Subgroups — C₃₁×#33903 #21490662

Order 7936.

470 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3336	C₂	1	2	yes	maximal
2	C₃₁×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2237	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2235	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2229	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2216	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2216	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#2158	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#366	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#552	—	2	2	no	—
13	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#499	C₂²	1	2	yes	—
17	C₃₁×#485	—	2	2	no	—
18	C₃₁×#483	—	2	2	no	—
19	C₃₁×#479	—	2	2	no	—
20	C₃₁×#467	C₂²	1	2	yes	—
21	C₃₁×#467	C₂²	1	2	yes	—
22	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
23	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#427	C₂²	1	2	yes	—
25	C₂×C₃₁×#107	C₂²	1	2	yes	—
26	C₃₁×#367	C₂²	1	2	yes	—
27	C₃₁×#367	C₂²	1	2	yes	—
28	C₃₁×#366	C₂²	1	1	yes	characteristic
29	C₃₁×#366	C₂²	1	1	yes	characteristic
30	C₃₁×#366	C₂²	1	2	yes	—
31	C₃₁×#366	C₂²	1	1	yes	characteristic
32	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₃₁×#358	C₂²	1	2	yes	—
34	C₃₁×#358	C₂²	1	2	yes	—
35	C₃₁×#358	C₂²	1	2	yes	—
36	C₃₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₃₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×#126	—	4	2	no	—
43	C₃₁×#125	—	2	2	no	—
44	C₃₁×#125	—	4	2	no	—
45	C₃₁×#123	—	4	2	no	—
46	C₃₁×C₂²⋊Q₈	—	2	2	no	—
47	C₃₁×C₂²⋊Q₈	—	2	2	no	—
48	C₄×Q₈×C₃₁	—	2	2	no	—
49	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
50	C₃₁×#117	—	2	2	no	—
51	C₃₁×#117	—	4	2	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₃₁×Q₃₂	—	2	2	no	—
54	C₂×C₁₆×C₃₁	—	2	2	no	—
55	C₂×C₁₆×C₃₁	—	2	2	no	—
56	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×#107	—	2	2	no	—
61	C₃₁×#107	C₂³	1	2	yes	—
62	C₃₁×#107	C₂³	1	2	yes	—
63	C₃₁×#107	C₂³	1	2	yes	—
64	C₃₁×#106	—	2	2	no	—
65	C₃₁×#106	—	2	2	no	—
66	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
67	C₃₁×#103	—	2	2	no	—
68	C₃₁×#103	—	2	2	no	—
69	C₃₁×#103	—	2	2	no	—
70	C₃₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
71	C₃₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
72	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×Q₈×C₃₁	—	2	2	no	—
74	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	—	4	2	no	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₃₁	—	2	2	no	—
79	Q₁₆×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	2	1	no	—
81	C₂×C₈×C₃₁	—	2	1	no	—
82	C₂×C₈×C₃₁	—	2	1	no	—
83	C₂×C₈×C₃₁	—	2	1	no	—
84	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₄⋊C₄×C₃₁	—	2	2	no	—
91	C₄⋊C₄×C₃₁	—	4	2	no	—
92	C₄⋊C₄×C₃₁	—	4	2	no	—
93	C₄⋊C₄×C₃₁	—	4	2	no	—
94	C₄⋊C₄×C₃₁	—	4	2	no	—
95	C₄⋊C₄×C₃₁	—	4	2	no	—
96	C₄⋊C₄×C₃₁	—	4	2	no	—
97	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
98	C₂²⋊C₄×C₃₁	—	4	2	no	—
99	C₂²⋊C₄×C₃₁	—	4	2	no	—
100	C₂²⋊C₄×C₃₁	—	4	2	no	—
101	C₂²⋊C₄×C₃₁	D₈	1	1	yes	characteristic
102	C₂²⋊C₄×C₃₁	D₈	1	1	yes	characteristic
103	C₄²×C₃₁	—	4	2	no	—
104	C₄²×C₃₁	—	4	2	no	—
105	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₁₆×C₃₁	—	2	1	no	—
107	C₁₆×C₃₁	—	2	1	no	—
108	C₁₆×C₃₁	—	2	1	no	—
109	C₁₆×C₃₁	#41	1	2	yes	—
110	#33903	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
112	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
113	Q₈×C₃₁	—	4	2	no	—
114	Q₈×C₃₁	—	4	2	no	—
115	D₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	1	no	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	8	4	no	—
125	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₃₁	—	2	1	no	—
131	C₈×C₃₁	—	2	1	no	—
132	C₈×C₃₁	—	2	1	no	—
133	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
134	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
135	#3336	C₂×C₃₁	1	2	yes	—
136	#2253	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#2237	C₂×C₃₁	1	2	yes	complemented
138	#2235	C₂×C₃₁	1	2	yes	complemented
139	#2229	C₂×C₃₁	1	2	yes	complemented
140	#2216	C₂×C₃₁	1	1	yes	characteristic, complemented
141	#2216	C₂×C₃₁	1	1	yes	characteristic, complemented
142	C₄×Q₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
143	D₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic
144	#2158	C₂×C₃₁	1	1	yes	characteristic, complemented
145	C₂×#366	C₂×C₃₁	1	1	yes	characteristic, complemented
146	C₂²×C₃₁	—	2	1	no	—
147	C₂²×C₃₁	—	2	1	no	—
148	C₂²×C₃₁	—	2	1	no	—
149	C₂²×C₃₁	—	2	1	no	—
150	C₂²×C₃₁	—	2	1	no	—
151	C₂²×C₃₁	#508	1	2	yes	—
152	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
153	C₄×C₃₁	—	2	1	no	—
154	C₄×C₃₁	—	2	1	no	—
155	C₄×C₃₁	—	2	1	no	—
156	C₄×C₃₁	—	2	1	no	—
157	C₄×C₃₁	—	8	2	no	—
158	C₄×C₃₁	—	8	2	no	—
159	C₄×C₃₁	—	8	2	no	—
