Subgroups — C₃₁×#19337 #21476096

Order 7936.

680 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3253	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1695	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1687	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1677	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1560	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#131	—	2	1	no	—
10	C₃₁×#532	C₂²	1	2	yes	—
11	C₃₁×#532	C₂²	1	1	yes	characteristic
12	C₃₁×#500	C₂²	1	1	yes	characteristic
13	C₃₁×#493	C₂²	1	1	yes	characteristic
14	C₃₁×#486	C₂²	1	2	yes	—
15	C₃₁×#476	—	2	2	no	—
16	C₃₁×#476	C₂²	1	4	yes	—
17	C₃₁×#469	—	2	4	no	—
18	C₃₁×#466	—	2	1	no	—
19	C₃₁×#466	—	2	1	no	—
20	C₃₁×#459	—	2	2	no	—
21	C₃₁×#459	C₂²	1	4	yes	—
22	C₄×D₈×C₃₁	—	2	1	no	—
23	C₄×D₈×C₃₁	—	2	1	no	—
24	C₃₁×#436	—	2	1	no	—
25	C₃₁×#436	C₂²	1	2	yes	—
26	C₈×D₄×C₃₁	—	2	1	no	—
27	C₈×D₄×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#423	C₂²	1	1	yes	characteristic
29	C₃₁×#423	C₂²	1	2	yes	—
30	C₃₁×#408	—	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×#333	C₂²	1	4	yes	—
35	C₃₁×#331	C₂²	1	4	yes	—
36	C₂×#41×C₃₁	—	2	1	no	—
37	C₂×#41×C₃₁	—	2	2	no	—
38	C₂×D₈×C₃₁	—	2	4	no	—
39	C₂×D₈×C₃₁	—	2	1	no	—
40	C₂×D₈×C₃₁	—	2	1	no	—
41	C₃₁×#131	—	4	4	no	—
42	C₂×M₁₆×C₃₁	—	2	1	no	—
43	C₂×M₁₆×C₃₁	—	2	2	no	—
44	C₂×M₁₆×C₃₁	—	2	2	no	—
45	C₂²×C₈×C₃₁	—	2	2	no	—
46	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
48	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
49	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₃₁×#127	D₄	1	1	yes	characteristic
51	C₃₁×#127	C₂³	1	2	yes	—
52	C₃₁×#124	—	2	1	no	—
53	C₃₁×#124	—	2	2	no	—
54	C₃₁×#121	—	2	1	no	—
55	C₃₁×#121	—	2	2	no	—
56	C₃₁×#121	—	2	2	no	—
57	C₃₁×#121	D₄	1	2	yes	—
58	C₄×D₄×C₃₁	—	2	1	no	—
59	C₄×D₄×C₃₁	—	2	2	no	—
60	C₄×D₄×C₃₁	—	2	2	no	—
61	C₄×D₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₃₁×#107	—	2	1	no	—
64	C₃₁×#107	—	2	1	no	—
65	C₃₁×#106	—	2	4	no	—
66	C₃₁×C₄⋊C₈	—	2	1	no	—
67	C₃₁×C₄⋊C₈	—	2	2	no	—
68	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
69	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
70	C₃₁×#102	—	2	4	no	—
71	C₃₁×#102	—	2	2	no	—
72	C₃₁×#102	—	2	2	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	#41×C₃₁	—	8	4	no	—
83	C₂×Q₈×C₃₁	—	2	1	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	—	2	2	no	—
89	C₂×D₄×C₃₁	—	2	2	no	—
90	C₂×D₄×C₃₁	—	2	2	no	—
91	C₂×D₄×C₃₁	—	4	2	no	—
92	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₃₁	—	2	2	no	—
94	C₂²×C₄×C₃₁	—	4	2	no	—
95	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
98	D₈×C₃₁	—	4	2	no	—
99	D₈×C₃₁	—	4	2	no	—
100	D₈×C₃₁	—	8	4	no	—
101	M₁₆×C₃₁	—	4	2	no	—
102	M₁₆×C₃₁	—	4	4	no	—
103	M₁₆×C₃₁	—	8	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	4	no	—
107	C₂×C₈×C₃₁	—	4	1	no	—
108	C₂×C₈×C₃₁	—	4	4	no	—
109	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
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	C₂×D₄	1	2	yes	—
117	C₂²⋊C₄×C₃₁	—	2	2	no	—
118	C₂²⋊C₄×C₃₁	—	2	2	no	—
119	C₂²⋊C₄×C₃₁	—	4	2	no	—
120	C₂²⋊C₄×C₃₁	—	4	2	no	—
121	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₃₁	#41	1	1	yes	characteristic
124	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
125	#19337	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂³×C₃₁	—	2	2	no	—
127	C₂³×C₃₁	—	4	2	no	—
128	C₂³×C₃₁	#121	1	1	yes	characteristic
129	Q₈×C₃₁	—	4	2	no	—
130	Q₈×C₃₁	—	8	2	no	—
131	D₄×C₃₁	—	2	4	no	—
132	D₄×C₃₁	—	4	2	no	—
133	D₄×C₃₁	—	4	4	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	—	4	2	no	—
139	D₄×C₃₁	—	8	2	no	—
140	D₄×C₃₁	—	8	4	no	—
