Subgroups — C₃₁×#19700 #21476459

Order 7936.

778 conjugacy classes of subgroups.

The subgroup lattice has 446 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×#3226	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3226	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3206	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1723	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1719	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1717	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1714	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1452	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	—	2	2	no	—
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃₁×#500	C₂²	1	2	yes	—
18	C₃₁×#500	C₂²	1	2	yes	—
19	C₃₁×#498	C₂²	1	2	yes	—
20	C₃₁×#495	—	2	2	no	—
21	C₃₁×#494	C₂²	1	1	yes	characteristic
22	C₃₁×#493	C₂²	1	1	yes	characteristic
23	C₃₁×#492	C₂²	1	1	yes	characteristic
24	C₃₁×#492	C₂²	1	1	yes	characteristic
25	C₃₁×#492	C₂²	1	2	yes	—
26	C₃₁×#487	—	2	2	no	—
27	C₃₁×#487	C₂²	1	2	yes	—
28	C₃₁×#487	C₂²	1	2	yes	—
29	C₃₁×#487	C₂²	1	2	yes	—
30	C₃₁×#486	C₂²	1	2	yes	—
31	C₃₁×#486	C₂²	1	2	yes	—
32	C₃₁×#486	C₂²	1	2	yes	—
33	C₃₁×#481	—	2	2	no	—
34	C₃₁×#480	—	2	2	no	—
35	C₃₁×#480	—	2	2	no	—
36	C₃₁×#476	—	2	2	no	—
37	C₃₁×#474	—	2	2	no	—
38	C₃₁×#474	—	2	2	no	—
39	C₃₁×#461	—	2	2	no	—
40	C₃₁×#448	—	2	2	no	—
41	C₃₁×#431	C₂²	1	1	yes	characteristic
42	C₃₁×#431	C₂²	1	1	yes	characteristic
43	C₂×C₃₁×#103	—	2	2	no	—
44	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#322	C₂²	1	2	yes	—
46	C₃₁×#322	C₂²	1	2	yes	—
47	C₂×#41×C₃₁	—	2	1	no	—
48	C₂×#41×C₃₁	—	2	1	no	—
49	C₂²×Q₈×C₃₁	—	2	1	no	—
50	C₂²×Q₈×C₃₁	—	2	1	no	—
51	C₃₁×#135	—	4	8	no	—
52	C₂×Q₁₆×C₃₁	—	2	2	no	—
53	C₂×SD₁₆×C₃₁	—	2	2	no	—
54	C₂×SD₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	4	no	—
56	C₂×D₈×C₃₁	—	2	2	no	—
57	C₂²×C₈×C₃₁	D₄	1	2	yes	—
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	—	2	1	no	—
60	C₃₁×C₄⋊Q₈	—	2	1	no	—
61	C₃₁×C₄⋊Q₈	—	2	1	no	—
62	C₃₁×C₄⋊Q₈	—	2	1	no	—
63	C₃₁×C₄⋊Q₈	—	2	1	no	—
64	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
65	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
66	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
67	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
68	C₃₁×#127	C₂³	1	1	yes	characteristic
69	C₃₁×#127	C₂³	1	1	yes	characteristic
70	C₃₁×#124	—	2	1	no	—
71	C₃₁×#124	—	2	1	no	—
72	C₃₁×#121	—	2	1	no	—
73	C₃₁×#121	—	2	1	no	—
74	C₃₁×#121	—	2	1	no	—
75	C₃₁×#121	—	2	1	no	—
76	C₄×D₄×C₃₁	—	2	1	no	—
77	C₄×D₄×C₃₁	—	2	1	no	—
78	C₄×D₄×C₃₁	—	2	1	no	—
79	C₄×D₄×C₃₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₃₁×#107	—	2	4	no	—
86	C₃₁×#107	—	2	4	no	—
87	C₃₁×#106	—	2	4	no	—
88	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
89	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
90	C₃₁×#103	—	2	4	no	—
91	C₃₁×#103	—	2	4	no	—
92	C₃₁×#103	—	2	4	no	—
93	C₃₁×#102	—	2	4	no	—
94	C₃₁×#102	—	2	4	no	—
95	C₃₁×#102	—	2	4	no	—
96	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
97	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
98	C₄×C₈×C₃₁	D₄	1	2	yes	—
