Subgroups — C₃₁×#10291 #21467050

Order 7936.

652 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3209	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2858	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#533	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#124	—	2	2	no	—
12	C₃₁×#494	—	2	1	no	—
13	C₃₁×#492	—	2	1	no	—
14	C₃₁×#487	—	2	1	no	—
15	C₃₁×#486	—	2	1	no	—
16	C₃₁×#462	—	2	2	no	—
17	C₃₁×#460	—	2	2	no	—
18	C₃₁×#451	—	2	2	no	—
19	C₃₁×#450	—	2	2	no	—
20	C₃₁×#431	C₂²	1	1	yes	characteristic
21	C₃₁×#431	C₂²	1	1	yes	characteristic
22	C₃₁×#423	C₂²	1	1	yes	characteristic
23	C₃₁×#390	—	2	2	no	—
24	C₃₁×#390	—	2	2	no	—
25	C₄³×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#339	—	2	2	no	—
27	C₃₁×#333	—	2	2	no	—
28	C₃₁×#331	—	2	2	no	—
29	C₂²×Q₈×C₃₁	D₄	1	2	yes	—
30	C₂²×D₄×C₃₁	—	2	1	no	—
31	C₂²×D₄×C₃₁	—	2	1	no	—
32	C₂²×D₄×C₃₁	D₄	1	2	yes	—
33	C₂×Q₁₆×C₃₁	—	2	4	no	—
34	C₂×SD₁₆×C₃₁	—	2	4	no	—
35	C₂×M₁₆×C₃₁	—	2	1	no	—
36	C₃₁×C₄⋊Q₈	—	2	1	no	—
37	C₃₁×C₄⋊Q₈	—	2	1	no	—
38	C₃₁×C₄⋊Q₈	—	2	1	no	—
39	C₃₁×C₄⋊Q₈	—	2	1	no	—
40	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
41	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
42	C₃₁×#127	—	2	1	no	—
43	C₃₁×#127	—	2	1	no	—
44	C₃₁×#127	—	2	1	no	—
45	C₃₁×#127	—	2	1	no	—
46	C₃₁×#125	—	2	1	no	—
47	C₃₁×#124	—	4	8	no	—
48	C₃₁×C₂²⋊Q₈	—	2	1	no	—
49	C₃₁×C₂²⋊Q₈	—	2	1	no	—
50	C₄×Q₈×C₃₁	—	2	1	no	—
51	C₄×Q₈×C₃₁	—	2	1	no	—
52	C₃₁×#117	—	2	1	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
56	C₂×C₄²×C₃₁	—	2	1	no	—
57	C₂×C₄²×C₃₁	—	2	1	no	—
58	C₂×C₄²×C₃₁	D₄	1	2	yes	—
59	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×C₄⋊C₈	—	2	1	no	—
61	C₃₁×C₄⋊C₈	—	2	1	no	—
62	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
63	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
64	C₃₁×#103	—	2	4	no	—
65	C₃₁×#102	—	2	4	no	—
66	C₃₁×C₂²⋊C₈	—	2	1	no	—
67	C₃₁×C₂²⋊C₈	—	2	1	no	—
68	C₄×C₈×C₃₁	—	2	1	no	—
69	C₄×C₈×C₃₁	—	2	1	no	—
70	C₂⁴×C₃₁	—	2	2	no	—
71	C₂×Q₈×C₃₁	—	2	2	no	—
72	C₂×Q₈×C₃₁	—	2	1	no	—
73	C₂×Q₈×C₃₁	—	2	1	no	—
74	C₂×Q₈×C₃₁	—	2	2	no	—
75	C₂×Q₈×C₃₁	—	4	4	no	—
76	C₂×D₄×C₃₁	—	2	2	no	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂×D₄×C₃₁	—	4	4	no	—
79	C₂×D₄×C₃₁	—	4	4	no	—
80	C₂×D₄×C₃₁	—	4	1	no	—
81	C₂×D₄×C₃₁	—	4	1	no	—
82	C₂×D₄×C₃₁	—	4	1	no	—
83	C₂×D₄×C₃₁	—	4	1	no	—
84	C₂×D₄×C₃₁	—	4	2	no	—
85	C₂×D₄×C₃₁	—	4	2	no	—
86	C₂²×C₄×C₃₁	—	2	2	no	—
87	C₂²×C₄×C₃₁	—	2	2	no	—
88	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	Q₁₆×C₃₁	—	8	4	no	—
92	SD₁₆×C₃₁	—	8	4	no	—
93	M₁₆×C₃₁	—	4	2	no	—
94	C₂×C₈×C₃₁	—	2	2	no	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	—	4	1	no	—
97	C₄⋊C₄×C₃₁	—	2	1	no	—
98	C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₄⋊C₄×C₃₁	—	2	1	no	—
100	C₄⋊C₄×C₃₁	—	2	1	no	—
101	C₄⋊C₄×C₃₁	—	4	1	no	—
102	C₄⋊C₄×C₃₁	—	4	1	no	—
103	C₄⋊C₄×C₃₁	—	4	1	no	—
104	C₄⋊C₄×C₃₁	—	4	1	no	—
105	C₄⋊C₄×C₃₁	—	4	1	no	—
106	C₄⋊C₄×C₃₁	—	4	1	no	—
107	C₂²⋊C₄×C₃₁	—	4	16	no	—
108	C₂²⋊C₄×C₃₁	—	4	1	no	—
109	C₄²×C₃₁	—	2	4	no	—
110	C₄²×C₃₁	—	2	2	no	—
111	C₄²×C₃₁	—	2	2	no	—
112	C₄²×C₃₁	—	4	2	no	—
113	C₄²×C₃₁	—	4	1	no	—
114	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₃₁	#41	1	1	yes	characteristic
118	#10291	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₃₁	—	4	4	no	—
