Subgroups — C₃₁×#21042 #21477801

Order 7936.

822 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2911	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2905	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2873	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1922	C₂	1	8	yes	maximal, complemented
5	C₃₁×#1738	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×#339	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×#339	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1728	C₂	1	1	yes	maximal, characteristic
9	C₂²×M₁₆×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	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#428	C₄	1	4	yes	complemented
17	C₃₁×#428	C₄	1	4	yes	complemented
18	C₃₁×#427	C₂²	1	4	yes	—
19	C₃₁×#424	—	2	2	no	—
20	C₃₁×#424	C₂²	1	4	yes	—
21	C₃₁×#423	—	2	2	no	—
22	C₃₁×#405	—	2	2	no	—
23	C₃₁×#405	C₂²	1	4	yes	—
24	C₄×M₁₆×C₃₁	—	2	2	no	—
25	C₃₁×#341	C₂²	1	4	yes	—
26	C₃₁×#339	C₂²	1	4	yes	—
27	C₃₁×#339	C₂²	1	4	yes	—
28	C₃₁×#336	C₂²	1	4	yes	—
29	C₂³×C₄×C₃₁	—	2	1	no	—
30	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₃₁	—	2	1	no	—
32	C₂×M₁₆×C₃₁	—	2	2	no	—
33	C₂×M₁₆×C₃₁	—	2	2	no	—
34	C₂×M₁₆×C₃₁	—	2	2	no	—
35	C₂×M₁₆×C₃₁	—	2	2	no	—
36	C₂×M₁₆×C₃₁	—	2	2	no	—
37	C₂×M₁₆×C₃₁	—	2	1	no	—
38	C₂×M₁₆×C₃₁	—	2	2	no	—
39	C₂×M₁₆×C₃₁	—	2	2	no	—
40	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
43	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
44	C₂×M₁₆×C₃₁	Q₈	1	4	yes	—
45	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
46	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	—	2	1	no	—
48	C₂²×C₈×C₃₁	—	2	1	no	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₃₁×#117	—	2	1	no	—
52	C₃₁×#117	—	2	1	no	—
53	C₃₁×#117	—	2	1	no	—
54	C₃₁×#117	—	2	1	no	—
55	C₃₁×#117	C₂×C₄	1	2	yes	—
56	C₃₁×#117	C₂×C₄	1	2	yes	—
57	C₃₁×#117	C₂³	1	2	yes	—
58	C₃₁×#117	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄²×C₃₁	—	2	1	no	—
64	C₂×C₄²×C₃₁	—	2	1	no	—
65	C₂×C₄²×C₃₁	—	2	2	no	—
66	C₂×C₄²×C₃₁	—	2	2	no	—
67	C₂×C₄²×C₃₁	—	2	1	no	—
68	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
69	C₃₁×#107	C₂×C₄	1	8	yes	complemented
70	C₃₁×#106	C₂×C₄	1	8	yes	complemented
71	C₃₁×C₄⋊C₈	—	2	8	no	—
72	C₃₁×#97	—	2	4	no	—
73	C₄×C₈×C₃₁	—	2	4	no	—
74	C₂⁴×C₃₁	#41	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	—	2	2	no	—
76	C₂²×C₄×C₃₁	—	2	2	no	—
77	C₂²×C₄×C₃₁	—	2	2	no	—
78	C₂²×C₄×C₃₁	—	4	4	no	—
79	C₂²×C₄×C₃₁	—	4	2	no	—
80	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₃₁	#41	1	2	yes	—
86	C₂²×C₄×C₃₁	#41	1	2	yes	—
87	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	#41	1	2	yes	—
89	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
90	M₁₆×C₃₁	—	2	4	no	—
91	M₁₆×C₃₁	—	2	4	no	—
92	M₁₆×C₃₁	—	2	4	no	—
93	M₁₆×C₃₁	—	2	4	no	—
94	M₁₆×C₃₁	—	2	4	no	—
95	M₁₆×C₃₁	—	2	2	no	—
96	M₁₆×C₃₁	—	2	2	no	—
97	M₁₆×C₃₁	—	2	2	no	—
98	M₁₆×C₃₁	—	4	2	no	—
99	M₁₆×C₃₁	C₂×Q₈	1	4	yes	—
100	C₂×C₈×C₃₁	—	2	4	no	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	4	2	no	—
108	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
109	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
110	C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
113	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
114	C₂²⋊C₄×C₃₁	—	2	2	no	—
115	C₂²⋊C₄×C₃₁	—	2	2	no	—
116	C₂²⋊C₄×C₃₁	—	2	2	no	—
117	C₂²⋊C₄×C₃₁	—	2	2	no	—
118	C₄²×C₃₁	—	2	2	no	—
119	C₄²×C₃₁	—	2	2	no	—
120	C₄²×C₃₁	—	2	2	no	—
121	C₄²×C₃₁	—	2	2	no	—
122	C₄²×C₃₁	—	2	2	no	—
123	C₄²×C₃₁	—	2	2	no	—
124	C₄²×C₃₁	—	4	2	no	—
125	#21042	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂³×C₃₁	—	4	2	no	—
127	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
128	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
129	C₂³×C₃₁	#125	1	2	yes	—
130	C₂³×C₃₁	#126	1	2	yes	—
131	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₃₁	—	2	4	no	—
133	C₂×C₄×C₃₁	—	2	4	no	—
134	C₂×C₄×C₃₁	—	2	4	no	—
135	C₂×C₄×C₃₁	—	4	2	no	—
136	C₂×C₄×C₃₁	—	4	8	no	—
137	C₂×C₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	8	2	no	—
