Subgroups — C₃₁×#19188 #21475947

Order 7936.

714 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×#3226	C₂	1	2	yes	maximal
2	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1689	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1669	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1546	C₂	1	2	yes	maximal, complemented
8	C₂²×Q₁₆×C₃₁	—	2	1	no	—
9	C₂²×Q₁₆×C₃₁	—	2	1	no	—
10	C₂×C₃₁×#131	—	2	1	no	—
11	C₃₁×#532	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₃₁×#500	—	2	2	no	—
14	C₃₁×#494	—	2	1	no	—
15	C₃₁×#494	—	2	1	no	—
16	C₃₁×#494	C₂²	1	2	yes	—
17	C₃₁×#494	C₂²	1	2	yes	—
18	C₃₁×#486	—	2	2	no	—
19	C₃₁×#480	—	2	4	no	—
20	C₃₁×#474	C₂²	1	8	yes	—
21	C₃₁×#436	—	2	1	no	—
22	C₃₁×#436	C₂²	1	2	yes	—
23	C₈×D₄×C₃₁	—	2	1	no	—
24	C₈×D₄×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#431	C₂²	1	2	yes	—
26	C₃₁×#423	C₂²	1	1	yes	characteristic
27	C₃₁×#408	—	2	1	no	—
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#335	C₂²	1	2	yes	—
30	C₃₁×#334	C₂²	1	2	yes	—
31	C₃₁×#332	C₂²	1	8	yes	—
32	C₂×#41×C₃₁	—	2	1	no	—
33	C₂²×Q₈×C₃₁	—	2	2	no	—
34	C₂×Q₁₆×C₃₁	—	2	2	no	—
35	C₂×Q₁₆×C₃₁	—	2	2	no	—
36	C₂×Q₁₆×C₃₁	—	2	2	no	—
37	C₂×Q₁₆×C₃₁	—	2	2	no	—
38	C₂×Q₁₆×C₃₁	—	4	4	no	—
39	C₂×Q₁₆×C₃₁	—	4	4	no	—
40	C₃₁×#131	—	4	4	no	—
41	C₂×M₁₆×C₃₁	—	2	1	no	—
42	C₂×M₁₆×C₃₁	—	2	2	no	—
43	C₂²×C₈×C₃₁	—	2	2	no	—
44	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
46	C₃₁×C₄⋊Q₈	—	2	2	no	—
47	C₃₁×C₄⋊Q₈	—	2	2	no	—
48	C₃₁×C₄⋊Q₈	—	2	2	no	—
49	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
50	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
51	C₃₁×#127	D₄	1	1	yes	characteristic
52	C₃₁×#124	—	2	1	no	—
53	C₃₁×#121	—	2	1	no	—
54	C₃₁×#121	D₄	1	2	yes	—
55	C₄×D₄×C₃₁	—	2	1	no	—
56	C₄×D₄×C₃₁	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×#107	—	2	8	no	—
61	C₃₁×C₄⋊C₈	—	2	1	no	—
62	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
63	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
64	C₃₁×#102	—	2	8	no	—
65	C₃₁×C₂²⋊C₈	—	2	2	no	—
66	C₃₁×C₂²⋊C₈	—	2	2	no	—
67	C₃₁×C₂²⋊C₈	—	2	2	no	—
68	C₄×C₈×C₃₁	—	2	2	no	—
69	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
70	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
71	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	#41×C₃₁	—	4	2	no	—
74	#41×C₃₁	—	4	2	no	—
75	C₂×Q₈×C₃₁	—	2	1	no	—
76	C₂×Q₈×C₃₁	—	2	4	no	—
77	C₂×Q₈×C₃₁	—	4	2	no	—
78	C₂×Q₈×C₃₁	—	4	4	no	—
79	C₂×Q₈×C₃₁	—	4	4	no	—
80	C₂×D₄×C₃₁	—	2	2	no	—
81	C₂×D₄×C₃₁	—	2	1	no	—
82	C₂×D₄×C₃₁	—	2	1	no	—
83	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	4	2	no	—
86	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
89	Q₁₆×C₃₁	—	4	4	no	—
90	Q₁₆×C₃₁	—	4	4	no	—
91	Q₁₆×C₃₁	—	4	4	no	—
92	Q₁₆×C₃₁	—	4	4	no	—
93	M₁₆×C₃₁	—	4	2	no	—
94	M₁₆×C₃₁	—	4	4	no	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	—	2	4	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	4	4	no	—
99	C₂×C₈×C₃₁	D₈	1	4	yes	—
100	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₃₁	—	2	1	no	—
105	C₄⋊C₄×C₃₁	—	2	4	no	—
106	C₄⋊C₄×C₃₁	—	2	4	no	—
107	C₄⋊C₄×C₃₁	—	4	2	no	—
108	C₄⋊C₄×C₃₁	—	4	2	no	—
109	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
