Subgroups — C₃₁×#19656 #21476415

Order 7936.

776 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3227	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3209	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1720	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1717	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1709	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1708	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#533	C₂²	1	1	yes	characteristic
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
16	C₃₁×#498	C₂²	1	2	yes	complemented
17	C₃₁×#494	C₂²	1	2	yes	complemented
18	C₃₁×#493	C₂²	1	2	yes	—
19	C₃₁×#492	C₂²	1	2	yes	—
20	C₃₁×#488	—	2	2	no	—
21	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
22	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#487	C₂²	1	2	yes	complemented
24	C₃₁×#486	—	2	2	no	—
25	C₃₁×#486	C₂²	1	2	yes	—
26	C₃₁×#486	C₂²	1	2	yes	—
27	C₃₁×#486	C₂²	1	2	yes	—
28	C₃₁×#486	C₂²	1	2	yes	—
29	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#486	C₂²	1	2	yes	complemented
32	C₃₁×#481	—	2	2	no	—
33	C₃₁×#476	—	2	2	no	—
34	C₃₁×#462	—	2	2	no	—
35	C₃₁×#460	—	2	2	no	—
36	C₃₁×#459	—	2	2	no	—
37	C₃₁×#451	—	2	2	no	—
38	C₃₁×#450	—	2	2	no	—
39	C₃₁×#447	—	2	2	no	—
40	C₃₁×#431	C₂²	1	1	yes	characteristic
41	C₃₁×#431	C₂²	1	1	yes	characteristic
42	C₃₁×#416	—	2	2	no	—
43	C₂×C₃₁×#102	—	2	2	no	—
44	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#322	C₂²	1	2	yes	complemented
46	C₈²×C₃₁	C₂²	1	2	yes	complemented
47	C₂×#41×C₃₁	—	2	1	no	—
48	C₂²×Q₈×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂×Q₁₆×C₃₁	—	2	4	no	—
53	C₂×SD₁₆×C₃₁	—	2	4	no	—
54	C₂×D₈×C₃₁	—	2	4	no	—
55	C₂²×C₈×C₃₁	D₄	1	2	yes	—
56	C₃₁×C₄⋊Q₈	—	2	1	no	—
57	C₃₁×C₄⋊Q₈	—	2	1	no	—
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
60	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
61	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₃₁×#127	—	2	1	no	—
64	C₃₁×#127	—	2	1	no	—
65	C₃₁×#127	—	2	1	no	—
66	C₃₁×#127	C₂³	1	2	yes	—
67	C₃₁×#127	C₂³	1	1	yes	characteristic
68	C₃₁×#125	—	2	1	no	—
69	C₃₁×#124	—	2	1	no	—
70	C₃₁×C₂²⋊Q₈	—	2	1	no	—
71	C₃₁×C₂²⋊Q₈	—	2	1	no	—
72	C₃₁×#121	—	2	1	no	—
73	C₃₁×#121	—	2	1	no	—
74	C₄×Q₈×C₃₁	—	2	1	no	—
75	C₄×Q₈×C₃₁	—	2	1	no	—
76	C₄×D₄×C₃₁	—	2	1	no	—
77	C₄×D₄×C₃₁	—	2	1	no	—
78	C₃₁×#117	—	2	1	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₃₁×#106	—	2	4	no	—
83	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
84	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
85	C₃₁×#103	—	2	2	no	—
86	C₃₁×#103	—	2	2	no	—
87	C₃₁×#103	—	2	4	no	—
88	C₃₁×#102	—	2	4	no	—
89	C₃₁×#102	—	2	4	no	—
90	C₃₁×#102	—	2	2	no	—
91	C₃₁×#102	—	2	2	no	—
92	C₃₁×#102	—	2	4	no	—
93	C₃₁×#102	—	2	4	no	—
94	C₃₁×#102	—	2	4	no	—
