Subgroups — C₃₁×#19619 #21476378

Order 7936.

820 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3167	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3167	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1723	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1721	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1720	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1719	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1717	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1717	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1715	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1714	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#135	—	2	2	no	—
17	C₃₁×#533	C₂²	1	1	yes	characteristic
18	C₃₁×#532	C₂²	1	1	yes	characteristic
19	C₃₁×#532	C₂²	1	1	yes	characteristic
20	C₃₁×#532	C₂²	1	1	yes	characteristic
21	C₃₁×#500	C₂²	1	1	yes	characteristic
22	C₃₁×#500	C₂²	1	1	yes	characteristic
23	C₃₁×#498	C₂²	1	1	yes	characteristic
24	C₃₁×#498	C₂²	1	1	yes	characteristic
25	C₃₁×#495	—	2	2	no	—
26	C₃₁×#494	C₂²	1	1	yes	characteristic
27	C₃₁×#493	C₂²	1	1	yes	characteristic
28	C₃₁×#493	C₂²	1	1	yes	characteristic
29	C₃₁×#493	C₂²	1	1	yes	characteristic
30	C₃₁×#492	C₂²	1	1	yes	characteristic
31	C₃₁×#492	C₂²	1	1	yes	characteristic
32	C₃₁×#492	C₂²	1	1	yes	characteristic
33	C₃₁×#492	C₂²	1	1	yes	characteristic
34	C₃₁×#488	—	2	2	no	—
35	C₃₁×#487	C₂²	1	1	yes	characteristic
36	C₃₁×#487	C₂²	1	1	yes	characteristic
37	C₃₁×#487	C₂²	1	1	yes	characteristic
38	C₃₁×#487	C₂²	1	1	yes	characteristic
39	C₃₁×#486	C₂²	1	1	yes	characteristic
40	C₃₁×#486	C₂²	1	1	yes	characteristic
41	C₃₁×#486	C₂²	1	1	yes	characteristic
42	C₃₁×#486	C₂²	1	1	yes	characteristic
43	C₃₁×#486	C₂²	1	1	yes	characteristic
44	C₃₁×#486	C₂²	1	1	yes	characteristic
45	C₃₁×#486	C₂²	1	1	yes	characteristic
46	C₃₁×#486	C₂²	1	1	yes	characteristic
47	C₃₁×#466	—	2	2	no	—
48	C₃₁×#466	—	2	2	no	—
49	C₃₁×#465	—	2	2	no	—
50	C₃₁×#465	—	2	2	no	—
51	C₄×SD₁₆×C₃₁	—	2	2	no	—
52	C₄×SD₁₆×C₃₁	—	2	2	no	—
53	C₄×D₈×C₃₁	—	2	2	no	—
54	C₄×D₈×C₃₁	—	2	2	no	—
55	C₃₁×#416	—	2	2	no	—
56	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
57	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
58	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
59	C₃₁×#322	C₂²	1	1	yes	characteristic
60	C₃₁×#322	C₂²	1	1	yes	characteristic
61	C₃₁×#322	C₂²	1	1	yes	characteristic
62	C₃₁×#322	C₂²	1	1	yes	characteristic
63	C₂×#41×C₃₁	—	2	1	no	—
64	C₂×#41×C₃₁	—	2	1	no	—
65	C₂×#41×C₃₁	—	2	1	no	—
66	C₃₁×#135	—	4	8	no	—
67	C₂×Q₁₆×C₃₁	—	2	2	no	—
