Subgroups — C₃₁×#11211 #21467970

Order 7936.

762 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2316	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#382	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#343	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1736	C₂	1	2	yes	maximal
5	C₂×C₃₁×#339	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂×C₃₁×#117	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#389	—	2	4	no	—
12	C₃₁×#382	—	2	2	no	—
13	C₃₁×#382	C₄	1	4	yes	complemented
14	C₃₁×#380	—	2	2	no	—
15	C₂×C₃₁×#95	—	2	2	no	—
16	C₃₁×#343	—	2	2	no	—
17	C₃₁×#340	—	2	4	no	—
18	C₃₁×#339	—	2	4	no	—
19	C₂³×C₄×C₃₁	—	2	1	no	—
20	C₂³×C₄×C₃₁	—	2	1	no	—
21	C₂³×C₄×C₃₁	D₄	1	2	yes	—
22	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
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	Q₈	1	4	yes	—
29	C₂²×C₈×C₃₁	—	2	2	no	—
30	C₃₁×#117	—	2	2	no	—
31	C₃₁×#117	—	2	2	no	—
32	C₃₁×#117	—	2	2	no	—
33	C₃₁×#117	—	2	2	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
40	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₃₁	D₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
44	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₄²×C₃₁	—	2	1	no	—
46	C₂×C₄²×C₃₁	—	2	1	no	—
47	C₂×C₄²×C₃₁	—	2	1	no	—
48	C₂×C₄²×C₃₁	—	2	2	no	—
49	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₃₁	D₄	1	2	yes	—
51	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
52	C₃₁×#95	—	2	4	no	—
53	C₃₁×#95	—	2	8	no	—
54	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	—	2	2	no	—
56	C₂²×C₄×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	—	2	2	no	—
58	C₂²×C₄×C₃₁	—	2	2	no	—
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	—	2	2	no	—
61	C₂²×C₄×C₃₁	—	2	4	no	—
62	C₂²×C₄×C₃₁	—	4	2	no	—
63	C₂²×C₄×C₃₁	—	4	2	no	—
64	C₂²×C₄×C₃₁	—	4	4	no	—
65	C₂²×C₄×C₃₁	—	4	4	no	—
66	C₂²×C₄×C₃₁	—	4	1	no	—
67	C₂²×C₄×C₃₁	—	4	1	no	—
68	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
69	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	#41	1	2	yes	—
78	C₂²×C₄×C₃₁	#41	1	2	yes	—
79	M₁₆×C₃₁	—	2	4	no	—
80	M₁₆×C₃₁	—	2	4	no	—
81	M₁₆×C₃₁	—	4	2	no	—
82	M₁₆×C₃₁	—	4	2	no	—
83	C₂×C₈×C₃₁	—	2	4	no	—
84	C₂×C₈×C₃₁	—	4	2	no	—
85	C₂×C₈×C₃₁	—	4	2	no	—
86	C₄⋊C₄×C₃₁	—	2	4	no	—
87	C₄⋊C₄×C₃₁	—	2	4	no	—
88	C₄⋊C₄×C₃₁	—	2	4	no	—
89	C₄⋊C₄×C₃₁	—	2	4	no	—
90	C₄⋊C₄×C₃₁	—	2	4	no	—
91	C₄⋊C₄×C₃₁	—	4	2	no	—
92	C₄⋊C₄×C₃₁	—	8	4	no	—
93	C₂²⋊C₄×C₃₁	—	4	2	no	—
94	C₂²⋊C₄×C₃₁	—	4	2	no	—
95	C₄²×C₃₁	—	2	2	no	—
96	C₄²×C₃₁	—	2	2	no	—
97	C₄²×C₃₁	—	2	2	no	—
98	C₄²×C₃₁	—	4	2	no	—
99	C₄²×C₃₁	—	4	2	no	—
100	C₄²×C₃₁	C₄⋊C₄	1	4	yes	complemented
101	#11211	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₃₁	—	2	2	no	—
103	C₂³×C₃₁	—	2	2	no	—
104	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₃₁	#120	1	1	yes	characteristic
106	C₂³×C₃₁	#121	1	1	yes	characteristic
107	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂³×C₃₁	#123	1	1	yes	characteristic
109	C₂³×C₃₁	#124	1	1	yes	characteristic
110	C₂³×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₃₁	—	2	4	no	—
112	C₂×C₄×C₃₁	—	2	4	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	4	2	no	—
116	C₂×C₄×C₃₁	—	4	4	no	—
117	C₂×C₄×C₃₁	—	4	4	no	—
118	C₂×C₄×C₃₁	—	4	4	no	—
