Subgroups — C₃₁×#33651 #21490410

Order 7936.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3059	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2913	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2237	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2237	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2236	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2235	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2159	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2158	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2141	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#358	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2113	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
14	C₃₁×#536	—	2	1	no	—
15	C₃₁×#520	—	2	1	no	—
16	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#470	—	2	2	no	—
19	C₃₁×#467	C₂²	1	2	yes	—
20	C₃₁×#467	C₂²	1	2	yes	—
21	C₃₁×#428	—	2	1	no	—
22	C₃₁×#428	—	2	2	no	—
23	C₃₁×#427	C₂²	1	1	yes	characteristic
24	C₃₁×#427	C₂²	1	2	yes	—
25	C₃₁×#427	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#107	C₂²	1	2	yes	—
28	C₃₁×#419	—	2	1	no	—
29	C₃₁×#367	C₂²	1	1	yes	characteristic
30	C₃₁×#367	C₂²	1	1	yes	characteristic
31	C₃₁×#367	C₂²	1	2	yes	—
32	C₃₁×#366	C₂²	1	1	yes	characteristic
33	C₃₁×#366	C₂²	1	1	yes	characteristic
34	C₃₁×#366	C₂²	1	2	yes	—
35	C₃₁×#358	C₂²	1	2	yes	—
36	C₃₁×#358	C₂²	1	2	yes	—
37	C₃₁×#358	C₂²	1	2	yes	—
38	C₃₁×#358	C₂²	1	2	yes	—
39	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
41	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₃₁	—	2	1	no	—
43	C₂×Q₁₆×C₃₁	—	2	1	no	—
44	C₂×Q₁₆×C₃₁	—	2	2	no	—
45	C₂×Q₁₆×C₃₁	—	2	2	no	—
46	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
47	C₃₁×#131	—	2	2	no	—
48	C₃₁×#131	—	2	2	no	—
49	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
50	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×C₄⋊Q₈	—	4	1	no	—
54	C₃₁×#124	—	4	1	no	—
55	C₃₁×#123	—	4	2	no	—
56	C₃₁×C₂²⋊Q₈	—	2	2	no	—
57	C₃₁×C₂²⋊Q₈	—	2	2	no	—
58	C₄×Q₈×C₃₁	—	4	1	no	—
59	C₄×D₄×C₃₁	—	4	2	no	—
60	C₄×D₄×C₃₁	—	4	1	no	—
61	C₃₁×#117	—	2	1	no	—
62	C₃₁×#117	—	2	2	no	—
63	C₃₁×#117	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₃₁×M₃₂	—	2	2	no	—
67	C₂×C₁₆×C₃₁	—	2	2	no	—
68	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
71	C₃₁×#107	—	2	2	no	—
72	C₃₁×#107	—	2	1	no	—
73	C₃₁×#107	C₂³	1	2	yes	—
74	C₃₁×#107	C₂³	1	1	yes	characteristic
75	C₃₁×#107	C₂³	1	1	yes	characteristic
76	C₃₁×#107	C₂³	1	1	yes	characteristic
77	C₃₁×#107	C₂³	1	1	yes	characteristic
78	C₃₁×#106	—	2	2	no	—
79	C₃₁×#106	—	2	1	no	—
80	C₃₁×#106	—	2	1	no	—
81	C₃₁×#103	—	2	2	no	—
82	C₃₁×#103	—	2	2	no	—
83	#41×C₃₁	—	2	2	no	—
84	#41×C₃₁	—	2	2	no	—
85	C₂×Q₈×C₃₁	—	2	2	no	—
86	C₂×Q₈×C₃₁	—	4	1	no	—
87	C₂×Q₈×C₃₁	—	4	2	no	—
88	C₂×Q₈×C₃₁	—	4	2	no	—
89	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
90	C₂×D₄×C₃₁	D₈	1	2	yes	—
91	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	—	4	1	no	—
93	C₂²×C₄×C₃₁	—	4	2	no	—
94	C₂²×C₄×C₃₁	—	4	1	no	—
95	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	Q₁₆×C₃₁	—	2	2	no	—
98	Q₁₆×C₃₁	—	2	2	no	—
99	Q₁₆×C₃₁	—	4	2	no	—
100	M₁₆×C₃₁	—	2	2	no	—
101	M₁₆×C₃₁	—	2	4	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₃₁	#41	1	2	yes	—
107	C₂×C₈×C₃₁	#41	1	2	yes	—
108	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₃₁	—	2	1	no	—
110	C₄⋊C₄×C₃₁	—	2	1	no	—
111	C₄⋊C₄×C₃₁	—	2	1	no	—
112	C₄⋊C₄×C₃₁	—	2	2	no	—
113	C₄⋊C₄×C₃₁	—	2	1	no	—
114	C₄⋊C₄×C₃₁	—	4	2	no	—
115	C₄⋊C₄×C₃₁	—	4	1	no	—
116	C₄⋊C₄×C₃₁	—	4	2	no	—
117	C₄⋊C₄×C₃₁	—	4	2	no	—
118	C₂²⋊C₄×C₃₁	—	4	2	no	—
119	C₂²⋊C₄×C₃₁	—	4	1	no	—
120	C₂²⋊C₄×C₃₁	—	4	1	no	—
121	C₂²⋊C₄×C₃₁	—	4	2	no	—
122	C₂²⋊C₄×C₃₁	—	4	2	no	—
123	C₄²×C₃₁	—	4	2	no	—
124	C₄²×C₃₁	—	4	1	no	—
125	C₄²×C₃₁	—	4	1	no	—
126	C₁₆×C₃₁	—	2	1	no	—
127	C₁₆×C₃₁	—	2	1	no	—
128	C₁₆×C₃₁	—	2	2	no	—
