Subgroups — C₃₁×#19729 #21476488

Order 7936.

788 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3210	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3205	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3184	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3138	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1721	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1720	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1717	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1709	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1453	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#533	C₂²	1	1	yes	characteristic
11	C₃₁×#532	C₂²	1	2	yes	—
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×#494	C₂²	1	2	yes	complemented
15	C₃₁×#493	C₂²	1	2	yes	—
16	C₃₁×#493	C₂²	1	2	yes	—
17	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
18	C₃₁×#492	C₂²	1	2	yes	complemented
19	C₃₁×#490	—	2	2	no	—
20	C₃₁×#487	C₂²	1	2	yes	complemented
21	C₃₁×#486	C₂²	1	2	yes	—
22	C₃₁×#486	C₂²	1	2	yes	—
23	C₃₁×#486	C₂²	1	2	yes	—
24	C₃₁×#486	C₂²	1	2	yes	—
25	C₃₁×#486	C₂²	1	2	yes	complemented
26	C₃₁×#486	C₂²	1	2	yes	complemented
27	C₃₁×#469	—	2	2	no	—
28	C₃₁×#464	—	2	4	no	—
29	C₃₁×#459	—	2	4	no	—
30	C₃₁×#452	—	2	4	no	—
31	C₃₁×#447	—	2	4	no	—
32	C₃₁×#442	—	2	2	no	—
33	C₃₁×#431	C₂²	1	2	yes	—
34	C₃₁×#418	—	2	2	no	—
35	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#322	C₂²	1	1	yes	characteristic, complemented
37	C₃₁×#322	C₂²	1	2	yes	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	C₂×Q₁₆×C₃₁	—	2	4	no	—
43	C₂×SD₁₆×C₃₁	—	2	4	no	—
44	C₂×D₈×C₃₁	—	2	2	no	—
45	C₂×D₈×C₃₁	—	2	4	no	—
46	C₂×D₈×C₃₁	—	2	4	no	—
47	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
48	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₃₁×#127	—	2	2	no	—
51	C₃₁×#127	—	2	1	no	—
52	C₃₁×#127	C₂³	1	2	yes	—
53	C₃₁×#127	C₂³	1	2	yes	—
54	C₃₁×#125	—	2	1	no	—
55	C₃₁×#124	—	2	2	no	—
56	C₃₁×C₂²⋊Q₈	—	2	2	no	—
57	C₃₁×#121	—	2	2	no	—
58	C₃₁×#121	—	2	2	no	—
59	C₄×Q₈×C₃₁	—	2	2	no	—
60	C₄×D₄×C₃₁	—	2	2	no	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₃₁×#117	—	2	1	no	—
63	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₃₁×#106	—	2	2	no	—
65	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
66	C₃₁×#103	—	2	4	no	—
67	C₃₁×#102	—	2	4	no	—
68	C₃₁×#102	—	2	4	no	—
69	C₃₁×#102	—	2	4	no	—
70	C₃₁×#102	—	2	4	no	—
71	C₃₁×#102	—	2	4	no	—
72	C₃₁×#102	—	2	4	no	—
73	C₃₁×C₂²⋊C₈	D₄	1	4	yes	—
74	C₃₁×#97	D₄	1	2	yes	—
75	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₃₁	C₂³	1	2	yes	—
78	C₄×C₈×C₃₁	C₂³	1	2	yes	—
79	C₂⁴×C₃₁	—	4	1	no	—
80	#41×C₃₁	—	8	4	no	—
81	C₂×Q₈×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	—	2	2	no	—
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	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	4	4	no	—
90	C₂×D₄×C₃₁	—	4	2	no	—
91	C₂×D₄×C₃₁	—	4	1	no	—
92	C₂×D₄×C₃₁	—	4	2	no	—
93	C₂×D₄×C₃₁	—	8	4	no	—
94	C₂²×C₄×C₃₁	—	4	2	no	—
95	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	Q₁₆×C₃₁	—	8	4	no	—
98	SD₁₆×C₃₁	—	8	4	no	—
99	D₈×C₃₁	—	4	4	no	—
100	D₈×C₃₁	—	8	4	no	—
101	D₈×C₃₁	—	8	4	no	—
102	M₁₆×C₃₁	—	4	2	no	—
103	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
106	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
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	—	4	2	no	—
112	C₂²⋊C₄×C₃₁	—	4	1	no	—
113	C₂²⋊C₄×C₃₁	—	4	2	no	—
114	C₂²⋊C₄×C₃₁	—	4	2	no	—
115	C₄²×C₃₁	—	4	1	no	—
116	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₃₁	C₂×D₄	1	2	yes	—
118	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