141	C₂×C₄×C₃₁	—	2	1	no	—
142	C₂×C₄×C₃₁	—	2	1	no	—
143	C₂×C₄×C₃₁	—	2	2	no	—
144	C₂×C₄×C₃₁	—	2	2	no	—
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	4	2	no	—
147	C₂×C₄×C₃₁	—	4	2	no	—
148	C₂×C₄×C₃₁	—	4	2	no	—
149	C₂×C₄×C₃₁	—	8	4	no	—
150	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
151	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
152	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
153	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
156	C₈×C₃₁	—	2	1	no	—
157	C₈×C₃₁	—	2	1	no	—
158	C₈×C₃₁	—	2	1	no	—
159	C₈×C₃₁	—	4	2	no	—
160	C₈×C₃₁	—	4	4	no	—
161	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
162	#3253	C₂×C₃₁	1	2	yes	complemented
163	#3169	C₂×C₃₁	1	1	yes	characteristic, complemented
164	#2931	C₂×C₃₁	1	1	yes	characteristic, complemented
165	#1695	C₂×C₃₁	1	2	yes	complemented
166	#1687	C₂×C₃₁	1	2	yes	complemented
167	#1677	C₂×C₃₁	1	4	yes	complemented
168	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
169	#1560	C₂×C₃₁	1	2	yes	complemented
170	C₂²×C₃₁	—	2	1	no	—
171	C₂²×C₃₁	—	2	1	no	—
172	C₂²×C₃₁	—	2	1	no	—
173	C₂²×C₃₁	—	4	2	no	—
174	C₂²×C₃₁	—	4	2	no	—
175	C₂²×C₃₁	—	4	2	no	—
176	C₂²×C₃₁	—	8	2	no	—
177	C₂²×C₃₁	—	8	2	no	—
178	C₂²×C₃₁	—	8	2	no	—
179	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
180	C₄×C₃₁	—	2	1	no	—
181	C₄×C₃₁	—	2	1	no	—
182	C₄×C₃₁	—	2	1	no	—
183	C₄×C₃₁	—	4	2	no	—
184	C₄×C₃₁	—	8	2	no	—
185	C₄×C₃₁	#472	1	2	yes	—
186	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₄×C₃₁	#575	1	1	yes	characteristic
188	C₄×C₃₁	#577	1	2	yes	—
189	C₂×#131	—	2	1	no	—
190	#532	C₂²×C₃₁	1	2	yes	—
191	#532	C₂²×C₃₁	1	1	yes	characteristic
192	#500	C₂²×C₃₁	1	1	yes	characteristic
193	#493	C₂²×C₃₁	1	1	yes	characteristic
194	#486	C₂²×C₃₁	1	2	yes	—
195	#476	—	2	2	no	—
196	#476	C₂²×C₃₁	1	4	yes	—
197	#469	—	2	4	no	—
198	#466	—	2	1	no	—
199	#466	—	2	1	no	—
200	#459	—	2	2	no	—
201	#459	C₂²×C₃₁	1	4	yes	—
202	C₄×D₈	—	2	1	no	—
203	C₄×D₈	—	2	1	no	—
204	#436	—	2	1	no	—
205	#436	C₂²×C₃₁	1	2	yes	—
206	C₈×D₄	—	2	1	no	—
207	C₈×D₄	C₂²×C₃₁	1	2	yes	—
208	#423	C₂²×C₃₁	1	1	yes	characteristic
209	#423	C₂²×C₃₁	1	2	yes	—
210	#408	—	2	1	no	—
211	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
212	#334	C₂²×C₃₁	1	2	yes	—
213	#334	C₂²×C₃₁	1	2	yes	—
214	#333	C₂²×C₃₁	1	4	yes	—
215	#331	C₂²×C₃₁	1	4	yes	—
216	C₂×C₃₁	—	4	1	no	—
217	C₂×C₃₁	—	8	2	no	—
218	C₂×C₃₁	—	16	2	no	—
219	C₂×C₃₁	#3036	1	1	yes	characteristic
220	C₂×C₃₁	#3060	1	1	yes	characteristic
221	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
222	C₂×#41	—	2	1	no	—
223	C₂×#41	—	2	2	no	—
224	C₂×D₈	—	2	4	no	—
225	C₂×D₈	—	2	1	no	—
226	C₂×D₈	—	2	1	no	—
227	#131	—	4	4	no	—
228	C₂×M₁₆	—	2	1	no	—
229	C₂×M₁₆	—	2	2	no	—
230	C₂×M₁₆	—	2	2	no	—
231	C₂²×C₈	—	2	2	no	—
232	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
233	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
234	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
235	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
236	#127	D₄×C₃₁	1	1	yes	characteristic
237	#127	C₂³×C₃₁	1	2	yes	—
238	#124	—	2	1	no	—
239	#124	—	2	2	no	—
240	#121	—	2	1	no	—
241	#121	—	2	2	no	—
242	#121	—	2	2	no	—
243	#121	D₄×C₃₁	1	2	yes	—
244	C₄×D₄	—	2	1	no	—
245	C₄×D₄	—	2	2	no	—
246	C₄×D₄	—	2	2	no	—
247	C₄×D₄	C₂³×C₃₁	1	2	yes	—
248	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
249	#107	—	2	1	no	—
250	#107	—	2	1	no	—
251	#106	—	2	4	no	—
252	C₄⋊C₈	—	2	1	no	—
253	C₄⋊C₈	—	2	2	no	—
254	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
255	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
256	#102	—	2	4	no	—
257	#102	—	2	2	no	—
