Subgroups — C₃₁×#33125 #21489884

Order 7936.

794 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3094	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#469	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2873	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1978	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1951	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1931	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1900	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1811	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#121	C₂²	1	2	yes	—
13	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
14	C₃₁×#482	—	2	4	no	—
15	C₃₁×#481	—	2	2	no	—
16	C₃₁×#480	—	2	2	no	—
17	C₃₁×#469	—	2	2	no	—
18	C₃₁×#469	—	2	2	no	—
19	C₂×C₃₁×#108	C₂²	1	2	yes	—
20	C₃₁×#428	C₂²	1	4	yes	—
21	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
22	C₃₁×#418	C₂²	1	4	yes	—
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₂×C₃₁×#100	C₂²	1	4	yes	—
25	C₃₁×#407	C₂²	1	2	yes	—
26	C₃₁×#405	—	2	2	no	—
27	C₃₁×#405	C₂²	1	2	yes	—
28	C₃₁×#405	C₂²	1	2	yes	—
29	C₄×M₁₆×C₃₁	—	2	2	no	—
30	C₃₁×#343	C₂²	1	4	yes	—
31	C₃₁×#341	C₂²	1	2	yes	—
32	C₂²×D₄×C₃₁	—	2	2	no	—
33	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
34	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂×D₈×C₃₁	—	2	2	no	—
36	C₂×D₈×C₃₁	—	2	4	no	—
37	C₂×M₁₆×C₃₁	—	2	1	no	—
38	C₂×M₁₆×C₃₁	—	2	4	no	—
39	C₂×M₁₆×C₃₁	—	2	4	no	—
40	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
41	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
42	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
43	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₃₁	—	2	1	no	—
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×#121	—	2	2	no	—
48	C₃₁×#121	—	2	2	no	—
49	C₃₁×#121	—	2	2	no	—
50	C₃₁×#121	—	2	2	no	—
51	C₃₁×#117	—	2	1	no	—
52	C₃₁×#117	—	2	1	no	—
53	C₃₁×#117	D₄	1	2	yes	—
54	C₃₁×#117	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
57	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄²×C₃₁	—	2	1	no	—
60	C₃₁×#108	—	2	4	no	—
61	C₃₁×#107	—	2	4	no	—
62	C₃₁×#106	—	2	2	no	—
63	C₃₁×#106	—	2	4	no	—
64	C₃₁×#102	—	2	4	no	—
65	C₃₁×#102	—	2	4	no	—
66	C₃₁×#102	—	2	4	no	—
67	C₃₁×#100	—	2	8	no	—
68	C₃₁×C₂²⋊C₈	—	2	2	no	—
69	C₃₁×C₂²⋊C₈	—	2	2	no	—
70	C₃₁×#97	—	2	2	no	—
71	C₃₁×#97	—	2	2	no	—
72	C₄×C₈×C₃₁	—	2	2	no	—
73	C₄×C₈×C₃₁	—	2	2	no	—
74	C₂⁴×C₃₁	—	2	2	no	—
75	C₂⁴×C₃₁	#41	1	1	yes	characteristic
76	C₂×D₄×C₃₁	—	2	2	no	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂×D₄×C₃₁	—	2	4	no	—
79	C₂×D₄×C₃₁	—	2	4	no	—
80	C₂×D₄×C₃₁	—	4	2	no	—
81	C₂×D₄×C₃₁	—	4	2	no	—
82	C₂×D₄×C₃₁	—	8	4	no	—
83	C₂²×C₄×C₃₁	—	2	2	no	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	2	no	—
86	C₂²×C₄×C₃₁	—	2	2	no	—
87	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
93	D₈×C₃₁	—	4	4	no	—
94	M₁₆×C₃₁	—	2	4	no	—
95	M₁₆×C₃₁	—	2	2	no	—
96	M₁₆×C₃₁	—	2	4	no	—
97	M₁₆×C₃₁	—	2	4	no	—
98	M₁₆×C₃₁	—	2	4	no	—
99	M₁₆×C₃₁	—	2	4	no	—
100	M₁₆×C₃₁	#41	1	4	yes	—
101	C₂×C₈×C₃₁	—	2	4	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	4	no	—
105	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₃₁	#41	1	4	yes	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
112	C₂²⋊C₄×C₃₁	—	2	2	no	—
113	C₂²⋊C₄×C₃₁	—	2	2	no	—
114	C₂²⋊C₄×C₃₁	—	4	2	no	—
115	C₂²⋊C₄×C₃₁	—	4	2	no	—
116	C₄²×C₃₁	—	2	2	no	—
117	C₄²×C₃₁	—	2	2	no	—
118	#33125	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₃₁	—	2	2	no	—
120	C₂³×C₃₁	—	4	2	no	—
121	C₂³×C₃₁	—	4	2	no	—
