Subgroups — C₃₁×#13040 #21469799

Order 7936.

508 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#509	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#360	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1972	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
9	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#131	—	2	1	no	—
11	C₃₁×#509	—	2	2	no	—
12	C₃₁×#509	—	2	2	no	—
13	C₂×C₃₁×SD₃₂	C₄	1	2	yes	complemented
14	C₂×C₃₁×D₁₆	C₄	1	2	yes	complemented
15	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
17	C₃₁×#444	—	2	2	no	—
18	C₃₁×#444	—	2	2	no	—
19	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#104	—	2	1	no	—
22	C₂×C₃₁×#102	—	2	1	no	—
23	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
24	C₃₁×#390	—	2	1	no	—
25	C₃₁×#361	—	2	2	no	—
26	C₃₁×#360	—	2	2	no	—
27	C₂×#41×C₃₁	—	2	1	no	—
28	C₂²×D₄×C₃₁	—	2	1	no	—
29	C₃₁×#135	—	2	1	no	—
30	C₃₁×#135	—	2	1	no	—
31	C₃₁×#135	D₄	1	2	yes	—
32	C₃₁×#135	D₄	1	2	yes	—
33	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×SD₁₆×C₃₁	—	2	1	no	—
35	C₂×D₈×C₃₁	—	2	1	no	—
36	C₂×D₈×C₃₁	—	2	1	no	—
37	C₂×D₈×C₃₁	—	2	1	no	—
38	C₂×D₈×C₃₁	—	2	1	no	—
39	C₂×D₈×C₃₁	—	2	1	no	—
40	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₃₁×#131	—	2	4	no	—
43	C₃₁×#131	—	4	2	no	—
44	C₂×M₁₆×C₃₁	—	2	1	no	—
45	C₂×M₁₆×C₃₁	—	4	1	no	—
46	C₂²×C₈×C₃₁	—	4	1	no	—
47	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
51	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
52	C₃₁×SD₃₂	—	2	4	no	—
53	C₃₁×D₁₆	—	2	4	no	—
54	C₃₁×M₃₂	—	2	1	no	—
55	C₃₁×M₃₂	—	2	1	no	—
56	C₃₁×M₃₂	—	2	1	no	—
57	C₃₁×M₃₂	—	2	1	no	—
58	C₂×C₁₆×C₃₁	—	2	1	no	—
59	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
60	C₃₁×#108	—	2	1	no	—
61	C₃₁×#108	D₄	1	2	yes	—
62	C₃₁×#106	—	2	1	no	—
63	C₃₁×#106	—	2	1	no	—
64	C₃₁×#104	—	2	4	no	—
65	C₃₁×#102	—	4	2	no	—
66	C₃₁×#97	—	2	1	no	—
67	C₃₁×#97	—	2	1	no	—
68	C₂⁴×C₃₁	—	4	1	no	—
69	#41×C₃₁	—	2	2	no	—
70	#41×C₃₁	—	2	2	no	—
71	#41×C₃₁	—	4	1	no	—
72	#41×C₃₁	—	4	1	no	—
73	C₂×Q₈×C₃₁	—	2	1	no	—
74	C₂×D₄×C₃₁	—	2	1	no	—
75	C₂×D₄×C₃₁	—	2	2	no	—
76	C₂×D₄×C₃₁	—	4	1	no	—
77	C₂×D₄×C₃₁	—	4	1	no	—
78	C₂×D₄×C₃₁	—	4	1	no	—
79	C₂×D₄×C₃₁	—	4	1	no	—
80	C₂×D₄×C₃₁	—	4	1	no	—
81	C₂×D₄×C₃₁	—	4	1	no	—
82	C₂²×C₄×C₃₁	—	2	1	no	—
83	C₂²×C₄×C₃₁	—	4	1	no	—
84	C₂²×C₄×C₃₁	—	4	1	no	—
85	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
86	Q₁₆×C₃₁	—	2	1	no	—
87	Q₁₆×C₃₁	C₂²⋊C₄	1	2	yes	complemented
88	SD₁₆×C₃₁	—	2	2	no	—
89	SD₁₆×C₃₁	—	2	2	no	—
90	D₈×C₃₁	—	2	2	no	—
91	D₈×C₃₁	—	2	1	no	—
92	D₈×C₃₁	—	2	2	no	—
93	D₈×C₃₁	—	4	1	no	—
94	D₈×C₃₁	—	4	1	no	—
95	D₈×C₃₁	C₂²⋊C₄	1	2	yes	complemented
96	M₁₆×C₃₁	—	2	2	no	—
97	M₁₆×C₃₁	—	4	4	no	—
98	M₁₆×C₃₁	—	4	1	no	—
