Subgroups — C₃₁×#18784 #21475543

Order 7936.

956 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3482	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3228	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3115	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3035	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1652	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1651	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1648	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1647	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1628	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1627	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1626	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1625	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1552	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1488	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1477	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#135	—	2	1	no	—
17	C₃₁×#557	C₂²	1	1	yes	characteristic
18	C₃₁×#556	—	2	1	no	—
19	C₃₁×#556	C₂²	1	1	yes	characteristic
20	C₃₁×#552	—	2	2	no	—
21	C₃₁×#552	—	2	1	no	—
22	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#548	—	2	1	no	—
24	C₃₁×#547	—	2	1	no	—
25	C₃₁×#547	C₂²	1	1	yes	characteristic
26	C₃₁×#533	—	2	1	no	—
27	C₃₁×#533	C₂²	1	1	yes	characteristic
28	C₃₁×#532	C₂²	1	1	yes	characteristic
29	C₃₁×#529	—	2	1	no	—
30	C₃₁×#529	—	2	2	no	—
31	C₄×#41×C₃₁	—	2	1	no	—
32	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
34	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
35	C₃₁×#500	C₂²	1	1	yes	characteristic
36	C₃₁×#498	C₂²	1	1	yes	characteristic
37	C₃₁×#486	C₂²	1	1	yes	characteristic
38	C₃₁×#486	C₂²	1	1	yes	characteristic
39	C₃₁×#482	—	2	1	no	—
40	C₃₁×#482	—	2	2	no	—
41	C₃₁×#478	—	2	2	no	—
42	C₃₁×#475	C₂²	1	1	yes	characteristic
43	C₃₁×#474	C₂²	1	1	yes	characteristic
44	C₃₁×#464	—	2	1	no	—
45	C₃₁×#463	—	2	1	no	—
46	C₃₁×#462	C₂²	1	1	yes	characteristic
47	C₃₁×#461	C₂²	1	1	yes	characteristic
48	C₃₁×#460	C₂²	1	1	yes	characteristic
49	C₃₁×#460	C₂²	1	1	yes	characteristic
50	C₃₁×#459	C₂²	1	1	yes	characteristic
51	C₃₁×#459	C₂²	1	1	yes	characteristic
52	C₃₁×#449	—	2	1	no	—
53	C₃₁×#442	—	2	1	no	—
54	C₃₁×#440	—	2	1	no	—
55	C₃₁×#432	C₂²	1	1	yes	characteristic
56	C₃₁×#431	C₂²	1	1	yes	characteristic
57	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
58	C₃₁×#416	—	2	1	no	—
59	C₃₁×#332	C₂²	1	1	yes	characteristic
60	C₃₁×#332	C₂²	1	1	yes	characteristic
61	C₃₁×#332	C₂²	1	1	yes	characteristic
62	C₃₁×#332	C₂²	1	1	yes	characteristic
63	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
66	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
67	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
68	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
69	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
70	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
71	C₂×#41×C₃₁	—	2	1	no	—
72	C₂×#41×C₃₁	—	2	1	no	—
73	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
74	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