120	C₂³×C₃₁	—	4	4	no	—
121	C₂³×C₃₁	—	4	2	no	—
122	C₂³×C₃₁	—	4	4	no	—
123	C₂³×C₃₁	#120	1	1	yes	characteristic
124	Q₈×C₃₁	—	4	2	no	—
125	Q₈×C₃₁	—	4	4	no	—
126	Q₈×C₃₁	—	4	2	no	—
127	Q₈×C₃₁	—	8	2	no	—
128	D₄×C₃₁	—	4	4	no	—
129	D₄×C₃₁	—	8	2	no	—
130	D₄×C₃₁	—	8	2	no	—
131	D₄×C₃₁	—	8	2	no	—
132	D₄×C₃₁	—	8	2	no	—
133	D₄×C₃₁	—	8	1	no	—
134	D₄×C₃₁	—	8	1	no	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	4	no	—
138	C₂×C₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	—	4	4	no	—
141	C₂×C₄×C₃₁	—	4	1	no	—
142	C₂×C₄×C₃₁	—	4	1	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
145	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
146	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
147	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
148	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
149	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
150	C₂×C₄×C₃₁	#135	1	2	yes	—
151	C₂×C₄×C₃₁	#137	1	2	yes	—
152	C₈×C₃₁	—	4	2	no	—
153	C₈×C₃₁	—	4	2	no	—
154	C₈×C₃₁	—	8	1	no	—
155	#3209	C₂×C₃₁	1	1	yes	characteristic, complemented
156	#3208	C₂×C₃₁	1	1	yes	characteristic
157	#2858	C₂×C₃₁	1	1	yes	characteristic
158	#1994	C₂×C₃₁	1	1	yes	characteristic, complemented
159	#1548	C₂×C₃₁	1	1	yes	characteristic, complemented
160	#1545	C₂×C₃₁	1	1	yes	characteristic
161	#1276	C₂×C₃₁	1	1	yes	characteristic, complemented
162	C₂²×C₃₁	—	2	1	no	—
163	C₂²×C₃₁	—	2	1	no	—
164	C₂²×C₃₁	—	8	8	no	—
165	C₂²×C₃₁	—	8	2	no	—
166	C₂²×C₃₁	—	8	2	no	—
167	C₂²×C₃₁	—	8	2	no	—
168	C₂²×C₃₁	#392	1	1	yes	characteristic
169	C₂²×C₃₁	#452	1	2	yes	—
170	C₄×C₃₁	—	2	1	no	—
171	C₄×C₃₁	—	2	1	no	—
172	C₄×C₃₁	—	4	4	no	—
173	C₄×C₃₁	—	8	2	no	—
174	C₄×C₃₁	—	8	1	no	—
175	C₄×C₃₁	—	8	1	no	—
176	C₄×C₃₁	#453	1	2	yes	—
177	C₄×C₃₁	#455	1	2	yes	—
178	C₄×C₃₁	#464	1	2	yes	—
179	C₄×C₃₁	#495	1	1	yes	characteristic
180	C₄×C₃₁	#497	1	1	yes	characteristic
181	#533	C₂²×C₃₁	1	1	yes	characteristic
182	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
183	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
184	C₂×#124	—	2	2	no	—
185	#494	—	2	1	no	—
186	#492	—	2	1	no	—
187	#487	—	2	1	no	—
188	#486	—	2	1	no	—
189	#462	—	2	2	no	—
190	#460	—	2	2	no	—
191	#451	—	2	2	no	—
192	#450	—	2	2	no	—
193	#431	C₂²×C₃₁	1	1	yes	characteristic
194	#431	C₂²×C₃₁	1	1	yes	characteristic
195	#423	C₂²×C₃₁	1	1	yes	characteristic
196	#390	—	2	2	no	—
197	#390	—	2	2	no	—
198	C₄³	C₂²×C₃₁	1	1	yes	characteristic
199	#339	—	2	2	no	—
200	#333	—	2	2	no	—
201	#331	—	2	2	no	—
202	C₂×C₃₁	—	2	2	no	—
203	C₂×C₃₁	—	16	2	no	—
204	C₂×C₃₁	#2002	1	1	yes	characteristic
205	C₂×C₃₁	#2004	1	1	yes	characteristic
206	C₂×C₃₁	#2015	1	1	yes	characteristic
207	C₂²×Q₈	D₄×C₃₁	1	2	yes	—
208	C₂²×D₄	—	2	1	no	—
209	C₂²×D₄	—	2	1	no	—
210	C₂²×D₄	D₄×C₃₁	1	2	yes	—
211	C₂×Q₁₆	—	2	4	no	—
212	C₂×SD₁₆	—	2	4	no	—
213	C₂×M₁₆	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
219	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
220	#127	—	2	1	no	—
221	#127	—	2	1	no	—
222	#127	—	2	1	no	—
223	#127	—	2	1	no	—
224	#125	—	2	1	no	—
225	#124	—	4	8	no	—
226	C₂²⋊Q₈	—	2	1	no	—
227	C₂²⋊Q₈	—	2	1	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	—	2	1	no	—
230	#117	—	2	1	no	—
231	C₂×C₄⋊C₄	—	2	1	no	—
