Subgroups — C₃₁×#19195 #21475954

Order 7936.

638 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×#3246	C₂	1	2	yes	maximal
2	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1691	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1675	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1673	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1550	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#131	—	2	1	no	—
10	C₃₁×#533	C₂²	1	2	yes	—
11	C₃₁×#532	C₂²	1	1	yes	characteristic
12	C₃₁×#500	C₂²	1	1	yes	characteristic
13	C₃₁×#500	C₂²	1	1	yes	characteristic
14	C₃₁×#499	—	2	1	no	—
15	C₃₁×#499	—	2	1	no	—
16	C₃₁×#498	C₂²	1	2	yes	—
17	C₃₁×#476	C₂²	1	4	yes	—
18	C₃₁×#475	—	2	2	no	—
19	C₃₁×#465	—	2	4	no	—
20	C₃₁×#461	C₂²	1	4	yes	—
21	C₃₁×#460	—	2	2	no	—
22	C₃₁×#436	—	2	1	no	—
23	C₃₁×#436	C₂²	1	2	yes	—
24	C₈×D₄×C₃₁	—	2	1	no	—
25	C₈×D₄×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#427	—	2	1	no	—
27	C₃₁×#427	—	2	1	no	—
28	C₃₁×#423	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
30	C₃₁×#408	—	2	1	no	—
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#335	C₂²	1	2	yes	—
33	C₃₁×#335	C₂²	1	2	yes	—
34	C₃₁×#333	C₂²	1	4	yes	—
35	C₃₁×#332	C₂²	1	4	yes	—
36	C₂×#41×C₃₁	—	2	1	no	—
37	C₂×SD₁₆×C₃₁	—	2	4	no	—
38	C₃₁×#131	—	4	4	no	—
39	C₂×M₁₆×C₃₁	—	2	1	no	—
40	C₂×M₁₆×C₃₁	—	2	2	no	—
41	C₂²×C₈×C₃₁	—	2	2	no	—
42	C₂²×C₈×C₃₁	—	2	2	no	—
43	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
45	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
46	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₃₁×#127	D₄	1	1	yes	characteristic
49	C₃₁×#125	—	2	2	no	—
50	C₃₁×#124	—	2	1	no	—
51	C₃₁×C₂²⋊Q₈	—	2	2	no	—
52	C₃₁×C₂²⋊Q₈	—	2	2	no	—
53	C₃₁×#121	—	2	1	no	—
54	C₃₁×#121	D₄	1	2	yes	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×Q₈×C₃₁	—	2	2	no	—
57	C₄×D₄×C₃₁	—	2	1	no	—
58	C₄×D₄×C₃₁	C₂³	1	2	yes	—
59	C₃₁×#117	—	2	2	no	—
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×#107	—	2	1	no	—
62	C₃₁×#107	—	2	1	no	—
63	C₃₁×#107	—	2	1	no	—
64	C₃₁×#107	—	2	1	no	—
65	C₃₁×#106	—	2	2	no	—
66	C₃₁×#106	—	2	2	no	—
67	C₃₁×#106	—	2	4	no	—
68	C₃₁×C₄⋊C₈	—	2	2	no	—
69	C₃₁×C₄⋊C₈	—	2	1	no	—
70	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
71	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
72	C₃₁×#103	—	2	4	no	—
73	C₃₁×#102	—	2	4	no	—
74	C₃₁×C₂²⋊C₈	—	2	2	no	—
75	C₃₁×C₂²⋊C₈	—	2	2	no	—
76	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
77	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
78	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	#41×C₃₁	—	4	2	no	—
81	#41×C₃₁	—	4	2	no	—
82	C₂×Q₈×C₃₁	—	2	1	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	2	1	no	—
87	C₂×D₄×C₃₁	—	2	1	no	—
88	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₃₁	—	2	2	no	—
90	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
93	SD₁₆×C₃₁	—	8	4	no	—
94	M₁₆×C₃₁	—	4	2	no	—
95	M₁₆×C₃₁	—	4	4	no	—
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	2	4	no	—
99	C₂×C₈×C₃₁	—	4	4	no	—
100	C₂×C₈×C₃₁	—	8	2	no	—
101	C₂×C₈×C₃₁	SD₁₆	1	4	yes	complemented
102	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₄⋊C₄×C₃₁	—	2	1	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₄⋊C₄×C₃₁	—	4	2	no	—
112	C₄⋊C₄×C₃₁	—	4	2	no	—
113	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
114	C₂²⋊C₄×C₃₁	—	2	2	no	—
115	C₂²⋊C₄×C₃₁	—	2	2	no	—
116	C₂²⋊C₄×C₃₁	—	4	2	no	—
117	C₄²×C₃₁	—	4	2	no	—
118	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₃₁	#41	1	1	yes	characteristic
121	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
