Subgroups — C₃₁×#19688 #21476447

Order 7936.

724 conjugacy classes of subgroups.

The subgroup lattice has 332 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×#3182	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1723	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1722	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1719	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1715	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#533	C₂²	1	2	yes	—
11	C₃₁×#532	C₂²	1	2	yes	—
12	C₃₁×#500	C₂²	1	1	yes	characteristic
13	C₃₁×#500	C₂²	1	1	yes	characteristic
14	C₃₁×#500	C₂²	1	1	yes	characteristic
15	C₃₁×#500	C₂²	1	2	yes	—
16	C₃₁×#499	—	2	2	no	—
17	C₃₁×#498	C₂²	1	2	yes	—
18	C₃₁×#494	C₂²	1	2	yes	—
19	C₃₁×#493	C₂²	1	1	yes	characteristic
20	C₃₁×#492	C₂²	1	2	yes	—
21	C₃₁×#492	C₂²	1	2	yes	—
22	C₃₁×#487	C₂²	1	2	yes	—
23	C₃₁×#487	C₂²	1	2	yes	—
24	C₃₁×#486	C₂²	1	2	yes	—
25	C₃₁×#486	C₂²	1	2	yes	—
26	C₃₁×#486	C₂²	1	2	yes	—
27	C₃₁×#468	—	2	4	no	—
28	C₃₁×#466	—	2	2	no	—
29	C₃₁×#464	—	2	4	no	—
30	C₃₁×#452	—	2	4	no	—
31	C₃₁×#440	—	2	4	no	—
32	C₄×D₈×C₃₁	—	2	2	no	—
33	C₃₁×#431	C₂²	1	1	yes	characteristic
34	C₃₁×#427	—	2	2	no	—
35	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#322	C₂²	1	2	yes	—
38	C₃₁×#322	C₂²	1	2	yes	—
39	C₂×#41×C₃₁	—	2	2	no	—
40	C₂×Q₁₆×C₃₁	—	2	4	no	—
41	C₂×SD₁₆×C₃₁	—	2	4	no	—
42	C₂×SD₁₆×C₃₁	—	2	4	no	—
43	C₂×D₈×C₃₁	—	2	2	no	—
44	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
45	C₂²×C₈×C₃₁	D₄	1	2	yes	—
46	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₃₁×#127	C₂³	1	2	yes	—
50	C₃₁×#125	—	2	2	no	—
51	C₃₁×#124	—	2	2	no	—
52	C₃₁×C₂²⋊Q₈	—	2	2	no	—
53	C₃₁×C₂²⋊Q₈	—	2	2	no	—
54	C₃₁×#121	—	2	2	no	—
55	C₃₁×#121	—	2	2	no	—
56	C₄×Q₈×C₃₁	—	2	2	no	—
57	C₄×Q₈×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	—	2	2	no	—
59	C₄×D₄×C₃₁	—	2	2	no	—
60	C₃₁×#117	—	2	2	no	—
61	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₃₁×#107	—	2	2	no	—
63	C₃₁×#107	—	2	2	no	—
64	C₃₁×#107	—	2	2	no	—
65	C₃₁×#107	—	2	4	no	—
66	C₃₁×#106	—	2	4	no	—
67	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
68	C₃₁×#103	—	2	4	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×C₂²⋊C₈	D₄	1	2	yes	—
74	C₃₁×#97	D₄	1	2	yes	—
75	C₄×C₈×C₃₁	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	#41×C₃₁	—	8	4	no	—
81	C₂×Q₈×C₃₁	—	2	2	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	—	4	2	no	—
88	C₂²×C₄×C₃₁	—	4	2	no	—
89	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₃₁	—	8	4	no	—
93	SD₁₆×C₃₁	—	4	8	no	—
94	SD₁₆×C₃₁	—	8	4	no	—
95	D₈×C₃₁	—	4	4	no	—
96	M₁₆×C₃₁	—	4	2	no	—
97	C₂×C₈×C₃₁	—	2	4	no	—
98	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
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	—	2	2	no	—
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	4	2	no	—
109	C₄⋊C₄×C₃₁	—	4	2	no	—
110	C₂²⋊C₄×C₃₁	—	4	2	no	—
111	C₂²⋊C₄×C₃₁	—	4	2	no	—
112	C₂²⋊C₄×C₃₁	—	4	2	no	—
113	C₄²×C₃₁	—	4	2	no	—