258	#102	—	2	2	no	—
259	#102	—	2	4	no	—
260	C₂²⋊C₈	—	2	2	no	—
261	C₂²⋊C₈	—	2	2	no	—
262	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
263	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
264	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
265	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
266	C₃₁	#19337	1	1	yes	characteristic, direct product factor, Sylow
267	#41	—	4	2	no	—
268	#41	—	4	2	no	—
269	#41	—	8	4	no	—
270	C₂×Q₈	—	2	1	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	4	2	no	—
282	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
283	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
284	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
285	D₈	—	4	2	no	—
286	D₈	—	4	2	no	—
287	D₈	—	8	4	no	—
288	M₁₆	—	4	2	no	—
289	M₁₆	—	4	4	no	—
290	M₁₆	—	8	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	4	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	—	4	4	no	—
296	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
297	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
298	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
299	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	—	2	2	no	—
306	C₂²⋊C₄	—	4	2	no	—
307	C₂²⋊C₄	—	4	2	no	—
308	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
309	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
310	C₄²	#41×C₃₁	1	1	yes	characteristic
311	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
312	C₂³	—	2	2	no	—
313	C₂³	—	4	2	no	—
314	C₂³	C₃₁×#121	1	1	yes	characteristic
315	Q₈	—	4	2	no	—
316	Q₈	—	8	2	no	—
317	D₄	—	2	4	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	4	no	—
320	D₄	—	4	2	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	2	no	—
323	D₄	—	4	2	no	—
324	D₄	—	4	2	no	—
325	D₄	—	8	2	no	—
326	D₄	—	8	4	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	8	4	no	—
336	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
337	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
338	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
339	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
340	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
341	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	4	2	no	—
346	C₈	—	4	4	no	—
347	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	4	2	no	—
352	C₂²	—	4	2	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	4	2	no	—
362	C₄	—	8	2	no	—
363	C₄	C₃₁×#472	1	2	yes	—
364	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
365	C₄	C₃₁×#575	1	1	yes	characteristic
366	C₄	C₃₁×#577	1	2	yes	—
367	C₂	—	4	1	no	—
368	C₂	—	8	2	no	—
369	C₂	—	16	2	no	—
370	C₂	C₃₁×#3036	1	1	yes	characteristic
371	C₂	C₃₁×#3060	1	1	yes	characteristic
372	C₂	C₂×C₃₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
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×#3253	C₂	semidirect
2. C₃₁×#3169	C₂	semidirect
3. C₃₁×#2931	C₂	semidirect
4. C₃₁×#1695	C₂	semidirect
5. C₃₁×#1687	C₂	semidirect
6. C₃₁×#1677	C₂	semidirect
7. C₃₁×#1566	C₂	semidirect
8. C₃₁×#1560	C₂	semidirect
125. #19337	C₃₁	direct (N × H)
162. #3253	C₂×C₃₁	semidirect
163. #3169	C₂×C₃₁	semidirect
164. #2931	C₂×C₃₁	semidirect
165. #1695	C₂×C₃₁	semidirect
166. #1687	C₂×C₃₁	semidirect
167. #1677	C₂×C₃₁	semidirect
168. #1566	C₂×C₃₁	semidirect
169. #1560	C₂×C₃₁	semidirect
266. C₃₁	#19337	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×#3253	2
C₃₁×#3169	2
C₃₁×#2931	2
C₃₁×#1695	2
C₃₁×#1687	2
C₃₁×#1677	2
C₃₁×#1566	2
C₃₁×#1560	2
#19337	31

Subgroups — C31×#19337 #21476096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19337 #21476096