119	C₂×C₄×C₃₁	—	4	4	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	4	no	—
122	C₂×C₄×C₃₁	—	4	4	no	—
123	C₂×C₄×C₃₁	—	4	4	no	—
124	C₂×C₄×C₃₁	—	8	2	no	—
125	C₂×C₄×C₃₁	—	8	2	no	—
126	C₂×C₄×C₃₁	—	8	2	no	—
127	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₃₁	#121	1	2	yes	—
132	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
133	C₂×C₄×C₃₁	#121	1	2	yes	—
134	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
136	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
137	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₃₁	#123	1	2	yes	—
139	C₂×C₄×C₃₁	#125	1	2	yes	—
140	C₂×C₄×C₃₁	#125	1	2	yes	—
141	C₂×C₄×C₃₁	#126	1	2	yes	—
142	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
143	C₈×C₃₁	—	4	4	no	—
144	#2316	C₂×C₃₁	1	1	yes	characteristic
145	C₂×#382	C₂×C₃₁	1	1	yes	characteristic
146	C₂×#343	C₂×C₃₁	1	1	yes	characteristic
147	#1736	C₂×C₃₁	1	2	yes	—
148	C₂×#339	C₂×C₃₁	1	2	yes	—
149	C₂²×C₃₁	—	2	2	no	—
150	C₂²×C₃₁	—	2	2	no	—
151	C₂²×C₃₁	—	2	1	no	—
152	C₂²×C₃₁	—	2	1	no	—
153	C₂²×C₃₁	—	2	1	no	—
154	C₂²×C₃₁	—	2	1	no	—
155	C₂²×C₃₁	—	2	2	no	—
156	C₂²×C₃₁	—	2	2	no	—
157	C₂²×C₃₁	#380	1	1	yes	characteristic
158	C₂²×C₃₁	#386	1	1	yes	characteristic
159	C₂²×C₃₁	#391	1	1	yes	characteristic
160	C₂²×C₃₁	#394	1	1	yes	characteristic
161	C₂²×C₃₁	#395	1	1	yes	characteristic
162	C₂²×C₃₁	#398	1	1	yes	characteristic
163	C₂²×C₃₁	#400	1	1	yes	characteristic
164	C₂²×C₃₁	#454	1	2	yes	—
165	C₂²×C₃₁	#455	1	2	yes	—
166	C₄×C₃₁	—	4	4	no	—
167	C₄×C₃₁	—	4	4	no	—
168	C₄×C₃₁	—	8	2	no	—
169	C₄×C₃₁	—	8	4	no	—
170	C₄×C₃₁	#387	1	2	yes	—
171	C₄×C₃₁	#389	1	2	yes	—
172	C₄×C₃₁	#396	1	2	yes	—
173	C₄×C₃₁	#398	1	2	yes	—
174	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
175	C₂×#117	C₂²×C₃₁	1	2	yes	—
176	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
177	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
178	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
179	#389	—	2	4	no	—
180	#382	—	2	2	no	—
181	#382	C₄×C₃₁	1	4	yes	complemented
182	#380	—	2	2	no	—
183	C₂×#95	—	2	2	no	—
184	#343	—	2	2	no	—
185	#340	—	2	4	no	—
186	#339	—	2	4	no	—
187	C₂×C₃₁	—	2	1	no	—
188	C₂×C₃₁	—	2	1	no	—
189	C₂×C₃₁	—	2	1	no	—
190	C₂×C₃₁	—	2	1	no	—
191	C₂×C₃₁	#1441	1	1	yes	characteristic
192	C₂×C₃₁	#1885	1	1	yes	characteristic
193	C₂×C₃₁	#1888	1	1	yes	characteristic
194	C₂×C₃₁	#2040	1	2	yes	—
195	C₂×C₃₁	#2061	1	2	yes	—
196	C₂³×C₄	—	2	1	no	—
197	C₂³×C₄	—	2	1	no	—
198	C₂³×C₄	D₄×C₃₁	1	2	yes	—
199	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	2	2	no	—
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	Q₈×C₃₁	1	4	yes	—
206	C₂²×C₈	—	2	2	no	—
207	#117	—	2	2	no	—
208	#117	—	2	2	no	—
209	#117	—	2	2	no	—
210	#117	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
218	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
219	C₂×C₄⋊C₄	D₄×C₃₁	1	4	yes	—
220	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
221	C₂×C₂²⋊C₄	—	2	2	no	—
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	—	2	1	no	—
225	C₂×C₄²	—	2	2	no	—
226	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
227	C₂×C₄²	D₄×C₃₁	1	2	yes	—
228	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
229	#95	—	2	4	no	—
230	#95	—	2	8	no	—