75	C₂²×Q₈×C₃₁	—	2	1	no	—
76	C₃₁×#135	—	4	4	no	—
77	C₂×Q₁₆×C₃₁	—	2	1	no	—
78	C₂×SD₁₆×C₃₁	—	2	1	no	—
79	C₂×SD₁₆×C₃₁	—	2	1	no	—
80	C₂×SD₁₆×C₃₁	—	2	1	no	—
81	C₂×D₈×C₃₁	—	2	1	no	—
82	C₂×D₈×C₃₁	—	2	1	no	—
83	C₂²×C₈×C₃₁	—	2	1	no	—
84	C₃₁×C₄⋊Q₈	—	2	1	no	—
85	C₃₁×C₄⋊Q₈	—	2	1	no	—
86	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₃₁×#127	C₂³	1	1	yes	characteristic
90	C₃₁×#126	—	2	2	no	—
91	C₃₁×#126	—	2	2	no	—
92	C₃₁×#126	—	2	2	no	—
93	C₃₁×#125	—	2	2	no	—
94	C₃₁×#125	—	2	2	no	—
95	C₃₁×#125	—	2	1	no	—
96	C₃₁×#125	D₄	1	2	yes	—
97	C₃₁×#124	—	2	2	no	—
98	C₃₁×#124	—	2	1	no	—
99	C₃₁×#124	—	2	1	no	—
100	C₃₁×#123	—	2	2	no	—
101	C₃₁×#123	—	2	2	no	—
102	C₃₁×#123	—	2	2	no	—
103	C₃₁×C₂²⋊Q₈	—	2	2	no	—
104	C₃₁×C₂²⋊Q₈	—	2	1	no	—
105	C₃₁×C₂²⋊Q₈	—	2	1	no	—
106	C₃₁×C₂²⋊Q₈	—	2	1	no	—
107	C₃₁×C₂²⋊Q₈	—	2	1	no	—
108	C₃₁×C₂²⋊Q₈	—	2	2	no	—
109	C₃₁×C₂²⋊Q₈	—	2	1	no	—
110	C₃₁×#121	—	2	2	no	—
111	C₃₁×#121	—	2	1	no	—
112	C₃₁×#121	—	2	1	no	—
113	C₃₁×#121	—	2	1	no	—
114	C₄×Q₈×C₃₁	—	2	2	no	—
115	C₄×Q₈×C₃₁	—	2	2	no	—
116	C₄×Q₈×C₃₁	—	2	1	no	—
117	C₄×Q₈×C₃₁	—	2	1	no	—
118	C₄×Q₈×C₃₁	—	2	1	no	—
119	C₄×Q₈×C₃₁	—	2	1	no	—
120	C₄×Q₈×C₃₁	—	2	1	no	—
121	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
122	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
123	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
124	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₄×D₄×C₃₁	—	2	2	no	—
126	C₄×D₄×C₃₁	—	2	2	no	—
127	C₄×D₄×C₃₁	—	2	1	no	—
128	C₄×D₄×C₃₁	—	2	1	no	—
129	C₄×D₄×C₃₁	—	2	1	no	—
130	C₄×D₄×C₃₁	—	2	1	no	—
131	C₄×D₄×C₃₁	—	2	1	no	—
132	C₄×D₄×C₃₁	—	2	1	no	—
133	C₄×D₄×C₃₁	—	2	1	no	—
134	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
135	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
136	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
137	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
138	C₃₁×#117	—	2	2	no	—
139	C₃₁×#117	—	2	2	no	—
140	C₃₁×#117	—	2	1	no	—
141	C₃₁×#117	—	2	1	no	—
142	C₃₁×#117	—	2	1	no	—
143	C₃₁×#117	D₄	1	1	yes	characteristic
144	C₃₁×#117	D₄	1	1	yes	characteristic
145	C₃₁×#117	D₄	1	2	yes	—
146	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₂×C₄²×C₃₁	—	2	2	no	—
149	C₂×C₄²×C₃₁	—	2	1	no	—
150	C₂×C₄²×C₃₁	—	2	1	no	—
151	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
152	C₃₁×#107	—	2	1	no	—
153	C₃₁×#107	—	2	2	no	—
154	C₃₁×#106	—	2	1	no	—
155	C₃₁×#106	—	2	2	no	—
156	C₃₁×C₄⋊C₈	—	2	1	no	—
157	C₃₁×C₄⋊C₈	—	2	1	no	—
158	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
159	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
160	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
162	C₃₁×#103	—	2	1	no	—
163	C₃₁×#103	—	2	1	no	—
164	C₃₁×#103	—	2	1	no	—
165	C₃₁×#102	—	2	1	no	—
166	C₃₁×#102	—	2	1	no	—
167	C₃₁×#102	—	2	1	no	—
168	C₃₁×#102	—	2	1	no	—
169	C₃₁×#102	—	2	1	no	—
170	C₃₁×#102	—	2	2	no	—
171	C₃₁×#102	—	2	2	no	—
172	C₃₁×C₂²⋊C₈	—	2	1	no	—
173	C₃₁×C₂²⋊C₈	—	2	1	no	—
174	C₃₁×#97	—	2	1	no	—
175	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
176	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
177	#41×C₃₁	—	2	2	no	—
178	#41×C₃₁	—	2	2	no	—
179	#41×C₃₁	—	4	1	no	—