129	#33651	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂³×C₃₁	C₂×D₈	1	2	yes	—
131	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
132	Q₈×C₃₁	—	2	2	no	—
133	Q₈×C₃₁	—	4	2	no	—
134	Q₈×C₃₁	—	4	2	no	—
135	Q₈×C₃₁	—	8	2	no	—
136	D₄×C₃₁	—	2	4	no	—
137	D₄×C₃₁	—	2	2	no	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	4	1	no	—
141	C₂×C₄×C₃₁	—	4	1	no	—
142	C₂×C₄×C₃₁	—	4	1	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	4	1	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	8	2	no	—
147	C₂×C₄×C₃₁	—	16	1	no	—
148	C₂×C₄×C₃₁	—	16	2	no	—
149	C₂×C₄×C₃₁	#123	1	2	yes	—
150	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
151	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₃₁	#137	1	2	yes	—
153	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
154	C₈×C₃₁	—	2	2	no	—
155	C₈×C₃₁	#123	1	2	yes	—
156	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
157	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
158	#3059	C₂×C₃₁	1	1	yes	characteristic
159	#2913	C₂×C₃₁	1	1	yes	characteristic
160	#2237	C₂×C₃₁	1	2	yes	complemented
161	#2237	C₂×C₃₁	1	2	yes	complemented
162	#2236	C₂×C₃₁	1	2	yes	complemented
163	#2235	C₂×C₃₁	1	2	yes	complemented
164	#2159	C₂×C₃₁	1	1	yes	characteristic, complemented
165	#2158	C₂×C₃₁	1	1	yes	characteristic, complemented
166	#2141	C₂×C₃₁	1	1	yes	characteristic, complemented
167	C₂×#358	C₂×C₃₁	1	1	yes	characteristic, complemented
168	#2113	C₂×C₃₁	1	1	yes	characteristic
169	C₂²×C₃₁	—	2	1	no	—
170	C₂²×C₃₁	—	2	1	no	—
171	C₂²×C₃₁	—	2	2	no	—
172	C₂²×C₃₁	—	2	2	no	—
173	C₂²×C₃₁	—	2	2	no	—
174	C₂²×C₃₁	#480	1	2	yes	—
175	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
176	C₄×C₃₁	—	2	2	no	—
177	C₄×C₃₁	—	8	2	no	—
178	C₄×C₃₁	—	8	1	no	—
179	C₄×C₃₁	—	8	2	no	—
180	C₄×C₃₁	—	8	1	no	—
181	C₄×C₃₁	—	8	1	no	—
182	C₄×C₃₁	—	8	1	no	—
183	C₄×C₃₁	#480	1	2	yes	—
184	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
185	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
186	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
187	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
188	#536	—	2	1	no	—
189	#520	—	2	1	no	—
190	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
191	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
192	#470	—	2	2	no	—
193	#467	C₂²×C₃₁	1	2	yes	—
194	#467	C₂²×C₃₁	1	2	yes	—
195	#428	—	2	1	no	—
196	#428	—	2	2	no	—
197	#427	C₂²×C₃₁	1	1	yes	characteristic
198	#427	C₂²×C₃₁	1	2	yes	—
199	#427	C₂²×C₃₁	1	1	yes	characteristic
200	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
201	C₂×#107	C₂²×C₃₁	1	2	yes	—
202	#419	—	2	1	no	—
203	#367	C₂²×C₃₁	1	1	yes	characteristic
204	#367	C₂²×C₃₁	1	1	yes	characteristic
205	#367	C₂²×C₃₁	1	2	yes	—
206	#366	C₂²×C₃₁	1	1	yes	characteristic
207	#366	C₂²×C₃₁	1	1	yes	characteristic
208	#366	C₂²×C₃₁	1	2	yes	—
209	#358	C₂²×C₃₁	1	2	yes	—
210	#358	C₂²×C₃₁	1	2	yes	—
211	#358	C₂²×C₃₁	1	2	yes	—
212	#358	C₂²×C₃₁	1	2	yes	—
213	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
214	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
215	C₂×C₃₁	—	2	2	no	—
216	C₂×C₃₁	—	4	2	no	—
217	C₂×C₃₁	C₂×#480	1	1	yes	characteristic
218	C₂×C₃₁	#3416	1	1	yes	characteristic
219	C₂×C₃₁	#3417	1	1	yes	characteristic
220	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
221	C₂²×Q₈	—	2	1	no	—
222	C₂×Q₁₆	—	2	1	no	—
223	C₂×Q₁₆	—	2	2	no	—
224	C₂×Q₁₆	—	2	2	no	—
225	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
226	#131	—	2	2	no	—
227	#131	—	2	2	no	—
228	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
229	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
230	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
231	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
232	C₄⋊Q₈	—	4	1	no	—
233	#124	—	4	1	no	—
234	#123	—	4	2	no	—
235	C₂²⋊Q₈	—	2	2	no	—
236	C₂²⋊Q₈	—	2	2	no	—