68	C₂×SD₁₆×C₃₁	—	2	2	no	—
69	C₂×SD₁₆×C₃₁	—	2	2	no	—
70	C₂×SD₁₆×C₃₁	—	2	2	no	—
71	C₂×SD₁₆×C₃₁	—	2	2	no	—
72	C₂×D₈×C₃₁	—	2	2	no	—
73	C₂×D₈×C₃₁	—	2	2	no	—
74	C₂×D₈×C₃₁	—	2	2	no	—
75	C₂²×C₈×C₃₁	D₄	1	2	yes	—
76	C₂²×C₈×C₃₁	D₄	1	2	yes	—
77	C₂²×C₈×C₃₁	D₄	1	2	yes	—
78	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₃₁×#127	C₂³	1	1	yes	characteristic
84	C₃₁×#127	C₂³	1	1	yes	characteristic
85	C₃₁×#127	C₂³	1	1	yes	characteristic
86	C₃₁×#125	—	2	1	no	—
87	C₃₁×#124	—	2	1	no	—
88	C₃₁×#124	—	2	1	no	—
89	C₃₁×#124	—	2	1	no	—
90	C₃₁×C₂²⋊Q₈	—	2	1	no	—
91	C₃₁×C₂²⋊Q₈	—	2	1	no	—
92	C₃₁×#121	—	2	1	no	—
93	C₃₁×#121	—	2	1	no	—
94	C₃₁×#121	—	2	1	no	—
95	C₃₁×#121	—	2	1	no	—
96	C₃₁×#121	—	2	1	no	—
97	C₃₁×#121	—	2	1	no	—
98	C₄×Q₈×C₃₁	—	2	1	no	—
99	C₄×Q₈×C₃₁	—	2	1	no	—
100	C₄×D₄×C₃₁	—	2	1	no	—
101	C₄×D₄×C₃₁	—	2	1	no	—
102	C₄×D₄×C₃₁	—	2	1	no	—
103	C₄×D₄×C₃₁	—	2	1	no	—
104	C₄×D₄×C₃₁	—	2	1	no	—
105	C₄×D₄×C₃₁	—	2	1	no	—
106	C₃₁×#117	—	2	1	no	—
107	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
108	C₃₁×#107	—	2	2	no	—
109	C₃₁×#107	—	2	2	no	—
110	C₃₁×#106	—	2	2	no	—
111	C₃₁×#106	—	2	2	no	—
112	C₃₁×#103	—	2	2	no	—
113	C₃₁×#103	—	2	2	no	—
114	C₃₁×#103	—	2	2	no	—
115	C₃₁×#103	—	2	2	no	—
116	C₃₁×#102	—	2	2	no	—
117	C₃₁×#102	—	2	2	no	—
118	C₃₁×#102	—	2	2	no	—
119	C₃₁×#102	—	2	2	no	—
120	C₃₁×#102	—	2	2	no	—
121	C₃₁×#102	—	2	2	no	—
122	C₃₁×#102	—	2	2	no	—
123	C₃₁×#102	—	2	2	no	—
124	C₄×C₈×C₃₁	D₄	1	2	yes	—
125	C₄×C₈×C₃₁	D₄	1	2	yes	—
126	C₄×C₈×C₃₁	D₄	1	2	yes	—
127	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
128	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
129	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
130	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
131	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
132	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
133	#41×C₃₁	—	4	4	no	—
134	#41×C₃₁	—	4	4	no	—
135	#41×C₃₁	—	4	4	no	—
136	C₂×Q₈×C₃₁	—	2	1	no	—
137	C₂×Q₈×C₃₁	—	2	1	no	—
138	C₂×Q₈×C₃₁	—	2	1	no	—
139	C₂×Q₈×C₃₁	—	2	1	no	—
140	C₂×Q₈×C₃₁	—	2	1	no	—
141	C₂×D₄×C₃₁	—	2	1	no	—
142	C₂×D₄×C₃₁	—	2	1	no	—
143	C₂×D₄×C₃₁	—	2	1	no	—
144	C₂×D₄×C₃₁	—	2	1	no	—
145	C₂×D₄×C₃₁	—	2	1	no	—
146	C₂×D₄×C₃₁	—	2	1	no	—
147	C₂×D₄×C₃₁	—	2	1	no	—
148	C₂×D₄×C₃₁	—	2	1	no	—
149	C₂×D₄×C₃₁	—	2	1	no	—
150	C₂×D₄×C₃₁	—	4	1	no	—
151	C₂×D₄×C₃₁	—	4	1	no	—
152	C₂×D₄×C₃₁	—	4	1	no	—
153	C₂²×C₄×C₃₁	—	4	1	no	—
154	C₂²×C₄×C₃₁	—	4	1	no	—
155	C₂²×C₄×C₃₁	—	4	1	no	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
159	Q₁₆×C₃₁	—	4	4	no	—
160	SD₁₆×C₃₁	—	4	4	no	—