180	#41×C₃₁	—	4	1	no	—
181	#41×C₃₁	—	4	2	no	—
182	#41×C₃₁	—	4	2	no	—
183	#41×C₃₁	—	4	2	no	—
184	#41×C₃₁	—	4	2	no	—
185	C₂×Q₈×C₃₁	—	2	2	no	—
186	C₂×Q₈×C₃₁	—	2	1	no	—
187	C₂×Q₈×C₃₁	—	2	1	no	—
188	C₂×Q₈×C₃₁	—	2	1	no	—
189	C₂×Q₈×C₃₁	—	4	2	no	—
190	C₂×Q₈×C₃₁	—	4	2	no	—
191	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
192	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
193	C₂×D₄×C₃₁	—	2	2	no	—
194	C₂×D₄×C₃₁	—	2	1	no	—
195	C₂×D₄×C₃₁	—	2	1	no	—
196	C₂×D₄×C₃₁	—	2	1	no	—
197	C₂×D₄×C₃₁	—	2	1	no	—
198	C₂×D₄×C₃₁	—	2	1	no	—
199	C₂×D₄×C₃₁	—	4	1	no	—
200	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
201	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
202	C₂²×C₄×C₃₁	—	2	2	no	—
203	C₂²×C₄×C₃₁	—	2	2	no	—
204	C₂²×C₄×C₃₁	—	2	1	no	—
205	C₂²×C₄×C₃₁	—	2	1	no	—
206	C₂²×C₄×C₃₁	—	2	1	no	—
207	C₂²×C₄×C₃₁	—	2	1	no	—
208	C₂²×C₄×C₃₁	—	2	1	no	—
209	C₂²×C₄×C₃₁	—	2	1	no	—
210	C₂²×C₄×C₃₁	—	4	1	no	—
211	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
212	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
213	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
214	Q₁₆×C₃₁	—	4	2	no	—
215	SD₁₆×C₃₁	—	4	2	no	—
216	SD₁₆×C₃₁	—	4	2	no	—
217	SD₁₆×C₃₁	—	4	2	no	—
218	D₈×C₃₁	—	4	2	no	—
219	D₈×C₃₁	—	4	2	no	—
220	C₂×C₈×C₃₁	—	2	1	no	—
221	C₂×C₈×C₃₁	—	2	1	no	—
222	C₂×C₈×C₃₁	—	2	1	no	—
223	C₂×C₈×C₃₁	—	2	1	no	—
224	C₂×C₈×C₃₁	—	2	1	no	—
225	C₂×C₈×C₃₁	—	2	1	no	—
226	C₂×C₈×C₃₁	—	4	2	no	—
227	C₄⋊C₄×C₃₁	—	2	2	no	—
228	C₄⋊C₄×C₃₁	—	2	2	no	—
229	C₄⋊C₄×C₃₁	—	2	2	no	—
230	C₄⋊C₄×C₃₁	—	2	2	no	—
231	C₄⋊C₄×C₃₁	—	2	2	no	—
232	C₄⋊C₄×C₃₁	—	2	2	no	—
233	C₄⋊C₄×C₃₁	—	2	1	no	—
234	C₄⋊C₄×C₃₁	—	2	2	no	—
235	C₄⋊C₄×C₃₁	—	2	2	no	—
236	C₄⋊C₄×C₃₁	—	2	1	no	—
237	C₄⋊C₄×C₃₁	—	2	1	no	—
238	C₄⋊C₄×C₃₁	—	2	1	no	—
239	C₄⋊C₄×C₃₁	—	2	1	no	—
240	C₄⋊C₄×C₃₁	—	2	1	no	—
241	C₄⋊C₄×C₃₁	—	2	1	no	—
242	C₄⋊C₄×C₃₁	—	2	1	no	—
243	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
244	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
245	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
246	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
247	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
248	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
249	C₂²⋊C₄×C₃₁	—	2	2	no	—
250	C₂²⋊C₄×C₃₁	—	2	2	no	—
251	C₂²⋊C₄×C₃₁	—	2	2	no	—
252	C₂²⋊C₄×C₃₁	—	2	2	no	—
253	C₂²⋊C₄×C₃₁	—	2	1	no	—
254	C₂²⋊C₄×C₃₁	—	2	1	no	—
255	C₂²⋊C₄×C₃₁	—	2	2	no	—
256	C₂²⋊C₄×C₃₁	—	2	1	no	—
257	C₂²⋊C₄×C₃₁	—	2	1	no	—
258	C₂²⋊C₄×C₃₁	—	2	1	no	—
259	C₂²⋊C₄×C₃₁	—	2	1	no	—
260	C₂²⋊C₄×C₃₁	—	4	1	no	—
261	C₂²⋊C₄×C₃₁	—	4	1	no	—
262	C₄²×C₃₁	—	2	2	no	—
263	C₄²×C₃₁	—	2	2	no	—
264	C₄²×C₃₁	—	2	1	no	—
265	C₄²×C₃₁	—	2	1	no	—
266	C₄²×C₃₁	—	2	1	no	—
267	C₄²×C₃₁	—	2	2	no	—
268	C₄²×C₃₁	—	2	1	no	—
269	C₄²×C₃₁	—	2	1	no	—
270	C₄²×C₃₁	—	2	1	no	—
271	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
272	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
273	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
274	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
275	#18784	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
276	C₂³×C₃₁	—	2	1	no	—
277	C₂³×C₃₁	—	2	1	no	—
278	C₂³×C₃₁	—	4	1	no	—
