Subgroups — C₃₁×#19359 #21476118

Order 7936.

584 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3238	C₂	1	2	yes	maximal
2	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2966	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1696	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1695	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1676	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1675	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1558	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₃₁×#529	C₂²	1	1	yes	characteristic
12	C₃₁×#501	—	2	2	no	—
13	C₃₁×#500	—	2	1	no	—
14	C₃₁×#500	—	2	1	no	—
15	C₃₁×#500	C₂²	1	2	yes	—
16	C₃₁×#500	C₂²	1	2	yes	—
17	C₃₁×#489	—	2	2	no	—
18	C₃₁×#483	—	2	2	no	—
19	C₃₁×#480	—	2	2	no	—
20	C₃₁×#477	C₂²	1	4	yes	—
21	C₃₁×#476	C₂²	1	4	yes	—
22	C₃₁×#446	C₂²	1	1	yes	characteristic
23	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#436	C₂²	1	1	yes	characteristic
25	C₃₁×#435	—	2	1	no	—
26	C₈×D₄×C₃₁	—	2	1	no	—
27	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#433	—	2	1	no	—
29	C₃₁×#432	C₂²	1	1	yes	characteristic
30	C₃₁×#432	C₂²	1	2	yes	—
31	C₂×C₃₁×#107	—	2	1	no	—
32	C₂×C₃₁×#107	—	2	1	no	—
33	C₃₁×#405	—	2	1	no	—
34	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#335	C₂²	1	2	yes	—
36	C₃₁×#334	C₂²	1	2	yes	—
37	C₃₁×#333	C₂²	1	4	yes	—
38	C₃₁×#333	C₂²	1	4	yes	—
39	C₂²×Q₈×C₃₁	—	2	2	no	—
40	C₂×M₁₆×C₃₁	—	2	1	no	—
41	C₂²×C₈×C₃₁	—	2	1	no	—
42	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
44	C₃₁×C₄⋊Q₈	—	2	2	no	—
45	C₃₁×C₄⋊Q₈	—	2	2	no	—
46	C₃₁×C₄⋊Q₈	—	2	2	no	—
47	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
48	C₃₁×#126	—	2	1	no	—
49	C₃₁×#125	D₄	1	1	yes	characteristic
50	C₃₁×#124	D₄	1	1	yes	characteristic
51	C₃₁×#123	—	2	1	no	—
52	C₃₁×C₂²⋊Q₈	D₄	1	1	yes	characteristic
53	C₃₁×#121	D₄	1	1	yes	characteristic
54	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₄×D₄×C₃₁	—	2	1	no	—
56	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₃₁×#117	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×#107	—	2	2	no	—
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	—	2	1	no	—
68	C₃₁×C₄⋊C₈	—	2	1	no	—
69	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
70	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
71	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
72	C₃₁×#103	—	2	4	no	—
73	C₃₁×#102	—	2	4	no	—
74	C₃₁×C₂²⋊C₈	—	2	1	no	—
75	C₃₁×C₂²⋊C₈	—	2	2	no	—
76	C₃₁×C₂²⋊C₈	—	2	1	no	—
77	C₃₁×C₂²⋊C₈	—	2	1	no	—
78	C₃₁×#97	—	2	1	no	—
79	C₃₁×#97	—	2	1	no	—
80	C₃₁×#97	—	2	2	no	—
81	C₄×C₈×C₃₁	—	2	1	no	—
82	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
83	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
84	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×Q₈×C₃₁	—	2	4	no	—
87	C₂×Q₈×C₃₁	—	4	2	no	—
88	C₂×Q₈×C₃₁	—	8	4	no	—
89	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×D₄×C₃₁	—	2	1	no	—
91	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	—	2	1	no	—
93	C₂²×C₄×C₃₁	—	4	2	no	—
94	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
97	M₁₆×C₃₁	—	4	2	no	—
98	C₂×C₈×C₃₁	—	2	2	no	—
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	—	2	1	no	—
101	C₂×C₈×C₃₁	—	2	1	no	—
102	C₂×C₈×C₃₁	—	2	4	no	—
103	C₂×C₈×C₃₁	—	4	2	no	—
104	C₂×C₈×C₃₁	C₂×D₄	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	C₄⋊C₄×C₃₁	—	2	1	no	—
109	C₄⋊C₄×C₃₁	—	2	1	no	—
110	C₄⋊C₄×C₃₁	—	2	1	no	—
111	C₄⋊C₄×C₃₁	—	2	1	no	—
112	C₄⋊C₄×C₃₁	—	2	4	no	—
113	C₄⋊C₄×C₃₁	—	2	4	no	—
114	C₄⋊C₄×C₃₁	—	4	2	no	—
115	C₄⋊C₄×C₃₁	—	4	2	no	—
116	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²⋊C₄×C₃₁	—	2	1	no	—
119	C₂²⋊C₄×C₃₁	—	2	1	no	—
120	C₂²⋊C₄×C₃₁	—	2	1	no	—
121	C₂²⋊C₄×C₃₁	—	2	1	no	—
122	C₂²⋊C₄×C₃₁	—	2	1	no	—
123	C₄²×C₃₁	—	2	1	no	—
124	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₃₁	#41	1	1	yes	characteristic
