Subgroups — C₃₁×#8255 #21465014

Order 7936.

470 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1540	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1502	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1303	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#131	—	2	1	no	—
9	C₃₁×#532	C₂²	1	1	yes	characteristic
10	C₃₁×#436	—	2	1	no	—
11	C₃₁×#436	C₄	1	2	yes	complemented
12	C₈×D₄×C₃₁	—	2	1	no	—
13	C₈×D₄×C₃₁	C₄	1	2	yes	complemented
14	C₃₁×#431	C₂²	1	1	yes	characteristic
15	C₃₁×#431	C₂²	1	1	yes	characteristic
16	C₃₁×#423	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₃₁×#408	—	2	1	no	—
19	C₂×C₃₁×#97	—	2	1	no	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#336	—	2	1	no	—
23	C₃₁×#332	C₄	1	2	yes	complemented
24	C₃₁×#331	C₄	1	2	yes	complemented
25	C₃₁×#327	C₄	1	4	yes	—
26	C₃₁×#322	—	2	1	no	—
27	C₈²×C₃₁	—	2	1	no	—
28	C₂×#41×C₃₁	—	2	1	no	—
29	C₃₁×#131	—	4	2	no	—
30	C₃₁×#131	—	4	2	no	—
31	C₂×M₁₆×C₃₁	—	2	1	no	—
32	C₂×M₁₆×C₃₁	—	2	2	no	—
33	C₂²×C₈×C₃₁	—	2	1	no	—
34	C₂²×C₈×C₃₁	—	2	2	no	—
35	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
37	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
38	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
39	C₃₁×#124	—	2	1	no	—
40	C₃₁×#121	—	2	1	no	—
41	C₃₁×#121	C₂×C₄	1	2	yes	—
42	C₄×D₄×C₃₁	—	2	1	no	—
43	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
45	C₂×C₄²×C₃₁	—	2	1	no	—
46	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×C₄⋊C₈	—	2	1	no	—
48	C₃₁×C₄⋊C₈	—	2	1	no	—
49	C₃₁×C₄⋊C₈	—	2	1	no	—
50	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
51	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
52	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
53	C₃₁×C₂²⋊C₈	—	2	1	no	—
54	C₃₁×C₂²⋊C₈	—	2	2	no	—
55	C₃₁×C₂²⋊C₈	—	2	2	no	—
56	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
57	C₃₁×#97	—	4	2	no	—
58	C₄×C₈×C₃₁	—	2	1	no	—
59	C₄×C₈×C₃₁	—	2	1	no	—
60	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
64	C₃₁×#95	C₂×C₄	1	2	yes	—
65	#41×C₃₁	—	4	1	no	—
66	#41×C₃₁	—	4	1	no	—
67	#41×C₃₁	—	4	1	no	—
68	#41×C₃₁	—	4	1	no	—
69	C₂×Q₈×C₃₁	—	2	1	no	—
70	C₂×D₄×C₃₁	—	2	2	no	—
71	C₂×D₄×C₃₁	—	2	1	no	—
72	C₂×D₄×C₃₁	—	2	1	no	—
73	C₂×D₄×C₃₁	C₄²	1	2	yes	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	—	2	2	no	—
76	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
79	M₁₆×C₃₁	—	4	1	no	—
80	M₁₆×C₃₁	—	4	1	no	—
81	M₁₆×C₃₁	—	4	4	no	—
82	C₂×C₈×C₃₁	—	2	1	no	—
83	C₂×C₈×C₃₁	—	2	1	no	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	—	4	1	no	—
89	C₂×C₈×C₃₁	—	4	1	no	—
90	C₂×C₈×C₃₁	—	4	4	no	—
91	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	D₈	1	2	yes	—
96	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
97	C₄⋊C₄×C₃₁	—	2	1	no	—
98	C₄⋊C₄×C₃₁	—	4	2	no	—
99	C₄⋊C₄×C₃₁	C₄²	1	2	yes	—
100	C₂²⋊C₄×C₃₁	—	2	2	no	—
101	C₂²⋊C₄×C₃₁	—	2	2	no	—
102	C₄²×C₃₁	—	4	1	no	—
103	C₄²×C₃₁	—	4	1	no	—
104	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₃₁	#41	1	1	yes	characteristic
108	#8255	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂³×C₃₁	—	2	2	no	—
110	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
111	Q₈×C₃₁	—	4	1	no	—
112	Q₈×C₃₁	—	4	1	no	—
113	D₄×C₃₁	—	4	2	no	—
114	D₄×C₃₁	—	4	2	no	—
115	D₄×C₃₁	—	4	2	no	—