279	C₂³×C₃₁	#120	1	1	yes	characteristic
280	Q₈×C₃₁	—	2	1	no	—
281	Q₈×C₃₁	—	2	1	no	—
282	Q₈×C₃₁	—	2	1	no	—
283	Q₈×C₃₁	—	4	4	no	—
284	Q₈×C₃₁	—	4	1	no	—
285	Q₈×C₃₁	—	4	1	no	—
286	Q₈×C₃₁	—	4	1	no	—
287	Q₈×C₃₁	—	4	1	no	—
288	Q₈×C₃₁	—	4	1	no	—
289	Q₈×C₃₁	—	4	1	no	—
290	Q₈×C₃₁	#135	1	2	yes	complemented
291	D₄×C₃₁	—	2	1	no	—
292	D₄×C₃₁	—	2	1	no	—
293	D₄×C₃₁	—	2	1	no	—
294	D₄×C₃₁	—	4	2	no	—
295	D₄×C₃₁	—	4	2	no	—
296	D₄×C₃₁	—	4	2	no	—
297	D₄×C₃₁	—	4	4	no	—
298	D₄×C₃₁	—	4	1	no	—
299	D₄×C₃₁	—	4	1	no	—
300	D₄×C₃₁	—	4	1	no	—
301	D₄×C₃₁	—	4	1	no	—
302	D₄×C₃₁	—	8	2	no	—
303	D₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
304	C₂×C₄×C₃₁	—	2	1	no	—
305	C₂×C₄×C₃₁	—	2	1	no	—
306	C₂×C₄×C₃₁	—	2	1	no	—
307	C₂×C₄×C₃₁	—	2	1	no	—
308	C₂×C₄×C₃₁	—	2	1	no	—
309	C₂×C₄×C₃₁	—	2	1	no	—
310	C₂×C₄×C₃₁	—	2	1	no	—
311	C₂×C₄×C₃₁	—	2	1	no	—
312	C₂×C₄×C₃₁	—	2	2	no	—
313	C₂×C₄×C₃₁	—	2	1	no	—
314	C₂×C₄×C₃₁	—	2	1	no	—
315	C₂×C₄×C₃₁	—	2	2	no	—
316	C₂×C₄×C₃₁	—	4	1	no	—
317	C₂×C₄×C₃₁	—	4	2	no	—
318	C₂×C₄×C₃₁	—	4	2	no	—
319	C₂×C₄×C₃₁	—	4	2	no	—
320	C₂×C₄×C₃₁	—	4	2	no	—
321	C₂×C₄×C₃₁	—	4	4	no	—
322	C₂×C₄×C₃₁	—	4	4	no	—
323	C₂×C₄×C₃₁	—	4	1	no	—
324	C₂×C₄×C₃₁	—	4	1	no	—
325	C₂×C₄×C₃₁	—	4	1	no	—
326	C₂×C₄×C₃₁	—	4	1	no	—
327	C₂×C₄×C₃₁	—	4	1	no	—
328	C₂×C₄×C₃₁	—	8	2	no	—
329	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
330	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
331	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
332	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
333	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
334	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
335	C₈×C₃₁	—	4	1	no	—
336	C₈×C₃₁	—	4	1	no	—
337	C₈×C₃₁	—	4	1	no	—
338	C₈×C₃₁	—	4	1	no	—
339	C₈×C₃₁	—	4	1	no	—
340	C₈×C₃₁	—	4	1	no	—
341	#3482	C₂×C₃₁	1	1	yes	characteristic, complemented
342	#3228	C₂×C₃₁	1	1	yes	characteristic, complemented
343	#3115	C₂×C₃₁	1	1	yes	characteristic, complemented
344	#3035	C₂×C₃₁	1	1	yes	characteristic, complemented
345	#1652	C₂×C₃₁	1	1	yes	characteristic, complemented
346	#1651	C₂×C₃₁	1	1	yes	characteristic, complemented
347	#1648	C₂×C₃₁	1	1	yes	characteristic, complemented
348	#1647	C₂×C₃₁	1	1	yes	characteristic, complemented
349	#1628	C₂×C₃₁	1	1	yes	characteristic, complemented
350	#1627	C₂×C₃₁	1	1	yes	characteristic, complemented
351	#1626	C₂×C₃₁	1	1	yes	characteristic, complemented
352	#1625	C₂×C₃₁	1	1	yes	characteristic, complemented
353	#1552	C₂×C₃₁	1	1	yes	characteristic, complemented
354	#1488	C₂×C₃₁	1	1	yes	characteristic, complemented
355	#1477	C₂×C₃₁	1	1	yes	characteristic, complemented
356	C₂²×C₃₁	—	2	1	no	—
357	C₂²×C₃₁	—	2	1	no	—
358	C₂²×C₃₁	—	2	2	no	—
359	C₂²×C₃₁	—	2	1	no	—
360	C₂²×C₃₁	—	4	1	no	—
361	C₂²×C₃₁	—	4	1	no	—
362	C₂²×C₃₁	—	4	1	no	—
363	C₂²×C₃₁	—	4	1	no	—
364	C₂²×C₃₁	—	4	1	no	—
365	C₂²×C₃₁	—	8	1	no	—
366	C₂²×C₃₁	—	8	1	no	—
367	C₂²×C₃₁	—	8	1	no	—
368	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
369	C₄×C₃₁	—	2	1	no	—
370	C₄×C₃₁	—	2	2	no	—
371	C₄×C₃₁	—	2	1	no	—
372	C₄×C₃₁	—	2	1	no	—
373	C₄×C₃₁	—	2	1	no	—
374	C₄×C₃₁	—	4	2	no	—
375	C₄×C₃₁	—	4	1	no	—