99	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₃₁	C₂³	1	2	yes	—
102	C₄×C₈×C₃₁	C₂³	1	2	yes	—
103	#41×C₃₁	—	4	4	no	—
104	#41×C₃₁	—	4	4	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	—	4	1	no	—
110	C₂×Q₈×C₃₁	—	4	1	no	—
111	C₂×Q₈×C₃₁	—	4	4	no	—
112	C₂×Q₈×C₃₁	—	8	2	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	1	no	—
119	C₂×D₄×C₃₁	—	4	1	no	—
120	C₂×D₄×C₃₁	—	4	1	no	—
121	C₂²×C₄×C₃₁	—	4	1	no	—
122	C₂²×C₄×C₃₁	—	4	1	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	—	4	4	no	—
129	SD₁₆×C₃₁	—	4	4	no	—
130	SD₁₆×C₃₁	—	4	4	no	—
131	SD₁₆×C₃₁	—	8	4	no	—
132	D₈×C₃₁	—	4	4	no	—
133	C₂×C₈×C₃₁	—	2	4	no	—
134	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
135	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
136	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
137	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	2	2	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	—	2	1	no	—
145	C₄⋊C₄×C₃₁	—	2	1	no	—
146	C₄⋊C₄×C₃₁	—	4	1	no	—
147	C₄⋊C₄×C₃₁	—	4	1	no	—
148	C₄⋊C₄×C₃₁	—	4	1	no	—
149	C₄⋊C₄×C₃₁	—	4	1	no	—
150	C₂²⋊C₄×C₃₁	—	4	1	no	—
151	C₂²⋊C₄×C₃₁	—	4	1	no	—
152	C₂²⋊C₄×C₃₁	—	4	1	no	—
153	C₂²⋊C₄×C₃₁	—	4	1	no	—
154	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
158	#19700	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂³×C₃₁	—	4	1	no	—
160	C₂³×C₃₁	—	4	1	no	—
161	C₂³×C₃₁	#127	1	1	yes	characteristic
162	Q₈×C₃₁	—	4	2	no	—
163	Q₈×C₃₁	—	4	2	no	—
164	Q₈×C₃₁	—	4	4	no	—
165	Q₈×C₃₁	—	4	4	no	—
166	Q₈×C₃₁	—	8	2	no	—
167	Q₈×C₃₁	—	8	2	no	—
168	D₄×C₃₁	—	4	2	no	—
169	D₄×C₃₁	—	4	2	no	—
170	D₄×C₃₁	—	4	2	no	—
171	D₄×C₃₁	—	4	2	no	—
172	D₄×C₃₁	—	4	2	no	—
173	D₄×C₃₁	—	4	2	no	—
174	D₄×C₃₁	—	8	2	no	—
175	D₄×C₃₁	—	8	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	4	1	no	—
179	C₂×C₄×C₃₁	—	4	1	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	8	2	no	—
183	C₂×C₄×C₃₁	—	8	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	AGL(1,ℤ/8)	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	#3226	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#3226	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#3206	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#3147	C₂×C₃₁	1	1	yes	characteristic
202	C₂×#487	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#1723	C₂×C₃₁	1	2	yes	complemented
204	#1719	C₂×C₃₁	1	2	yes	complemented
205	#1717	C₂×C₃₁	1	2	yes	complemented
206	#1714	C₂×C₃₁	1	2	yes	complemented
207	#1452	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	1	no	—
212	C₂²×C₃₁	—	8	1	no	—
213	C₂²×C₃₁	—	8	1	no	—
214	C₂²×C₃₁	—	8	1	no	—
215	C₂²×C₃₁	—	8	1	no	—
216	C₂²×C₃₁	#496	1	2	yes	—
217	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
218	C₄×C₃₁	—	2	1	no	—
219	C₄×C₃₁	—	2	1	no	—
220	C₄×C₃₁	—	8	1	no	—
221	C₄×C₃₁	—	8	1	no	—
222	C₄×C₃₁	—	8	2	no	—
223	C₄×C₃₁	—	8	2	no	—
224	C₄×C₃₁	#496	1	2	yes	—
225	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
226	C₄×C₃₁	#577	1	2	yes	—
227	C₄×C₃₁	#578	1	2	yes	—
228	C₂×#135	—	2	2	no	—
229	#532	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×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
233	#500	C₂²×C₃₁	1	2	yes	—