116	D₄×C₃₁	—	4	1	no	—
117	D₄×C₃₁	—	4	1	no	—
118	D₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	4	1	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
136	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
137	C₈×C₃₁	—	2	1	no	—
138	C₈×C₃₁	—	2	1	no	—
139	C₈×C₃₁	—	4	2	no	—
140	C₈×C₃₁	—	4	2	no	—
141	C₈×C₃₁	—	4	2	no	—
142	C₈×C₃₁	—	4	2	no	—
143	C₈×C₃₁	#102	1	2	yes	—
144	C₈×C₃₁	#102	1	2	yes	—
145	#2931	C₂×C₃₁	1	1	yes	characteristic
146	#1770	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#1540	C₂×C₃₁	1	1	yes	characteristic
148	#1502	C₂×C₃₁	1	1	yes	characteristic
149	#1450	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#1303	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#1301	C₂×C₃₁	1	1	yes	characteristic, complemented
152	C₂²×C₃₁	—	2	1	no	—
153	C₂²×C₃₁	—	2	1	no	—
154	C₂²×C₃₁	—	4	2	no	—
155	C₂²×C₃₁	—	4	2	no	—
156	C₂²×C₃₁	—	4	2	no	—
157	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₃₁	C₄×D₈	1	1	yes	characteristic
159	C₂²×C₃₁	C₄×SD₁₆	1	1	yes	characteristic
160	C₄×C₃₁	—	2	1	no	—
161	C₄×C₃₁	—	2	1	no	—
162	C₄×C₃₁	—	2	1	no	—
163	C₄×C₃₁	—	4	2	no	—
164	C₄×C₃₁	—	8	2	no	—
165	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
166	C₄×C₃₁	#416	1	1	yes	characteristic
167	C₄×C₃₁	C₄×D₈	1	1	yes	characteristic
168	C₄×C₃₁	C₄×SD₁₆	1	1	yes	characteristic
169	C₄×C₃₁	#444	1	2	yes	—
170	C₂×#131	—	2	1	no	—
171	#532	C₂²×C₃₁	1	1	yes	characteristic
172	#436	—	2	1	no	—
173	#436	C₄×C₃₁	1	2	yes	complemented
174	C₈×D₄	—	2	1	no	—
175	C₈×D₄	C₄×C₃₁	1	2	yes	complemented
176	#431	C₂²×C₃₁	1	1	yes	characteristic
177	#431	C₂²×C₃₁	1	1	yes	characteristic
178	#423	C₂²×C₃₁	1	1	yes	characteristic
179	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
180	#408	—	2	1	no	—
181	C₂×#97	—	2	1	no	—
182	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
183	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
184	#336	—	2	1	no	—
185	#332	C₄×C₃₁	1	2	yes	complemented
186	#331	C₄×C₃₁	1	2	yes	complemented
187	#327	C₄×C₃₁	1	4	yes	—
188	#322	—	2	1	no	—
189	C₈²	—	2	1	no	—
190	C₂×C₃₁	—	2	1	no	—
191	C₂×C₃₁	—	2	1	no	—
192	C₂×C₃₁	—	8	2	no	—
193	C₂×C₃₁	#1757	1	1	yes	characteristic
194	C₂×C₃₁	C₄×#102	1	1	yes	characteristic
195	C₂×C₃₁	#1765	1	1	yes	characteristic
196	C₂×#41	—	2	1	no	—
197	#131	—	4	2	no	—
198	#131	—	4	2	no	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
204	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
205	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
206	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
207	#124	—	2	1	no	—
208	#121	—	2	1	no	—
209	#121	C₂×C₄×C₃₁	1	2	yes	—
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
219	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
220	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
221	C₂²⋊C₈	—	2	1	no	—
222	C₂²⋊C₈	—	2	2	no	—
223	C₂²⋊C₈	—	2	2	no	—
224	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
225	#97	—	4	2	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
229	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
230	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
231	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
232	#95	C₂×C₄×C₃₁	1	2	yes	—