237	C₄×Q₈	—	4	1	no	—
238	C₄×D₄	—	4	2	no	—
239	C₄×D₄	—	4	1	no	—
240	#117	—	2	2	no	—
241	#117	—	2	1	no	—
242	#117	—	2	1	no	—
243	C₂×C₄⋊C₄	—	2	2	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	M₃₂	—	2	2	no	—
246	C₂×C₁₆	—	2	2	no	—
247	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
248	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
249	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
250	#107	—	2	2	no	—
251	#107	—	2	1	no	—
252	#107	C₂³×C₃₁	1	2	yes	—
253	#107	C₂³×C₃₁	1	1	yes	characteristic
254	#107	C₂³×C₃₁	1	1	yes	characteristic
255	#107	C₂³×C₃₁	1	1	yes	characteristic
256	#107	C₂³×C₃₁	1	1	yes	characteristic
257	#106	—	2	2	no	—
258	#106	—	2	1	no	—
259	#106	—	2	1	no	—
260	#103	—	2	2	no	—
261	#103	—	2	2	no	—
262	C₃₁	#33651	1	1	yes	characteristic, direct product factor, Sylow
263	#41	—	2	2	no	—
264	#41	—	2	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
270	C₂×D₄	D₈×C₃₁	1	2	yes	—
271	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
272	C₂²×C₄	—	4	1	no	—
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
276	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
277	Q₁₆	—	2	2	no	—
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	4	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	2	4	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
285	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
286	C₂×C₈	#41×C₃₁	1	2	yes	—
287	C₂×C₈	#41×C₃₁	1	2	yes	—
288	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	2	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₂²⋊C₄	—	4	1	no	—
300	C₂²⋊C₄	—	4	1	no	—
301	C₂²⋊C₄	—	4	2	no	—
302	C₂²⋊C₄	—	4	2	no	—
303	C₄²	—	4	2	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	4	1	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	2	2	no	—
309	C₂³	C₂×D₈×C₃₁	1	2	yes	—
310	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
311	Q₈	—	2	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	8	2	no	—
315	D₄	—	2	4	no	—
316	D₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	2	no	—
326	C₂×C₄	—	16	1	no	—
327	C₂×C₄	—	16	2	no	—
328	C₂×C₄	C₃₁×#123	1	2	yes	—
329	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
330	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
331	C₂×C₄	C₃₁×#137	1	2	yes	—
332	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	C₃₁×#123	1	2	yes	—
335	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
336	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	C₃₁×#480	1	2	yes	—
343	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	1	no	—
349	C₄	—	8	1	no	—
350	C₄	—	8	1	no	—
351	C₄	C₃₁×#480	1	2	yes	—
352	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
353	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
354	C₂	—	2	2	no	—
355	C₂	—	4	2	no	—
356	C₂	C₂×C₃₁×#480	1	1	yes	characteristic
357	C₂	C₃₁×#3416	1	1	yes	characteristic
358	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×#41×C₃₁	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×#41×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2237	C₂	semidirect
4. C₃₁×#2237	C₂	semidirect
5. C₃₁×#2236	C₂	semidirect
6. C₃₁×#2235	C₂	semidirect
7. C₃₁×#2159	C₂	semidirect
8. C₃₁×#2158	C₂	semidirect
9. C₃₁×#2141	C₂	semidirect
10. C₂×C₃₁×#358	C₂	semidirect
129. #33651	C₃₁	direct (N × H)
160. #2237	C₂×C₃₁	semidirect
161. #2237	C₂×C₃₁	semidirect
162. #2236	C₂×C₃₁	semidirect
163. #2235	C₂×C₃₁	semidirect
164. #2159	C₂×C₃₁	semidirect
165. #2158	C₂×C₃₁	semidirect
166. #2141	C₂×C₃₁	semidirect
167. C₂×#358	C₂×C₃₁	semidirect
262. C₃₁	#33651	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×#3059	2
C₃₁×#2913	2
C₃₁×#2237	2
C₃₁×#2236	2
C₃₁×#2235	2
C₃₁×#2159	2
C₃₁×#2158	2
C₃₁×#2141	2
C₂×C₃₁×#358	2
C₃₁×#2113	2
#33651	31

Subgroups — C31×#33651 #21490410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33651 #21490410