119	#19729	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₃₁	—	4	2	no	—
121	C₂³×C₃₁	—	4	2	no	—
122	C₂³×C₃₁	—	8	2	no	—
123	C₂³×C₃₁	—	8	4	no	—
124	C₂³×C₃₁	#127	1	1	yes	characteristic
125	Q₈×C₃₁	—	4	4	no	—
126	Q₈×C₃₁	—	4	4	no	—
127	D₄×C₃₁	—	4	4	no	—
128	D₄×C₃₁	—	4	4	no	—
129	D₄×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	—	8	2	no	—
134	D₄×C₃₁	—	8	4	no	—
135	D₄×C₃₁	—	8	4	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	4	no	—
138	C₂×C₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	—	8	4	no	—
141	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
142	C₂×C₄×C₃₁	#127	1	2	yes	—
143	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
145	C₈×C₃₁	—	2	2	no	—
146	C₈×C₃₁	—	2	2	no	—
147	C₈×C₃₁	—	2	4	no	—
148	C₈×C₃₁	—	2	4	no	—
149	#3210	C₂×C₃₁	1	2	yes	complemented
150	#3205	C₂×C₃₁	1	2	yes	complemented
151	#3184	C₂×C₃₁	1	1	yes	characteristic, complemented
152	#3138	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#1721	C₂×C₃₁	1	2	yes	complemented
154	#1720	C₂×C₃₁	1	2	yes	complemented
155	#1717	C₂×C₃₁	1	2	yes	complemented
156	#1709	C₂×C₃₁	1	2	yes	complemented
157	#1453	C₂×C₃₁	1	1	yes	characteristic, complemented
158	C₂²×C₃₁	—	2	1	no	—
159	C₂²×C₃₁	—	2	2	no	—
160	C₂²×C₃₁	—	8	2	no	—
161	C₂²×C₃₁	—	8	2	no	—
162	C₂²×C₃₁	—	8	2	no	—
163	C₂²×C₃₁	—	8	2	no	—
164	C₂²×C₃₁	—	8	2	no	—
165	C₂²×C₃₁	—	8	2	no	—
166	C₂²×C₃₁	—	16	4	no	—
167	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
168	C₄×C₃₁	—	2	1	no	—
169	C₄×C₃₁	—	2	2	no	—
170	C₄×C₃₁	—	8	2	no	—
171	C₄×C₃₁	—	8	2	no	—
172	C₄×C₃₁	#575	1	2	yes	—
173	C₄×C₃₁	#576	1	4	yes	—
174	#533	C₂²×C₃₁	1	1	yes	characteristic
175	#532	C₂²×C₃₁	1	2	yes	—
176	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
177	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
178	#494	C₂²×C₃₁	1	2	yes	complemented
179	#493	C₂²×C₃₁	1	2	yes	—
180	#493	C₂²×C₃₁	1	2	yes	—
181	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
182	#492	C₂²×C₃₁	1	2	yes	complemented
183	#490	—	2	2	no	—
184	#487	C₂²×C₃₁	1	2	yes	complemented
185	#486	C₂²×C₃₁	1	2	yes	—
186	#486	C₂²×C₃₁	1	2	yes	—
187	#486	C₂²×C₃₁	1	2	yes	—
188	#486	C₂²×C₃₁	1	2	yes	—
189	#486	C₂²×C₃₁	1	2	yes	complemented
190	#486	C₂²×C₃₁	1	2	yes	complemented
191	#469	—	2	2	no	—
192	#464	—	2	4	no	—
193	#459	—	2	4	no	—
194	#452	—	2	4	no	—
195	#447	—	2	4	no	—
196	#442	—	2	2	no	—
197	#431	C₂²×C₃₁	1	2	yes	—
198	#418	—	2	2	no	—
199	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
200	#322	C₂²×C₃₁	1	1	yes	characteristic, complemented
201	#322	C₂²×C₃₁	1	2	yes	complemented
202	C₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
203	C₂×C₃₁	—	4	1	no	—
204	C₂×C₃₁	—	16	2	no	—
205	C₂×C₃₁	—	16	2	no	—
206	C₂×C₃₁	#3153	1	1	yes	characteristic
207	C₂×C₃₁	#3155	1	2	yes	—
208	C₂×#41	—	2	2	no	—
209	C₂²×D₄	—	2	1	no	—
210	C₂²×D₄	—	2	2	no	—
211	C₂×Q₁₆	—	2	4	no	—
212	C₂×SD₁₆	—	2	4	no	—
213	C₂×D₈	—	2	2	no	—
214	C₂×D₈	—	2	4	no	—
215	C₂×D₈	—	2	4	no	—
216	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
217	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
218	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
219	#127	—	2	2	no	—
220	#127	—	2	1	no	—
221	#127	C₂³×C₃₁	1	2	yes	—
222	#127	C₂³×C₃₁	1	2	yes	—
223	#125	—	2	1	no	—
224	#124	—	2	2	no	—
225	C₂²⋊Q₈	—	2	2	no	—
226	#121	—	2	2	no	—
227	#121	—	2	2	no	—
228	C₄×Q₈	—	2	2	no	—
229	C₄×D₄	—	2	2	no	—
230	C₄×D₄	—	2	2	no	—
231	#117	—	2	1	no	—
232	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
233	#106	—	2	2	no	—
234	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
