Subgroups — C₃₁×#33750 #21490509

Order 7936.

516 conjugacy classes of subgroups.

The subgroup lattice has 308 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	2	yes	maximal, complemented
2	C₃₁×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2242	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2220	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2219	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2218	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#2177	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2176	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2175	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#529	—	2	2	no	—
13	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
15	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
18	C₃₁×#499	C₂²	1	2	yes	—
19	C₃₁×#495	C₂²	1	2	yes	—
20	C₃₁×#488	—	2	2	no	—
21	C₃₁×#467	C₂²	1	2	yes	—
22	C₃₁×#466	C₂²	1	2	yes	—
23	C₃₁×#465	—	2	2	no	—
24	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
25	C₄×SD₁₆×C₃₁	—	2	2	no	—
26	C₄×D₈×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#367	C₂²	1	1	yes	characteristic
29	C₃₁×#367	C₂²	1	1	yes	characteristic
30	C₃₁×#366	C₂²	1	2	yes	—
31	C₃₁×#358	C₂²	1	2	yes	—
32	C₃₁×#358	C₂²	1	2	yes	—
33	C₃₁×#357	C₂²	1	2	yes	—
34	C₃₁×#357	C₂²	1	2	yes	—
35	C₃₁×#346	C₂²	1	1	yes	characteristic
36	C₃₁×#346	C₂²	1	1	yes	characteristic
37	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
40	C₂×SD₁₆×C₃₁	—	2	2	no	—
41	C₂×D₈×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×#126	—	4	2	no	—
45	C₃₁×#125	—	2	2	no	—
46	C₃₁×#124	—	2	2	no	—
47	C₃₁×#123	—	4	2	no	—
48	C₃₁×C₂²⋊Q₈	—	2	2	no	—
49	C₃₁×#121	—	2	2	no	—
50	C₄×Q₈×C₃₁	—	2	2	no	—
51	C₄×D₄×C₃₁	—	2	2	no	—
52	C₄×D₄×C₃₁	—	4	2	no	—
53	C₃₁×#117	—	4	2	no	—
54	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
55	C₃₁×Q₃₂	—	2	2	no	—
56	C₃₁×SD₃₂	—	2	4	no	—
57	C₃₁×D₁₆	—	2	2	no	—
58	C₃₁×M₃₂	—	2	2	no	—
59	C₃₁×M₃₂	—	2	2	no	—
60	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
64	C₃₁×#107	C₂³	1	2	yes	—
65	C₃₁×#107	C₂³	1	2	yes	—
66	C₃₁×#106	—	2	2	no	—
67	C₃₁×#103	—	2	2	no	—
68	C₃₁×#103	—	2	2	no	—
69	C₃₁×#102	—	2	2	no	—
70	C₃₁×#102	—	2	2	no	—
71	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
72	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
73	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₃₁	—	2	2	no	—
76	C₂×D₄×C₃₁	—	2	2	no	—
77	C₂×D₄×C₃₁	—	4	2	no	—
78	C₂²×C₄×C₃₁	—	4	2	no	—
79	C₂²×C₄×C₃₁	D₈	1	2	yes	—
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	Q₁₆×C₃₁	—	2	2	no	—
82	Q₁₆×C₃₁	—	2	2	no	—
83	SD₁₆×C₃₁	—	4	4	no	—
84	D₈×C₃₁	—	2	2	no	—
85	D₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	1	no	—
88	C₂×C₈×C₃₁	—	2	1	no	—
89	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₄⋊C₄×C₃₁	—	4	2	no	—
96	C₄⋊C₄×C₃₁	—	4	2	no	—
97	C₄⋊C₄×C₃₁	—	4	2	no	—
98	C₄⋊C₄×C₃₁	—	4	2	no	—
99	C₂²⋊C₄×C₃₁	—	4	2	no	—
100	C₂²⋊C₄×C₃₁	—	4	2	no	—
101	C₂²⋊C₄×C₃₁	—	4	2	no	—
102	C₂²⋊C₄×C₃₁	—	4	2	no	—
103	C₂²⋊C₄×C₃₁	—	4	2	no	—
104	C₄²×C₃₁	—	4	2	no	—
105	C₄²×C₃₁	D₈	1	2	yes	—
106	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₁₆×C₃₁	—	2	1	no	—
109	C₁₆×C₃₁	—	2	1	no	—