234	#500	C₂²×C₃₁	1	2	yes	—
235	#498	C₂²×C₃₁	1	2	yes	—
236	#495	—	2	2	no	—
237	#494	C₂²×C₃₁	1	1	yes	characteristic
238	#493	C₂²×C₃₁	1	1	yes	characteristic
239	#492	C₂²×C₃₁	1	1	yes	characteristic
240	#492	C₂²×C₃₁	1	1	yes	characteristic
241	#492	C₂²×C₃₁	1	2	yes	—
242	#487	—	2	2	no	—
243	#487	C₂²×C₃₁	1	2	yes	—
244	#487	C₂²×C₃₁	1	2	yes	—
245	#487	C₂²×C₃₁	1	2	yes	—
246	#486	C₂²×C₃₁	1	2	yes	—
247	#486	C₂²×C₃₁	1	2	yes	—
248	#486	C₂²×C₃₁	1	2	yes	—
249	#481	—	2	2	no	—
250	#480	—	2	2	no	—
251	#480	—	2	2	no	—
252	#476	—	2	2	no	—
253	#474	—	2	2	no	—
254	#474	—	2	2	no	—
255	#461	—	2	2	no	—
256	#448	—	2	2	no	—
257	#431	C₂²×C₃₁	1	1	yes	characteristic
258	#431	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×#103	—	2	2	no	—
260	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
261	#322	C₂²×C₃₁	1	2	yes	—
262	#322	C₂²×C₃₁	1	2	yes	—
263	C₂×C₃₁	—	2	2	no	—
264	C₂×C₃₁	—	16	1	no	—
265	C₂×C₃₁	—	16	1	no	—
266	C₂×C₃₁	C₂×#496	1	1	yes	characteristic
267	C₂×C₃₁	#3154	1	1	yes	characteristic
268	C₂×C₃₁	#3156	1	1	yes	characteristic
269	C₂×#41	—	2	1	no	—
270	C₂×#41	—	2	1	no	—
271	C₂²×Q₈	—	2	1	no	—
272	C₂²×Q₈	—	2	1	no	—
273	#135	—	4	8	no	—
274	C₂×Q₁₆	—	2	2	no	—
275	C₂×SD₁₆	—	2	2	no	—
276	C₂×SD₁₆	—	2	2	no	—
277	C₂×SD₁₆	—	2	4	no	—
278	C₂×D₈	—	2	2	no	—
279	C₂²×C₈	D₄×C₃₁	1	2	yes	—
280	C₄⋊Q₈	—	2	1	no	—
281	C₄⋊Q₈	—	2	1	no	—
282	C₄⋊Q₈	—	2	1	no	—
283	C₄⋊Q₈	—	2	1	no	—
284	C₄⋊Q₈	—	2	1	no	—
285	C₄⋊Q₈	—	2	1	no	—
286	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
287	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
288	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
289	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
290	#127	C₂³×C₃₁	1	1	yes	characteristic
291	#127	C₂³×C₃₁	1	1	yes	characteristic
292	#124	—	2	1	no	—
293	#124	—	2	1	no	—
294	#121	—	2	1	no	—
295	#121	—	2	1	no	—
296	#121	—	2	1	no	—
297	#121	—	2	1	no	—
298	C₄×D₄	—	2	1	no	—
299	C₄×D₄	—	2	1	no	—
300	C₄×D₄	—	2	1	no	—
301	C₄×D₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
307	#107	—	2	4	no	—
308	#107	—	2	4	no	—
309	#106	—	2	4	no	—
310	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
311	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
312	#103	—	2	4	no	—
313	#103	—	2	4	no	—
314	#103	—	2	4	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	1	yes	characteristic
323	C₄×C₈	C₂³×C₃₁	1	2	yes	—
324	C₄×C₈	C₂³×C₃₁	1	2	yes	—
325	C₃₁	#19700	1	1	yes	characteristic, direct product factor, Sylow
326	#41	—	4	4	no	—
327	#41	—	4	4	no	—
328	C₂×Q₈	—	2	2	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	—	4	1	no	—
334	C₂×Q₈	—	4	4	no	—
335	C₂×Q₈	—	8	2	no	—
336	C₂×D₄	—	2	1	no	—
337	C₂×D₄	—	2	1	no	—
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	4	1	no	—
343	C₂×D₄	—	4	1	no	—
344	C₂²×C₄	—	4	1	no	—
345	C₂²×C₄	—	4	1	no	—
346	C₂²×C₄	—	4	1	no	—
347	C₂²×C₄	—	4	1	no	—
348	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
349	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
351	Q₁₆	—	4	4	no	—
352	SD₁₆	—	4	4	no	—
353	SD₁₆	—	4	4	no	—
354	SD₁₆	—	8	4	no	—
355	D₈	—	4	4	no	—
356	C₂×C₈	—	2	4	no	—
357	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