139	C₂×C₄×C₃₁	—	8	2	no	—
140	C₂×C₄×C₃₁	#117	1	2	yes	—
141	C₂×C₄×C₃₁	#117	1	2	yes	—
142	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
145	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
146	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
147	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
148	C₂×C₄×C₃₁	#123	1	2	yes	—
149	C₂×C₄×C₃₁	#123	1	2	yes	—
150	C₂×C₄×C₃₁	#125	1	2	yes	—
151	C₂×C₄×C₃₁	#126	1	2	yes	—
152	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
155	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
158	C₈×C₃₁	—	2	4	no	—
159	C₈×C₃₁	—	2	4	no	—
160	C₈×C₃₁	—	2	4	no	—
161	C₈×C₃₁	—	4	2	no	—
162	#2911	C₂×C₃₁	1	1	yes	characteristic
163	#2905	C₂×C₃₁	1	1	yes	characteristic
164	#2873	C₂×C₃₁	1	1	yes	characteristic
165	#1922	C₂×C₃₁	1	8	yes	complemented
166	#1738	C₂×C₃₁	1	1	yes	characteristic
167	C₂×#339	C₂×C₃₁	1	1	yes	characteristic
168	C₂×#339	C₂×C₃₁	1	1	yes	characteristic
169	#1728	C₂×C₃₁	1	1	yes	characteristic
170	C₂²×C₃₁	—	2	2	no	—
171	C₂²×C₃₁	—	2	2	no	—
172	C₂²×C₃₁	—	2	2	no	—
173	C₂²×C₃₁	—	4	2	no	—
174	C₂²×C₃₁	—	4	2	no	—
175	C₂²×C₃₁	#387	1	2	yes	—
176	C₂²×C₃₁	#387	1	2	yes	—
177	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂²×C₃₁	C₂×#125	1	1	yes	characteristic
179	C₂²×C₃₁	C₂×#126	1	1	yes	characteristic
180	C₄×C₃₁	—	4	4	no	—
181	C₄×C₃₁	—	4	4	no	—
182	C₄×C₃₁	—	4	4	no	—
183	C₄×C₃₁	#387	1	2	yes	—
184	C₄×C₃₁	#387	1	2	yes	—
185	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
186	C₄×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
187	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
188	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
189	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
190	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
191	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
192	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
193	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
194	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
195	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
196	#428	C₄×C₃₁	1	4	yes	complemented
197	#428	C₄×C₃₁	1	4	yes	complemented
198	#427	C₂²×C₃₁	1	4	yes	—
199	#424	—	2	2	no	—
200	#424	C₂²×C₃₁	1	4	yes	—
201	#423	—	2	2	no	—
202	#405	—	2	2	no	—
203	#405	C₂²×C₃₁	1	4	yes	—
204	C₄×M₁₆	—	2	2	no	—
205	#341	C₂²×C₃₁	1	4	yes	—
206	#339	C₂²×C₃₁	1	4	yes	—
207	#339	C₂²×C₃₁	1	4	yes	—
208	#336	C₂²×C₃₁	1	4	yes	—
209	C₂×C₃₁	—	2	1	no	—
210	C₂×C₃₁	—	2	1	no	—
211	C₂×C₃₁	—	2	2	no	—
212	C₂×C₃₁	—	2	2	no	—
213	C₂×C₃₁	C₂×#387	1	1	yes	characteristic
214	C₂×C₃₁	#2956	1	2	yes	—
215	C₂³×C₄	—	2	1	no	—
216	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	2	2	no	—
219	C₂×M₁₆	—	2	2	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	—	2	2	no	—
222	C₂×M₁₆	—	2	2	no	—
223	C₂×M₁₆	—	2	1	no	—
224	C₂×M₁₆	—	2	2	no	—
225	C₂×M₁₆	—	2	2	no	—
226	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
227	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
228	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
229	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
230	C₂×M₁₆	Q₈×C₃₁	1	4	yes	—
231	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
232	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
233	C₂²×C₈	—	2	1	no	—
234	C₂²×C₈	—	2	1	no	—
235	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
236	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
237	#117	—	2	1	no	—
238	#117	—	2	1	no	—
239	#117	—	2	1	no	—
240	#117	—	2	1	no	—
241	#117	C₂×C₄×C₃₁	1	2	yes	—
242	#117	C₂×C₄×C₃₁	1	2	yes	—
243	#117	C₂³×C₃₁	1	2	yes	—
244	#117	C₂³×C₃₁	1	2	yes	—
245	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
246	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
247	C₂×C₂²⋊C₄	—	2	1	no	—
248	C₂×C₂²⋊C₄	—	2	1	no	—
249	C₂×C₄²	—	2	1	no	—
250	C₂×C₄²	—	2	1	no	—
251	C₂×C₄²	—	2	2	no	—
252	C₂×C₄²	—	2	2	no	—
253	C₂×C₄²	—	2	1	no	—
254	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
255	#107	C₂×C₄×C₃₁	1	8	yes	complemented