122	#19195	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂³×C₃₁	—	2	2	no	—
124	C₂³×C₃₁	#121	1	1	yes	characteristic
125	Q₈×C₃₁	—	4	2	no	—
126	Q₈×C₃₁	—	4	2	no	—
127	Q₈×C₃₁	—	4	2	no	—
128	Q₈×C₃₁	—	8	2	no	—
129	D₄×C₃₁	—	2	2	no	—
130	D₄×C₃₁	—	2	2	no	—
131	D₄×C₃₁	—	4	2	no	—
132	D₄×C₃₁	—	4	4	no	—
133	D₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	—	2	1	no	—
135	C₂×C₄×C₃₁	—	2	1	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	4	1	no	—
138	C₂×C₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	—	4	2	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
143	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
144	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
145	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
146	C₂×C₄×C₃₁	#137	1	2	yes	—
147	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
150	C₈×C₃₁	—	2	1	no	—
151	C₈×C₃₁	—	2	1	no	—
152	C₈×C₃₁	—	4	2	no	—
153	C₈×C₃₁	—	4	4	no	—
154	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
155	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
156	#3246	C₂×C₃₁	1	2	yes	—
157	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
158	#2931	C₂×C₃₁	1	1	yes	characteristic
159	#1691	C₂×C₃₁	1	4	yes	complemented
160	#1675	C₂×C₃₁	1	2	yes	complemented
161	#1673	C₂×C₃₁	1	2	yes	complemented
162	C₂×#335	C₂×C₃₁	1	1	yes	characteristic, complemented
163	#1550	C₂×C₃₁	1	2	yes	complemented
164	C₂²×C₃₁	—	2	1	no	—
165	C₂²×C₃₁	—	2	1	no	—
166	C₂²×C₃₁	—	4	2	no	—
167	C₂²×C₃₁	—	4	2	no	—
168	C₂²×C₃₁	—	4	2	no	—
169	C₂²×C₃₁	#461	1	2	yes	—
170	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
171	C₄×C₃₁	—	2	1	no	—
172	C₄×C₃₁	—	2	1	no	—
173	C₄×C₃₁	—	2	1	no	—
174	C₄×C₃₁	—	4	2	no	—
175	C₄×C₃₁	—	8	2	no	—
176	C₄×C₃₁	—	8	2	no	—
177	C₄×C₃₁	#461	1	2	yes	—
178	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
179	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
180	C₄×C₃₁	#576	1	1	yes	characteristic
181	C₄×C₃₁	#578	1	1	yes	characteristic
182	C₂×#131	—	2	1	no	—
183	#533	C₂²×C₃₁	1	2	yes	—
184	#532	C₂²×C₃₁	1	1	yes	characteristic
185	#500	C₂²×C₃₁	1	1	yes	characteristic
186	#500	C₂²×C₃₁	1	1	yes	characteristic
187	#499	—	2	1	no	—
188	#499	—	2	1	no	—
189	#498	C₂²×C₃₁	1	2	yes	—
190	#476	C₂²×C₃₁	1	4	yes	—
191	#475	—	2	2	no	—
192	#465	—	2	4	no	—
193	#461	C₂²×C₃₁	1	4	yes	—
194	#460	—	2	2	no	—
195	#436	—	2	1	no	—
196	#436	C₂²×C₃₁	1	2	yes	—
197	C₈×D₄	—	2	1	no	—
198	C₈×D₄	C₂²×C₃₁	1	2	yes	—
199	#427	—	2	1	no	—
200	#427	—	2	1	no	—
201	#423	C₂²×C₃₁	1	1	yes	characteristic
202	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
203	#408	—	2	1	no	—
204	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
205	#335	C₂²×C₃₁	1	2	yes	—
206	#335	C₂²×C₃₁	1	2	yes	—
207	#333	C₂²×C₃₁	1	4	yes	—
208	#332	C₂²×C₃₁	1	4	yes	—
209	C₂×C₃₁	—	2	2	no	—
210	C₂×C₃₁	—	8	2	no	—
211	C₂×C₃₁	C₂×#461	1	1	yes	characteristic
212	C₂×C₃₁	#3061	1	1	yes	characteristic
213	C₂×C₃₁	#3067	1	1	yes	characteristic
214	C₂×#41	—	2	1	no	—
215	C₂×SD₁₆	—	2	4	no	—
216	#131	—	4	4	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	2	2	no	—
219	C₂²×C₈	—	2	2	no	—
220	C₂²×C₈	—	2	2	no	—
221	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
223	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
224	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
225	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
226	#127	D₄×C₃₁	1	1	yes	characteristic
227	#125	—	2	2	no	—
228	#124	—	2	1	no	—
229	C₂²⋊Q₈	—	2	2	no	—
230	C₂²⋊Q₈	—	2	2	no	—
231	#121	—	2	1	no	—
232	#121	D₄×C₃₁	1	2	yes	—
233	C₄×Q₈	—	2	2	no	—
234	C₄×Q₈	—	2	2	no	—
235	C₄×D₄	—	2	1	no	—