122	C₂³×C₃₁	—	4	4	no	—
123	C₂³×C₃₁	—	8	2	no	—
124	C₂³×C₃₁	—	8	2	no	—
125	C₂³×C₃₁	#121	1	1	yes	characteristic
126	C₂³×C₃₁	#121	1	1	yes	characteristic
127	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
128	D₄×C₃₁	—	4	2	no	—
129	D₄×C₃₁	—	4	2	no	—
130	D₄×C₃₁	—	4	4	no	—
131	D₄×C₃₁	—	8	4	no	—
132	D₄×C₃₁	—	8	4	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	1	no	—
135	C₂×C₄×C₃₁	—	2	1	no	—
136	C₂×C₄×C₃₁	—	2	1	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	4	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	—	8	2	no	—
143	C₂×C₄×C₃₁	—	8	2	no	—
144	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
145	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
146	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
149	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
150	C₈×C₃₁	—	2	2	no	—
151	C₈×C₃₁	—	2	4	no	—
152	C₈×C₃₁	—	4	4	no	—
153	C₈×C₃₁	#123	1	4	yes	—
154	#3094	C₂×C₃₁	1	2	yes	complemented
155	C₂×#469	C₂×C₃₁	1	1	yes	characteristic
156	#2873	C₂×C₃₁	1	1	yes	characteristic, complemented
157	#1978	C₂×C₃₁	1	2	yes	complemented
158	#1951	C₂×C₃₁	1	2	yes	complemented
159	#1931	C₂×C₃₁	1	4	yes	complemented
160	#1900	C₂×C₃₁	1	2	yes	complemented
161	#1811	C₂×C₃₁	1	1	yes	characteristic, complemented
162	C₂²×C₃₁	—	2	2	no	—
163	C₂²×C₃₁	—	2	1	no	—
164	C₂²×C₃₁	—	2	1	no	—
165	C₂²×C₃₁	—	2	2	no	—
166	C₂²×C₃₁	—	2	2	no	—
167	C₂²×C₃₁	—	8	2	no	—
168	C₂²×C₃₁	—	8	2	no	—
169	C₂²×C₃₁	—	8	2	no	—
170	C₂²×C₃₁	—	8	2	no	—
171	C₂²×C₃₁	—	16	4	no	—
172	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
173	C₂²×C₃₁	#529	1	2	yes	—
174	C₄×C₃₁	—	2	1	no	—
175	C₄×C₃₁	—	2	1	no	—
176	C₄×C₃₁	—	4	4	no	—
177	C₄×C₃₁	—	8	2	no	—
178	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
179	C₄×C₃₁	#525	1	1	yes	characteristic
180	C₄×C₃₁	#532	1	2	yes	—
181	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
182	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
183	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
184	C₂×#121	C₂²×C₃₁	1	2	yes	—
185	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
186	#482	—	2	4	no	—
187	#481	—	2	2	no	—
188	#480	—	2	2	no	—
189	#469	—	2	2	no	—
190	#469	—	2	2	no	—
191	C₂×#108	C₂²×C₃₁	1	2	yes	—
192	#428	C₂²×C₃₁	1	4	yes	—
193	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
194	#418	C₂²×C₃₁	1	4	yes	—
195	C₂×#102	C₂²×C₃₁	1	2	yes	—
196	C₂×#100	C₂²×C₃₁	1	4	yes	—
197	#407	C₂²×C₃₁	1	2	yes	—
198	#405	—	2	2	no	—
199	#405	C₂²×C₃₁	1	2	yes	—
200	#405	C₂²×C₃₁	1	2	yes	—
201	C₄×M₁₆	—	2	2	no	—
202	#343	C₂²×C₃₁	1	4	yes	—
203	#341	C₂²×C₃₁	1	2	yes	—
204	C₂×C₃₁	—	2	2	no	—
205	C₂×C₃₁	—	4	1	no	—
206	C₂×C₃₁	—	4	1	no	—
207	C₂×C₃₁	—	16	2	no	—
208	C₂×C₃₁	#2854	1	1	yes	characteristic
209	C₂×C₃₁	#3068	1	2	yes	—
210	C₂²×D₄	—	2	2	no	—
211	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
212	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
213	C₂×D₈	—	2	2	no	—
214	C₂×D₈	—	2	4	no	—
215	C₂×M₁₆	—	2	1	no	—
216	C₂×M₁₆	—	2	4	no	—
217	C₂×M₁₆	—	2	4	no	—
218	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
219	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
220	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
221	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
224	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
225	#121	—	2	2	no	—
226	#121	—	2	2	no	—
227	#121	—	2	2	no	—
228	#121	—	2	2	no	—
229	#117	—	2	1	no	—
230	#117	—	2	1	no	—
231	#117	D₄×C₃₁	1	2	yes	—
232	#117	C₂³×C₃₁	1	2	yes	—
233	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