231	C₃₁	#11211	1	1	yes	characteristic, direct product factor, Sylow
232	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	4	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
247	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
252	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
253	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
254	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
255	C₂²×C₄	#41×C₃₁	1	2	yes	—
256	C₂²×C₄	#41×C₃₁	1	2	yes	—
257	M₁₆	—	2	4	no	—
258	M₁₆	—	2	4	no	—
259	M₁₆	—	4	2	no	—
260	M₁₆	—	4	2	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	8	4	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	4	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	C₄⋊C₄×C₃₁	1	4	yes	complemented
279	C₂³	—	2	2	no	—
280	C₂³	—	2	2	no	—
281	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
282	C₂³	C₃₁×#120	1	1	yes	characteristic
283	C₂³	C₃₁×#121	1	1	yes	characteristic
284	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
285	C₂³	C₃₁×#123	1	1	yes	characteristic
286	C₂³	C₃₁×#124	1	1	yes	characteristic
287	C₂³	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
305	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
306	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
307	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
308	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
309	C₂×C₄	C₃₁×#121	1	2	yes	—
310	C₂×C₄	C₃₁×#121	1	2	yes	—
311	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
312	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
313	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
314	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
315	C₂×C₄	C₃₁×#123	1	2	yes	—
316	C₂×C₄	C₃₁×#125	1	2	yes	—
317	C₂×C₄	C₃₁×#125	1	2	yes	—
318	C₂×C₄	C₃₁×#126	1	2	yes	—
319	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
320	C₈	—	4	4	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	C₃₁×#380	1	1	yes	characteristic
330	C₂²	C₃₁×#386	1	1	yes	characteristic
331	C₂²	C₃₁×#391	1	1	yes	characteristic
332	C₂²	C₃₁×#394	1	1	yes	characteristic
333	C₂²	C₃₁×#395	1	1	yes	characteristic
334	C₂²	C₃₁×#398	1	1	yes	characteristic
335	C₂²	C₃₁×#400	1	1	yes	characteristic
336	C₂²	C₃₁×#454	1	2	yes	—
337	C₂²	C₃₁×#455	1	2	yes	—
338	C₄	—	4	4	no	—
339	C₄	—	4	4	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	4	no	—
342	C₄	C₃₁×#387	1	2	yes	—
343	C₄	C₃₁×#389	1	2	yes	—
344	C₄	C₃₁×#396	1	2	yes	—
345	C₄	C₃₁×#398	1	2	yes	—
346	C₂	—	2	1	no	—
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	C₃₁×#1441	1	1	yes	characteristic
351	C₂	C₃₁×#1885	1	1	yes	characteristic
352	C₂	C₃₁×#1888	1	1	yes	characteristic
353	C₂	C₃₁×#2040	1	2	yes	—
354	C₂	C₃₁×#2061	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	#120
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₃₁×#382	C₄	semidirect
100. C₄²×C₃₁	C₄⋊C₄	semidirect
101. #11211	C₃₁	direct (N × H)
181. #382	C₄×C₃₁	semidirect
231. C₃₁	#11211	direct (N × H)
278. C₄²	C₄⋊C₄×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#2316	2
C₂×C₃₁×#382	2
C₂×C₃₁×#343	2
C₃₁×#1736	2
C₂×C₃₁×#339	2
#11211	31

Subgroups — C31×#11211 #21467970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11211 #21467970