Subgroups — C₃₁×#19938 #21476697

Order 7936.

558 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3383	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3372	C₂	1	2	yes	maximal
3	C₃₁×#3338	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1710	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1704	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1676	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1673	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1643	C₂	1	2	yes	maximal
9	C₃₁×#1636	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1591	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#557	C₂²	1	2	yes	—
13	C₃₁×#554	C₂²	1	2	yes	—
14	C₃₁×#500	C₂²	1	1	yes	characteristic
15	C₃₁×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#499	—	2	1	no	—
17	C₃₁×#491	—	2	2	no	—
18	C₃₁×#487	C₂²	1	2	yes	—
19	C₃₁×#487	C₂²	1	2	yes	—
20	C₃₁×#484	—	2	2	no	—
21	C₃₁×#479	—	2	2	no	—
22	C₃₁×#479	—	2	2	no	—
23	C₃₁×#477	C₂²	1	2	yes	—
24	C₃₁×#477	C₂²	1	2	yes	—
25	C₃₁×#475	C₂²	1	2	yes	—
26	C₃₁×#467	—	2	1	no	—
27	C₃₁×#461	—	2	2	no	—
28	C₃₁×#461	C₂²	1	2	yes	—
29	C₃₁×#446	C₂²	1	2	yes	—
30	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#440	—	2	2	no	—
32	C₄×Q₁₆×C₃₁	—	2	1	no	—
33	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
34	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
35	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#107	—	2	1	no	—
37	C₃₁×#335	C₂²	1	1	yes	characteristic
38	C₃₁×#334	C₂²	1	1	yes	characteristic
39	C₃₁×#333	C₂²	1	2	yes	—
40	C₃₁×#332	C₂²	1	2	yes	—
41	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
43	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic
45	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
46	C₂×SD₁₆×C₃₁	—	2	2	no	—
47	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	—	2	1	no	—
49	C₃₁×C₄⋊Q₈	—	2	2	no	—
50	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
51	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
52	C₃₁×#125	—	2	2	no	—
53	C₃₁×#125	—	2	2	no	—
54	C₃₁×#125	—	4	2	no	—
55	C₃₁×C₂²⋊Q₈	—	2	2	no	—
56	C₃₁×C₂²⋊Q₈	—	4	2	no	—
57	C₄×Q₈×C₃₁	—	2	2	no	—
58	C₄×Q₈×C₃₁	—	2	2	no	—
59	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₃₁×#107	—	2	1	no	—
65	C₃₁×#107	—	2	1	no	—
66	C₃₁×#107	—	2	1	no	—
67	C₃₁×#107	—	2	1	no	—
68	C₃₁×#107	—	2	2	no	—
69	C₃₁×#107	—	2	2	no	—
70	C₃₁×#107	D₄	1	1	yes	characteristic
71	C₃₁×#106	—	2	2	no	—
72	C₃₁×#106	D₄	1	1	yes	characteristic
73	C₃₁×C₄⋊C₈	—	2	1	no	—
74	C₃₁×C₄⋊C₈	—	2	2	no	—
75	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
76	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
77	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
78	C₃₁×#103	—	2	2	no	—
79	C₃₁×#103	—	2	2	no	—
80	C₃₁×#103	—	2	2	no	—
81	C₃₁×#103	—	2	2	no	—
82	C₃₁×#103	—	2	2	no	—
83	C₃₁×#103	—	2	2	no	—
84	C₃₁×#103	—	4	2	no	—
85	C₃₁×#103	D₄	1	1	yes	characteristic
86	C₃₁×#103	D₄	1	2	yes	—
87	C₃₁×#102	D₄	1	1	yes	characteristic
88	C₃₁×C₂²⋊C₈	—	2	1	no	—
89	C₃₁×#97	—	2	2	no	—
90	C₄×C₈×C₃₁	—	2	1	no	—
91	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₃₁	—	2	2	no	—
95	C₂×Q₈×C₃₁	—	2	2	no	—
96	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
97	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	4	2	no	—
101	Q₁₆×C₃₁	—	2	2	no	—
102	SD₁₆×C₃₁	—	2	2	no	—
103	SD₁₆×C₃₁	—	4	4	no	—
104	C₂×C₈×C₃₁	—	2	1	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	1	no	—
107	C₂×C₈×C₃₁	—	2	1	no	—
108	C₂×C₈×C₃₁	—	4	2	no	—
109	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₄⋊C₄×C₃₁	—	2	1	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	4	2	no	—
120	C₄⋊C₄×C₃₁	—	4	2	no	—
121	C₄⋊C₄×C₃₁	—	4	2	no	—
122	C₄⋊C₄×C₃₁	—	4	2	no	—
123	C₄⋊C₄×C₃₁	—	4	2	no	—
124	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
127	C₂²⋊C₄×C₃₁	—	2	1	no	—
128	C₂²⋊C₄×C₃₁	—	4	2	no	—
129	C₄²×C₃₁	—	2	1	no	—
130	C₄²×C₃₁	—	2	2	no	—