114	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
118	#19688	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₃₁	—	4	2	no	—
120	C₂³×C₃₁	#127	1	1	yes	characteristic
121	Q₈×C₃₁	—	4	4	no	—
122	Q₈×C₃₁	—	4	4	no	—
123	Q₈×C₃₁	—	4	4	no	—
124	D₄×C₃₁	—	4	4	no	—
125	D₄×C₃₁	—	4	4	no	—
126	D₄×C₃₁	—	4	4	no	—
127	D₄×C₃₁	—	8	4	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	2	2	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	—	8	4	no	—
135	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
136	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
137	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₃₁	—	2	2	no	—
142	C₈×C₃₁	—	2	4	no	—
143	C₈×C₃₁	—	2	4	no	—
144	C₈×C₃₁	#137	1	4	yes	—
145	#3210	C₂×C₃₁	1	2	yes	complemented
146	#3182	C₂×C₃₁	1	2	yes	complemented
147	#3169	C₂×C₃₁	1	1	yes	characteristic
148	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
149	#1723	C₂×C₃₁	1	2	yes	complemented
150	#1722	C₂×C₃₁	1	2	yes	complemented
151	#1719	C₂×C₃₁	1	2	yes	complemented
152	#1715	C₂×C₃₁	1	2	yes	complemented
153	#1451	C₂×C₃₁	1	1	yes	characteristic, complemented
154	C₂²×C₃₁	—	2	1	no	—
155	C₂²×C₃₁	—	2	1	no	—
156	C₂²×C₃₁	—	2	1	no	—
157	C₂²×C₃₁	—	8	2	no	—
158	C₂²×C₃₁	—	8	2	no	—
159	C₂²×C₃₁	—	8	2	no	—
160	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
161	C₄×C₃₁	—	2	1	no	—
162	C₄×C₃₁	—	2	1	no	—
163	C₄×C₃₁	—	2	1	no	—
164	C₄×C₃₁	—	8	2	no	—
165	C₄×C₃₁	—	8	2	no	—
166	C₄×C₃₁	—	8	2	no	—
167	C₄×C₃₁	C₂×#137	1	2	yes	—
168	C₄×C₃₁	#575	1	2	yes	—
169	C₄×C₃₁	#576	1	2	yes	—
170	#533	C₂²×C₃₁	1	2	yes	—
171	#532	C₂²×C₃₁	1	2	yes	—
172	#500	C₂²×C₃₁	1	1	yes	characteristic
173	#500	C₂²×C₃₁	1	1	yes	characteristic
174	#500	C₂²×C₃₁	1	1	yes	characteristic
175	#500	C₂²×C₃₁	1	2	yes	—
176	#499	—	2	2	no	—
177	#498	C₂²×C₃₁	1	2	yes	—
178	#494	C₂²×C₃₁	1	2	yes	—
179	#493	C₂²×C₃₁	1	1	yes	characteristic
180	#492	C₂²×C₃₁	1	2	yes	—
181	#492	C₂²×C₃₁	1	2	yes	—
182	#487	C₂²×C₃₁	1	2	yes	—
183	#487	C₂²×C₃₁	1	2	yes	—
184	#486	C₂²×C₃₁	1	2	yes	—
185	#486	C₂²×C₃₁	1	2	yes	—
186	#486	C₂²×C₃₁	1	2	yes	—
187	#468	—	2	4	no	—
188	#466	—	2	2	no	—
189	#464	—	2	4	no	—
190	#452	—	2	4	no	—
191	#440	—	2	4	no	—
192	C₄×D₈	—	2	2	no	—
193	#431	C₂²×C₃₁	1	1	yes	characteristic
194	#427	—	2	2	no	—
195	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
197	#322	C₂²×C₃₁	1	2	yes	—
198	#322	C₂²×C₃₁	1	2	yes	—
199	C₂×C₃₁	—	4	1	no	—
200	C₂×C₃₁	—	16	2	no	—
201	C₂×C₃₁	#3150	1	1	yes	characteristic
202	C₂×C₃₁	#3152	1	1	yes	characteristic
203	C₂×C₃₁	#3155	1	1	yes	characteristic
204	C₂×#41	—	2	2	no	—
205	C₂×Q₁₆	—	2	4	no	—
206	C₂×SD₁₆	—	2	4	no	—
207	C₂×SD₁₆	—	2	4	no	—
208	C₂×D₈	—	2	2	no	—
209	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
210	C₂²×C₈	D₄×C₃₁	1	2	yes	—
211	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
212	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
213	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
214	#127	C₂³×C₃₁	1	2	yes	—