232	C₂×C₄⋊C₄	—	2	1	no	—
233	C₂×C₂²⋊C₄	—	2	8	no	—
234	C₂×C₄²	—	2	1	no	—
235	C₂×C₄²	—	2	1	no	—
236	C₂×C₄²	D₄×C₃₁	1	2	yes	—
237	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
241	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
242	#103	—	2	4	no	—
243	#102	—	2	4	no	—
244	C₂²⋊C₈	—	2	1	no	—
245	C₂²⋊C₈	—	2	1	no	—
246	C₄×C₈	—	2	1	no	—
247	C₄×C₈	—	2	1	no	—
248	C₃₁	#10291	1	1	yes	characteristic, direct product factor, Sylow
249	C₂⁴	—	2	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	1	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	4	4	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	4	4	no	—
258	C₂×D₄	—	4	4	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
270	Q₁₆	—	8	4	no	—
271	SD₁₆	—	8	4	no	—
272	M₁₆	—	4	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	4	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	4	16	no	—
287	C₂²⋊C₄	—	4	1	no	—
288	C₄²	—	2	4	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	4	2	no	—
292	C₄²	—	4	1	no	—
293	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
294	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
295	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
296	C₄²	#41×C₃₁	1	1	yes	characteristic
297	C₂³	—	4	4	no	—
298	C₂³	—	4	4	no	—
299	C₂³	—	4	4	no	—
300	C₂³	—	4	2	no	—
301	C₂³	C₃₁×#120	1	1	yes	characteristic
302	Q₈	—	4	2	no	—
303	Q₈	—	4	4	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	8	2	no	—
306	D₄	—	4	4	no	—
307	D₄	—	8	2	no	—
308	D₄	—	8	2	no	—
309	D₄	—	8	2	no	—
310	D₄	—	8	2	no	—
311	D₄	—	8	1	no	—
312	D₄	—	8	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
323	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
324	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
325	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
326	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
327	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
328	C₂×C₄	C₃₁×#135	1	2	yes	—
329	C₂×C₄	C₃₁×#137	1	2	yes	—
330	C₈	—	4	2	no	—
331	C₈	—	4	2	no	—
332	C₈	—	8	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	8	8	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	C₃₁×#392	1	1	yes	characteristic
340	C₂²	C₃₁×#452	1	2	yes	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	4	4	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	C₃₁×#453	1	2	yes	—
348	C₄	C₃₁×#455	1	2	yes	—
349	C₄	C₃₁×#464	1	2	yes	—
350	C₄	C₃₁×#495	1	1	yes	characteristic
351	C₄	C₃₁×#497	1	1	yes	characteristic
352	C₂	—	2	2	no	—
353	C₂	—	16	2	no	—
354	C₂	C₃₁×#2002	1	1	yes	characteristic
355	C₂	C₃₁×#2004	1	1	yes	characteristic
356	C₂	C₃₁×#2015	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂×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
1. C₃₁×#3209	C₂	semidirect
4. C₃₁×#1994	C₂	semidirect
5. C₃₁×#1548	C₂	semidirect
7. C₃₁×#1276	C₂	semidirect
118. #10291	C₃₁	direct (N × H)
155. #3209	C₂×C₃₁	semidirect
158. #1994	C₂×C₃₁	semidirect
159. #1548	C₂×C₃₁	semidirect
161. #1276	C₂×C₃₁	semidirect
248. C₃₁	#10291	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×#3209	2
C₃₁×#3208	2
C₃₁×#2858	2
C₃₁×#1994	2
C₃₁×#1548	2
C₃₁×#1545	2
C₃₁×#1276	2
#10291	31

Subgroups — C31×#10291 #21467050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10291 #21467050