Subgroups — C₃₁×#33804 #21490563

Order 7936.

496 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3168	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2255	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2243	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2235	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2232	C₂	1	2	yes	maximal
7	C₃₁×#2227	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2226	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2163	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
11	C₃₁×#529	—	2	1	no	—
12	C₃₁×#500	—	2	1	no	—
13	C₃₁×#500	—	2	1	no	—
14	C₃₁×#500	C₂²	1	2	yes	—
15	C₃₁×#500	C₂²	1	2	yes	—
16	C₃₁×#499	C₂²	1	1	yes	characteristic
17	C₃₁×#495	C₂²	1	1	yes	characteristic
18	C₃₁×#488	—	2	1	no	—
19	C₃₁×#467	C₂²	1	1	yes	characteristic
20	C₃₁×#466	C₂²	1	1	yes	characteristic
21	C₃₁×#465	—	2	1	no	—
22	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
23	C₄×SD₁₆×C₃₁	—	2	1	no	—
24	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#107	—	2	1	no	—
26	C₂×C₃₁×#107	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#367	C₂²	1	4	yes	—
29	C₃₁×#366	C₂²	1	4	yes	—
30	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₃₁×#358	C₂²	1	4	yes	—
32	C₃₁×#357	C₂²	1	4	yes	—
33	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₃₁×#346	C₂²	1	2	yes	—
35	C₂²×Q₈×C₃₁	—	4	1	no	—
36	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
37	C₂×SD₁₆×C₃₁	—	2	1	no	—
38	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×C₄⋊Q₈	—	2	2	no	—
42	C₃₁×C₄⋊Q₈	—	2	2	no	—
43	C₃₁×C₄⋊Q₈	—	4	1	no	—
44	C₃₁×C₄⋊Q₈	—	4	1	no	—
45	C₃₁×#126	—	4	1	no	—
46	C₃₁×#125	—	2	1	no	—
47	C₃₁×#124	—	2	1	no	—
48	C₃₁×#123	—	4	1	no	—
49	C₃₁×C₂²⋊Q₈	—	2	1	no	—
50	C₃₁×#121	—	2	1	no	—
51	C₄×Q₈×C₃₁	—	2	1	no	—
52	C₄×D₄×C₃₁	—	2	1	no	—
53	C₄×D₄×C₃₁	—	4	1	no	—
54	C₃₁×#117	—	4	1	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
57	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
58	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
59	C₃₁×#107	—	2	2	no	—
60	C₃₁×#107	—	2	2	no	—
61	C₃₁×#107	—	2	2	no	—
62	C₃₁×#107	C₂³	1	4	yes	—
63	C₃₁×#107	C₂³	1	1	yes	characteristic
64	C₃₁×#107	C₂³	1	1	yes	characteristic
65	C₃₁×#106	—	2	1	no	—
66	C₃₁×#103	—	2	1	no	—
67	C₃₁×#103	—	2	1	no	—
68	C₃₁×#102	—	2	1	no	—
69	C₃₁×#102	—	2	1	no	—
70	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
71	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
72	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×Q₈×C₃₁	—	2	1	no	—
75	C₂×Q₈×C₃₁	—	4	2	no	—
76	C₂×Q₈×C₃₁	—	4	2	no	—
77	C₂×Q₈×C₃₁	—	8	4	no	—
78	C₂×D₄×C₃₁	—	2	1	no	—
79	C₂×D₄×C₃₁	—	4	1	no	—
80	C₂²×C₄×C₃₁	—	4	2	no	—
81	C₂²×C₄×C₃₁	—	4	1	no	—
82	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
85	Q₁₆×C₃₁	—	2	2	no	—
86	SD₁₆×C₃₁	—	4	2	no	—
87	D₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	—	2	2	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₃₁	—	2	1	no	—
95	C₄⋊C₄×C₃₁	—	2	1	no	—
96	C₄⋊C₄×C₃₁	—	2	4	no	—
97	C₄⋊C₄×C₃₁	—	4	2	no	—
98	C₄⋊C₄×C₃₁	—	4	2	no	—
99	C₄⋊C₄×C₃₁	—	4	2	no	—
100	C₄⋊C₄×C₃₁	—	4	1	no	—
101	C₄⋊C₄×C₃₁	—	4	1	no	—
102	C₄⋊C₄×C₃₁	—	4	1	no	—