131	C₄²×C₃₁	—	4	2	no	—
132	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
133	#19938	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂³×C₃₁	—	2	1	no	—
135	Q₈×C₃₁	—	2	1	no	—
136	Q₈×C₃₁	—	2	2	no	—
137	Q₈×C₃₁	—	2	2	no	—
138	Q₈×C₃₁	—	2	1	no	—
139	Q₈×C₃₁	—	4	2	no	—
140	Q₈×C₃₁	—	4	2	no	—
141	Q₈×C₃₁	—	8	2	no	—
142	D₄×C₃₁	—	2	1	no	—
143	D₄×C₃₁	#135	1	2	yes	—
144	C₂×C₄×C₃₁	—	2	1	no	—
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	2	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	4	1	no	—
150	C₂×C₄×C₃₁	—	4	1	no	—
151	C₂×C₄×C₃₁	—	4	2	no	—
152	C₂×C₄×C₃₁	—	4	2	no	—
153	C₂×C₄×C₃₁	—	16	2	no	—
154	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
157	C₈×C₃₁	—	2	1	no	—
158	C₈×C₃₁	—	2	1	no	—
159	C₈×C₃₁	—	2	1	no	—
160	C₈×C₃₁	—	4	1	no	—
161	C₈×C₃₁	—	4	2	no	—
162	C₈×C₃₁	—	4	1	no	—
163	C₈×C₃₁	—	4	1	no	—
164	C₈×C₃₁	#137	1	2	yes	—
165	#3383	C₂×C₃₁	1	1	yes	characteristic, complemented
166	#3372	C₂×C₃₁	1	2	yes	—
167	#3338	C₂×C₃₁	1	1	yes	characteristic
168	#1710	C₂×C₃₁	1	1	yes	characteristic, complemented
169	#1704	C₂×C₃₁	1	2	yes	complemented
170	#1676	C₂×C₃₁	1	1	yes	characteristic, complemented
171	#1673	C₂×C₃₁	1	1	yes	characteristic
172	#1643	C₂×C₃₁	1	2	yes	—
173	#1636	C₂×C₃₁	1	2	yes	complemented
174	#1591	C₂×C₃₁	1	1	yes	characteristic, complemented
175	C₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
176	C₂²×C₃₁	—	2	2	no	—
177	C₂²×C₃₁	—	4	1	no	—
178	C₂²×C₃₁	—	4	1	no	—
179	C₂²×C₃₁	D₄²	1	1	yes	characteristic
180	C₄×C₃₁	—	2	1	no	—
181	C₄×C₃₁	—	4	1	no	—
182	C₄×C₃₁	—	4	2	no	—
183	C₄×C₃₁	—	4	1	no	—
184	C₄×C₃₁	—	4	1	no	—
185	C₄×C₃₁	—	4	2	no	—
186	C₄×C₃₁	—	8	2	no	—
187	C₄×C₃₁	—	8	2	no	—
188	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
189	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
190	C₄×C₃₁	#578	1	1	yes	characteristic
191	C₄×C₃₁	#578	1	1	yes	characteristic
192	#557	C₂²×C₃₁	1	2	yes	—
193	#554	C₂²×C₃₁	1	2	yes	—
194	#500	C₂²×C₃₁	1	1	yes	characteristic
195	#500	C₂²×C₃₁	1	1	yes	characteristic
196	#499	—	2	1	no	—
197	#491	—	2	2	no	—
198	#487	C₂²×C₃₁	1	2	yes	—
199	#487	C₂²×C₃₁	1	2	yes	—
200	#484	—	2	2	no	—
201	#479	—	2	2	no	—
202	#479	—	2	2	no	—
203	#477	C₂²×C₃₁	1	2	yes	—
204	#477	C₂²×C₃₁	1	2	yes	—
205	#475	C₂²×C₃₁	1	2	yes	—
206	#467	—	2	1	no	—
207	#461	—	2	2	no	—
208	#461	C₂²×C₃₁	1	2	yes	—
209	#446	C₂²×C₃₁	1	2	yes	—
210	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
211	#440	—	2	2	no	—
212	C₄×Q₁₆	—	2	1	no	—
213	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
214	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
215	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
216	C₂×#107	—	2	1	no	—
217	#335	C₂²×C₃₁	1	1	yes	characteristic
218	#334	C₂²×C₃₁	1	1	yes	characteristic
219	#333	C₂²×C₃₁	1	2	yes	—
220	#332	C₂²×C₃₁	1	2	yes	—
221	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
222	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
223	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
224	C₈²	C₂²×C₃₁	1	1	yes	characteristic
225	C₂×C₃₁	—	4	2	no	—
226	C₂×C₃₁	#3276	1	1	yes	characteristic
227	C₂×C₃₁	#3285	1	1	yes	characteristic
228	C₂×C₃₁	#3290	1	1	yes	characteristic
229	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
230	C₂×SD₁₆	—	2	2	no	—
231	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	2	no	—
234	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
235	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
236	#125	—	2	2	no	—
237	#125	—	2	2	no	—
238	#125	—	4	2	no	—
239	C₂²⋊Q₈	—	2	2	no	—
240	C₂²⋊Q₈	—	4	2	no	—
241	C₄×Q₈	—	2	2	no	—
242	C₄×Q₈	—	2	2	no	—
243	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
244	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
245	C₄×D₄	—	2	2	no	—
246	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
247	C₂×C₄⋊C₄	—	2	2	no	—
248	#107	—	2	1	no	—
249	#107	—	2	1	no	—
250	#107	—	2	1	no	—