161	SD₁₆×C₃₁	—	4	4	no	—
162	SD₁₆×C₃₁	—	4	4	no	—
163	SD₁₆×C₃₁	—	4	4	no	—
164	D₈×C₃₁	—	4	4	no	—
165	D₈×C₃₁	—	4	4	no	—
166	D₈×C₃₁	—	4	4	no	—
167	C₂×C₈×C₃₁	—	2	4	no	—
168	C₂×C₈×C₃₁	—	2	4	no	—
169	C₂×C₈×C₃₁	—	4	2	no	—
170	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
171	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
172	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
173	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
174	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
175	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
176	C₄⋊C₄×C₃₁	—	2	1	no	—
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	2	1	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	—	2	1	no	—
181	C₄⋊C₄×C₃₁	—	2	1	no	—
182	C₄⋊C₄×C₃₁	—	2	1	no	—
183	C₄⋊C₄×C₃₁	—	2	1	no	—
184	C₄⋊C₄×C₃₁	—	2	1	no	—
185	C₄⋊C₄×C₃₁	—	2	1	no	—
186	C₄⋊C₄×C₃₁	—	4	1	no	—
187	C₄⋊C₄×C₃₁	—	4	1	no	—
188	C₂²⋊C₄×C₃₁	—	4	1	no	—
189	C₂²⋊C₄×C₃₁	—	4	1	no	—
190	C₂²⋊C₄×C₃₁	—	4	1	no	—
191	C₂²⋊C₄×C₃₁	—	4	1	no	—
192	C₂²⋊C₄×C₃₁	—	4	1	no	—
193	C₂²⋊C₄×C₃₁	—	4	1	no	—
194	C₂²⋊C₄×C₃₁	—	4	1	no	—
195	C₄²×C₃₁	—	4	1	no	—
196	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
200	#19619	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂³×C₃₁	—	4	1	no	—
202	C₂³×C₃₁	—	4	1	no	—
203	C₂³×C₃₁	—	4	1	no	—
204	C₂³×C₃₁	#127	1	1	yes	characteristic
205	Q₈×C₃₁	—	4	2	no	—
206	Q₈×C₃₁	—	4	2	no	—
207	Q₈×C₃₁	—	4	2	no	—
208	Q₈×C₃₁	—	4	2	no	—
209	Q₈×C₃₁	—	4	2	no	—
210	D₄×C₃₁	—	4	2	no	—
211	D₄×C₃₁	—	4	2	no	—
212	D₄×C₃₁	—	4	2	no	—
213	D₄×C₃₁	—	4	2	no	—
214	D₄×C₃₁	—	4	2	no	—
215	D₄×C₃₁	—	4	2	no	—
216	D₄×C₃₁	—	4	2	no	—
217	D₄×C₃₁	—	4	2	no	—
218	D₄×C₃₁	—	4	2	no	—
219	D₄×C₃₁	—	8	2	no	—
220	D₄×C₃₁	—	8	2	no	—
221	D₄×C₃₁	—	8	2	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	4	1	no	—
225	C₂×C₄×C₃₁	—	4	1	no	—
226	C₂×C₄×C₃₁	—	4	1	no	—
227	C₂×C₄×C₃₁	—	4	1	no	—
228	C₂×C₄×C₃₁	—	4	1	no	—
229	C₂×C₄×C₃₁	—	8	2	no	—
230	C₂×C₄×C₃₁	—	8	2	no	—
231	C₂×C₄×C₃₁	—	8	2	no	—
232	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
233	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
234	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
235	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
239	C₈×C₃₁	—	2	2	no	—
240	C₈×C₃₁	—	2	2	no	—
241	C₈×C₃₁	—	2	2	no	—
242	C₈×C₃₁	—	2	2	no	—
243	C₈×C₃₁	—	2	2	no	—
244	C₈×C₃₁	#135	1	4	yes	complemented
245	#3169	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#3169	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#3167	C₂×C₃₁	1	1	yes	characteristic, complemented