95	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
96	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
97	C₄×C₈×C₃₁	D₄	1	2	yes	—
98	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₃₁	C₂³	1	2	yes	—
101	C₄×C₈×C₃₁	C₂³	1	2	yes	—
102	C₂⁴×C₃₁	—	4	1	no	—
103	#41×C₃₁	—	4	4	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	1	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	4	1	no	—
109	C₂×Q₈×C₃₁	—	8	2	no	—
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	4	1	no	—
117	C₂×D₄×C₃₁	—	4	1	no	—
118	C₂×D₄×C₃₁	—	4	1	no	—
119	C₂×D₄×C₃₁	—	4	1	no	—
120	C₂×D₄×C₃₁	—	4	4	no	—
121	C₂×D₄×C₃₁	—	8	2	no	—
122	C₂×D₄×C₃₁	—	8	2	no	—
123	C₂²×C₄×C₃₁	—	4	1	no	—
124	C₂²×C₄×C₃₁	—	4	1	no	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	Q₁₆×C₃₁	—	8	4	no	—
129	SD₁₆×C₃₁	—	8	4	no	—
130	D₈×C₃₁	—	8	4	no	—
131	C₂×C₈×C₃₁	—	2	4	no	—
132	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
135	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	1	no	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₄⋊C₄×C₃₁	—	2	1	no	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	4	1	no	—
145	C₄⋊C₄×C₃₁	—	4	1	no	—
146	C₄⋊C₄×C₃₁	—	4	1	no	—
147	C₄⋊C₄×C₃₁	—	4	1	no	—
148	C₂²⋊C₄×C₃₁	—	4	1	no	—
149	C₂²⋊C₄×C₃₁	—	4	1	no	—
150	C₂²⋊C₄×C₃₁	—	4	1	no	—
151	C₄²×C₃₁	—	4	1	no	—
152	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
156	#19656	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂³×C₃₁	—	4	1	no	—
158	C₂³×C₃₁	—	4	2	no	—
159	C₂³×C₃₁	—	8	2	no	—
160	C₂³×C₃₁	—	8	2	no	—
161	C₂³×C₃₁	—	8	2	no	—
162	C₂³×C₃₁	#127	1	1	yes	characteristic
163	Q₈×C₃₁	—	4	2	no	—
164	Q₈×C₃₁	—	4	2	no	—
165	Q₈×C₃₁	—	4	4	no	—
166	Q₈×C₃₁	—	4	2	no	—
167	Q₈×C₃₁	—	8	2	no	—
168	D₄×C₃₁	—	4	4	no	—
169	D₄×C₃₁	—	4	2	no	—
170	D₄×C₃₁	—	4	2	no	—
171	D₄×C₃₁	—	4	4	no	—
172	D₄×C₃₁	—	4	4	no	—
173	D₄×C₃₁	—	4	2	no	—
174	D₄×C₃₁	—	8	2	no	—
175	D₄×C₃₁	—	8	2	no	—
176	D₄×C₃₁	—	8	2	no	—
177	D₄×C₃₁	—	8	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	—	4	1	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	8	2	no	—
185	C₂×C₄×C₃₁	—	8	2	no	—
186	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
187	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
188	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
189	C₂×C₄×C₃₁	#135	1	4	yes	—
190	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
193	C₈×C₃₁	—	2	2	no	—
194	C₈×C₃₁	—	2	2	no	—
195	C₈×C₃₁	—	2	4	no	—
196	C₈×C₃₁	—	2	4	no	—
197	#3227	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#3209	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#3208	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#3205	C₂×C₃₁	1	1	yes	characteristic
201	#3131	C₂×C₃₁	1	1	yes	characteristic, complemented