256	#106	C₂×C₄×C₃₁	1	8	yes	complemented
257	C₄⋊C₈	—	2	8	no	—
258	#97	—	2	4	no	—
259	C₄×C₈	—	2	4	no	—
260	C₃₁	#21042	1	1	yes	characteristic, direct product factor, Sylow
261	C₂⁴	#41×C₃₁	1	1	yes	characteristic
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	4	4	no	—
266	C₂²×C₄	—	4	2	no	—
267	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
268	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
271	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
272	C₂²×C₄	#41×C₃₁	1	2	yes	—
273	C₂²×C₄	#41×C₃₁	1	2	yes	—
274	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
275	C₂²×C₄	#41×C₃₁	1	2	yes	—
276	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
277	M₁₆	—	2	4	no	—
278	M₁₆	—	2	4	no	—
279	M₁₆	—	2	4	no	—
280	M₁₆	—	2	4	no	—
281	M₁₆	—	2	4	no	—
282	M₁₆	—	2	2	no	—
283	M₁₆	—	2	2	no	—
284	M₁₆	—	2	2	no	—
285	M₁₆	—	4	2	no	—
286	M₁₆	C₂×Q₈×C₃₁	1	4	yes	—
287	C₂×C₈	—	2	4	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	4	2	no	—
295	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
296	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
300	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	2	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	4	2	no	—
312	C₂³	—	4	2	no	—
313	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
314	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
315	C₂³	C₃₁×#125	1	2	yes	—
316	C₂³	C₃₁×#126	1	2	yes	—
317	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	8	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	8	2	no	—
326	C₂×C₄	C₃₁×#117	1	2	yes	—
327	C₂×C₄	C₃₁×#117	1	2	yes	—
328	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
329	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
330	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
331	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
332	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
333	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
334	C₂×C₄	C₃₁×#123	1	2	yes	—
335	C₂×C₄	C₃₁×#123	1	2	yes	—
336	C₂×C₄	C₃₁×#125	1	2	yes	—
337	C₂×C₄	C₃₁×#126	1	2	yes	—
338	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
339	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
340	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
341	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
342	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
343	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
344	C₈	—	2	4	no	—
345	C₈	—	2	4	no	—
346	C₈	—	2	4	no	—
347	C₈	—	4	2	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	—	4	2	no	—
352	C₂²	—	4	2	no	—
353	C₂²	C₃₁×#387	1	2	yes	—
354	C₂²	C₃₁×#387	1	2	yes	—
355	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
356	C₂²	C₂×C₃₁×#125	1	1	yes	characteristic
357	C₂²	C₂×C₃₁×#126	1	1	yes	characteristic
358	C₄	—	4	4	no	—
359	C₄	—	4	4	no	—
360	C₄	—	4	4	no	—
361	C₄	C₃₁×#387	1	2	yes	—
362	C₄	C₃₁×#387	1	2	yes	—
363	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
364	C₄	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
365	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
366	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
367	C₂	—	2	1	no	—
368	C₂	—	2	1	no	—
369	C₂	—	2	2	no	—
370	C₂	—	2	2	no	—
371	C₂	C₂×C₃₁×#387	1	1	yes	characteristic
372	C₂	C₃₁×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂⁴×C₃₁	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1922	C₂	semidirect
16. C₃₁×#428	C₄	semidirect
17. C₃₁×#428	C₄	semidirect
69. C₃₁×#107	C₂×C₄	semidirect
70. C₃₁×#106	C₂×C₄	semidirect
125. #21042	C₃₁	direct (N × H)
165. #1922	C₂×C₃₁	semidirect
196. #428	C₄×C₃₁	semidirect
197. #428	C₄×C₃₁	semidirect
255. #107	C₂×C₄×C₃₁	semidirect
256. #106	C₂×C₄×C₃₁	semidirect
260. C₃₁	#21042	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×#2911	2
C₃₁×#2905	2
C₃₁×#2873	2
C₃₁×#1922	2
C₃₁×#1738	2
C₂×C₃₁×#339	2
C₃₁×#1728	2
#21042	31

Subgroups — C31×#21042 #21477801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21042 #21477801