248	#3167	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#3147	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#3131	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#1723	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1721	C₂×C₃₁	1	1	yes	characteristic, complemented
253	#1720	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#1719	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#1717	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#1717	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#1715	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#1714	C₂×C₃₁	1	1	yes	characteristic, complemented
259	C₂×#322	C₂×C₃₁	1	1	yes	characteristic, complemented
260	C₂²×C₃₁	—	2	1	no	—
261	C₂²×C₃₁	—	2	1	no	—
262	C₂²×C₃₁	—	8	1	no	—
263	C₂²×C₃₁	—	8	1	no	—
264	C₂²×C₃₁	—	8	1	no	—
265	C₂²×C₃₁	—	8	1	no	—
266	C₂²×C₃₁	—	8	1	no	—
267	C₂²×C₃₁	—	8	1	no	—
268	C₂²×C₃₁	—	8	1	no	—
269	C₂²×C₃₁	—	8	1	no	—
270	C₂²×C₃₁	—	8	1	no	—
271	C₂²×C₃₁	#496	1	2	yes	—
272	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
273	C₄×C₃₁	—	2	1	no	—
274	C₄×C₃₁	—	2	1	no	—
275	C₄×C₃₁	—	8	1	no	—
276	C₄×C₃₁	—	8	1	no	—
277	C₄×C₃₁	—	8	1	no	—
278	C₄×C₃₁	—	8	1	no	—
279	C₄×C₃₁	—	8	1	no	—
280	C₄×C₃₁	#496	1	2	yes	—
281	C₄×C₃₁	C₂×#135	1	2	yes	—
282	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
283	C₄×C₃₁	#575	1	2	yes	—
284	C₂×#135	—	2	2	no	—
285	#533	C₂²×C₃₁	1	1	yes	characteristic
286	#532	C₂²×C₃₁	1	1	yes	characteristic
287	#532	C₂²×C₃₁	1	1	yes	characteristic
288	#532	C₂²×C₃₁	1	1	yes	characteristic
289	#500	C₂²×C₃₁	1	1	yes	characteristic
290	#500	C₂²×C₃₁	1	1	yes	characteristic
291	#498	C₂²×C₃₁	1	1	yes	characteristic
292	#498	C₂²×C₃₁	1	1	yes	characteristic
293	#495	—	2	2	no	—
294	#494	C₂²×C₃₁	1	1	yes	characteristic
295	#493	C₂²×C₃₁	1	1	yes	characteristic
296	#493	C₂²×C₃₁	1	1	yes	characteristic
297	#493	C₂²×C₃₁	1	1	yes	characteristic
298	#492	C₂²×C₃₁	1	1	yes	characteristic
299	#492	C₂²×C₃₁	1	1	yes	characteristic
300	#492	C₂²×C₃₁	1	1	yes	characteristic
301	#492	C₂²×C₃₁	1	1	yes	characteristic
302	#488	—	2	2	no	—
303	#487	C₂²×C₃₁	1	1	yes	characteristic
304	#487	C₂²×C₃₁	1	1	yes	characteristic
305	#487	C₂²×C₃₁	1	1	yes	characteristic
306	#487	C₂²×C₃₁	1	1	yes	characteristic
307	#486	C₂²×C₃₁	1	1	yes	characteristic
308	#486	C₂²×C₃₁	1	1	yes	characteristic
309	#486	C₂²×C₃₁	1	1	yes	characteristic
310	#486	C₂²×C₃₁	1	1	yes	characteristic
311	#486	C₂²×C₃₁	1	1	yes	characteristic
312	#486	C₂²×C₃₁	1	1	yes	characteristic
313	#486	C₂²×C₃₁	1	1	yes	characteristic
314	#486	C₂²×C₃₁	1	1	yes	characteristic