358	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
359	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
360	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	4	1	no	—
370	C₄⋊C₄	—	4	1	no	—
371	C₄⋊C₄	—	4	1	no	—
372	C₄⋊C₄	—	4	1	no	—
373	C₂²⋊C₄	—	4	1	no	—
374	C₂²⋊C₄	—	4	1	no	—
375	C₂²⋊C₄	—	4	1	no	—
376	C₂²⋊C₄	—	4	1	no	—
377	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
379	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
380	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
381	C₂³	—	4	1	no	—
382	C₂³	—	4	1	no	—
383	C₂³	C₃₁×#127	1	1	yes	characteristic
384	Q₈	—	4	2	no	—
385	Q₈	—	4	2	no	—
386	Q₈	—	4	4	no	—
387	Q₈	—	4	4	no	—
388	Q₈	—	8	2	no	—
389	Q₈	—	8	2	no	—
390	D₄	—	4	2	no	—
391	D₄	—	4	2	no	—
392	D₄	—	4	2	no	—
393	D₄	—	4	2	no	—
394	D₄	—	4	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	8	2	no	—
397	D₄	—	8	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	8	2	no	—
405	C₂×C₄	—	8	2	no	—
406	C₂×C₄	—	8	2	no	—
407	C₂×C₄	—	8	2	no	—
408	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
409	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
410	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
411	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
412	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
413	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
414	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
415	C₈	—	2	2	no	—
416	C₈	—	2	2	no	—
417	C₈	—	2	4	no	—
418	C₈	—	2	4	no	—
419	C₂²	—	2	1	no	—
420	C₂²	—	2	1	no	—
421	C₂²	—	8	1	no	—
422	C₂²	—	8	1	no	—
423	C₂²	—	8	1	no	—
424	C₂²	—	8	1	no	—
425	C₂²	—	8	1	no	—
426	C₂²	—	8	1	no	—
427	C₂²	C₃₁×#496	1	2	yes	—
428	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
429	C₄	—	2	1	no	—
430	C₄	—	2	1	no	—
431	C₄	—	8	1	no	—
432	C₄	—	8	1	no	—
433	C₄	—	8	2	no	—
434	C₄	—	8	2	no	—
435	C₄	C₃₁×#496	1	2	yes	—
436	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
437	C₄	C₃₁×#577	1	2	yes	—
438	C₄	C₃₁×#578	1	2	yes	—
439	C₂	—	2	2	no	—
440	C₂	—	16	1	no	—
441	C₂	—	16	1	no	—
442	C₂	C₂×C₃₁×#496	1	1	yes	characteristic
443	C₂	C₃₁×#3154	1	1	yes	characteristic
444	C₂	C₃₁×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3147	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×#3226	C₂	semidirect
3. C₃₁×#3226	C₂	semidirect
4. C₃₁×#3206	C₂	semidirect
6. C₂×C₃₁×#487	C₂	semidirect
7. C₃₁×#1723	C₂	semidirect
8. C₃₁×#1719	C₂	semidirect
9. C₃₁×#1717	C₂	semidirect
10. C₃₁×#1714	C₂	semidirect
11. C₃₁×#1452	C₂	semidirect
158. #19700	C₃₁	direct (N × H)
197. #3227	C₂×C₃₁	semidirect
198. #3226	C₂×C₃₁	semidirect
199. #3226	C₂×C₃₁	semidirect
200. #3206	C₂×C₃₁	semidirect
202. C₂×#487	C₂×C₃₁	semidirect
203. #1723	C₂×C₃₁	semidirect
204. #1719	C₂×C₃₁	semidirect
205. #1717	C₂×C₃₁	semidirect
206. #1714	C₂×C₃₁	semidirect
207. #1452	C₂×C₃₁	semidirect
325. C₃₁	#19700	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×#3226	2
C₃₁×#3206	2
C₃₁×#3147	2
C₂×C₃₁×#487	2
C₃₁×#1723	2
C₃₁×#1719	2
C₃₁×#1717	2
C₃₁×#1714	2
C₃₁×#1452	2
#19700	31

Subgroups — C31×#19700 #21476459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19700 #21476459