202	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#1720	C₂×C₃₁	1	2	yes	complemented
204	#1717	C₂×C₃₁	1	2	yes	complemented
205	#1709	C₂×C₃₁	1	2	yes	complemented
206	#1708	C₂×C₃₁	1	2	yes	complemented
207	#1450	C₂×C₃₁	1	1	yes	characteristic, complemented
208	C₂²×C₃₁	—	2	1	no	—
209	C₂²×C₃₁	—	2	1	no	—
210	C₂²×C₃₁	—	8	1	no	—
211	C₂²×C₃₁	—	8	2	no	—
212	C₂²×C₃₁	—	8	2	no	—
213	C₂²×C₃₁	—	8	1	no	—
214	C₂²×C₃₁	—	8	2	no	—
215	C₂²×C₃₁	—	8	1	no	—
216	C₂²×C₃₁	—	16	4	no	—
217	C₂²×C₃₁	#495	1	2	yes	—
218	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
219	C₄×C₃₁	—	2	1	no	—
220	C₄×C₃₁	—	2	1	no	—
221	C₄×C₃₁	—	8	1	no	—
222	C₄×C₃₁	—	8	1	no	—
223	C₄×C₃₁	—	8	2	no	—
224	C₄×C₃₁	—	8	1	no	—
225	C₄×C₃₁	#495	1	2	yes	—
226	C₄×C₃₁	C₂×#135	1	2	yes	—
227	C₄×C₃₁	#576	1	2	yes	—
228	C₄×C₃₁	#577	1	2	yes	—
229	#533	C₂²×C₃₁	1	1	yes	characteristic
230	#532	C₂²×C₃₁	1	1	yes	characteristic
231	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
232	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
233	#498	C₂²×C₃₁	1	2	yes	complemented
234	#494	C₂²×C₃₁	1	2	yes	complemented
235	#493	C₂²×C₃₁	1	2	yes	—
236	#492	C₂²×C₃₁	1	2	yes	—
237	#488	—	2	2	no	—
238	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
239	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
240	#487	C₂²×C₃₁	1	2	yes	complemented
241	#486	—	2	2	no	—
242	#486	C₂²×C₃₁	1	2	yes	—
243	#486	C₂²×C₃₁	1	2	yes	—
244	#486	C₂²×C₃₁	1	2	yes	—
245	#486	C₂²×C₃₁	1	2	yes	—
246	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
247	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
248	#486	C₂²×C₃₁	1	2	yes	complemented
249	#481	—	2	2	no	—
250	#476	—	2	2	no	—
251	#462	—	2	2	no	—
252	#460	—	2	2	no	—
253	#459	—	2	2	no	—
254	#451	—	2	2	no	—
255	#450	—	2	2	no	—
256	#447	—	2	2	no	—
257	#431	C₂²×C₃₁	1	1	yes	characteristic
258	#431	C₂²×C₃₁	1	1	yes	characteristic
259	#416	—	2	2	no	—
260	C₂×#102	—	2	2	no	—
261	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
262	#322	C₂²×C₃₁	1	2	yes	complemented
263	C₈²	C₂²×C₃₁	1	2	yes	complemented
264	C₂×C₃₁	—	2	2	no	—
265	C₂×C₃₁	—	16	1	no	—
266	C₂×C₃₁	—	16	2	no	—
267	C₂×C₃₁	C₂×#495	1	1	yes	characteristic
268	C₂×C₃₁	#3154	1	1	yes	characteristic
269	C₂×C₃₁	#3155	1	1	yes	characteristic
270	C₂×#41	—	2	1	no	—
271	C₂²×Q₈	—	2	1	no	—
272	C₂²×D₄	—	2	1	no	—
273	C₂²×D₄	—	2	1	no	—
274	C₂²×D₄	—	2	1	no	—
275	C₂×Q₁₆	—	2	4	no	—
276	C₂×SD₁₆	—	2	4	no	—
277	C₂×D₈	—	2	4	no	—
278	C₂²×C₈	D₄×C₃₁	1	2	yes	—
279	C₄⋊Q₈	—	2	1	no	—
280	C₄⋊Q₈	—	2	1	no	—
281	C₄⋊Q₈	—	2	1	no	—
282	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
283	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
284	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
285	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
286	#127	—	2	1	no	—
287	#127	—	2	1	no	—