376	C₄×C₃₁	—	4	1	no	—
377	C₄×C₃₁	—	4	1	no	—
378	C₄×C₃₁	—	4	2	no	—
379	C₄×C₃₁	—	4	1	no	—
380	C₄×C₃₁	—	4	1	no	—
381	C₄×C₃₁	—	8	1	no	—
382	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
383	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
384	C₄×C₃₁	#575	1	1	yes	characteristic
385	C₄×C₃₁	#575	1	1	yes	characteristic
386	C₂×#135	—	2	1	no	—
387	#557	C₂²×C₃₁	1	1	yes	characteristic
388	#556	—	2	1	no	—
389	#556	C₂²×C₃₁	1	1	yes	characteristic
390	#552	—	2	2	no	—
391	#552	—	2	1	no	—
392	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
393	#548	—	2	1	no	—
394	#547	—	2	1	no	—
395	#547	C₂²×C₃₁	1	1	yes	characteristic
396	#533	—	2	1	no	—
397	#533	C₂²×C₃₁	1	1	yes	characteristic
398	#532	C₂²×C₃₁	1	1	yes	characteristic
399	#529	—	2	1	no	—
400	#529	—	2	2	no	—
401	C₄×#41	—	2	1	no	—
402	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
403	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
404	C₂×C₄×Q₈	—	2	1	no	—
405	#500	C₂²×C₃₁	1	1	yes	characteristic
406	#498	C₂²×C₃₁	1	1	yes	characteristic
407	#486	C₂²×C₃₁	1	1	yes	characteristic
408	#486	C₂²×C₃₁	1	1	yes	characteristic
409	#482	—	2	1	no	—
410	#482	—	2	2	no	—
411	#478	—	2	2	no	—
412	#475	C₂²×C₃₁	1	1	yes	characteristic
413	#474	C₂²×C₃₁	1	1	yes	characteristic
414	#464	—	2	1	no	—
415	#463	—	2	1	no	—
416	#462	C₂²×C₃₁	1	1	yes	characteristic
417	#461	C₂²×C₃₁	1	1	yes	characteristic
418	#460	C₂²×C₃₁	1	1	yes	characteristic
419	#460	C₂²×C₃₁	1	1	yes	characteristic
420	#459	C₂²×C₃₁	1	1	yes	characteristic
421	#459	C₂²×C₃₁	1	1	yes	characteristic
422	#449	—	2	1	no	—
423	#442	—	2	1	no	—
424	#440	—	2	1	no	—
425	#432	C₂²×C₃₁	1	1	yes	characteristic
426	#431	C₂²×C₃₁	1	1	yes	characteristic
427	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
428	#416	—	2	1	no	—
429	#332	C₂²×C₃₁	1	1	yes	characteristic
430	#332	C₂²×C₃₁	1	1	yes	characteristic
431	#332	C₂²×C₃₁	1	1	yes	characteristic
432	#332	C₂²×C₃₁	1	1	yes	characteristic
433	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
434	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
435	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
436	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
437	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
438	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
439	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
440	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
441	C₂×C₃₁	—	4	1	no	—
442	C₂×C₃₁	—	4	2	no	—
443	C₂×C₃₁	—	8	1	no	—
444	C₂×C₃₁	—	16	1	no	—
445	C₂×C₃₁	#3002	1	1	yes	characteristic
446	C₂×C₃₁	#3006	1	1	yes	characteristic
447	C₂×C₃₁	#3020	1	1	yes	characteristic
448	C₂×#41	—	2	1	no	—
449	C₂×#41	—	2	1	no	—
450	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
451	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
452	C₂²×Q₈	—	2	1	no	—
453	#135	—	4	4	no	—
454	C₂×Q₁₆	—	2	1	no	—
455	C₂×SD₁₆	—	2	1	no	—
456	C₂×SD₁₆	—	2	1	no	—
457	C₂×SD₁₆	—	2	1	no	—
458	C₂×D₈	—	2	1	no	—
459	C₂×D₈	—	2	1	no	—
460	C₂²×C₈	—	2	1	no	—
461	C₄⋊Q₈	—	2	1	no	—
462	C₄⋊Q₈	—	2	1	no	—
463	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