315	#466	—	2	2	no	—
316	#466	—	2	2	no	—
317	#465	—	2	2	no	—
318	#465	—	2	2	no	—
319	C₄×SD₁₆	—	2	2	no	—
320	C₄×SD₁₆	—	2	2	no	—
321	C₄×D₈	—	2	2	no	—
322	C₄×D₈	—	2	2	no	—
323	#416	—	2	2	no	—
324	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
325	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
326	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
327	#322	C₂²×C₃₁	1	1	yes	characteristic
328	#322	C₂²×C₃₁	1	1	yes	characteristic
329	#322	C₂²×C₃₁	1	1	yes	characteristic
330	#322	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×C₃₁	—	2	2	no	—
332	C₂×C₃₁	—	16	1	no	—
333	C₂×C₃₁	—	16	1	no	—
334	C₂×C₃₁	—	16	1	no	—
335	C₂×C₃₁	#3147	1	1	yes	characteristic
336	C₂×C₃₁	C₂×#496	1	1	yes	characteristic
337	C₂×C₃₁	#3150	1	1	yes	characteristic
338	C₂×#41	—	2	1	no	—
339	C₂×#41	—	2	1	no	—
340	C₂×#41	—	2	1	no	—
341	#135	—	4	8	no	—
342	C₂×Q₁₆	—	2	2	no	—
343	C₂×SD₁₆	—	2	2	no	—
344	C₂×SD₁₆	—	2	2	no	—
345	C₂×SD₁₆	—	2	2	no	—
346	C₂×SD₁₆	—	2	2	no	—
347	C₂×D₈	—	2	2	no	—
348	C₂×D₈	—	2	2	no	—
349	C₂×D₈	—	2	2	no	—
350	C₂²×C₈	D₄×C₃₁	1	2	yes	—
351	C₂²×C₈	D₄×C₃₁	1	2	yes	—
352	C₂²×C₈	D₄×C₃₁	1	2	yes	—
353	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
354	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
355	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
356	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
357	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
358	#127	C₂³×C₃₁	1	1	yes	characteristic
359	#127	C₂³×C₃₁	1	1	yes	characteristic
360	#127	C₂³×C₃₁	1	1	yes	characteristic
361	#125	—	2	1	no	—
362	#124	—	2	1	no	—
363	#124	—	2	1	no	—
364	#124	—	2	1	no	—
365	C₂²⋊Q₈	—	2	1	no	—
366	C₂²⋊Q₈	—	2	1	no	—
367	#121	—	2	1	no	—
368	#121	—	2	1	no	—
369	#121	—	2	1	no	—
370	#121	—	2	1	no	—
371	#121	—	2	1	no	—
372	#121	—	2	1	no	—
373	C₄×Q₈	—	2	1	no	—
374	C₄×Q₈	—	2	1	no	—
375	C₄×D₄	—	2	1	no	—
376	C₄×D₄	—	2	1	no	—
377	C₄×D₄	—	2	1	no	—
378	C₄×D₄	—	2	1	no	—
379	C₄×D₄	—	2	1	no	—
380	C₄×D₄	—	2	1	no	—
381	#117	—	2	1	no	—
382	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
383	#107	—	2	2	no	—
384	#107	—	2	2	no	—
385	#106	—	2	2	no	—
386	#106	—	2	2	no	—
387	#103	—	2	2	no	—
388	#103	—	2	2	no	—
389	#103	—	2	2	no	—
390	#103	—	2	2	no	—
391	#102	—	2	2	no	—
392	#102	—	2	2	no	—
393	#102	—	2	2	no	—
394	#102	—	2	2	no	—
395	#102	—	2	2	no	—
396	#102	—	2	2	no	—
397	#102	—	2	2	no	—
398	#102	—	2	2	no	—
399	C₄×C₈	D₄×C₃₁	1	2	yes	—
400	C₄×C₈	D₄×C₃₁	1	2	yes	—
401	C₄×C₈	D₄×C₃₁	1	2	yes	—
402	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
403	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