103	C₄⋊C₄×C₃₁	—	4	1	no	—
104	C₂²⋊C₄×C₃₁	—	4	1	no	—
105	C₂²⋊C₄×C₃₁	—	4	1	no	—
106	C₂²⋊C₄×C₃₁	—	4	1	no	—
107	C₂²⋊C₄×C₃₁	—	4	1	no	—
108	C₂²⋊C₄×C₃₁	—	4	1	no	—
109	C₄²×C₃₁	—	4	1	no	—
110	C₄²×C₃₁	D₈	1	1	yes	characteristic
111	C₄²×C₃₁	D₈	1	1	yes	characteristic
112	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₁₆×C₃₁	—	2	4	no	—
115	#33804	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₃₁	—	4	1	no	—
117	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
118	Q₈×C₃₁	—	4	1	no	—
119	Q₈×C₃₁	—	4	1	no	—
120	Q₈×C₃₁	—	8	4	no	—
121	Q₈×C₃₁	—	8	4	no	—
122	D₄×C₃₁	—	4	1	no	—
123	D₄×C₃₁	—	4	1	no	—
124	D₄×C₃₁	—	8	2	no	—
125	C₂×C₄×C₃₁	—	2	2	no	—
126	C₂×C₄×C₃₁	—	2	1	no	—
127	C₂×C₄×C₃₁	—	2	1	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	4	1	no	—
130	C₂×C₄×C₃₁	—	4	1	no	—
131	C₂×C₄×C₃₁	—	4	1	no	—
132	C₂×C₄×C₃₁	—	4	4	no	—
133	C₂×C₄×C₃₁	—	8	4	no	—
134	C₂×C₄×C₃₁	—	8	2	no	—
135	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₃₁	—	2	1	no	—
142	C₈×C₃₁	—	2	1	no	—
143	C₈×C₃₁	#137	1	2	yes	—
144	C₈×C₃₁	2₋¹⁺⁴	1	2	yes	—
145	#3168	C₂×C₃₁	1	1	yes	characteristic
146	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#2255	C₂×C₃₁	1	2	yes	complemented
148	#2243	C₂×C₃₁	1	2	yes	complemented
149	#2235	C₂×C₃₁	1	2	yes	complemented
150	#2232	C₂×C₃₁	1	2	yes	—
151	#2227	C₂×C₃₁	1	2	yes	complemented
152	#2226	C₂×C₃₁	1	2	yes	complemented
153	#2163	C₂×C₃₁	1	1	yes	characteristic, complemented
154	C₂²×C₃₁	—	2	1	no	—
155	C₂²×C₃₁	—	2	1	no	—
156	C₂²×C₃₁	—	8	1	no	—
157	C₂²×C₃₁	—	8	1	no	—
158	C₂²×C₃₁	—	8	1	no	—
159	C₂²×C₃₁	#509	1	2	yes	—
160	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₃₁	—	2	1	no	—
162	C₄×C₃₁	—	2	1	no	—
163	C₄×C₃₁	—	2	1	no	—
164	C₄×C₃₁	—	2	1	no	—
165	C₄×C₃₁	—	8	4	no	—
166	C₄×C₃₁	—	8	1	no	—
167	C₄×C₃₁	—	8	1	no	—
168	C₄×C₃₁	—	8	1	no	—
169	C₄×C₃₁	#510	1	2	yes	—
170	C₄×C₃₁	#536	1	1	yes	characteristic
171	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
172	C₂×C₄⋊Q₈	—	2	1	no	—
173	#529	—	2	1	no	—
174	#500	—	2	1	no	—
175	#500	—	2	1	no	—
176	#500	C₂²×C₃₁	1	2	yes	—
177	#500	C₂²×C₃₁	1	2	yes	—
178	#499	C₂²×C₃₁	1	1	yes	characteristic
179	#495	C₂²×C₃₁	1	1	yes	characteristic
180	#488	—	2	1	no	—
181	#467	C₂²×C₃₁	1	1	yes	characteristic
182	#466	C₂²×C₃₁	1	1	yes	characteristic
183	#465	—	2	1	no	—
184	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
185	C₄×SD₁₆	—	2	1	no	—
186	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
187	C₂×#107	—	2	1	no	—
188	C₂×#107	C₂²×C₃₁	1	2	yes	—
189	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
190	#367	C₂²×C₃₁	1	4	yes	—
191	#366	C₂²×C₃₁	1	4	yes	—
192	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
193	#358	C₂²×C₃₁	1	4	yes	—
194	#357	C₂²×C₃₁	1	4	yes	—
195	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
196	#346	C₂²×C₃₁	1	2	yes	—
197	C₂×C₃₁	—	2	2	no	—
198	C₂×C₃₁	—	16	1	no	—
199	C₂×C₃₁	#3226	1	1	yes	characteristic
200	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
201	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
202	C₂²×Q₈	—	4	1	no	—
203	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
204	C₂×SD₁₆	—	2	1	no	—
205	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