464	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
465	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
466	#127	C₂³×C₃₁	1	1	yes	characteristic
467	#126	—	2	2	no	—
468	#126	—	2	2	no	—
469	#126	—	2	2	no	—
470	#125	—	2	2	no	—
471	#125	—	2	2	no	—
472	#125	—	2	1	no	—
473	#125	D₄×C₃₁	1	2	yes	—
474	#124	—	2	2	no	—
475	#124	—	2	1	no	—
476	#124	—	2	1	no	—
477	#123	—	2	2	no	—
478	#123	—	2	2	no	—
479	#123	—	2	2	no	—
480	C₂²⋊Q₈	—	2	2	no	—
481	C₂²⋊Q₈	—	2	1	no	—
482	C₂²⋊Q₈	—	2	1	no	—
483	C₂²⋊Q₈	—	2	1	no	—
484	C₂²⋊Q₈	—	2	1	no	—
485	C₂²⋊Q₈	—	2	2	no	—
486	C₂²⋊Q₈	—	2	1	no	—
487	#121	—	2	2	no	—
488	#121	—	2	1	no	—
489	#121	—	2	1	no	—
490	#121	—	2	1	no	—
491	C₄×Q₈	—	2	2	no	—
492	C₄×Q₈	—	2	2	no	—
493	C₄×Q₈	—	2	1	no	—
494	C₄×Q₈	—	2	1	no	—
495	C₄×Q₈	—	2	1	no	—
496	C₄×Q₈	—	2	1	no	—
497	C₄×Q₈	—	2	1	no	—
498	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
499	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
500	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
501	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
502	C₄×D₄	—	2	2	no	—
503	C₄×D₄	—	2	2	no	—
504	C₄×D₄	—	2	1	no	—
505	C₄×D₄	—	2	1	no	—
506	C₄×D₄	—	2	1	no	—
507	C₄×D₄	—	2	1	no	—
508	C₄×D₄	—	2	1	no	—
509	C₄×D₄	—	2	1	no	—
510	C₄×D₄	—	2	1	no	—
511	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
512	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
513	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
514	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
515	#117	—	2	2	no	—
516	#117	—	2	2	no	—
517	#117	—	2	1	no	—
518	#117	—	2	1	no	—
519	#117	—	2	1	no	—
520	#117	D₄×C₃₁	1	1	yes	characteristic
521	#117	D₄×C₃₁	1	1	yes	characteristic
522	#117	D₄×C₃₁	1	2	yes	—
523	C₂×C₄⋊C₄	—	2	2	no	—
524	C₂×C₄⋊C₄	—	2	1	no	—
525	C₂×C₄²	—	2	2	no	—
526	C₂×C₄²	—	2	1	no	—
527	C₂×C₄²	—	2	1	no	—
528	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
529	#107	—	2	1	no	—
530	#107	—	2	2	no	—
531	#106	—	2	1	no	—
532	#106	—	2	2	no	—
533	C₄⋊C₈	—	2	1	no	—
534	C₄⋊C₈	—	2	1	no	—
535	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
536	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
537	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
538	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
539	#103	—	2	1	no	—
540	#103	—	2	1	no	—
541	#103	—	2	1	no	—
542	#102	—	2	1	no	—
543	#102	—	2	1	no	—
544	#102	—	2	1	no	—
545	#102	—	2	1	no	—
546	#102	—	2	1	no	—
547	#102	—	2	2	no	—
548	#102	—	2	2	no	—
549	C₂²⋊C₈	—	2	1	no	—
550	C₂²⋊C₈	—	2	1	no	—
551	#97	—	2	1	no	—
552	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
553	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
554	C₃₁	#18784	1	1	yes	characteristic, direct product factor, Sylow
555	#41	—	2	2	no	—
556	#41	—	2	2	no	—
557	#41	—	4	1	no	—
558	#41	—	4	1	no	—
559	#41	—	4	2	no	—
560	#41	—	4	2	no	—
561	#41	—	4	2	no	—
562	#41	—	4	2	no	—
563	C₂×Q₈	—	2	2	no	—
564	C₂×Q₈	—	2	1	no	—
565	C₂×Q₈	—	2	1	no	—
566	C₂×Q₈	—	2	1	no	—
567	C₂×Q₈	—	4	2	no	—
568	C₂×Q₈	—	4	2	no	—
569	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
570	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
571	C₂×D₄	—	2	2	no	—
572	C₂×D₄	—	2	1	no	—