233	C₃₁	#8255	1	1	yes	characteristic, direct product factor, Sylow
234	#41	—	4	1	no	—
235	#41	—	4	1	no	—
236	#41	—	4	1	no	—
237	#41	—	4	1	no	—
238	C₂×Q₈	—	2	1	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	C₄²×C₃₁	1	2	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
247	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
248	M₁₆	—	4	1	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	4	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	4	no	—
260	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
261	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
262	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
263	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
264	C₂×C₈	D₈×C₃₁	1	2	yes	—
265	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	C₄²×C₃₁	1	2	yes	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	2	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
274	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
275	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
276	C₄²	#41×C₃₁	1	1	yes	characteristic
277	C₂³	—	2	2	no	—
278	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
279	Q₈	—	4	1	no	—
280	Q₈	—	4	1	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	4	1	no	—
285	D₄	—	4	1	no	—
286	D₄	—	4	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
296	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
297	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
298	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
299	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
300	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
304	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	4	2	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	C₃₁×#102	1	2	yes	—
312	C₈	C₃₁×#102	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
319	C₂²	C₄×D₈×C₃₁	1	1	yes	characteristic
320	C₂²	C₄×SD₁₆×C₃₁	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	2	no	—
325	C₄	—	8	2	no	—
326	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
327	C₄	C₃₁×#416	1	1	yes	characteristic
328	C₄	C₄×D₈×C₃₁	1	1	yes	characteristic
329	C₄	C₄×SD₁₆×C₃₁	1	1	yes	characteristic
330	C₄	C₃₁×#444	1	2	yes	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	8	2	no	—
334	C₂	C₃₁×#1757	1	1	yes	characteristic
335	C₂	C₄×C₃₁×#102	1	1	yes	characteristic
336	C₂	C₃₁×#1765	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	C₄×C₂²⋊C₄
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1770	C₂	semidirect
5. C₃₁×#1450	C₂	semidirect
6. C₃₁×#1303	C₂	semidirect
7. C₃₁×#1301	C₂	semidirect
11. C₃₁×#436	C₄	semidirect
13. C₈×D₄×C₃₁	C₄	semidirect
23. C₃₁×#332	C₄	semidirect
24. C₃₁×#331	C₄	semidirect
108. #8255	C₃₁	direct (N × H)
146. #1770	C₂×C₃₁	semidirect
149. #1450	C₂×C₃₁	semidirect
150. #1303	C₂×C₃₁	semidirect
151. #1301	C₂×C₃₁	semidirect
173. #436	C₄×C₃₁	semidirect
175. C₈×D₄	C₄×C₃₁	semidirect
185. #332	C₄×C₃₁	semidirect
186. #331	C₄×C₃₁	semidirect
233. C₃₁	#8255	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×#2931	2
C₃₁×#1770	2
C₃₁×#1540	2
C₃₁×#1502	2
C₃₁×#1450	2
C₃₁×#1303	2
C₃₁×#1301	2
#8255	31

Subgroups — C31×#8255 #21465014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8255 #21465014