127	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
128	#19359	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂³×C₃₁	—	2	1	no	—
130	C₂³×C₃₁	#121	1	1	yes	characteristic
131	Q₈×C₃₁	—	2	2	no	—
132	Q₈×C₃₁	—	8	4	no	—
133	Q₈×C₃₁	—	8	4	no	—
134	D₄×C₃₁	—	2	2	no	—
135	D₄×C₃₁	—	4	2	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	2	1	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	1	no	—
141	C₂×C₄×C₃₁	—	2	1	no	—
142	C₂×C₄×C₃₁	—	2	1	no	—
143	C₂×C₄×C₃₁	—	4	1	no	—
144	C₂×C₄×C₃₁	—	4	1	no	—
145	C₂×C₄×C₃₁	—	4	4	no	—
146	C₂×C₄×C₃₁	—	8	4	no	—
147	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
148	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
149	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
150	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
153	C₈×C₃₁	—	2	1	no	—
154	C₈×C₃₁	—	2	1	no	—
155	C₈×C₃₁	—	4	1	no	—
156	C₈×C₃₁	—	4	1	no	—
157	C₈×C₃₁	—	4	4	no	—
158	C₈×C₃₁	#137	1	2	yes	—
159	C₈×C₃₁	#137	1	2	yes	—
160	#3238	C₂×C₃₁	1	2	yes	—
161	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
162	#2966	C₂×C₃₁	1	1	yes	characteristic
163	#1696	C₂×C₃₁	1	2	yes	complemented
164	#1695	C₂×C₃₁	1	2	yes	complemented
165	#1676	C₂×C₃₁	1	2	yes	complemented
166	#1675	C₂×C₃₁	1	2	yes	complemented
167	#1568	C₂×C₃₁	1	1	yes	characteristic, complemented
168	#1558	C₂×C₃₁	1	2	yes	complemented
169	C₂²×C₃₁	—	2	1	no	—
170	C₂²×C₃₁	—	2	1	no	—
171	C₂²×C₃₁	—	4	1	no	—
172	C₂²×C₃₁	—	4	1	no	—
173	C₂²×C₃₁	—	4	1	no	—
174	C₂²×C₃₁	#472	1	2	yes	—
175	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
176	C₄×C₃₁	—	2	1	no	—
177	C₄×C₃₁	—	2	1	no	—
178	C₄×C₃₁	—	2	1	no	—
179	C₄×C₃₁	—	2	1	no	—
180	C₄×C₃₁	—	4	1	no	—
181	C₄×C₃₁	—	4	1	no	—
182	C₄×C₃₁	—	4	1	no	—
183	C₄×C₃₁	—	8	4	no	—
184	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
185	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
186	C₄×C₃₁	#578	1	2	yes	—
187	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
188	#529	C₂²×C₃₁	1	1	yes	characteristic
189	#501	—	2	2	no	—
190	#500	—	2	1	no	—
191	#500	—	2	1	no	—
192	#500	C₂²×C₃₁	1	2	yes	—
193	#500	C₂²×C₃₁	1	2	yes	—
194	#489	—	2	2	no	—
195	#483	—	2	2	no	—
196	#480	—	2	2	no	—
197	#477	C₂²×C₃₁	1	4	yes	—
198	#476	C₂²×C₃₁	1	4	yes	—
199	#446	C₂²×C₃₁	1	1	yes	characteristic
200	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
201	#436	C₂²×C₃₁	1	1	yes	characteristic
202	#435	—	2	1	no	—
203	C₈×D₄	—	2	1	no	—
204	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
205	#433	—	2	1	no	—
206	#432	C₂²×C₃₁	1	1	yes	characteristic
207	#432	C₂²×C₃₁	1	2	yes	—
208	C₂×#107	—	2	1	no	—
209	C₂×#107	—	2	1	no	—
210	#405	—	2	1	no	—
211	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
212	#335	C₂²×C₃₁	1	2	yes	—
213	#334	C₂²×C₃₁	1	2	yes	—
214	#333	C₂²×C₃₁	1	4	yes	—
215	#333	C₂²×C₃₁	1	4	yes	—
216	C₂×C₃₁	—	2	2	no	—
217	C₂×C₃₁	—	8	1	no	—
218	C₂×C₃₁	#3039	1	1	yes	characteristic
219	C₂×C₃₁	#3059	1	1	yes	characteristic
220	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
221	C₂²×Q₈	—	2	2	no	—
222	C₂×M₁₆	—	2	1	no	—
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
225	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
226	C₄⋊Q₈	—	2	2	no	—
227	C₄⋊Q₈	—	2	2	no	—
228	C₄⋊Q₈	—	2	2	no	—
229	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
230	#126	—	2	1	no	—
231	#125	D₄×C₃₁	1	1	yes	characteristic
232	#124	D₄×C₃₁	1	1	yes	characteristic
233	#123	—	2	1	no	—
234	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
235	#121	D₄×C₃₁	1	1	yes	characteristic
236	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
237	C₄×D₄	—	2	1	no	—
238	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
239	#117	—	2	1	no	—
240	C₂×C₄⋊C₄	—	2	2	no	—
241	C₂×C₄⋊C₄	—	2	2	no	—
242	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