99	C₂×C₈×C₃₁	—	2	1	no	—
100	C₂×C₈×C₃₁	—	2	1	no	—
101	C₂×C₈×C₃₁	—	4	4	no	—
102	C₂×C₈×C₃₁	—	4	1	no	—
103	C₂×C₈×C₃₁	D₈	1	2	yes	—
104	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
109	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
110	C₄⋊C₄×C₃₁	—	4	1	no	—
111	C₄⋊C₄×C₃₁	—	4	1	no	—
112	C₂²⋊C₄×C₃₁	—	8	4	no	—
113	C₄²×C₃₁	—	2	1	no	—
114	C₄²×C₃₁	—	2	1	no	—
115	C₁₆×C₃₁	—	2	2	no	—
116	C₁₆×C₃₁	—	4	1	no	—
117	#13040	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂³×C₃₁	—	4	1	no	—
119	C₂³×C₃₁	—	4	2	no	—
120	C₂³×C₃₁	—	8	2	no	—
121	C₂³×C₃₁	—	8	2	no	—
122	C₂³×C₃₁	—	8	1	no	—
123	C₂³×C₃₁	—	8	1	no	—
124	C₂³×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
125	Q₈×C₃₁	—	2	2	no	—
126	Q₈×C₃₁	—	4	1	no	—
127	D₄×C₃₁	—	2	2	no	—
128	D₄×C₃₁	—	4	2	no	—
129	D₄×C₃₁	—	4	2	no	—
130	D₄×C₃₁	—	4	2	no	—
131	D₄×C₃₁	—	4	1	no	—
132	D₄×C₃₁	—	4	2	no	—
133	D₄×C₃₁	—	8	1	no	—
134	D₄×C₃₁	—	8	1	no	—
135	C₂×C₄×C₃₁	—	4	1	no	—
136	C₂×C₄×C₃₁	—	4	2	no	—
137	C₂×C₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	—	4	1	no	—
140	C₂×C₄×C₃₁	—	8	2	no	—
141	C₂×C₄×C₃₁	—	8	1	no	—
142	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
144	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
145	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
147	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₈×C₃₁	—	2	2	no	—
149	C₈×C₃₁	—	4	2	no	—
150	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
152	C₈×C₃₁	#121	1	1	yes	characteristic
153	C₈×C₃₁	#123	1	1	yes	characteristic
154	C₂×#509	C₂×C₃₁	1	1	yes	characteristic
155	C₂×#444	C₂×C₃₁	1	1	yes	characteristic, complemented
156	C₂×#361	C₂×C₃₁	1	1	yes	characteristic, complemented
157	C₂×#360	C₂×C₃₁	1	1	yes	characteristic, complemented
158	#1972	C₂×C₃₁	1	1	yes	characteristic, complemented
159	#1383	C₂×C₃₁	1	1	yes	characteristic, complemented
160	#1375	C₂×C₃₁	1	1	yes	characteristic, complemented
161	C₂²×C₃₁	—	2	2	no	—
162	C₂²×C₃₁	—	4	2	no	—
163	C₂²×C₃₁	—	8	2	no	—
164	C₂²×C₃₁	—	8	2	no	—
165	C₂²×C₃₁	—	8	4	no	—
166	C₂²×C₃₁	—	16	2	no	—
167	C₂²×C₃₁	—	16	1	no	—
168	C₂²×C₃₁	—	16	1	no	—
169	C₂²×C₃₁	#442	1	1	yes	characteristic
170	C₂²×C₃₁	#449	1	1	yes	characteristic
171	C₂²×C₃₁	#459	1	1	yes	characteristic
172	C₄×C₃₁	—	4	2	no	—
173	C₄×C₃₁	—	8	1	no	—
174	C₄×C₃₁	—	16	1	no	—
175	C₄×C₃₁	#386	1	1	yes	characteristic
176	C₄×C₃₁	C₄×D₈	1	1	yes	characteristic
177	C₄×C₃₁	#447	1	1	yes	characteristic
178	C₄×C₃₁	#463	1	1	yes	characteristic
179	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
180	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
181	C₂×#131	—	2	1	no	—
182	#509	—	2	2	no	—
183	#509	—	2	2	no	—
184	C₂×SD₃₂	C₄×C₃₁	1	2	yes	complemented
185	C₂×D₁₆	C₄×C₃₁	1	2	yes	complemented
