Subgroups — C₃₁×#10830 #21467589

Order 7936.

678 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2274	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#344	C₂	1	2	yes	maximal
3	C₂×C₃₁×#344	C₂	1	2	yes	maximal
4	C₂×C₃₁×#343	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#343	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
12	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
13	C₄×C₂²⋊C₄×C₃₁	C₄	1	4	yes	—
14	C₃₁×#344	—	2	4	no	—
15	C₃₁×#344	C₄	1	8	yes	complemented
16	C₃₁×#343	—	2	2	no	—
17	C₃₁×#343	—	2	2	no	—
18	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₃₁	—	2	2	no	—
21	C₂×M₁₆×C₃₁	—	2	2	no	—
22	C₂×M₁₆×C₃₁	—	2	2	no	—
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	2	no	—
26	C₂×M₁₆×C₃₁	—	2	2	no	—
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂×M₁₆×C₃₁	—	2	2	no	—
30	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
31	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₃₁	—	2	2	no	—
33	C₂²×C₈×C₃₁	—	2	2	no	—
34	C₃₁×#117	—	2	2	no	—
35	C₃₁×#117	—	2	1	no	—
36	C₃₁×#117	—	2	1	no	—
37	C₃₁×#117	C₂×C₄	1	4	yes	—
38	C₃₁×#117	D₄	1	2	yes	—
39	C₃₁×#117	Q₈	1	2	yes	—
40	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₃₁	—	2	1	no	—
49	C₂×C₄²×C₃₁	—	2	1	no	—
50	C₂×C₄²×C₃₁	—	2	4	no	—
51	C₃₁×#95	—	2	4	no	—
52	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	—	2	2	no	—
56	C₂²×C₄×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	—	8	2	no	—
58	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	#41	1	2	yes	—
67	C₂²×C₄×C₃₁	#41	1	2	yes	—
68	C₂²×C₄×C₃₁	#41	1	2	yes	—
69	M₁₆×C₃₁	—	2	4	no	—
70	M₁₆×C₃₁	—	2	4	no	—
71	M₁₆×C₃₁	—	2	4	no	—
72	M₁₆×C₃₁	—	2	4	no	—
73	M₁₆×C₃₁	—	4	2	no	—
74	M₁₆×C₃₁	—	4	2	no	—
75	M₁₆×C₃₁	—	4	2	no	—
76	M₁₆×C₃₁	—	4	2	no	—
77	C₂×C₈×C₃₁	—	2	4	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	4	2	no	—
80	C₂×C₈×C₃₁	—	4	2	no	—
81	C₂×C₈×C₃₁	—	4	2	no	—
82	C₂×C₈×C₃₁	—	4	2	no	—
83	C₄⋊C₄×C₃₁	—	2	4	no	—
84	C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₄⋊C₄×C₃₁	—	2	4	no	—
87	C₄⋊C₄×C₃₁	—	2	4	no	—
88	C₂²⋊C₄×C₃₁	—	2	2	no	—
89	C₂²⋊C₄×C₃₁	—	2	2	no	—
90	C₂²⋊C₄×C₃₁	C₄²	1	4	yes	—
91	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
92	C₄²×C₃₁	—	2	4	no	—
93	C₄²×C₃₁	—	2	2	no	—
94	C₄²×C₃₁	—	2	2	no	—
95	C₄²×C₃₁	—	8	4	no	—
96	#10830	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₃₁	—	4	1	no	—
98	C₂³×C₃₁	—	4	1	no	—
99	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
100	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₃₁	#117	1	1	yes	characteristic
103	C₂³×C₃₁	#117	1	1	yes	characteristic
104	C₂³×C₃₁	#117	1	1	yes	characteristic
105	C₂³×C₃₁	#117	1	1	yes	characteristic
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	2	4	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	—	8	2	no	—
115	C₂×C₄×C₃₁	—	8	2	no	—
116	C₂×C₄×C₃₁	#117	1	2	yes	—
117	C₂×C₄×C₃₁	#117	1	2	yes	—
118	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
123	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#123	1	2	yes	—
127	C₂×C₄×C₃₁	#123	1	2	yes	—
128	C₂×C₄×C₃₁	#124	1	2	yes	—
129	C₂×C₄×C₃₁	#125	1	2	yes	—
130	C₂×C₄×C₃₁	#126	1	2	yes	—
131	C₂×C₄×C₃₁	#126	1	2	yes	—
132	C₈×C₃₁	—	4	4	no	—
133	C₈×C₃₁	—	4	4	no	—
134	#2274	C₂×C₃₁	1	1	yes	characteristic
135	C₂×#344	C₂×C₃₁	1	2	yes	—
136	C₂×#344	C₂×C₃₁	1	2	yes	—
137	C₂×#343	C₂×C₃₁	1	1	yes	characteristic
138	C₂×#343	C₂×C₃₁	1	1	yes	characteristic
139	C₂²×C₃₁	—	2	1	no	—
140	C₂²×C₃₁	—	2	1	no	—
141	C₂²×C₃₁	—	2	1	no	—
142	C₂²×C₃₁	—	2	1	no	—
143	C₂²×C₃₁	—	2	1	no	—
144	C₂²×C₃₁	—	2	1	no	—
145	C₂²×C₃₁	—	4	1	no	—
146	C₂²×C₃₁	—	4	1	no	—
147	C₂²×C₃₁	—	4	1	no	—
148	C₂²×C₃₁	—	4	1	no	—
149	C₂²×C₃₁	#376	1	1	yes	characteristic
150	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₃₁	#380	1	1	yes	characteristic
153	C₂²×C₃₁	#381	1	1	yes	characteristic
154	C₂²×C₃₁	#382	1	1	yes	characteristic
155	C₂²×C₃₁	#383	1	1	yes	characteristic
156	C₄×C₃₁	—	4	4	no	—
157	C₄×C₃₁	—	4	4	no	—
158	C₄×C₃₁	—	4	4	no	—
159	C₄×C₃₁	#387	1	2	yes	—
160	C₄×C₃₁	#387	1	2	yes	—
161	C₄×C₃₁	#388	1	2	yes	—
162	C₄×C₃₁	#388	1	2	yes	—
163	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