573	C₂×D₄	—	2	1	no	—
574	C₂×D₄	—	2	1	no	—
575	C₂×D₄	—	2	1	no	—
576	C₂×D₄	—	2	1	no	—
577	C₂×D₄	—	4	1	no	—
578	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
579	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
580	C₂²×C₄	—	2	2	no	—
581	C₂²×C₄	—	2	2	no	—
582	C₂²×C₄	—	2	1	no	—
583	C₂²×C₄	—	2	1	no	—
584	C₂²×C₄	—	2	1	no	—
585	C₂²×C₄	—	2	1	no	—
586	C₂²×C₄	—	2	1	no	—
587	C₂²×C₄	—	2	1	no	—
588	C₂²×C₄	—	4	1	no	—
589	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
590	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
591	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
592	Q₁₆	—	4	2	no	—
593	SD₁₆	—	4	2	no	—
594	SD₁₆	—	4	2	no	—
595	SD₁₆	—	4	2	no	—
596	D₈	—	4	2	no	—
597	D₈	—	4	2	no	—
598	C₂×C₈	—	2	1	no	—
599	C₂×C₈	—	2	1	no	—
600	C₂×C₈	—	2	1	no	—
601	C₂×C₈	—	2	1	no	—
602	C₂×C₈	—	2	1	no	—
603	C₂×C₈	—	2	1	no	—
604	C₂×C₈	—	4	2	no	—
605	C₄⋊C₄	—	2	2	no	—
606	C₄⋊C₄	—	2	2	no	—
607	C₄⋊C₄	—	2	2	no	—
608	C₄⋊C₄	—	2	2	no	—
609	C₄⋊C₄	—	2	2	no	—
610	C₄⋊C₄	—	2	2	no	—
611	C₄⋊C₄	—	2	1	no	—
612	C₄⋊C₄	—	2	2	no	—
613	C₄⋊C₄	—	2	2	no	—
614	C₄⋊C₄	—	2	1	no	—
615	C₄⋊C₄	—	2	1	no	—
616	C₄⋊C₄	—	2	1	no	—
617	C₄⋊C₄	—	2	1	no	—
618	C₄⋊C₄	—	2	1	no	—
619	C₄⋊C₄	—	2	1	no	—
620	C₄⋊C₄	—	2	1	no	—
621	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
622	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
623	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
624	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
625	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
626	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
627	C₂²⋊C₄	—	2	2	no	—
628	C₂²⋊C₄	—	2	2	no	—
629	C₂²⋊C₄	—	2	2	no	—
630	C₂²⋊C₄	—	2	2	no	—
631	C₂²⋊C₄	—	2	1	no	—
632	C₂²⋊C₄	—	2	1	no	—
633	C₂²⋊C₄	—	2	2	no	—
634	C₂²⋊C₄	—	2	1	no	—
635	C₂²⋊C₄	—	2	1	no	—
636	C₂²⋊C₄	—	2	1	no	—
637	C₂²⋊C₄	—	2	1	no	—
638	C₂²⋊C₄	—	4	1	no	—
639	C₂²⋊C₄	—	4	1	no	—
640	C₄²	—	2	2	no	—
641	C₄²	—	2	2	no	—
642	C₄²	—	2	1	no	—
643	C₄²	—	2	1	no	—
644	C₄²	—	2	1	no	—
645	C₄²	—	2	2	no	—
646	C₄²	—	2	1	no	—
647	C₄²	—	2	1	no	—
648	C₄²	—	2	1	no	—
649	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
650	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
651	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
652	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
653	C₂³	—	2	1	no	—
654	C₂³	—	2	1	no	—
655	C₂³	—	4	1	no	—
656	C₂³	C₃₁×#120	1	1	yes	characteristic
657	Q₈	—	2	1	no	—
658	Q₈	—	2	1	no	—
659	Q₈	—	2	1	no	—
660	Q₈	—	4	4	no	—
661	Q₈	—	4	1	no	—
662	Q₈	—	4	1	no	—
663	Q₈	—	4	1	no	—
664	Q₈	—	4	1	no	—
665	Q₈	—	4	1	no	—
666	Q₈	—	4	1	no	—
667	Q₈	C₃₁×#135	1	2	yes	complemented
668	D₄	—	2	1	no	—
669	D₄	—	2	1	no	—
670	D₄	—	2	1	no	—
671	D₄	—	4	2	no	—
672	D₄	—	4	2	no	—
673	D₄	—	4	2	no	—
674	D₄	—	4	4	no	—
675	D₄	—	4	1	no	—
676	D₄	—	4	1	no	—
677	D₄	—	4	1	no	—
678	D₄	—	4	1	no	—
679	D₄	—	8	2	no	—
680	D₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
681	C₂×C₄	—	2	1	no	—
682	C₂×C₄	—	2	1	no	—
683	C₂×C₄	—	2	1	no	—
684	C₂×C₄	—	2	1	no	—
685	C₂×C₄	—	2	1	no	—
686	C₂×C₄	—	2	1	no	—
687	C₂×C₄	—	2	1	no	—
688	C₂×C₄	—	2	1	no	—
689	C₂×C₄	—	2	2	no	—
690	C₂×C₄	—	2	1	no	—