186	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
187	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
188	#444	—	2	2	no	—
189	#444	—	2	2	no	—
190	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
191	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
192	C₂×#104	—	2	1	no	—
193	C₂×#102	—	2	1	no	—
194	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
195	#390	—	2	1	no	—
196	#361	—	2	2	no	—
197	#360	—	2	2	no	—
198	C₂×C₃₁	—	2	1	no	—
199	C₂×C₃₁	—	2	1	no	—
200	C₂×C₃₁	—	8	2	no	—
201	C₂×C₃₁	—	16	2	no	—
202	C₂×C₃₁	#1879	1	1	yes	characteristic
203	C₂×C₃₁	AGL(1,ℤ/16)	1	2	yes	—
204	C₂×#41	—	2	1	no	—
205	C₂²×D₄	—	2	1	no	—
206	#135	—	2	1	no	—
207	#135	—	2	1	no	—
208	#135	D₄×C₃₁	1	2	yes	—
209	#135	D₄×C₃₁	1	2	yes	—
210	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
211	C₂×SD₁₆	—	2	1	no	—
212	C₂×D₈	—	2	1	no	—
213	C₂×D₈	—	2	1	no	—
214	C₂×D₈	—	2	1	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
218	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
219	#131	—	2	4	no	—
220	#131	—	4	2	no	—
221	C₂×M₁₆	—	2	1	no	—
222	C₂×M₁₆	—	4	1	no	—
223	C₂²×C₈	—	4	1	no	—
224	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
225	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₂²⋊C₄	—	4	2	no	—
228	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
229	SD₃₂	—	2	4	no	—
230	D₁₆	—	2	4	no	—
231	M₃₂	—	2	1	no	—
232	M₃₂	—	2	1	no	—
233	M₃₂	—	2	1	no	—
234	M₃₂	—	2	1	no	—
235	C₂×C₁₆	—	2	1	no	—
236	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
237	#108	—	2	1	no	—
238	#108	D₄×C₃₁	1	2	yes	—
239	#106	—	2	1	no	—
240	#106	—	2	1	no	—
241	#104	—	2	4	no	—
242	#102	—	4	2	no	—
243	#97	—	2	1	no	—
244	#97	—	2	1	no	—
245	C₃₁	#13040	1	1	yes	characteristic, direct product factor, Sylow
246	C₂⁴	—	4	1	no	—
247	#41	—	2	2	no	—
248	#41	—	2	2	no	—
249	#41	—	4	1	no	—
250	#41	—	4	1	no	—
251	C₂×Q₈	—	2	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
264	Q₁₆	—	2	1	no	—
265	Q₁₆	C₂²⋊C₄×C₃₁	1	2	yes	complemented
266	SD₁₆	—	2	2	no	—
267	SD₁₆	—	2	2	no	—
268	D₈	—	2	2	no	—
269	D₈	—	2	1	no	—
270	D₈	—	2	2	no	—
271	D₈	—	4	1	no	—
272	D₈	—	4	1	no	—
273	D₈	C₂²⋊C₄×C₃₁	1	2	yes	complemented
274	M₁₆	—	2	2	no	—
275	M₁₆	—	4	4	no	—
276	M₁₆	—	4	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	4	4	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	D₈×C₃₁	1	2	yes	—
282	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
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	#41×C₃₁	1	1	yes	characteristic
287	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	8	4	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	1	no	—
293	C₁₆	—	2	2	no	—
294	C₁₆	—	4	1	no	—