110	C₂²⋊C₄×C₃₁	—	2	2	no	—
111	C₂²⋊C₄×C₃₁	—	2	2	no	—
112	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₃₁	#41	1	1	yes	characteristic
115	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
116	#19188	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₃₁	—	2	2	no	—
118	C₂³×C₃₁	#121	1	1	yes	characteristic
119	Q₈×C₃₁	—	4	2	no	—
120	Q₈×C₃₁	—	4	4	no	—
121	Q₈×C₃₁	—	4	4	no	—
122	Q₈×C₃₁	—	8	4	no	—
123	D₄×C₃₁	—	2	4	no	—
124	D₄×C₃₁	—	4	4	no	—
125	D₄×C₃₁	—	4	2	no	—
126	D₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	4	no	—
133	C₂×C₄×C₃₁	—	8	4	no	—
134	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
135	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
136	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
139	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
142	C₈×C₃₁	—	2	1	no	—
143	C₈×C₃₁	—	2	1	no	—
144	C₈×C₃₁	—	4	2	no	—
145	C₈×C₃₁	—	4	4	no	—
146	C₈×C₃₁	C₂×D₈	1	2	yes	—
147	C₈×C₃₁	C₂×D₈	1	2	yes	—
148	#3226	C₂×C₃₁	1	2	yes	—
149	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#2931	C₂×C₃₁	1	1	yes	characteristic
151	#1689	C₂×C₃₁	1	4	yes	complemented
152	#1669	C₂×C₃₁	1	4	yes	complemented
153	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
154	#1546	C₂×C₃₁	1	2	yes	complemented
155	C₂²×C₃₁	—	2	1	no	—
156	C₂²×C₃₁	—	2	1	no	—
157	C₂²×C₃₁	—	4	2	no	—
158	C₂²×C₃₁	—	4	2	no	—
159	C₂²×C₃₁	—	4	2	no	—
160	C₂²×C₃₁	#459	1	2	yes	—
161	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
162	C₄×C₃₁	—	2	1	no	—
163	C₄×C₃₁	—	2	1	no	—
164	C₄×C₃₁	—	4	2	no	—
165	C₄×C₃₁	—	8	4	no	—
166	C₄×C₃₁	#459	1	2	yes	—
167	C₄×C₃₁	#473	1	2	yes	—
168	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
169	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₄×C₃₁	#578	1	2	yes	—
171	C₂²×Q₁₆	—	2	1	no	—
172	C₂²×Q₁₆	—	2	1	no	—
173	C₂×#131	—	2	1	no	—
174	#532	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
176	#500	—	2	2	no	—
177	#494	—	2	1	no	—
178	#494	—	2	1	no	—
179	#494	C₂²×C₃₁	1	2	yes	—
180	#494	C₂²×C₃₁	1	2	yes	—
181	#486	—	2	2	no	—
182	#480	—	2	4	no	—
183	#474	C₂²×C₃₁	1	8	yes	—
184	#436	—	2	1	no	—
185	#436	C₂²×C₃₁	1	2	yes	—
186	C₈×D₄	—	2	1	no	—
187	C₈×D₄	C₂²×C₃₁	1	2	yes	—
188	#431	C₂²×C₃₁	1	2	yes	—
189	#423	C₂²×C₃₁	1	1	yes	characteristic
190	#408	—	2	1	no	—
191	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
192	#335	C₂²×C₃₁	1	2	yes	—
193	#334	C₂²×C₃₁	1	2	yes	—
194	#332	C₂²×C₃₁	1	8	yes	—
195	C₂×C₃₁	—	2	2	no	—
196	C₂×C₃₁	—	8	2	no	—
197	C₂×C₃₁	C₂×#459	1	1	yes	characteristic
198	C₂×C₃₁	#3059	1	1	yes	characteristic
199	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
200	C₂×#41	—	2	1	no	—
201	C₂²×Q₈	—	2	2	no	—
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×Q₁₆	—	2	2	no	—
204	C₂×Q₁₆	—	2	2	no	—
205	C₂×Q₁₆	—	2	2	no	—
206	C₂×Q₁₆	—	4	4	no	—
207	C₂×Q₁₆	—	4	4	no	—
208	#131	—	4	4	no	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	—	2	2	no	—
211	C₂²×C₈	—	2	2	no	—
212	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
213	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
214	C₄⋊Q₈	—	2	2	no	—
215	C₄⋊Q₈	—	2	2	no	—
216	C₄⋊Q₈	—	2	2	no	—
