Subgroups — C₃₁×#19958 #21476717

Order 7936.

570 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3360	C₂	1	2	yes	maximal
2	C₃₁×#3356	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1707	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1691	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1684	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1650	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1632	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1589	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#556	C₂²	1	2	yes	—
11	C₃₁×#554	C₂²	1	2	yes	—
12	C₃₁×#500	C₂²	1	1	yes	characteristic
13	C₃₁×#498	C₂²	1	2	yes	—
14	C₃₁×#494	C₂²	1	1	yes	characteristic
15	C₃₁×#488	—	2	2	no	—
16	C₃₁×#487	C₂²	1	2	yes	—
17	C₃₁×#485	—	2	2	no	—
18	C₃₁×#483	—	2	2	no	—
19	C₃₁×#479	—	2	2	no	—
20	C₃₁×#477	C₂²	1	2	yes	—
21	C₃₁×#476	C₂²	1	2	yes	—
22	C₃₁×#475	C₂²	1	2	yes	—
23	C₃₁×#470	—	2	2	no	—
24	C₃₁×#463	—	2	2	no	—
25	C₃₁×#461	C₂²	1	2	yes	—
26	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
27	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
28	C₄×SD₁₆×C₃₁	—	2	2	no	—
29	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#436	C₂²	1	2	yes	—
31	C₃₁×#428	—	2	2	no	—
32	C₃₁×#335	C₂²	1	1	yes	characteristic
33	C₃₁×#335	C₂²	1	1	yes	characteristic
34	C₃₁×#333	C₂²	1	2	yes	—
35	C₃₁×#332	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×Q₁₆×C₃₁	—	4	1	no	—
40	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
41	C₂×SD₁₆×C₃₁	—	2	2	no	—
42	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
43	C₂×M₁₆×C₃₁	—	2	2	no	—
44	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
45	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
46	C₃₁×#126	—	4	2	no	—
47	C₃₁×#125	—	2	2	no	—
48	C₃₁×#124	—	2	2	no	—
49	C₃₁×C₂²⋊Q₈	—	2	2	no	—
50	C₃₁×C₂²⋊Q₈	—	2	2	no	—
51	C₃₁×C₂²⋊Q₈	—	4	2	no	—
52	C₄×Q₈×C₃₁	—	2	2	no	—
53	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
54	C₄×D₄×C₃₁	—	2	2	no	—
55	C₄×D₄×C₃₁	C₂³	1	2	yes	—
56	C₃₁×#117	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₃₁×#107	—	2	2	no	—
59	C₃₁×#107	—	4	1	no	—
60	C₃₁×#107	D₄	1	1	yes	characteristic
61	C₃₁×#107	D₄	1	1	yes	characteristic
62	C₃₁×#106	—	2	2	no	—
63	C₃₁×#106	—	2	2	no	—
64	C₃₁×#106	—	2	2	no	—
65	C₃₁×#106	—	2	2	no	—
66	C₃₁×C₄⋊C₈	—	2	2	no	—
67	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
68	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
69	C₃₁×#103	—	2	2	no	—
70	C₃₁×#103	—	2	2	no	—
71	C₃₁×#103	—	2	2	no	—
72	C₃₁×#103	—	2	2	no	—
73	C₃₁×#103	—	2	2	no	—
74	C₃₁×#103	D₄	1	2	yes	—
75	C₃₁×#102	—	2	2	no	—
76	C₃₁×#102	D₄	1	2	yes	—
77	C₃₁×C₂²⋊C₈	—	2	2	no	—
78	C₄×C₈×C₃₁	—	2	2	no	—
79	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
85	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₃₁	—	2	2	no	—
87	C₂²×C₄×C₃₁	—	4	2	no	—
88	Q₁₆×C₃₁	—	2	2	no	—
89	Q₁₆×C₃₁	—	8	2	no	—
90	SD₁₆×C₃₁	—	4	4	no	—
91	D₈×C₃₁	—	4	1	no	—
92	M₁₆×C₃₁	—	4	4	no	—
93	C₂×C₈×C₃₁	—	2	1	no	—
94	C₂×C₈×C₃₁	—	2	2	no	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₄⋊C₄×C₃₁	—	4	2	no	—