404	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
405	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
406	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
407	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
408	C₃₁	#19619	1	1	yes	characteristic, direct product factor, Sylow
409	#41	—	4	4	no	—
410	#41	—	4	4	no	—
411	#41	—	4	4	no	—
412	C₂×Q₈	—	2	1	no	—
413	C₂×Q₈	—	2	1	no	—
414	C₂×Q₈	—	2	1	no	—
415	C₂×Q₈	—	2	1	no	—
416	C₂×Q₈	—	2	1	no	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	—	2	1	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	—	2	1	no	—
423	C₂×D₄	—	2	1	no	—
424	C₂×D₄	—	2	1	no	—
425	C₂×D₄	—	2	1	no	—
426	C₂×D₄	—	4	1	no	—
427	C₂×D₄	—	4	1	no	—
428	C₂×D₄	—	4	1	no	—
429	C₂²×C₄	—	4	1	no	—
430	C₂²×C₄	—	4	1	no	—
431	C₂²×C₄	—	4	1	no	—
432	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
433	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
434	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
435	Q₁₆	—	4	4	no	—
436	SD₁₆	—	4	4	no	—
437	SD₁₆	—	4	4	no	—
438	SD₁₆	—	4	4	no	—
439	SD₁₆	—	4	4	no	—
440	D₈	—	4	4	no	—
441	D₈	—	4	4	no	—
442	D₈	—	4	4	no	—
443	C₂×C₈	—	2	4	no	—
444	C₂×C₈	—	2	4	no	—
445	C₂×C₈	—	4	2	no	—
446	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
447	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
448	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
449	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
450	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
451	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	4	1	no	—
463	C₄⋊C₄	—	4	1	no	—
464	C₂²⋊C₄	—	4	1	no	—
465	C₂²⋊C₄	—	4	1	no	—
466	C₂²⋊C₄	—	4	1	no	—
467	C₂²⋊C₄	—	4	1	no	—
468	C₂²⋊C₄	—	4	1	no	—
469	C₂²⋊C₄	—	4	1	no	—
470	C₂²⋊C₄	—	4	1	no	—
471	C₄²	—	4	1	no	—
472	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
473	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
474	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
476	C₂³	—	4	1	no	—
477	C₂³	—	4	1	no	—
478	C₂³	—	4	1	no	—
479	C₂³	C₃₁×#127	1	1	yes	characteristic
480	Q₈	—	4	2	no	—
481	Q₈	—	4	2	no	—
482	Q₈	—	4	2	no	—
483	Q₈	—	4	2	no	—
484	Q₈	—	4	2	no	—
485	D₄	—	4	2	no	—
486	D₄	—	4	2	no	—
487	D₄	—	4	2	no	—
488	D₄	—	4	2	no	—
489	D₄	—	4	2	no	—
490	D₄	—	4	2	no	—
491	D₄	—	4	2	no	—
492	D₄	—	4	2	no	—
493	D₄	—	4	2	no	—
494	D₄	—	8	2	no	—
495	D₄	—	8	2	no	—
496	D₄	—	8	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	8	2	no	—
505	C₂×C₄	—	8	2	no	—
506	C₂×C₄	—	8	2	no	—
507	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