164	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
165	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
167	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
168	C₄×C₄⋊C₄	—	2	2	no	—
169	C₄×C₄⋊C₄	—	2	2	no	—
170	C₄×C₂²⋊C₄	C₄×C₃₁	1	4	yes	—
171	#344	—	2	4	no	—
172	#344	C₄×C₃₁	1	8	yes	complemented
173	#343	—	2	2	no	—
174	#343	—	2	2	no	—
175	C₂×C₃₁	—	2	1	no	—
176	C₂×C₃₁	—	2	1	no	—
177	C₂×C₃₁	—	2	1	no	—
178	C₂×C₃₁	—	2	1	no	—
179	C₂×C₃₁	—	2	1	no	—
180	C₂×C₃₁	—	2	1	no	—
181	C₂×C₃₁	#1434	1	1	yes	characteristic
182	C₂×C₃₁	#1829	1	1	yes	characteristic
183	C₂×C₃₁	#1830	1	1	yes	characteristic
184	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
185	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
197	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	—	2	1	no	—
202	#117	—	2	1	no	—
203	#117	C₂×C₄×C₃₁	1	4	yes	—
204	#117	D₄×C₃₁	1	2	yes	—
205	#117	Q₈×C₃₁	1	2	yes	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
211	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
212	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
213	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	—	2	4	no	—
217	#95	—	2	4	no	—
218	C₃₁	#10830	1	1	yes	characteristic, direct product factor, Sylow
219	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	8	2	no	—
225	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
226	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
233	C₂²×C₄	#41×C₃₁	1	2	yes	—
234	C₂²×C₄	#41×C₃₁	1	2	yes	—
235	C₂²×C₄	#41×C₃₁	1	2	yes	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	4	2	no	—
241	M₁₆	—	4	2	no	—
242	M₁₆	—	4	2	no	—
243	M₁₆	—	4	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	C₄²×C₃₁	1	4	yes	—
258	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	8	4	no	—
263	C₂³	—	4	1	no	—
264	C₂³	—	4	1	no	—
265	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
266	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
267	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
268	C₂³	C₃₁×#117	1	1	yes	characteristic
269	C₂³	C₃₁×#117	1	1	yes	characteristic
270	C₂³	C₃₁×#117	1	1	yes	characteristic
271	C₂³	C₃₁×#117	1	1	yes	characteristic
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	—	8	2	no	—
282	C₂×C₄	C₃₁×#117	1	2	yes	—
283	C₂×C₄	C₃₁×#117	1	2	yes	—
284	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
285	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
286	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
287	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
288	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
289	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
290	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
291	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
292	C₂×C₄	C₃₁×#123	1	2	yes	—
293	C₂×C₄	C₃₁×#123	1	2	yes	—
294	C₂×C₄	C₃₁×#124	1	2	yes	—
295	C₂×C₄	C₃₁×#125	1	2	yes	—
296	C₂×C₄	C₃₁×#126	1	2	yes	—
297	C₂×C₄	C₃₁×#126	1	2	yes	—
298	C₈	—	4	4	no	—
299	C₈	—	4	4	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	C₃₁×#376	1	1	yes	characteristic
311	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
312	C₂²	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
313	C₂²	C₃₁×#380	1	1	yes	characteristic
314	C₂²	C₃₁×#381	1	1	yes	characteristic
315	C₂²	C₃₁×#382	1	1	yes	characteristic
316	C₂²	C₃₁×#383	1	1	yes	characteristic
317	C₄	—	4	4	no	—
318	C₄	—	4	4	no	—
319	C₄	—	4	4	no	—
320	C₄	C₃₁×#387	1	2	yes	—
321	C₄	C₃₁×#387	1	2	yes	—
322	C₄	C₃₁×#388	1	2	yes	—
323	C₄	C₃₁×#388	1	2	yes	—
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	C₃₁×#1434	1	1	yes	characteristic
331	C₂	C₃₁×#1829	1	1	yes	characteristic
332	C₂	C₃₁×#1830	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#381
commutator	C₂³	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
15. C₃₁×#344	C₄	semidirect
96. #10830	C₃₁	direct (N × H)
172. #344	C₄×C₃₁	semidirect
218. C₃₁	#10830	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×#2274	2
C₂×C₃₁×#344	2
C₂×C₃₁×#343	2
#10830	31

Subgroups — C31×#10830 #21467589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10830 #21467589