206	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
207	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
208	C₄⋊Q₈	—	2	2	no	—
209	C₄⋊Q₈	—	2	2	no	—
210	C₄⋊Q₈	—	4	1	no	—
211	C₄⋊Q₈	—	4	1	no	—
212	#126	—	4	1	no	—
213	#125	—	2	1	no	—
214	#124	—	2	1	no	—
215	#123	—	4	1	no	—
216	C₂²⋊Q₈	—	2	1	no	—
217	#121	—	2	1	no	—
218	C₄×Q₈	—	2	1	no	—
219	C₄×D₄	—	2	1	no	—
220	C₄×D₄	—	4	1	no	—
221	#117	—	4	1	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	4	1	no	—
224	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
225	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
226	#107	—	2	2	no	—
227	#107	—	2	2	no	—
228	#107	—	2	2	no	—
229	#107	C₂³×C₃₁	1	4	yes	—
230	#107	C₂³×C₃₁	1	1	yes	characteristic
231	#107	C₂³×C₃₁	1	1	yes	characteristic
232	#106	—	2	1	no	—
233	#103	—	2	1	no	—
234	#103	—	2	1	no	—
235	#102	—	2	1	no	—
236	#102	—	2	1	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	#33804	1	1	yes	characteristic, direct product factor, Sylow
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	8	4	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
251	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
253	Q₁₆	—	2	2	no	—
254	SD₁₆	—	4	2	no	—
255	D₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	4	1	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	—	4	1	no	—
277	C₄²	—	4	1	no	—
278	C₄²	D₈×C₃₁	1	1	yes	characteristic
279	C₄²	D₈×C₃₁	1	1	yes	characteristic
280	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₁₆	—	2	4	no	—
283	C₂³	—	4	1	no	—
284	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
285	Q₈	—	4	1	no	—
286	Q₈	—	4	1	no	—
287	Q₈	—	8	4	no	—
288	Q₈	—	8	4	no	—
289	D₄	—	4	1	no	—
290	D₄	—	4	1	no	—
291	D₄	—	8	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	C₃₁×#137	1	2	yes	—
311	C₈	C₃₁×2₋¹⁺⁴	1	2	yes	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	8	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	—	8	1	no	—
317	C₂²	C₃₁×#509	1	2	yes	—
318	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	4	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	1	no	—
326	C₄	—	8	1	no	—
327	C₄	C₃₁×#510	1	2	yes	—
328	C₄	C₃₁×#536	1	1	yes	characteristic
329	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	16	1	no	—
332	C₂	C₃₁×#3226	1	1	yes	characteristic
333	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
334	C₂	C₂×C₃₁×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₃₁×#500	C₂	semidirect
3. C₃₁×#2255	C₂	semidirect
4. C₃₁×#2243	C₂	semidirect
5. C₃₁×#2235	C₂	semidirect
7. C₃₁×#2227	C₂	semidirect
8. C₃₁×#2226	C₂	semidirect
9. C₃₁×#2163	C₂	semidirect
115. #33804	C₃₁	direct (N × H)
146. C₂×#500	C₂×C₃₁	semidirect
147. #2255	C₂×C₃₁	semidirect
148. #2243	C₂×C₃₁	semidirect
149. #2235	C₂×C₃₁	semidirect
151. #2227	C₂×C₃₁	semidirect
152. #2226	C₂×C₃₁	semidirect
153. #2163	C₂×C₃₁	semidirect
241. C₃₁	#33804	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×#3168	2
C₂×C₃₁×#500	2
C₃₁×#2255	2
C₃₁×#2243	2
C₃₁×#2235	2
C₃₁×#2232	2
C₃₁×#2227	2
C₃₁×#2226	2
C₃₁×#2163	2
#33804	31

Subgroups — C31×#33804 #21490563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33804 #21490563