243	#107	—	2	2	no	—
244	#107	—	2	2	no	—
245	#107	—	2	2	no	—
246	#107	—	2	2	no	—
247	#107	—	2	4	no	—
248	#106	—	2	4	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
252	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
253	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
254	#103	—	2	4	no	—
255	#102	—	2	4	no	—
256	C₂²⋊C₈	—	2	1	no	—
257	C₂²⋊C₈	—	2	2	no	—
258	C₂²⋊C₈	—	2	1	no	—
259	C₂²⋊C₈	—	2	1	no	—
260	#97	—	2	1	no	—
261	#97	—	2	1	no	—
262	#97	—	2	2	no	—
263	C₄×C₈	—	2	1	no	—
264	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
265	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
266	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
267	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
268	C₃₁	#19359	1	1	yes	characteristic, direct product factor, Sylow
269	C₂×Q₈	—	2	4	no	—
270	C₂×Q₈	—	4	2	no	—
271	C₂×Q₈	—	8	4	no	—
272	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	4	2	no	—
277	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
279	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
280	M₁₆	—	4	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	4	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
288	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
289	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
290	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	4	no	—
296	C₄⋊C₄	—	2	4	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₄⋊C₄	—	4	2	no	—
299	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
300	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
301	C₂²⋊C₄	—	2	1	no	—
302	C₂²⋊C₄	—	2	1	no	—
303	C₂²⋊C₄	—	2	1	no	—
304	C₂²⋊C₄	—	2	1	no	—
305	C₂²⋊C₄	—	2	1	no	—
306	C₄²	—	2	1	no	—
307	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
308	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
309	C₄²	#41×C₃₁	1	1	yes	characteristic
310	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
311	C₂³	—	2	1	no	—
312	C₂³	C₃₁×#121	1	1	yes	characteristic
313	Q₈	—	2	2	no	—
314	Q₈	—	8	4	no	—
315	Q₈	—	8	4	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	—	8	4	no	—
329	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
330	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
331	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
332	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
334	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	4	1	no	—
338	C₈	—	4	1	no	—
339	C₈	—	4	4	no	—
340	C₈	C₃₁×#137	1	2	yes	—
341	C₈	C₃₁×#137	1	2	yes	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	C₃₁×#472	1	2	yes	—
348	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	8	4	no	—
357	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
358	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
359	C₄	C₃₁×#578	1	2	yes	—
360	C₂	—	2	2	no	—
361	C₂	—	8	1	no	—
362	C₂	C₃₁×#3039	1	1	yes	characteristic
363	C₂	C₃₁×#3059	1	1	yes	characteristic
364	C₂	C₂×C₃₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#121	D₄
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
2. C₂×C₃₁×#500	C₂	semidirect
4. C₃₁×#1696	C₂	semidirect
5. C₃₁×#1695	C₂	semidirect
6. C₃₁×#1676	C₂	semidirect
7. C₃₁×#1675	C₂	semidirect
8. C₃₁×#1568	C₂	semidirect
9. C₃₁×#1558	C₂	semidirect
128. #19359	C₃₁	direct (N × H)
161. C₂×#500	C₂×C₃₁	semidirect
163. #1696	C₂×C₃₁	semidirect
164. #1695	C₂×C₃₁	semidirect
165. #1676	C₂×C₃₁	semidirect
166. #1675	C₂×C₃₁	semidirect
167. #1568	C₂×C₃₁	semidirect
168. #1558	C₂×C₃₁	semidirect
268. C₃₁	#19359	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×#3238	2
C₂×C₃₁×#500	2
C₃₁×#2966	2
C₃₁×#1696	2
C₃₁×#1695	2
C₃₁×#1676	2
C₃₁×#1675	2
C₃₁×#1568	2
C₃₁×#1558	2
#19359	31

Subgroups — C31×#19359 #21476118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19359 #21476118