Subgroups — C₃₁×#18688 #21475447

Order 7936.

954 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3550	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3254	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1633	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1630	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1623	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1620	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1560	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1493	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×#135	—	2	2	no	—
11	C₃₁×#563	—	2	2	no	—
12	C₃₁×#559	C₂²	1	2	yes	—
13	Q₈²×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#556	—	2	2	no	—
15	D₄×Q₈×C₃₁	—	2	2	no	—
16	C₃₁×#542	—	2	4	no	—
17	C₃₁×#536	—	2	2	no	—
18	C₃₁×#533	C₂²	1	2	yes	—
19	C₃₁×#532	C₂²	1	1	yes	characteristic
20	C₃₁×#532	C₂²	1	1	yes	characteristic
21	C₃₁×#495	—	2	2	no	—
22	C₃₁×#494	C₂²	1	1	yes	characteristic
23	C₃₁×#493	C₂²	1	1	yes	characteristic
24	C₃₁×#492	C₂²	1	1	yes	characteristic
25	C₃₁×#492	C₂²	1	1	yes	characteristic
26	C₃₁×#477	C₂²	1	4	yes	—
27	C₃₁×#474	—	2	2	no	—
28	C₃₁×#468	—	2	4	no	—
29	C₃₁×#461	—	2	2	no	—
30	C₃₁×#460	C₂²	1	4	yes	—
31	C₃₁×#423	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#333	C₂²	1	2	yes	—
34	C₃₁×#331	C₂²	1	2	yes	—
35	C₃₁×Q₈⋊C₈	C₂²	1	4	yes	—
36	C₃₁×Q₈⋊C₈	C₂²	1	4	yes	—
37	C₂×#41×C₃₁	—	2	1	no	—
38	C₂×#41×C₃₁	D₄	1	2	yes	—
39	C₂²×Q₈×C₃₁	—	2	2	no	—
40	C₃₁×#135	—	4	8	no	—
41	C₂×Q₁₆×C₃₁	—	2	2	no	—
42	C₂×SD₁₆×C₃₁	—	2	2	no	—
43	C₂×SD₁₆×C₃₁	—	2	2	no	—
44	C₂×SD₁₆×C₃₁	—	2	4	no	—
45	C₂×D₈×C₃₁	—	2	2	no	—
46	C₂×M₁₆×C₃₁	—	2	2	no	—
47	C₂²×C₈×C₃₁	—	2	1	no	—
48	C₃₁×C₄⋊Q₈	—	2	4	no	—
49	C₃₁×C₄⋊Q₈	—	2	4	no	—
50	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
51	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
52	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
53	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
54	C₃₁×#127	C₂³	1	1	yes	characteristic
55	C₃₁×#127	C₂³	1	1	yes	characteristic
56	C₃₁×#126	—	2	8	no	—
57	C₃₁×#125	—	2	2	no	—
58	C₃₁×#124	—	2	4	no	—
59	C₃₁×#124	—	2	4	no	—
60	C₃₁×#124	—	2	1	no	—
61	C₃₁×#124	D₄	1	2	yes	—
62	C₃₁×#123	—	2	4	no	—
63	C₃₁×#123	—	2	4	no	—
64	C₃₁×C₂²⋊Q₈	—	2	4	no	—
65	C₃₁×C₂²⋊Q₈	—	2	8	no	—
66	C₃₁×C₂²⋊Q₈	—	2	2	no	—
67	C₃₁×C₂²⋊Q₈	D₄	1	4	yes	—
68	C₃₁×#121	—	2	2	no	—
69	C₃₁×#121	—	2	2	no	—
70	C₄×Q₈×C₃₁	—	2	4	no	—
71	C₄×Q₈×C₃₁	—	2	2	no	—
72	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
73	C₄×D₄×C₃₁	—	2	4	no	—
74	C₄×D₄×C₃₁	—	2	2	no	—
75	C₄×D₄×C₃₁	—	2	2	no	—
76	C₃₁×#117	—	2	2	no	—
77	C₃₁×#117	—	2	2	no	—
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#107	—	2	4	no	—
80	C₃₁×C₄⋊C₈	—	2	2	no	—
81	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
82	C₃₁×#103	—	2	4	no	—
83	C₃₁×#102	—	2	4	no	—
84	C₄×C₈×C₃₁	—	2	1	no	—
85	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
87	#41×C₃₁	—	4	4	no	—