217	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
218	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
219	#127	D₄×C₃₁	1	1	yes	characteristic
220	#124	—	2	1	no	—
221	#121	—	2	1	no	—
222	#121	D₄×C₃₁	1	2	yes	—
223	C₄×D₄	—	2	1	no	—
224	C₄×D₄	C₂³×C₃₁	1	2	yes	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
228	#107	—	2	8	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
231	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
232	#102	—	2	8	no	—
233	C₂²⋊C₈	—	2	2	no	—
234	C₂²⋊C₈	—	2	2	no	—
235	C₂²⋊C₈	—	2	2	no	—
236	C₄×C₈	—	2	2	no	—
237	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
238	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
241	C₃₁	#19188	1	1	yes	characteristic, direct product factor, Sylow
242	#41	—	4	2	no	—
243	#41	—	4	2	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	—	4	4	no	—
248	C₂×Q₈	—	4	4	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
257	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
258	Q₁₆	—	4	4	no	—
259	Q₁₆	—	4	4	no	—
260	Q₁₆	—	4	4	no	—
261	Q₁₆	—	4	4	no	—
262	M₁₆	—	4	2	no	—
263	M₁₆	—	4	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	4	4	no	—
268	C₂×C₈	D₈×C₃₁	1	4	yes	—
269	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
283	C₄²	#41×C₃₁	1	1	yes	characteristic
284	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
285	C₂³	—	2	2	no	—
286	C₂³	C₃₁×#121	1	1	yes	characteristic
287	Q₈	—	4	2	no	—
288	Q₈	—	4	4	no	—
289	Q₈	—	4	4	no	—
290	Q₈	—	8	4	no	—
291	D₄	—	2	4	no	—
292	D₄	—	4	4	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
303	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
304	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
305	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
306	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
307	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
309	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	2	no	—
313	C₈	—	4	4	no	—
314	C₈	C₂×D₈×C₃₁	1	2	yes	—
315	C₈	C₂×D₈×C₃₁	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	C₃₁×#459	1	2	yes	—
322	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	8	4	no	—
327	C₄	C₃₁×#459	1	2	yes	—
328	C₄	C₃₁×#473	1	2	yes	—
329	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
330	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
331	C₄	C₃₁×#578	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×C₃₁×#459	1	1	yes	characteristic
335	C₂	C₃₁×#3059	1	1	yes	characteristic
336	C₂	C₂×C₃₁×#473	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. C₂×C₃₁×#494	C₂	semidirect
4. C₃₁×#1689	C₂	semidirect
5. C₃₁×#1669	C₂	semidirect
6. C₃₁×#1564	C₂	semidirect
7. C₃₁×#1546	C₂	semidirect
116. #19188	C₃₁	direct (N × H)
149. C₂×#494	C₂×C₃₁	semidirect
151. #1689	C₂×C₃₁	semidirect
152. #1669	C₂×C₃₁	semidirect
153. #1564	C₂×C₃₁	semidirect
154. #1546	C₂×C₃₁	semidirect
241. C₃₁	#19188	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×#3226	2
C₂×C₃₁×#494	2
C₃₁×#2931	2
C₃₁×#1689	2
C₃₁×#1669	2
C₃₁×#1564	2
C₃₁×#1546	2
#19188	31

Subgroups — C31×#19188 #21475947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19188 #21475947