236	C₄×D₄	C₂³×C₃₁	1	2	yes	—
237	#117	—	2	2	no	—
238	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
239	#107	—	2	1	no	—
240	#107	—	2	1	no	—
241	#107	—	2	1	no	—
242	#107	—	2	1	no	—
243	#106	—	2	2	no	—
244	#106	—	2	2	no	—
245	#106	—	2	4	no	—
246	C₄⋊C₈	—	2	2	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
249	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
250	#103	—	2	4	no	—
251	#102	—	2	4	no	—
252	C₂²⋊C₈	—	2	2	no	—
253	C₂²⋊C₈	—	2	2	no	—
254	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
255	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
257	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
258	C₃₁	#19195	1	1	yes	characteristic, direct product factor, Sylow
259	#41	—	4	2	no	—
260	#41	—	4	2	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
271	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
272	SD₁₆	—	8	4	no	—
273	M₁₆	—	4	2	no	—
274	M₁₆	—	4	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	4	4	no	—
279	C₂×C₈	—	8	2	no	—
280	C₂×C₈	SD₁₆×C₃₁	1	4	yes	complemented
281	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
283	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
284	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
293	C₂²⋊C₄	—	2	2	no	—
294	C₂²⋊C₄	—	2	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₄²	—	4	2	no	—
297	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
298	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
299	C₄²	#41×C₃₁	1	1	yes	characteristic
300	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
301	C₂³	—	2	2	no	—
302	C₂³	C₃₁×#121	1	1	yes	characteristic
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	8	2	no	—
307	D₄	—	2	2	no	—
308	D₄	—	2	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
321	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
322	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
323	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
324	C₂×C₄	C₃₁×#137	1	2	yes	—
325	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
326	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
327	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	4	2	no	—
331	C₈	—	4	4	no	—
332	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
333	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	C₃₁×#461	1	2	yes	—
340	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	C₃₁×#461	1	2	yes	—
348	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
349	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
350	C₄	C₃₁×#576	1	1	yes	characteristic
351	C₄	C₃₁×#578	1	1	yes	characteristic
352	C₂	—	2	2	no	—
353	C₂	—	8	2	no	—
354	C₂	C₂×C₃₁×#461	1	1	yes	characteristic
355	C₂	C₃₁×#3061	1	1	yes	characteristic
356	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#532	C₂²
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
2. C₃₁×#3160	C₂	semidirect
4. C₃₁×#1691	C₂	semidirect
5. C₃₁×#1675	C₂	semidirect
6. C₃₁×#1673	C₂	semidirect
7. C₂×C₃₁×#335	C₂	semidirect
8. C₃₁×#1550	C₂	semidirect
101. C₂×C₈×C₃₁	SD₁₆	semidirect
122. #19195	C₃₁	direct (N × H)
157. #3160	C₂×C₃₁	semidirect
159. #1691	C₂×C₃₁	semidirect
160. #1675	C₂×C₃₁	semidirect
161. #1673	C₂×C₃₁	semidirect
162. C₂×#335	C₂×C₃₁	semidirect
163. #1550	C₂×C₃₁	semidirect
258. C₃₁	#19195	direct (N × H)
280. C₂×C₈	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3246	2
C₃₁×#3160	2
C₃₁×#2931	2
C₃₁×#1691	2
C₃₁×#1675	2
C₃₁×#1673	2
C₂×C₃₁×#335	2
C₃₁×#1550	2
#19195	31

Subgroups — C31×#19195 #21475954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19195 #21475954