235	#103	—	2	4	no	—
236	#102	—	2	4	no	—
237	#102	—	2	4	no	—
238	#102	—	2	4	no	—
239	#102	—	2	4	no	—
240	#102	—	2	4	no	—
241	#102	—	2	4	no	—
242	C₂²⋊C₈	D₄×C₃₁	1	4	yes	—
243	#97	D₄×C₃₁	1	2	yes	—
244	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
245	C₄×C₈	C₂³×C₃₁	1	2	yes	—
246	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₃₁	1	2	yes	—
248	C₃₁	#19729	1	1	yes	characteristic, direct product factor, Sylow
249	C₂⁴	—	4	1	no	—
250	#41	—	8	4	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	4	4	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	8	4	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
266	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
267	Q₁₆	—	8	4	no	—
268	SD₁₆	—	8	4	no	—
269	D₈	—	4	4	no	—
270	D₈	—	8	4	no	—
271	D₈	—	8	4	no	—
272	M₁₆	—	4	2	no	—
273	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
274	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
275	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
276	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₄²	—	4	1	no	—
286	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₃₁	1	2	yes	—
288	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
289	C₂³	—	4	2	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	8	2	no	—
292	C₂³	—	8	4	no	—
293	C₂³	C₃₁×#127	1	1	yes	characteristic
294	Q₈	—	4	4	no	—
295	Q₈	—	4	4	no	—
296	D₄	—	4	4	no	—
297	D₄	—	4	4	no	—
298	D₄	—	4	4	no	—
299	D₄	—	4	4	no	—
300	D₄	—	4	4	no	—
301	D₄	—	4	4	no	—
302	D₄	—	8	2	no	—
303	D₄	—	8	4	no	—
304	D₄	—	8	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	4	no	—
310	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
311	C₂×C₄	C₃₁×#127	1	2	yes	—
312	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
314	C₈	—	2	2	no	—
315	C₈	—	2	2	no	—
316	C₈	—	2	4	no	—
317	C₈	—	2	4	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	16	4	no	—
327	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	C₃₁×#575	1	2	yes	—
333	C₄	C₃₁×#576	1	4	yes	—
334	C₂	—	4	1	no	—
335	C₂	—	16	2	no	—
336	C₂	—	16	2	no	—
337	C₂	C₃₁×#3153	1	1	yes	characteristic
338	C₂	C₃₁×#3155	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×#3210	C₂	semidirect
2. C₃₁×#3205	C₂	semidirect
3. C₃₁×#3184	C₂	semidirect
4. C₃₁×#3138	C₂	semidirect
5. C₃₁×#1721	C₂	semidirect
6. C₃₁×#1720	C₂	semidirect
7. C₃₁×#1717	C₂	semidirect
8. C₃₁×#1709	C₂	semidirect
9. C₃₁×#1453	C₂	semidirect
13. C₃₁×#498	C₂²	semidirect
14. C₃₁×#494	C₂²	semidirect
17. C₃₁×#493	C₂²	semidirect
18. C₃₁×#492	C₂²	semidirect
20. C₃₁×#487	C₂²	semidirect
25. C₃₁×#486	C₂²	semidirect
26. C₃₁×#486	C₂²	semidirect
36. C₃₁×#322	C₂²	semidirect
37. C₃₁×#322	C₂²	semidirect
38. C₈²×C₃₁	C₂²	semidirect
119. #19729	C₃₁	direct (N × H)
149. #3210	C₂×C₃₁	semidirect
150. #3205	C₂×C₃₁	semidirect
151. #3184	C₂×C₃₁	semidirect
152. #3138	C₂×C₃₁	semidirect
153. #1721	C₂×C₃₁	semidirect
154. #1720	C₂×C₃₁	semidirect
155. #1717	C₂×C₃₁	semidirect
156. #1709	C₂×C₃₁	semidirect
157. #1453	C₂×C₃₁	semidirect
177. #498	C₂²×C₃₁	semidirect
178. #494	C₂²×C₃₁	semidirect
181. #493	C₂²×C₃₁	semidirect
182. #492	C₂²×C₃₁	semidirect
184. #487	C₂²×C₃₁	semidirect
189. #486	C₂²×C₃₁	semidirect
190. #486	C₂²×C₃₁	semidirect
200. #322	C₂²×C₃₁	semidirect
201. #322	C₂²×C₃₁	semidirect
202. C₈²	C₂²×C₃₁	semidirect
248. C₃₁	#19729	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×#3210	2
C₃₁×#3205	2
C₃₁×#3184	2
C₃₁×#3138	2
C₃₁×#1721	2
C₃₁×#1720	2
C₃₁×#1717	2
C₃₁×#1709	2
C₃₁×#1453	2
#19729	31

Subgroups — C31×#19729 #21476488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19729 #21476488