508	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
509	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
510	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
511	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
512	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
513	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
514	C₈	—	2	2	no	—
515	C₈	—	2	2	no	—
516	C₈	—	2	2	no	—
517	C₈	—	2	2	no	—
518	C₈	—	2	2	no	—
519	C₈	C₃₁×#135	1	4	yes	complemented
520	C₂²	—	2	1	no	—
521	C₂²	—	2	1	no	—
522	C₂²	—	8	1	no	—
523	C₂²	—	8	1	no	—
524	C₂²	—	8	1	no	—
525	C₂²	—	8	1	no	—
526	C₂²	—	8	1	no	—
527	C₂²	—	8	1	no	—
528	C₂²	—	8	1	no	—
529	C₂²	—	8	1	no	—
530	C₂²	—	8	1	no	—
531	C₂²	C₃₁×#496	1	2	yes	—
532	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
533	C₄	—	2	1	no	—
534	C₄	—	2	1	no	—
535	C₄	—	8	1	no	—
536	C₄	—	8	1	no	—
537	C₄	—	8	1	no	—
538	C₄	—	8	1	no	—
539	C₄	—	8	1	no	—
540	C₄	C₃₁×#496	1	2	yes	—
541	C₄	C₂×C₃₁×#135	1	2	yes	—
542	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
543	C₄	C₃₁×#575	1	2	yes	—
544	C₂	—	2	2	no	—
545	C₂	—	16	1	no	—
546	C₂	—	16	1	no	—
547	C₂	—	16	1	no	—
548	C₂	C₃₁×#3147	1	1	yes	characteristic
549	C₂	C₂×C₃₁×#496	1	1	yes	characteristic
550	C₂	C₃₁×#3150	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3169	C₂	semidirect
2. C₃₁×#3169	C₂	semidirect
3. C₃₁×#3167	C₂	semidirect
4. C₃₁×#3167	C₂	semidirect
5. C₃₁×#3147	C₂	semidirect
6. C₃₁×#3131	C₂	semidirect
7. C₃₁×#1723	C₂	semidirect
8. C₃₁×#1721	C₂	semidirect
9. C₃₁×#1720	C₂	semidirect
10. C₃₁×#1719	C₂	semidirect
11. C₃₁×#1717	C₂	semidirect
12. C₃₁×#1717	C₂	semidirect
13. C₃₁×#1715	C₂	semidirect
14. C₃₁×#1714	C₂	semidirect
15. C₂×C₃₁×#322	C₂	semidirect
200. #19619	C₃₁	direct (N × H)
244. C₈×C₃₁	#135	semidirect
245. #3169	C₂×C₃₁	semidirect
246. #3169	C₂×C₃₁	semidirect
247. #3167	C₂×C₃₁	semidirect
248. #3167	C₂×C₃₁	semidirect
249. #3147	C₂×C₃₁	semidirect
250. #3131	C₂×C₃₁	semidirect
251. #1723	C₂×C₃₁	semidirect
252. #1721	C₂×C₃₁	semidirect
253. #1720	C₂×C₃₁	semidirect
254. #1719	C₂×C₃₁	semidirect
255. #1717	C₂×C₃₁	semidirect
256. #1717	C₂×C₃₁	semidirect
257. #1715	C₂×C₃₁	semidirect
258. #1714	C₂×C₃₁	semidirect
259. C₂×#322	C₂×C₃₁	semidirect
408. C₃₁	#19619	direct (N × H)
519. C₈	C₃₁×#135	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3169	2
C₃₁×#3167	2
C₃₁×#3147	2
C₃₁×#3131	2
C₃₁×#1723	2
C₃₁×#1721	2
C₃₁×#1720	2
C₃₁×#1719	2
C₃₁×#1717	2
C₃₁×#1715	2
C₃₁×#1714	2
C₂×C₃₁×#322	2
#19619	31

Subgroups — C31×#19619 #21476378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19619 #21476378