251	#107	—	2	1	no	—
252	#107	—	2	2	no	—
253	#107	—	2	2	no	—
254	#107	D₄×C₃₁	1	1	yes	characteristic
255	#106	—	2	2	no	—
256	#106	D₄×C₃₁	1	1	yes	characteristic
257	C₄⋊C₈	—	2	1	no	—
258	C₄⋊C₈	—	2	2	no	—
259	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
260	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
261	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
262	#103	—	2	2	no	—
263	#103	—	2	2	no	—
264	#103	—	2	2	no	—
265	#103	—	2	2	no	—
266	#103	—	2	2	no	—
267	#103	—	2	2	no	—
268	#103	—	4	2	no	—
269	#103	D₄×C₃₁	1	1	yes	characteristic
270	#103	D₄×C₃₁	1	2	yes	—
271	#102	D₄×C₃₁	1	1	yes	characteristic
272	C₂²⋊C₈	—	2	1	no	—
273	#97	—	2	2	no	—
274	C₄×C₈	—	2	1	no	—
275	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
276	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
277	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
278	C₃₁	#19938	1	1	yes	characteristic, direct product factor, Sylow
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
282	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
283	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
284	C₂²×C₄	—	2	1	no	—
285	C₂²×C₄	—	4	2	no	—
286	Q₁₆	—	2	2	no	—
287	SD₁₆	—	2	2	no	—
288	SD₁₆	—	4	4	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	4	2	no	—
294	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
295	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
296	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
297	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	—	4	2	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	—	4	2	no	—
308	C₄⋊C₄	—	4	2	no	—
309	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
310	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
311	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
312	C₂²⋊C₄	—	2	1	no	—
313	C₂²⋊C₄	—	4	2	no	—
314	C₄²	—	2	1	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	4	2	no	—
317	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
318	C₂³	—	2	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	2	2	no	—
321	Q₈	—	2	2	no	—
322	Q₈	—	2	1	no	—
323	Q₈	—	4	2	no	—
324	Q₈	—	4	2	no	—
325	Q₈	—	8	2	no	—
326	D₄	—	2	1	no	—
327	D₄	C₃₁×#135	1	2	yes	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	16	2	no	—
338	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
339	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
340	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	4	1	no	—
345	C₈	—	4	2	no	—
346	C₈	—	4	1	no	—
347	C₈	—	4	1	no	—
348	C₈	C₃₁×#137	1	2	yes	—
349	C₂²	—	2	2	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	D₄²×C₃₁	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	2	no	—
358	C₄	—	4	1	no	—
359	C₄	—	8	2	no	—
360	C₄	—	8	2	no	—
361	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
362	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
363	C₄	C₃₁×#578	1	1	yes	characteristic
364	C₄	C₃₁×#578	1	1	yes	characteristic
365	C₂	—	4	2	no	—
366	C₂	C₃₁×#3276	1	1	yes	characteristic
367	C₂	C₃₁×#3285	1	1	yes	characteristic
368	C₂	C₃₁×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×D₄×C₃₁	C₂×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
1. C₃₁×#3383	C₂	semidirect
4. C₃₁×#1710	C₂	semidirect
5. C₃₁×#1704	C₂	semidirect
6. C₃₁×#1676	C₂	semidirect
9. C₃₁×#1636	C₂	semidirect
10. C₃₁×#1591	C₂	semidirect
133. #19938	C₃₁	direct (N × H)
165. #3383	C₂×C₃₁	semidirect
168. #1710	C₂×C₃₁	semidirect
169. #1704	C₂×C₃₁	semidirect
170. #1676	C₂×C₃₁	semidirect
173. #1636	C₂×C₃₁	semidirect
174. #1591	C₂×C₃₁	semidirect
278. C₃₁	#19938	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×#3383	2
C₃₁×#3372	2
C₃₁×#3338	2
C₃₁×#1710	2
C₃₁×#1704	2
C₃₁×#1676	2
C₃₁×#1673	2
C₃₁×#1643	2
C₃₁×#1636	2
C₃₁×#1591	2
C₈×SD₁₆×C₃₁	2
#19938	31

Subgroups — C31×#19938 #21476697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19938 #21476697