691	C₂×C₄	—	2	1	no	—
692	C₂×C₄	—	2	2	no	—
693	C₂×C₄	—	4	1	no	—
694	C₂×C₄	—	4	2	no	—
695	C₂×C₄	—	4	2	no	—
696	C₂×C₄	—	4	2	no	—
697	C₂×C₄	—	4	2	no	—
698	C₂×C₄	—	4	4	no	—
699	C₂×C₄	—	4	4	no	—
700	C₂×C₄	—	4	1	no	—
701	C₂×C₄	—	4	1	no	—
702	C₂×C₄	—	4	1	no	—
703	C₂×C₄	—	4	1	no	—
704	C₂×C₄	—	4	1	no	—
705	C₂×C₄	—	8	2	no	—
706	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
707	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
708	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
709	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
710	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
711	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
712	C₈	—	4	1	no	—
713	C₈	—	4	1	no	—
714	C₈	—	4	1	no	—
715	C₈	—	4	1	no	—
716	C₈	—	4	1	no	—
717	C₈	—	4	1	no	—
718	C₂²	—	2	1	no	—
719	C₂²	—	2	1	no	—
720	C₂²	—	2	2	no	—
721	C₂²	—	2	1	no	—
722	C₂²	—	4	1	no	—
723	C₂²	—	4	1	no	—
724	C₂²	—	4	1	no	—
725	C₂²	—	4	1	no	—
726	C₂²	—	4	1	no	—
727	C₂²	—	8	1	no	—
728	C₂²	—	8	1	no	—
729	C₂²	—	8	1	no	—
730	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
731	C₄	—	2	1	no	—
732	C₄	—	2	2	no	—
733	C₄	—	2	1	no	—
734	C₄	—	2	1	no	—
735	C₄	—	2	1	no	—
736	C₄	—	4	2	no	—
737	C₄	—	4	1	no	—
738	C₄	—	4	1	no	—
739	C₄	—	4	1	no	—
740	C₄	—	4	2	no	—
741	C₄	—	4	1	no	—
742	C₄	—	4	1	no	—
743	C₄	—	8	1	no	—
744	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
745	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
746	C₄	C₃₁×#575	1	1	yes	characteristic
747	C₄	C₃₁×#575	1	1	yes	characteristic
748	C₂	—	4	1	no	—
749	C₂	—	4	2	no	—
750	C₂	—	8	1	no	—
751	C₂	—	16	1	no	—
752	C₂	C₃₁×#3002	1	1	yes	characteristic
753	C₂	C₃₁×#3006	1	1	yes	characteristic
754	C₂	C₃₁×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
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×#3482	C₂	semidirect
2. C₃₁×#3228	C₂	semidirect
3. C₃₁×#3115	C₂	semidirect
4. C₃₁×#3035	C₂	semidirect
5. C₃₁×#1652	C₂	semidirect
6. C₃₁×#1651	C₂	semidirect
7. C₃₁×#1648	C₂	semidirect
8. C₃₁×#1647	C₂	semidirect
9. C₃₁×#1628	C₂	semidirect
10. C₃₁×#1627	C₂	semidirect
11. C₃₁×#1626	C₂	semidirect
12. C₃₁×#1625	C₂	semidirect
13. C₃₁×#1552	C₂	semidirect
14. C₃₁×#1488	C₂	semidirect
15. C₃₁×#1477	C₂	semidirect
275. #18784	C₃₁	direct (N × H)
290. Q₈×C₃₁	#135	semidirect
341. #3482	C₂×C₃₁	semidirect
342. #3228	C₂×C₃₁	semidirect
343. #3115	C₂×C₃₁	semidirect
344. #3035	C₂×C₃₁	semidirect
345. #1652	C₂×C₃₁	semidirect
346. #1651	C₂×C₃₁	semidirect
347. #1648	C₂×C₃₁	semidirect
348. #1647	C₂×C₃₁	semidirect
349. #1628	C₂×C₃₁	semidirect
350. #1627	C₂×C₃₁	semidirect
351. #1626	C₂×C₃₁	semidirect
352. #1625	C₂×C₃₁	semidirect
353. #1552	C₂×C₃₁	semidirect
354. #1488	C₂×C₃₁	semidirect
355. #1477	C₂×C₃₁	semidirect
554. C₃₁	#18784	direct (N × H)
667. Q₈	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×#3482	2
C₃₁×#3228	2
C₃₁×#3115	2
C₃₁×#3035	2
C₃₁×#1652	2
C₃₁×#1651	2
C₃₁×#1648	2
C₃₁×#1647	2
C₃₁×#1628	2
C₃₁×#1627	2
C₃₁×#1626	2
C₃₁×#1625	2
C₃₁×#1552	2
C₃₁×#1488	2
C₃₁×#1477	2
#18784	31

Subgroups — C31×#18784 #21475543

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18784 #21475543