288	#127	—	2	1	no	—
289	#127	C₂³×C₃₁	1	2	yes	—
290	#127	C₂³×C₃₁	1	1	yes	characteristic
291	#125	—	2	1	no	—
292	#124	—	2	1	no	—
293	C₂²⋊Q₈	—	2	1	no	—
294	C₂²⋊Q₈	—	2	1	no	—
295	#121	—	2	1	no	—
296	#121	—	2	1	no	—
297	C₄×Q₈	—	2	1	no	—
298	C₄×Q₈	—	2	1	no	—
299	C₄×D₄	—	2	1	no	—
300	C₄×D₄	—	2	1	no	—
301	#117	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
305	#106	—	2	4	no	—
306	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
307	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
308	#103	—	2	2	no	—
309	#103	—	2	2	no	—
310	#103	—	2	4	no	—
311	#102	—	2	4	no	—
312	#102	—	2	4	no	—
313	#102	—	2	2	no	—
314	#102	—	2	2	no	—
315	#102	—	2	4	no	—
316	#102	—	2	4	no	—
317	#102	—	2	4	no	—
318	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
319	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
320	C₄×C₈	D₄×C₃₁	1	2	yes	—
321	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
322	C₄×C₈	C₂³×C₃₁	1	2	yes	—
323	C₄×C₈	C₂³×C₃₁	1	2	yes	—
324	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
325	C₃₁	#19656	1	1	yes	characteristic, direct product factor, Sylow
326	C₂⁴	—	4	1	no	—
327	#41	—	4	4	no	—
328	C₂×Q₈	—	2	1	no	—
329	C₂×Q₈	—	2	2	no	—
330	C₂×Q₈	—	2	1	no	—
331	C₂×Q₈	—	2	1	no	—
332	C₂×Q₈	—	4	1	no	—
333	C₂×Q₈	—	8	2	no	—
334	C₂×D₄	—	2	1	no	—
335	C₂×D₄	—	2	1	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	—	4	1	no	—
341	C₂×D₄	—	4	1	no	—
342	C₂×D₄	—	4	1	no	—
343	C₂×D₄	—	4	1	no	—
344	C₂×D₄	—	4	4	no	—
345	C₂×D₄	—	8	2	no	—
346	C₂×D₄	—	8	2	no	—
347	C₂²×C₄	—	4	1	no	—
348	C₂²×C₄	—	4	1	no	—
349	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
352	Q₁₆	—	8	4	no	—
353	SD₁₆	—	8	4	no	—
354	D₈	—	8	4	no	—
355	C₂×C₈	—	2	4	no	—
356	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
357	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
358	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
359	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	4	1	no	—
369	C₄⋊C₄	—	4	1	no	—
370	C₄⋊C₄	—	4	1	no	—
371	C₄⋊C₄	—	4	1	no	—
372	C₂²⋊C₄	—	4	1	no	—
373	C₂²⋊C₄	—	4	1	no	—
374	C₂²⋊C₄	—	4	1	no	—
375	C₄²	—	4	1	no	—
376	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
377	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
379	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
380	C₂³	—	4	1	no	—
381	C₂³	—	4	2	no	—
382	C₂³	—	8	2	no	—
383	C₂³	—	8	2	no	—
384	C₂³	—	8	2	no	—
385	C₂³	C₃₁×#127	1	1	yes	characteristic
386	Q₈	—	4	2	no	—
387	Q₈	—	4	2	no	—
388	Q₈	—	4	4	no	—
389	Q₈	—	4	2	no	—
390	Q₈	—	8	2	no	—
391	D₄	—	4	4	no	—
392	D₄	—	4	2	no	—
393	D₄	—	4	2	no	—
394	D₄	—	4	4	no	—
395	D₄	—	4	4	no	—
396	D₄	—	4	2	no	—
397	D₄	—	8	2	no	—
398	D₄	—	8	2	no	—
399	D₄	—	8	2	no	—
400	D₄	—	8	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	8	2	no	—