88	#41×C₃₁	—	4	4	no	—
89	C₂×Q₈×C₃₁	—	2	4	no	—
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	—	2	1	no	—
92	C₂×Q₈×C₃₁	—	8	4	no	—
93	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
94	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	—	2	1	no	—
99	C₂×D₄×C₃₁	—	4	1	no	—
100	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₃₁	—	2	4	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	4	1	no	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	Q₁₆×C₃₁	—	4	4	no	—
107	SD₁₆×C₃₁	—	4	4	no	—
108	SD₁₆×C₃₁	—	4	4	no	—
109	SD₁₆×C₃₁	—	8	4	no	—
110	D₈×C₃₁	—	4	4	no	—
111	M₁₆×C₃₁	—	8	2	no	—
112	C₂×C₈×C₃₁	—	2	4	no	—
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₂×C₈×C₃₁	—	2	1	no	—
115	C₂×C₈×C₃₁	—	2	4	no	—
116	C₄⋊C₄×C₃₁	—	2	8	no	—
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₄⋊C₄×C₃₁	—	2	4	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	4	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
125	C₂²⋊C₄×C₃₁	—	2	4	no	—
126	C₂²⋊C₄×C₃₁	—	2	8	no	—
127	C₂²⋊C₄×C₃₁	—	2	4	no	—
128	C₂²⋊C₄×C₃₁	—	2	4	no	—
129	C₂²⋊C₄×C₃₁	—	4	2	no	—
130	C₄²×C₃₁	—	2	4	no	—
131	C₄²×C₃₁	—	2	4	no	—
132	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₃₁	C₂×D₄	1	2	yes	—
134	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
135	#18688	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂³×C₃₁	—	2	2	no	—
137	C₂³×C₃₁	—	4	1	no	—
138	C₂³×C₃₁	#120	1	1	yes	characteristic
139	Q₈×C₃₁	—	2	4	no	—
140	Q₈×C₃₁	—	2	4	no	—
141	Q₈×C₃₁	—	4	2	no	—
142	Q₈×C₃₁	—	4	2	no	—
143	Q₈×C₃₁	—	4	4	no	—
144	Q₈×C₃₁	—	8	4	no	—
145	D₄×C₃₁	—	4	2	no	—
146	D₄×C₃₁	—	4	2	no	—
147	D₄×C₃₁	—	4	2	no	—
148	D₄×C₃₁	—	4	2	no	—
149	D₄×C₃₁	—	4	2	no	—
150	D₄×C₃₁	—	4	2	no	—
151	D₄×C₃₁	—	8	2	no	—
152	D₄×C₃₁	—	8	2	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	4	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	4	no	—
159	C₂×C₄×C₃₁	—	4	4	no	—
160	C₂×C₄×C₃₁	—	4	1	no	—
161	C₂×C₄×C₃₁	—	8	4	no	—
162	C₂×C₄×C₃₁	—	8	2	no	—
163	C₂×C₄×C₃₁	#120	1	2	yes	—
164	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
167	C₈×C₃₁	—	2	2	no	—
168	C₈×C₃₁	—	2	2	no	—
169	C₈×C₃₁	—	4	4	no	—
170	#3550	C₂×C₃₁	1	1	yes	characteristic, complemented
171	#3254	C₂×C₃₁	1	2	yes	complemented
172	#3147	C₂×C₃₁	1	1	yes	characteristic
173	#1633	C₂×C₃₁	1	2	yes	complemented
174	#1630	C₂×C₃₁	1	2	yes	complemented
175	#1623	C₂×C₃₁	1	2	yes	complemented
176	#1620	C₂×C₃₁	1	2	yes	complemented
177	#1560	C₂×C₃₁	1	1	yes	characteristic, complemented
178	#1493	C₂×C₃₁	1	2	yes	complemented
179	C₂²×C₃₁	—	2	2	no	—
180	C₂²×C₃₁	—	2	1	no	—
181	C₂²×C₃₁	—	4	4	no	—
182	C₂²×C₃₁	—	4	2	no	—
183	C₂²×C₃₁	—	8	2	no	—
184	C₂²×C₃₁	—	8	1	no	—
185	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
186	C₄×C₃₁	—	2	2	no	—
187	C₄×C₃₁	—	2	1	no	—
188	C₄×C₃₁	—	4	4	no	—
189	C₄×C₃₁	—	4	2	no	—
190	C₄×C₃₁	—	4	4	no	—
191	C₄×C₃₁	—	8	1	no	—
192	C₄×C₃₁	#453	1	2	yes	—
193	C₄×C₃₁	#577	1	2	yes	—
194	C₄×C₃₁	#578	1	2	yes	—
