Subgroups — C₃₁×#19205 #21475964

Order 7936.

678 conjugacy classes of subgroups.

The subgroup lattice has 364 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, complemented
2	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1690	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1688	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1666	C₂	1	4	yes	maximal, complemented
7	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1547	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#135	—	2	1	no	—
10	C₂×C₃₁×#135	—	2	1	no	—
11	C₃₁×#533	C₂²	1	1	yes	characteristic
12	C₃₁×#532	C₂²	1	2	yes	—
13	C₃₁×#495	—	2	1	no	—
14	C₃₁×#495	—	2	1	no	—
15	C₃₁×#494	C₂²	1	1	yes	characteristic
16	C₃₁×#493	C₂²	1	1	yes	characteristic
17	C₃₁×#492	C₂²	1	2	yes	—
18	C₃₁×#476	—	2	2	no	—
19	C₃₁×#475	C₂²	1	4	yes	—
20	C₃₁×#465	—	2	4	no	—
21	C₃₁×#461	—	2	2	no	—
22	C₃₁×#460	C₂²	1	4	yes	—
23	C₃₁×#446	—	2	1	no	—
24	C₃₁×#446	C₂²	1	2	yes	—
25	C₈×Q₈×C₃₁	—	2	1	no	—
26	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
27	C₃₁×#424	—	2	1	no	—
28	C₃₁×#423	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
30	C₃₁×#405	—	2	1	no	—
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#334	C₂²	1	2	yes	—
33	C₃₁×#334	C₂²	1	2	yes	—
34	C₃₁×#332	C₂²	1	4	yes	—
35	C₃₁×#331	C₂²	1	4	yes	—
36	C₂×#41×C₃₁	—	2	2	no	—
37	C₃₁×#135	—	4	4	no	—
38	C₃₁×#135	—	4	4	no	—
39	C₂×Q₁₆×C₃₁	—	2	1	no	—
40	C₂×Q₁₆×C₃₁	—	2	1	no	—
41	C₂×SD₁₆×C₃₁	—	2	4	no	—
42	C₂×SD₁₆×C₃₁	—	2	2	no	—
43	C₂×SD₁₆×C₃₁	—	2	2	no	—
44	C₂×D₈×C₃₁	—	2	1	no	—
45	C₂×D₈×C₃₁	—	2	1	no	—
46	C₂×M₁₆×C₃₁	—	2	1	no	—
47	C₂²×C₈×C₃₁	—	2	2	no	—
48	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
50	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
51	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
52	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
53	C₃₁×#127	C₂³	1	2	yes	—
54	C₃₁×#125	—	2	1	no	—
55	C₃₁×#124	—	2	2	no	—
56	C₃₁×C₂²⋊Q₈	—	2	1	no	—
57	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
58	C₃₁×#121	—	2	2	no	—
59	C₃₁×#121	—	2	2	no	—
60	C₄×Q₈×C₃₁	—	2	1	no	—
61	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
62	C₄×D₄×C₃₁	—	2	2	no	—
63	C₄×D₄×C₃₁	—	2	2	no	—
64	C₃₁×#117	—	2	1	no	—
65	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×#106	—	2	4	no	—
67	C₃₁×C₄⋊C₈	—	2	2	no	—
68	C₃₁×C₄⋊C₈	—	2	2	no	—
69	C₃₁×C₄⋊C₈	—	2	2	no	—
70	C₃₁×C₄⋊C₈	—	2	1	no	—
71	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
72	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
73	C₃₁×#103	—	2	4	no	—
74	C₃₁×#102	—	2	4	no	—
75	C₃₁×#97	—	2	2	no	—
76	C₄×C₈×C₃₁	—	2	2	no	—
77	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
78	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
79	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	#41×C₃₁	—	4	4	no	—
82	#41×C₃₁	—	4	4	no	—
83	C₂×Q₈×C₃₁	—	2	1	no	—
84	C₂×Q₈×C₃₁	—	2	2	no	—
85	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	4	2	no	—
90	C₂²×C₄×C₃₁	—	4	2	no	—
91	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
94	Q₁₆×C₃₁	—	4	2	no	—
95	Q₁₆×C₃₁	—	4	2	no	—
96	SD₁₆×C₃₁	—	4	4	no	—
97	SD₁₆×C₃₁	—	4	4	no	—
98	SD₁₆×C₃₁	—	8	4	no	—
99	D₈×C₃₁	—	4	2	no	—
100	D₈×C₃₁	—	4	2	no	—
101	M₁₆×C₃₁	—	4	2	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	4	no	—
105	C₂×C₈×C₃₁	—	8	2	no	—
106	C₂×C₈×C₃₁	SD₁₆	1	4	yes	complemented
107	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₄⋊C₄×C₃₁	—	2	1	no	—
112	C₄⋊C₄×C₃₁	—	2	1	no	—
113	C₄⋊C₄×C₃₁	—	2	1	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
119	C₂²⋊C₄×C₃₁	—	2	2	no	—
120	C₂²⋊C₄×C₃₁	—	4	2	no	—
121	C₂²⋊C₄×C₃₁	—	4	2	no	—
122	C₄²×C₃₁	—	2	2	no	—
123	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₃₁	#41	1	1	yes	characteristic