110	C₁₆×C₃₁	#41	1	2	yes	—
111	C₁₆×C₃₁	#41	1	2	yes	—
112	#33750	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₃₁	—	4	2	no	—
114	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
115	Q₈×C₃₁	—	4	2	no	—
116	Q₈×C₃₁	—	4	2	no	—
117	D₄×C₃₁	—	4	2	no	—
118	D₄×C₃₁	—	4	2	no	—
119	D₄×C₃₁	—	8	4	no	—
120	C₂×C₄×C₃₁	—	2	1	no	—
121	C₂×C₄×C₃₁	—	2	1	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	—	8	4	no	—
127	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
131	C₈×C₃₁	—	2	1	no	—
132	C₈×C₃₁	—	2	1	no	—
133	C₈×C₃₁	#135	1	2	yes	—
134	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
135	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
136	#3168	C₂×C₃₁	1	2	yes	complemented
137	#2253	C₂×C₃₁	1	1	yes	characteristic, complemented
138	#2249	C₂×C₃₁	1	1	yes	characteristic, complemented
139	#2242	C₂×C₃₁	1	2	yes	complemented
140	#2220	C₂×C₃₁	1	2	yes	complemented
141	#2219	C₂×C₃₁	1	1	yes	characteristic, complemented
142	#2218	C₂×C₃₁	1	1	yes	characteristic
143	#2177	C₂×C₃₁	1	1	yes	characteristic, complemented
144	#2176	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#2175	C₂×C₃₁	1	2	yes	complemented
146	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
147	C₂²×C₃₁	—	2	2	no	—
148	C₂²×C₃₁	—	2	1	no	—
149	C₂²×C₃₁	—	8	2	no	—
150	C₂²×C₃₁	—	8	2	no	—
151	C₂²×C₃₁	—	8	2	no	—
152	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
153	C₄×C₃₁	—	2	2	no	—
154	C₄×C₃₁	—	2	2	no	—
155	C₄×C₃₁	—	2	1	no	—
156	C₄×C₃₁	—	8	2	no	—
157	C₄×C₃₁	—	8	2	no	—
158	C₄×C₃₁	—	8	2	no	—
159	C₄×C₃₁	#532	1	1	yes	characteristic
160	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
161	#529	—	2	2	no	—
162	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
164	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
167	#499	C₂²×C₃₁	1	2	yes	—
168	#495	C₂²×C₃₁	1	2	yes	—
169	#488	—	2	2	no	—
170	#467	C₂²×C₃₁	1	2	yes	—
171	#466	C₂²×C₃₁	1	2	yes	—
172	#465	—	2	2	no	—
173	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
174	C₄×SD₁₆	—	2	2	no	—
175	C₄×D₈	C₂²×C₃₁	1	2	yes	—
176	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
177	#367	C₂²×C₃₁	1	1	yes	characteristic
178	#367	C₂²×C₃₁	1	1	yes	characteristic
179	#366	C₂²×C₃₁	1	2	yes	—
180	#358	C₂²×C₃₁	1	2	yes	—
181	#358	C₂²×C₃₁	1	2	yes	—
182	#357	C₂²×C₃₁	1	2	yes	—
183	#357	C₂²×C₃₁	1	2	yes	—
184	#346	C₂²×C₃₁	1	1	yes	characteristic
185	#346	C₂²×C₃₁	1	1	yes	characteristic
186	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
187	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
188	C₂×C₃₁	—	4	1	no	—
189	C₂×C₃₁	—	16	2	no	—
190	C₂×C₃₁	#3169	1	1	yes	characteristic
191	C₂×C₃₁	#3414	1	2	yes	—
192	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
193	C₂×SD₁₆	—	2	2	no	—
194	C₂×D₈	C₂³×C₃₁	1	2	yes	—
195	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
197	#126	—	4	2	no	—
198	#125	—	2	2	no	—
199	#124	—	2	2	no	—
200	#123	—	4	2	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	#121	—	2	2	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×D₄	—	2	2	no	—
205	C₄×D₄	—	4	2	no	—
206	#117	—	4	2	no	—
207	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
208	Q₃₂	—	2	2	no	—
209	SD₃₂	—	2	4	no	—
210	D₁₆	—	2	2	no	—