106	C₄⋊C₄×C₃₁	—	4	2	no	—
107	C₄⋊C₄×C₃₁	—	4	2	no	—
108	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
109	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
110	C₂²⋊C₄×C₃₁	—	2	2	no	—
111	C₂²⋊C₄×C₃₁	—	4	2	no	—
112	C₂²⋊C₄×C₃₁	—	4	2	no	—
113	C₂²⋊C₄×C₃₁	—	4	2	no	—
114	C₄²×C₃₁	—	2	2	no	—
115	C₄²×C₃₁	—	4	2	no	—
116	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
117	#19958	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂³×C₃₁	—	2	2	no	—
119	Q₈×C₃₁	—	2	2	no	—
120	Q₈×C₃₁	—	2	2	no	—
121	Q₈×C₃₁	—	4	2	no	—
122	Q₈×C₃₁	—	4	2	no	—
123	Q₈×C₃₁	—	8	2	no	—
124	D₄×C₃₁	—	2	2	no	—
125	D₄×C₃₁	—	2	2	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	16	2	no	—
134	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
137	C₈×C₃₁	—	2	2	no	—
138	C₈×C₃₁	—	2	1	no	—
139	C₈×C₃₁	—	2	1	no	—
140	C₈×C₃₁	—	2	1	no	—
141	C₈×C₃₁	—	4	2	no	—
142	C₈×C₃₁	—	4	2	no	—
143	C₈×C₃₁	#135	1	2	yes	—
144	#3360	C₂×C₃₁	1	2	yes	—
145	#3356	C₂×C₃₁	1	2	yes	complemented
146	#1707	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#1691	C₂×C₃₁	1	2	yes	complemented
148	#1684	C₂×C₃₁	1	2	yes	complemented
149	#1650	C₂×C₃₁	1	2	yes	complemented
150	#1632	C₂×C₃₁	1	2	yes	complemented
151	#1589	C₂×C₃₁	1	1	yes	characteristic
152	C₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
153	C₂²×C₃₁	—	4	2	no	—
154	C₂²×C₃₁	—	4	2	no	—
155	C₂²×C₃₁	—	4	2	no	—
156	C₂²×C₃₁	D₄²	1	1	yes	characteristic
157	C₄×C₃₁	—	2	1	no	—
158	C₄×C₃₁	—	4	2	no	—
159	C₄×C₃₁	—	4	2	no	—
160	C₄×C₃₁	—	4	2	no	—
161	C₄×C₃₁	—	8	2	no	—
162	C₄×C₃₁	—	8	2	no	—
163	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
164	C₄×C₃₁	#575	1	1	yes	characteristic
165	C₄×C₃₁	#578	1	1	yes	characteristic
166	C₄×C₃₁	#578	1	1	yes	characteristic
167	#556	C₂²×C₃₁	1	2	yes	—
168	#554	C₂²×C₃₁	1	2	yes	—
169	#500	C₂²×C₃₁	1	1	yes	characteristic
170	#498	C₂²×C₃₁	1	2	yes	—
171	#494	C₂²×C₃₁	1	1	yes	characteristic
172	#488	—	2	2	no	—
173	#487	C₂²×C₃₁	1	2	yes	—
174	#485	—	2	2	no	—
175	#483	—	2	2	no	—
176	#479	—	2	2	no	—
177	#477	C₂²×C₃₁	1	2	yes	—
178	#476	C₂²×C₃₁	1	2	yes	—
179	#475	C₂²×C₃₁	1	2	yes	—
180	#470	—	2	2	no	—
181	#463	—	2	2	no	—
182	#461	C₂²×C₃₁	1	2	yes	—
183	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
184	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
185	C₄×SD₁₆	—	2	2	no	—
186	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
187	#436	C₂²×C₃₁	1	2	yes	—
188	#428	—	2	2	no	—
189	#335	C₂²×C₃₁	1	1	yes	characteristic
190	#335	C₂²×C₃₁	1	1	yes	characteristic
191	#333	C₂²×C₃₁	1	2	yes	—
192	#332	C₂²×C₃₁	1	2	yes	—
193	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
194	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
195	C₈²	C₂²×C₃₁	1	1	yes	characteristic
196	C₂×C₃₁	—	8	2	no	—
197	C₂×C₃₁	#3277	1	1	yes	characteristic
198	C₂×C₃₁	#3283	1	1	yes	characteristic
199	C₂×C₃₁	#3291	1	1	yes	characteristic
200	C₂×Q₁₆	—	4	1	no	—
201	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
202	C₂×SD₁₆	—	2	2	no	—
203	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
204	C₂×M₁₆	—	2	2	no	—
205	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