160	C₄×C₃₁	—	8	2	no	—
161	C₄×C₃₁	#547	1	1	yes	characteristic
162	C₄×C₃₁	#578	1	1	yes	characteristic
163	#552	—	2	2	no	—
164	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
165	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
166	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
167	#499	C₂²×C₃₁	1	2	yes	—
168	#485	—	2	2	no	—
169	#483	—	2	2	no	—
170	#479	—	2	2	no	—
171	#467	C₂²×C₃₁	1	2	yes	—
172	#467	C₂²×C₃₁	1	2	yes	—
173	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
174	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
175	#427	C₂²×C₃₁	1	2	yes	—
176	C₂×#107	C₂²×C₃₁	1	2	yes	—
177	#367	C₂²×C₃₁	1	2	yes	—
178	#367	C₂²×C₃₁	1	2	yes	—
179	#366	C₂²×C₃₁	1	1	yes	characteristic
180	#366	C₂²×C₃₁	1	1	yes	characteristic
181	#366	C₂²×C₃₁	1	2	yes	—
182	#366	C₂²×C₃₁	1	1	yes	characteristic
183	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
184	#358	C₂²×C₃₁	1	2	yes	—
185	#358	C₂²×C₃₁	1	2	yes	—
186	#358	C₂²×C₃₁	1	2	yes	—
187	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
188	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
189	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
190	C₂×C₃₁	—	2	2	no	—
191	C₂×C₃₁	—	4	1	no	—
192	C₂×C₃₁	#3334	1	1	yes	characteristic
193	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
194	C₂×C₃₁	#3417	1	1	yes	characteristic
195	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
196	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
198	#126	—	4	2	no	—
199	#125	—	2	2	no	—
200	#125	—	4	2	no	—
201	#123	—	4	2	no	—
202	C₂²⋊Q₈	—	2	2	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
206	#117	—	2	2	no	—
207	#117	—	4	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	Q₃₂	—	2	2	no	—
210	C₂×C₁₆	—	2	2	no	—
211	C₂×C₁₆	—	2	2	no	—
212	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
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	#107	—	2	2	no	—
217	#107	C₂³×C₃₁	1	2	yes	—
218	#107	C₂³×C₃₁	1	2	yes	—
219	#107	C₂³×C₃₁	1	2	yes	—
220	#106	—	2	2	no	—
221	#106	—	2	2	no	—
222	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
223	#103	—	2	2	no	—
224	#103	—	2	2	no	—
225	#103	—	2	2	no	—
226	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
227	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
229	C₃₁	#33903	1	1	yes	characteristic, direct product factor, Sylow
230	C₂×Q₈	—	2	2	no	—
231	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	2	2	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
244	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	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	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	D₈×C₃₁	1	1	yes	characteristic
259	C₂²⋊C₄	D₈×C₃₁	1	1	yes	characteristic
260	C₄²	—	4	2	no	—
261	C₄²	—	4	2	no	—
262	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
263	C₁₆	—	2	1	no	—
264	C₁₆	—	2	1	no	—
265	C₁₆	—	2	1	no	—
266	C₁₆	#41×C₃₁	1	2	yes	—
267	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
268	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	D₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	4	no	—
281	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
282	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
283	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
284	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
285	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
290	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	C₃₁×#508	1	2	yes	—
297	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	C₃₁×#547	1	1	yes	characteristic
307	C₄	C₃₁×#578	1	1	yes	characteristic
308	C₂	—	2	2	no	—
309	C₂	—	4	1	no	—
310	C₂	C₃₁×#3334	1	1	yes	characteristic
311	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
312	C₂	C₃₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×D₄×C₃₁	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ C₈×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2253	C₂	semidirect
3. C₃₁×#2237	C₂	semidirect
4. C₃₁×#2235	C₂	semidirect
5. C₃₁×#2229	C₂	semidirect
6. C₃₁×#2216	C₂	semidirect
7. C₃₁×#2216	C₂	semidirect
8. C₄×C₃₁×Q₃₂	C₂	semidirect
10. C₃₁×#2158	C₂	semidirect
11. C₂×C₃₁×#366	C₂	semidirect
110. #33903	C₃₁	direct (N × H)
136. #2253	C₂×C₃₁	semidirect
137. #2237	C₂×C₃₁	semidirect
138. #2235	C₂×C₃₁	semidirect
139. #2229	C₂×C₃₁	semidirect
140. #2216	C₂×C₃₁	semidirect
141. #2216	C₂×C₃₁	semidirect
142. C₄×Q₃₂	C₂×C₃₁	semidirect
144. #2158	C₂×C₃₁	semidirect
145. C₂×#366	C₂×C₃₁	semidirect
229. C₃₁	#33903	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×#3336	2
C₃₁×#2253	2
C₃₁×#2237	2
C₃₁×#2235	2
C₃₁×#2229	2
C₃₁×#2216	2
C₄×C₃₁×Q₃₂	2
D₄×C₁₆×C₃₁	2
C₃₁×#2158	2
C₂×C₃₁×#366	2
#33903	31

Subgroups — C31×#33903 #21490662

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33903 #21490662