234	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
235	C₂×C₂²⋊C₄	—	2	1	no	—
236	C₂×C₂²⋊C₄	—	2	2	no	—
237	C₂×C₄²	—	2	1	no	—
238	#108	—	2	4	no	—
239	#107	—	2	4	no	—
240	#106	—	2	2	no	—
241	#106	—	2	4	no	—
242	#102	—	2	4	no	—
243	#102	—	2	4	no	—
244	#102	—	2	4	no	—
245	#100	—	2	8	no	—
246	C₂²⋊C₈	—	2	2	no	—
247	C₂²⋊C₈	—	2	2	no	—
248	#97	—	2	2	no	—
249	#97	—	2	2	no	—
250	C₄×C₈	—	2	2	no	—
251	C₄×C₈	—	2	2	no	—
252	C₃₁	#33125	1	1	yes	characteristic, direct product factor, Sylow
253	C₂⁴	—	2	2	no	—
254	C₂⁴	#41×C₃₁	1	1	yes	characteristic
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	4	no	—
258	C₂×D₄	—	2	4	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	8	4	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
270	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
271	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
272	D₈	—	4	4	no	—
273	M₁₆	—	2	4	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	4	no	—
276	M₁₆	—	2	4	no	—
277	M₁₆	—	2	4	no	—
278	M₁₆	—	2	4	no	—
279	M₁₆	#41×C₃₁	1	4	yes	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
285	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
286	C₂×C₈	#41×C₃₁	1	4	yes	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
291	C₂²⋊C₄	—	2	2	no	—
292	C₂²⋊C₄	—	2	2	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₂³	—	2	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	4	no	—
301	C₂³	—	8	2	no	—
302	C₂³	—	8	2	no	—
303	C₂³	C₃₁×#121	1	1	yes	characteristic
304	C₂³	C₃₁×#121	1	1	yes	characteristic
305	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	4	no	—
309	D₄	—	8	4	no	—
310	D₄	—	8	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	8	2	no	—
321	C₂×C₄	—	8	2	no	—
322	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
323	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
324	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
325	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
326	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
327	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
328	C₈	—	2	2	no	—
329	C₈	—	2	4	no	—
330	C₈	—	4	4	no	—
331	C₈	C₃₁×#123	1	4	yes	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	16	4	no	—
342	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
343	C₂²	C₃₁×#529	1	2	yes	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	4	4	no	—
347	C₄	—	8	2	no	—
348	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
349	C₄	C₃₁×#525	1	1	yes	characteristic
350	C₄	C₃₁×#532	1	2	yes	—
351	C₂	—	2	2	no	—
352	C₂	—	4	1	no	—
353	C₂	—	4	1	no	—
354	C₂	—	16	2	no	—
355	C₂	C₃₁×#2854	1	1	yes	characteristic
356	C₂	C₃₁×#3068	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#469	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3094	C₂	semidirect
3. C₃₁×#2873	C₂	semidirect
4. C₃₁×#1978	C₂	semidirect
5. C₃₁×#1951	C₂	semidirect
6. C₃₁×#1931	C₂	semidirect
7. C₃₁×#1900	C₂	semidirect
8. C₃₁×#1811	C₂	semidirect
118. #33125	C₃₁	direct (N × H)
154. #3094	C₂×C₃₁	semidirect
156. #2873	C₂×C₃₁	semidirect
157. #1978	C₂×C₃₁	semidirect
158. #1951	C₂×C₃₁	semidirect
159. #1931	C₂×C₃₁	semidirect
160. #1900	C₂×C₃₁	semidirect
161. #1811	C₂×C₃₁	semidirect
252. C₃₁	#33125	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×#3094	2
C₂×C₃₁×#469	2
C₃₁×#2873	2
C₃₁×#1978	2
C₃₁×#1951	2
C₃₁×#1931	2
C₃₁×#1900	2
C₃₁×#1811	2
#33125	31

Subgroups — C31×#33125 #21489884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33125 #21489884