195	C₂×#135	—	2	2	no	—
196	#563	—	2	2	no	—
197	#559	C₂²×C₃₁	1	2	yes	—
198	Q₈²	C₂²×C₃₁	1	2	yes	—
199	#556	—	2	2	no	—
200	D₄×Q₈	—	2	2	no	—
201	#542	—	2	4	no	—
202	#536	—	2	2	no	—
203	#533	C₂²×C₃₁	1	2	yes	—
204	#532	C₂²×C₃₁	1	1	yes	characteristic
205	#532	C₂²×C₃₁	1	1	yes	characteristic
206	#495	—	2	2	no	—
207	#494	C₂²×C₃₁	1	1	yes	characteristic
208	#493	C₂²×C₃₁	1	1	yes	characteristic
209	#492	C₂²×C₃₁	1	1	yes	characteristic
210	#492	C₂²×C₃₁	1	1	yes	characteristic
211	#477	C₂²×C₃₁	1	4	yes	—
212	#474	—	2	2	no	—
213	#468	—	2	4	no	—
214	#461	—	2	2	no	—
215	#460	C₂²×C₃₁	1	4	yes	—
216	#423	C₂²×C₃₁	1	2	yes	—
217	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
218	#333	C₂²×C₃₁	1	2	yes	—
219	#331	C₂²×C₃₁	1	2	yes	—
220	Q₈⋊C₈	C₂²×C₃₁	1	4	yes	—
221	Q₈⋊C₈	C₂²×C₃₁	1	4	yes	—
222	C₂×C₃₁	—	4	1	no	—
223	C₂×C₃₁	—	8	2	no	—
224	C₂×C₃₁	—	16	1	no	—
225	C₂×C₃₁	#3013	1	2	yes	—
226	C₂×C₃₁	C₂×#453	1	1	yes	characteristic
227	C₂×#41	—	2	1	no	—
228	C₂×#41	D₄×C₃₁	1	2	yes	—
229	C₂²×Q₈	—	2	2	no	—
230	#135	—	4	8	no	—
231	C₂×Q₁₆	—	2	2	no	—
232	C₂×SD₁₆	—	2	2	no	—
233	C₂×SD₁₆	—	2	2	no	—
234	C₂×SD₁₆	—	2	4	no	—
235	C₂×D₈	—	2	2	no	—
236	C₂×M₁₆	—	2	2	no	—
237	C₂²×C₈	—	2	1	no	—
238	C₄⋊Q₈	—	2	4	no	—
239	C₄⋊Q₈	—	2	4	no	—
240	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
241	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
242	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
243	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
244	#127	C₂³×C₃₁	1	1	yes	characteristic
245	#127	C₂³×C₃₁	1	1	yes	characteristic
246	#126	—	2	8	no	—
247	#125	—	2	2	no	—
248	#124	—	2	4	no	—
249	#124	—	2	4	no	—
250	#124	—	2	1	no	—
251	#124	D₄×C₃₁	1	2	yes	—
252	#123	—	2	4	no	—
253	#123	—	2	4	no	—
254	C₂²⋊Q₈	—	2	4	no	—
255	C₂²⋊Q₈	—	2	8	no	—
256	C₂²⋊Q₈	—	2	2	no	—
257	C₂²⋊Q₈	D₄×C₃₁	1	4	yes	—
258	#121	—	2	2	no	—
259	#121	—	2	2	no	—
260	C₄×Q₈	—	2	4	no	—
261	C₄×Q₈	—	2	2	no	—
262	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
263	C₄×D₄	—	2	4	no	—
264	C₄×D₄	—	2	2	no	—
265	C₄×D₄	—	2	2	no	—
266	#117	—	2	2	no	—
267	#117	—	2	2	no	—
268	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
269	#107	—	2	4	no	—
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
272	#103	—	2	4	no	—
273	#102	—	2	4	no	—
274	C₄×C₈	—	2	1	no	—
275	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
276	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
277	C₃₁	#18688	1	1	yes	characteristic, direct product factor, Sylow
278	#41	—	4	4	no	—
279	#41	—	4	4	no	—
280	C₂×Q₈	—	2	4	no	—
281	C₂×Q₈	—	2	2	no	—
282	C₂×Q₈	—	2	1	no	—
283	C₂×Q₈	—	8	4	no	—
284	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
285	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	2	no	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	—	2	1	no	—
290	C₂×D₄	—	4	1	no	—
291	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
292	C₂²×C₄	—	2	4	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	4	1	no	—