408	C₂×C₄	—	8	2	no	—
409	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
410	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
411	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
412	C₂×C₄	C₃₁×#135	1	4	yes	—
413	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
414	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
415	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
416	C₈	—	2	2	no	—
417	C₈	—	2	2	no	—
418	C₈	—	2	4	no	—
419	C₈	—	2	4	no	—
420	C₂²	—	2	1	no	—
421	C₂²	—	2	1	no	—
422	C₂²	—	8	1	no	—
423	C₂²	—	8	2	no	—
424	C₂²	—	8	2	no	—
425	C₂²	—	8	1	no	—
426	C₂²	—	8	2	no	—
427	C₂²	—	8	1	no	—
428	C₂²	—	16	4	no	—
429	C₂²	C₃₁×#495	1	2	yes	—
430	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
431	C₄	—	2	1	no	—
432	C₄	—	2	1	no	—
433	C₄	—	8	1	no	—
434	C₄	—	8	1	no	—
435	C₄	—	8	2	no	—
436	C₄	—	8	1	no	—
437	C₄	C₃₁×#495	1	2	yes	—
438	C₄	C₂×C₃₁×#135	1	2	yes	—
439	C₄	C₃₁×#576	1	2	yes	—
440	C₄	C₃₁×#577	1	2	yes	—
441	C₂	—	2	2	no	—
442	C₂	—	16	2	no	—
443	C₂	—	16	1	no	—
444	C₂	C₂×C₃₁×#495	1	1	yes	characteristic
445	C₂	C₃₁×#3154	1	1	yes	characteristic
446	C₂	C₃₁×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3205	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3227	C₂	semidirect
2. C₃₁×#3209	C₂	semidirect
3. C₃₁×#3208	C₂	semidirect
5. C₃₁×#3131	C₂	semidirect
6. C₂×C₃₁×#486	C₂	semidirect
7. C₃₁×#1720	C₂	semidirect
8. C₃₁×#1717	C₂	semidirect
9. C₃₁×#1709	C₂	semidirect
10. C₃₁×#1708	C₂	semidirect
11. C₃₁×#1450	C₂	semidirect
16. C₃₁×#498	C₂²	semidirect
17. C₃₁×#494	C₂²	semidirect
21. C₃₁×#487	C₂²	semidirect
22. C₃₁×#487	C₂²	semidirect
23. C₃₁×#487	C₂²	semidirect
29. C₃₁×#486	C₂²	semidirect
30. C₃₁×#486	C₂²	semidirect
31. C₃₁×#486	C₂²	semidirect
45. C₃₁×#322	C₂²	semidirect
46. C₈²×C₃₁	C₂²	semidirect
156. #19656	C₃₁	direct (N × H)
197. #3227	C₂×C₃₁	semidirect
198. #3209	C₂×C₃₁	semidirect
199. #3208	C₂×C₃₁	semidirect
201. #3131	C₂×C₃₁	semidirect
202. C₂×#486	C₂×C₃₁	semidirect
203. #1720	C₂×C₃₁	semidirect
204. #1717	C₂×C₃₁	semidirect
205. #1709	C₂×C₃₁	semidirect
206. #1708	C₂×C₃₁	semidirect
207. #1450	C₂×C₃₁	semidirect
233. #498	C₂²×C₃₁	semidirect
234. #494	C₂²×C₃₁	semidirect
238. #487	C₂²×C₃₁	semidirect
239. #487	C₂²×C₃₁	semidirect
240. #487	C₂²×C₃₁	semidirect
246. #486	C₂²×C₃₁	semidirect
247. #486	C₂²×C₃₁	semidirect
248. #486	C₂²×C₃₁	semidirect
262. #322	C₂²×C₃₁	semidirect
263. C₈²	C₂²×C₃₁	semidirect
325. C₃₁	#19656	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×#3227	2
C₃₁×#3209	2
C₃₁×#3208	2
C₃₁×#3205	2
C₃₁×#3131	2
C₂×C₃₁×#486	2
C₃₁×#1720	2
C₃₁×#1717	2
C₃₁×#1709	2
C₃₁×#1708	2
C₃₁×#1450	2
#19656	31

Subgroups — C31×#19656 #21476415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19656 #21476415