295	C₂³	—	4	1	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	8	2	no	—
298	C₂³	—	8	2	no	—
299	C₂³	—	8	1	no	—
300	C₂³	—	8	1	no	—
301	C₂³	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
302	Q₈	—	2	2	no	—
303	Q₈	—	4	1	no	—
304	D₄	—	2	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	1	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	8	1	no	—
311	D₄	—	8	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	8	2	no	—
318	C₂×C₄	—	8	1	no	—
319	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
320	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
321	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
322	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
323	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
324	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	4	2	no	—
327	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
328	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
329	C₈	C₃₁×#121	1	1	yes	characteristic
330	C₈	C₃₁×#123	1	1	yes	characteristic
331	C₂²	—	2	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	4	no	—
336	C₂²	—	16	2	no	—
337	C₂²	—	16	1	no	—
338	C₂²	—	16	1	no	—
339	C₂²	C₃₁×#442	1	1	yes	characteristic
340	C₂²	C₃₁×#449	1	1	yes	characteristic
341	C₂²	C₃₁×#459	1	1	yes	characteristic
342	C₄	—	4	2	no	—
343	C₄	—	8	1	no	—
344	C₄	—	16	1	no	—
345	C₄	C₃₁×#386	1	1	yes	characteristic
346	C₄	C₄×D₈×C₃₁	1	1	yes	characteristic
347	C₄	C₃₁×#447	1	1	yes	characteristic
348	C₄	C₃₁×#463	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	8	2	no	—
352	C₂	—	16	2	no	—
353	C₂	C₃₁×#1879	1	1	yes	characteristic
354	C₂	C₃₁×AGL(1,ℤ/16)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	#442
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₃₁×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#444	C₂	semidirect
3. C₂×C₃₁×#361	C₂	semidirect
4. C₂×C₃₁×#360	C₂	semidirect
5. C₃₁×#1972	C₂	semidirect
6. C₃₁×#1383	C₂	semidirect
7. C₃₁×#1375	C₂	semidirect
13. C₂×C₃₁×SD₃₂	C₄	semidirect
14. C₂×C₃₁×D₁₆	C₄	semidirect
87. Q₁₆×C₃₁	C₂²⋊C₄	semidirect
95. D₈×C₃₁	C₂²⋊C₄	semidirect
117. #13040	C₃₁	direct (N × H)
155. C₂×#444	C₂×C₃₁	semidirect
156. C₂×#361	C₂×C₃₁	semidirect
157. C₂×#360	C₂×C₃₁	semidirect
158. #1972	C₂×C₃₁	semidirect
159. #1383	C₂×C₃₁	semidirect
160. #1375	C₂×C₃₁	semidirect
184. C₂×SD₃₂	C₄×C₃₁	semidirect
185. C₂×D₁₆	C₄×C₃₁	semidirect
245. C₃₁	#13040	direct (N × H)
265. Q₁₆	C₂²⋊C₄×C₃₁	semidirect
273. D₈	C₂²⋊C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#509	2
C₂×C₃₁×#444	2
C₂×C₃₁×#361	2
C₂×C₃₁×#360	2
C₃₁×#1972	2
C₃₁×#1383	2
C₃₁×#1375	2
#13040	31

Subgroups — C31×#13040 #21469799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13040 #21469799