126	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
127	#19205	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₃₁	—	4	2	no	—
129	C₂³×C₃₁	#121	1	1	yes	characteristic
130	Q₈×C₃₁	—	2	4	no	—
131	Q₈×C₃₁	—	4	2	no	—
132	Q₈×C₃₁	—	4	2	no	—
133	Q₈×C₃₁	—	4	2	no	—
134	D₄×C₃₁	—	4	2	no	—
135	D₄×C₃₁	—	4	2	no	—
136	D₄×C₃₁	—	4	2	no	—
137	D₄×C₃₁	—	4	2	no	—
138	D₄×C₃₁	—	8	2	no	—
139	D₄×C₃₁	—	8	4	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	2	2	no	—
143	C₂×C₄×C₃₁	—	4	1	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	8	4	no	—
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	C₂×SD₁₆	1	2	yes	—
150	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
151	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
154	C₈×C₃₁	—	2	1	no	—
155	C₈×C₃₁	—	2	1	no	—
156	C₈×C₃₁	—	4	2	no	—
157	C₈×C₃₁	—	4	4	no	—
158	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
159	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
160	#3246	C₂×C₃₁	1	2	yes	complemented
161	#3147	C₂×C₃₁	1	1	yes	characteristic
162	#2960	C₂×C₃₁	1	1	yes	characteristic, complemented
163	#1690	C₂×C₃₁	1	2	yes	complemented
164	#1688	C₂×C₃₁	1	2	yes	complemented
165	#1666	C₂×C₃₁	1	4	yes	complemented
166	C₂×#334	C₂×C₃₁	1	1	yes	characteristic, complemented
167	#1547	C₂×C₃₁	1	2	yes	complemented
168	C₂²×C₃₁	—	2	1	no	—
169	C₂²×C₃₁	—	2	1	no	—
170	C₂²×C₃₁	—	8	2	no	—
171	C₂²×C₃₁	—	8	2	no	—
172	C₂²×C₃₁	—	8	2	no	—
173	C₂²×C₃₁	#460	1	2	yes	—
174	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
175	C₄×C₃₁	—	2	1	no	—
176	C₄×C₃₁	—	2	1	no	—
177	C₄×C₃₁	—	2	1	no	—
178	C₄×C₃₁	—	4	2	no	—
179	C₄×C₃₁	—	4	2	no	—
180	C₄×C₃₁	—	8	2	no	—
181	C₄×C₃₁	#460	1	2	yes	—
182	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
183	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₄×C₃₁	#577	1	2	yes	—
185	C₂×#135	—	2	1	no	—
186	C₂×#135	—	2	1	no	—
187	#533	C₂²×C₃₁	1	1	yes	characteristic
188	#532	C₂²×C₃₁	1	2	yes	—
189	#495	—	2	1	no	—
190	#495	—	2	1	no	—
191	#494	C₂²×C₃₁	1	1	yes	characteristic
192	#493	C₂²×C₃₁	1	1	yes	characteristic
193	#492	C₂²×C₃₁	1	2	yes	—
194	#476	—	2	2	no	—
195	#475	C₂²×C₃₁	1	4	yes	—
196	#465	—	2	4	no	—
197	#461	—	2	2	no	—
198	#460	C₂²×C₃₁	1	4	yes	—
199	#446	—	2	1	no	—
200	#446	C₂²×C₃₁	1	2	yes	—
201	C₈×Q₈	—	2	1	no	—
202	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
203	#424	—	2	1	no	—
204	#423	C₂²×C₃₁	1	1	yes	characteristic
205	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
206	#405	—	2	1	no	—
207	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
208	#334	C₂²×C₃₁	1	2	yes	—
209	#334	C₂²×C₃₁	1	2	yes	—
210	#332	C₂²×C₃₁	1	4	yes	—
211	#331	C₂²×C₃₁	1	4	yes	—
212	C₂×C₃₁	—	2	2	no	—
213	C₂×C₃₁	—	16	2	no	—
214	C₂×C₃₁	C₂×#460	1	1	yes	characteristic
215	C₂×C₃₁	#3060	1	1	yes	characteristic
216	C₂×C₃₁	#3067	1	1	yes	characteristic
217	C₂×#41	—	2	2	no	—
218	#135	—	4	4	no	—
219	#135	—	4	4	no	—
220	C₂×Q₁₆	—	2	1	no	—
221	C₂×Q₁₆	—	2	1	no	—
222	C₂×SD₁₆	—	2	4	no	—
223	C₂×SD₁₆	—	2	2	no	—
224	C₂×SD₁₆	—	2	2	no	—
225	C₂×D₈	—	2	1	no	—
226	C₂×D₈	—	2	1	no	—
227	C₂×M₁₆	—	2	1	no	—
228	C₂²×C₈	—	2	2	no	—
229	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
230	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
231	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
232	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
233	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
234	#127	C₂³×C₃₁	1	2	yes	—
235	#125	—	2	1	no	—
236	#124	—	2	2	no	—
237	C₂²⋊Q₈	—	2	1	no	—
238	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
239	#121	—	2	2	no	—
240	#121	—	2	2	no	—
241	C₄×Q₈	—	2	1	no	—
242	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
243	C₄×D₄	—	2	2	no	—
244	C₄×D₄	—	2	2	no	—
245	#117	—	2	1	no	—