211	M₃₂	—	2	2	no	—
212	M₃₂	—	2	2	no	—
213	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
214	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
215	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
216	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
217	#107	C₂³×C₃₁	1	2	yes	—
218	#107	C₂³×C₃₁	1	2	yes	—
219	#106	—	2	2	no	—
220	#103	—	2	2	no	—
221	#103	—	2	2	no	—
222	#102	—	2	2	no	—
223	#102	—	2	2	no	—
224	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
225	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
226	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
227	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
228	C₃₁	#33750	1	1	yes	characteristic, direct product factor, Sylow
229	C₂×Q₈	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	D₈×C₃₁	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	2	2	no	—
237	SD₁₆	—	4	4	no	—
238	D₈	—	2	2	no	—
239	D₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
244	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
245	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
246	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₄²	—	4	2	no	—
259	C₄²	D₈×C₃₁	1	2	yes	—
260	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
262	C₁₆	—	2	1	no	—
263	C₁₆	—	2	1	no	—
264	C₁₆	#41×C₃₁	1	2	yes	—
265	C₁₆	#41×C₃₁	1	2	yes	—
266	C₂³	—	4	2	no	—
267	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	8	4	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	8	4	no	—
280	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
281	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
282	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
283	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	C₃₁×#135	1	2	yes	—
287	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
288	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
289	C₂²	—	2	2	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	2	no	—
301	C₄	C₃₁×#532	1	1	yes	characteristic
302	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	—	16	2	no	—
305	C₂	C₃₁×#3169	1	1	yes	characteristic
306	C₂	C₃₁×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#2218	C₂
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
1. C₃₁×#3168	C₂	semidirect
2. C₃₁×#2253	C₂	semidirect
3. C₃₁×#2249	C₂	semidirect
4. C₃₁×#2242	C₂	semidirect
5. C₃₁×#2220	C₂	semidirect
6. C₃₁×#2219	C₂	semidirect
8. C₃₁×#2177	C₂	semidirect
9. C₃₁×#2176	C₂	semidirect
10. C₃₁×#2175	C₂	semidirect
11. C₄×C₃₁×M₃₂	C₂	semidirect
112. #33750	C₃₁	direct (N × H)
136. #3168	C₂×C₃₁	semidirect
137. #2253	C₂×C₃₁	semidirect
138. #2249	C₂×C₃₁	semidirect
139. #2242	C₂×C₃₁	semidirect
140. #2220	C₂×C₃₁	semidirect
141. #2219	C₂×C₃₁	semidirect
143. #2177	C₂×C₃₁	semidirect
144. #2176	C₂×C₃₁	semidirect
145. #2175	C₂×C₃₁	semidirect
146. C₄×M₃₂	C₂×C₃₁	semidirect
228. C₃₁	#33750	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
C₃₁×#2253	2
C₃₁×#2249	2
C₃₁×#2242	2
C₃₁×#2220	2
C₃₁×#2219	2
C₃₁×#2218	2
C₃₁×#2177	2
C₃₁×#2176	2
C₃₁×#2175	2
C₄×C₃₁×M₃₂	2
#33750	31

Subgroups — C31×#33750 #21490509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33750 #21490509