁	1	1	yes	characteristic, complemented
205	C₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
206	C₂×C₃₁	—	4	1	no	—
207	C₂×C₃₁	—	16	2	no	—
208	C₂×C₃₁	—	16	2	no	—
209	C₂×C₃₁	#3151	1	2	yes	—
210	C₂×C₃₁	#3153	1	1	yes	characteristic
211	C₂×#41	—	2	2	no	—
212	C₂²×D₄	—	2	2	no	—
213	C₂²×D₄	—	2	1	no	—
214	AGL(1,ℤ/8)	—	4	16	no	—
215	C₂×SD₁₆	—	2	4	no	—
216	C₂×SD₁₆	—	2	4	no	—
217	C₂×SD₁₆	—	2	4	no	—
218	C₂×D₈	—	2	4	no	—
219	C₂×D₈	—	2	4	no	—
220	C₂×D₈	—	2	2	no	—
221	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
222	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
223	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
224	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
225	#127	—	2	1	no	—
226	#127	—	2	2	no	—
227	#127	C₂³×C₃₁	1	2	yes	—
228	#127	C₂³×C₃₁	1	2	yes	—
229	#125	—	2	1	no	—
230	#124	—	2	2	no	—
231	C₂²⋊Q₈	—	2	2	no	—
232	#121	—	2	2	no	—
233	#121	—	2	2	no	—
234	C₄×Q₈	—	2	2	no	—
235	C₄×D₄	—	2	2	no	—
236	C₄×D₄	—	2	2	no	—
237	#117	—	2	1	no	—
238	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
239	#107	—	2	4	no	—
240	#106	—	2	2	no	—
241	#103	—	2	4	no	—
242	#102	—	2	4	no	—
243	#102	—	2	4	no	—
244	#102	—	2	4	no	—
245	#102	—	2	4	no	—
246	#97	D₄×C₃₁	1	4	yes	—
247	#97	D₄×C₃₁	1	2	yes	—
248	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₃₁	1	2	yes	—
250	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
251	C₄×C₈	C₂³×C₃₁	1	2	yes	—
252	C₃₁	#19747	1	1	yes	characteristic, direct product factor, Sylow
253	C₂⁴	—	4	1	no	—
254	#41	—	4	8	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	4	8	no	—
264	C₂×D₄	—	4	4	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
271	SD₁₆	—	4	8	no	—
272	SD₁₆	—	4	8	no	—
273	SD₁₆	—	4	8	no	—
274	D₈	—	4	4	no	—
275	D₈	—	4	8	no	—
276	D₈	—	4	8	no	—
277	M₁₆	—	2	8	no	—
278	M₁₆	—	4	2	no	—
279	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
280	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
281	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
282	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₄²	—	4	1	no	—
292	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
293	C₄²	C₂×D₄×C₃₁	1	2	yes	—
294	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
295	C₂³	—	4	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	8	4	no	—
298	C₂³	—	8	2	no	—
299	C₂³	C₃₁×#127	1	1	yes	characteristic
300	Q₈	—	4	4	no	—
301	Q₈	—	4	4	no	—
302	D₄	—	4	4	no	—
303	D₄	—	4	4	no	—
304	D₄	—	4	4	no	—
305	D₄	—	4	4	no	—
306	D₄	—	4	4	no	—
307	D₄	—	4	4	no	—
308	D₄	—	8	4	no	—
309	D₄	—	8	2	no	—
310	D₄	—	8	4	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	8	4	no	—
316	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
317	C₂×C₄	C₃₁×#127	1	2	yes	—
318	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
319	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
320	C₈	—	2	4	no	—
321	C₈	—	2	2	no	—
322	C₈	—	2	2	no	—
323	C₈	C₃₁×AGL(1,ℤ/8)	1	8	yes	complemented
324	C₂²	—	2	2	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	16	4	no	—
333	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
339	C₄	C₃₁×#575	1	2	yes	—
340	C₂	—	4	1	no	—
341	C₂	—	16	2	no	—
342	C₂	—	16	2	no	—
343	C₂	C₃₁×#3151	1	2	yes	—
344	C₂	C₃₁×#3153	1	1	yes	characteristic

	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×#3184	C₂	semidirect
2. C₃₁×#3182	C₂	semidirect
3. C₃₁×#3151	C₂	semidirect
4. C₃₁×#3138	C₂	semidirect
5. C₃₁×#1721	C₂	semidirect
6. C₃₁×#1717	C₂	semidirect
7. C₃₁×#1714	C₂	semidirect
8. C₃₁×#1706	C₂	semidirect
9. C₃₁×#1455	C₂	semidirect
14. C₃₁×#500	C₂²	semidirect
15. C₃₁×#498	C₂²	semidirect
19. C₃₁×#493	C₂²	semidirect
22. C₃₁×#492	C₂²	semidirect
24. C₃₁×#487	C₂²	semidirect
27. C₃₁×#486	C₂²	semidirect
28. C₃₁×#486	C₂²	semidirect
36. C₃₁×#322	C₂²	semidirect
37. C₃₁×#322	C₂²	semidirect
38. C₈²×C₃₁	C₂²	semidirect
122. #19747	C₃₁	direct (N × H)
151. C₈×C₃₁	AGL(1,ℤ/8)	semidirect
152. #3184	C₂×C₃₁	semidirect
153. #3182	C₂×C₃₁	semidirect
154. #3151	C₂×C₃₁	semidirect
155. #3138	C₂×C₃₁	semidirect
156. #1721	C₂×C₃₁	semidirect
157. #1717	C₂×C₃₁	semidirect
158. #1714	C₂×C₃₁	semidirect
159. #1706	C₂×C₃₁	semidirect
160. #1455	C₂×C₃₁	semidirect
181. #500	C₂²×C₃₁	semidirect
182. #498	C₂²×C₃₁	semidirect
186. #493	C₂²×C₃₁	semidirect
189. #492	C₂²×C₃₁	semidirect
191. #487	C₂²×C₃₁	semidirect
194. #486	C₂²×C₃₁	semidirect
195. #486	C₂²×C₃₁	semidirect
203. #322	C₂²×C₃₁	semidirect
204. #322	C₂²×C₃₁	semidirect
205. C₈²	C₂²×C₃₁	semidirect
252. C₃₁	#19747	direct (N × H)
323. C₈	C₃₁×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3184	2
C₃₁×#3182	2
C₃₁×#3151	2
C₃₁×#3138	2
C₃₁×#1721	2
C₃₁×#1717	2
C₃₁×#1714	2
C₃₁×#1706	2
C₃₁×#1455	2
#19747	31

Subgroups — C31×#19747 #21476506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19747 #21476506