206	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
207	#126	—	4	2	no	—
208	#125	—	2	2	no	—
209	#124	—	2	2	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	C₂²⋊Q₈	—	2	2	no	—
212	C₂²⋊Q₈	—	4	2	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	C₂³×C₃₁	1	2	yes	—
217	#117	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	#107	—	2	2	no	—
220	#107	—	4	1	no	—
221	#107	D₄×C₃₁	1	1	yes	characteristic
222	#107	D₄×C₃₁	1	1	yes	characteristic
223	#106	—	2	2	no	—
224	#106	—	2	2	no	—
225	#106	—	2	2	no	—
226	#106	—	2	2	no	—
227	C₄⋊C₈	—	2	2	no	—
228	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
229	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
230	#103	—	2	2	no	—
231	#103	—	2	2	no	—
232	#103	—	2	2	no	—
233	#103	—	2	2	no	—
234	#103	—	2	2	no	—
235	#103	D₄×C₃₁	1	2	yes	—
236	#102	—	2	2	no	—
237	#102	D₄×C₃₁	1	2	yes	—
238	C₂²⋊C₈	—	2	2	no	—
239	C₄×C₈	—	2	2	no	—
240	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
243	C₃₁	#19958	1	1	yes	characteristic, direct product factor, Sylow
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
247	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	4	2	no	—
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	8	2	no	—
252	SD₁₆	—	4	4	no	—
253	D₈	—	4	1	no	—
254	M₁₆	—	4	4	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
271	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
279	C₂³	—	2	2	no	—
280	Q₈	—	2	2	no	—
281	Q₈	—	2	2	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	8	2	no	—
285	D₄	—	2	2	no	—
286	D₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	16	2	no	—
295	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
297	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	4	2	no	—
303	C₈	—	4	2	no	—
304	C₈	C₃₁×#135	1	2	yes	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	D₄²×C₃₁	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	4	2	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
316	C₄	C₃₁×#575	1	1	yes	characteristic
317	C₄	C₃₁×#578	1	1	yes	characteristic
318	C₄	C₃₁×#578	1	1	yes	characteristic
319	C₂	—	8	2	no	—
320	C₂	C₃₁×#3277	1	1	yes	characteristic
321	C₂	C₃₁×#3283	1	1	yes	characteristic
322	C₂	C₃₁×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×D₈×C₃₁	D₄
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
2. C₃₁×#3356	C₂	semidirect
3. C₃₁×#1707	C₂	semidirect
4. C₃₁×#1691	C₂	semidirect
5. C₃₁×#1684	C₂	semidirect
6. C₃₁×#1650	C₂	semidirect
7. C₃₁×#1632	C₂	semidirect
9. C₈×Q₁₆×C₃₁	C₂	semidirect
117. #19958	C₃₁	direct (N × H)
145. #3356	C₂×C₃₁	semidirect
146. #1707	C₂×C₃₁	semidirect
147. #1691	C₂×C₃₁	semidirect
148. #1684	C₂×C₃₁	semidirect
149. #1650	C₂×C₃₁	semidirect
150. #1632	C₂×C₃₁	semidirect
152. C₈×Q₁₆	C₂×C₃₁	semidirect
243. C₃₁	#19958	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×#3360	2
C₃₁×#3356	2
C₃₁×#1707	2
C₃₁×#1691	2
C₃₁×#1684	2
C₃₁×#1650	2
C₃₁×#1632	2
C₃₁×#1589	2
C₈×Q₁₆×C₃₁	2
#19958	31

Subgroups — C31×#19958 #21476717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19958 #21476717