246	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
247	#106	—	2	4	no	—
248	C₄⋊C₈	—	2	2	no	—
249	C₄⋊C₈	—	2	2	no	—
250	C₄⋊C₈	—	2	2	no	—
251	C₄⋊C₈	—	2	1	no	—
252	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
253	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
254	#103	—	2	4	no	—
255	#102	—	2	4	no	—
256	#97	—	2	2	no	—
257	C₄×C₈	—	2	2	no	—
258	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
259	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
260	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
261	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
262	C₃₁	#19205	1	1	yes	characteristic, direct product factor, Sylow
263	#41	—	4	4	no	—
264	#41	—	4	4	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	4	2	no	—
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
275	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
276	Q₁₆	—	4	2	no	—
277	Q₁₆	—	4	2	no	—
278	SD₁₆	—	4	4	no	—
279	SD₁₆	—	4	4	no	—
280	SD₁₆	—	8	4	no	—
281	D₈	—	4	2	no	—
282	D₈	—	4	2	no	—
283	M₁₆	—	4	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	4	no	—
287	C₂×C₈	—	8	2	no	—
288	C₂×C₈	SD₁₆×C₃₁	1	4	yes	complemented
289	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
290	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
291	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
292	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	4	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
306	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
307	C₄²	#41×C₃₁	1	1	yes	characteristic
308	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
309	C₂³	—	4	2	no	—
310	C₂³	C₃₁×#121	1	1	yes	characteristic
311	Q₈	—	2	4	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	4	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	8	2	no	—
320	D₄	—	8	4	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	8	4	no	—
327	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
328	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
329	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
330	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
331	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
332	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
334	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	4	2	no	—
338	C₈	—	4	4	no	—
339	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
340	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	C₃₁×#460	1	2	yes	—
347	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	8	2	no	—
354	C₄	C₃₁×#460	1	2	yes	—
355	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
356	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
357	C₄	C₃₁×#577	1	2	yes	—
358	C₂	—	2	2	no	—
359	C₂	—	16	2	no	—
360	C₂	C₂×C₃₁×#460	1	1	yes	characteristic
361	C₂	C₃₁×#3060	1	1	yes	characteristic
362	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×#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×#3246	C₂	semidirect
3. C₃₁×#2960	C₂	semidirect
4. C₃₁×#1690	C₂	semidirect
5. C₃₁×#1688	C₂	semidirect
6. C₃₁×#1666	C₂	semidirect
7. C₂×C₃₁×#334	C₂	semidirect
8. C₃₁×#1547	C₂	semidirect
106. C₂×C₈×C₃₁	SD₁₆	semidirect
127. #19205	C₃₁	direct (N × H)
160. #3246	C₂×C₃₁	semidirect
162. #2960	C₂×C₃₁	semidirect
163. #1690	C₂×C₃₁	semidirect
164. #1688	C₂×C₃₁	semidirect
165. #1666	C₂×C₃₁	semidirect
166. C₂×#334	C₂×C₃₁	semidirect
167. #1547	C₂×C₃₁	semidirect
262. C₃₁	#19205	direct (N × H)
288. 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×#3147	2
C₃₁×#2960	2
C₃₁×#1690	2
C₃₁×#1688	2
C₃₁×#1666	2
C₂×C₃₁×#334	2
C₃₁×#1547	2
#19205	31

Subgroups — C31×#19205 #21475964

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19205 #21475964