215	#125	—	2	2	no	—
216	#124	—	2	2	no	—
217	C₂²⋊Q₈	—	2	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	#121	—	2	2	no	—
220	#121	—	2	2	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×D₄	—	2	2	no	—
224	C₄×D₄	—	2	2	no	—
225	#117	—	2	2	no	—
226	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
227	#107	—	2	2	no	—
228	#107	—	2	2	no	—
229	#107	—	2	2	no	—
230	#107	—	2	4	no	—
231	#106	—	2	4	no	—
232	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
233	#103	—	2	4	no	—
234	#103	—	2	4	no	—
235	#102	—	2	4	no	—
236	#102	—	2	4	no	—
237	#102	—	2	4	no	—
238	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
239	#97	D₄×C₃₁	1	2	yes	—
240	C₄×C₈	D₄×C₃₁	1	2	yes	—
241	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₃₁	1	2	yes	—
244	C₄×C₈	C₂³×C₃₁	1	2	yes	—
245	C₃₁	#19688	1	1	yes	characteristic, direct product factor, Sylow
246	#41	—	8	4	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
258	Q₁₆	—	8	4	no	—
259	SD₁₆	—	4	8	no	—
260	SD₁₆	—	8	4	no	—
261	D₈	—	4	4	no	—
262	M₁₆	—	4	2	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
265	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
266	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
267	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	4	2	no	—
280	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
283	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
284	C₂³	—	4	2	no	—
285	C₂³	C₃₁×#127	1	1	yes	characteristic
286	Q₈	—	4	4	no	—
287	Q₈	—	4	4	no	—
288	Q₈	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	8	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
301	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
302	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	4	no	—
308	C₈	—	2	4	no	—
309	C₈	C₃₁×#137	1	4	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	C₂×C₃₁×#137	1	2	yes	—
324	C₄	C₃₁×#575	1	2	yes	—
325	C₄	C₃₁×#576	1	2	yes	—
326	C₂	—	4	1	no	—
327	C₂	—	16	2	no	—
328	C₂	C₃₁×#3150	1	1	yes	characteristic
329	C₂	C₃₁×#3152	1	1	yes	characteristic
330	C₂	C₃₁×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
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×#3210	C₂	semidirect
2. C₃₁×#3182	C₂	semidirect
4. C₃₁×#3160	C₂	semidirect
5. C₃₁×#1723	C₂	semidirect
6. C₃₁×#1722	C₂	semidirect
7. C₃₁×#1719	C₂	semidirect
8. C₃₁×#1715	C₂	semidirect
9. C₃₁×#1451	C₂	semidirect
118. #19688	C₃₁	direct (N × H)
145. #3210	C₂×C₃₁	semidirect
146. #3182	C₂×C₃₁	semidirect
148. #3160	C₂×C₃₁	semidirect
149. #1723	C₂×C₃₁	semidirect
150. #1722	C₂×C₃₁	semidirect
151. #1719	C₂×C₃₁	semidirect
152. #1715	C₂×C₃₁	semidirect
153. #1451	C₂×C₃₁	semidirect
245. C₃₁	#19688	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×#3182	2
C₃₁×#3169	2
C₃₁×#3160	2
C₃₁×#1723	2
C₃₁×#1722	2
C₃₁×#1719	2
C₃₁×#1715	2
C₃₁×#1451	2
#19688	31

Subgroups — C31×#19688 #21476447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19688 #21476447