295	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
296	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
297	Q₁₆	—	4	4	no	—
298	SD₁₆	—	4	4	no	—
299	SD₁₆	—	4	4	no	—
300	SD₁₆	—	8	4	no	—
301	D₈	—	4	4	no	—
302	M₁₆	—	8	2	no	—
303	C₂×C₈	—	2	4	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	4	no	—
307	C₄⋊C₄	—	2	8	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	4	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	4	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
316	C₂²⋊C₄	—	2	4	no	—
317	C₂²⋊C₄	—	2	8	no	—
318	C₂²⋊C₄	—	2	4	no	—
319	C₂²⋊C₄	—	2	4	no	—
320	C₂²⋊C₄	—	4	2	no	—
321	C₄²	—	2	4	no	—
322	C₄²	—	2	4	no	—
323	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
324	C₄²	C₂×D₄×C₃₁	1	2	yes	—
325	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
326	C₂³	—	2	2	no	—
327	C₂³	—	4	1	no	—
328	C₂³	C₃₁×#120	1	1	yes	characteristic
329	Q₈	—	2	4	no	—
330	Q₈	—	2	4	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	4	2	no	—
333	Q₈	—	4	4	no	—
334	Q₈	—	8	4	no	—
335	D₄	—	4	2	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	2	no	—
339	D₄	—	4	2	no	—
340	D₄	—	4	2	no	—
341	D₄	—	8	2	no	—
342	D₄	—	8	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	4	no	—
349	C₂×C₄	—	4	4	no	—
350	C₂×C₄	—	4	1	no	—
351	C₂×C₄	—	8	4	no	—
352	C₂×C₄	—	8	2	no	—
353	C₂×C₄	C₃₁×#120	1	2	yes	—
354	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
357	C₈	—	2	2	no	—
358	C₈	—	2	2	no	—
359	C₈	—	4	4	no	—
360	C₂²	—	2	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	4	4	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	8	1	no	—
366	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
367	C₄	—	2	2	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	4	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	4	no	—
372	C₄	—	8	1	no	—
373	C₄	C₃₁×#453	1	2	yes	—
374	C₄	C₃₁×#577	1	2	yes	—
375	C₄	C₃₁×#578	1	2	yes	—
376	C₂	—	4	1	no	—
377	C₂	—	8	2	no	—
378	C₂	—	16	1	no	—
379	C₂	C₃₁×#3013	1	2	yes	—
380	C₂	C₂×C₃₁×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3147	C₂
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
1. C₃₁×#3550	C₂	semidirect
2. C₃₁×#3254	C₂	semidirect
4. C₃₁×#1633	C₂	semidirect
5. C₃₁×#1630	C₂	semidirect
6. C₃₁×#1623	C₂	semidirect
7. C₃₁×#1620	C₂	semidirect
8. C₃₁×#1560	C₂	semidirect
9. C₃₁×#1493	C₂	semidirect
135. #18688	C₃₁	direct (N × H)
170. #3550	C₂×C₃₁	semidirect
171. #3254	C₂×C₃₁	semidirect
173. #1633	C₂×C₃₁	semidirect
174. #1630	C₂×C₃₁	semidirect
175. #1623	C₂×C₃₁	semidirect
176. #1620	C₂×C₃₁	semidirect
177. #1560	C₂×C₃₁	semidirect
178. #1493	C₂×C₃₁	semidirect
277. C₃₁	#18688	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×#3550	2
C₃₁×#3254	2
C₃₁×#3147	2
C₃₁×#1633	2
C₃₁×#1630	2
C₃₁×#1623	2
C₃₁×#1620	2
C₃₁×#1560	2
C₃₁×#1493	2
#18688	31

Subgroups — C31×#18688 #21475447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18688 #21475447