Subgroups — C₃₁×#61890 #21518649

Order 7936.

2042 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3507	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3399	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3397	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3368	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3366	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3217	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3114	C₂	1	4	yes	maximal, complemented
11	C₃₁×#3101	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3100	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2983	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₃₁	—	2	1	no	—
15	C₃₁×#559	C₂²	1	2	yes	complemented
16	C₃₁×#557	C₂²	1	2	yes	complemented
17	C₃₁×#556	—	2	2	no	—
18	C₃₁×#556	C₂²	1	2	yes	complemented
19	C₃₁×#554	C₂²	1	2	yes	complemented
20	C₃₁×#552	—	2	2	no	—
21	C₃₁×#550	—	2	2	no	—
22	C₃₁×#550	—	2	2	no	—
23	C₃₁×#550	C₂²	1	4	yes	—
24	C₃₁×#550	C₂²	1	4	yes	—
25	C₃₁×#539	—	2	2	no	—
26	C₃₁×#536	—	2	1	no	—
27	C₃₁×#533	C₂²	1	1	yes	characteristic
28	C₃₁×#532	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#127	C₂²	1	2	yes	—
31	C₃₁×#529	C₂²	1	4	yes	—
32	C₂×C₃₁×#121	—	2	1	no	—
33	C₂×C₃₁×#121	—	2	2	no	—
34	C₃₁×#520	—	2	1	no	—
35	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
38	C₃₁×#496	C₂²	1	8	yes	complemented
39	C₃₁×#493	C₂²	1	2	yes	complemented
40	C₃₁×#492	C₂²	1	2	yes	complemented
41	C₃₁×#490	C₂²	1	8	yes	complemented
42	C₃₁×#486	C₂²	1	2	yes	complemented
43	C₃₁×#486	C₂²	1	2	yes	complemented
44	C₃₁×#483	C₂²	1	2	yes	—
45	C₃₁×#480	C₂²	1	2	yes	—
46	C₃₁×#478	C₂²	1	8	yes	complemented
47	C₃₁×#476	C₂²	1	2	yes	complemented
48	C₃₁×#474	C₂²	1	2	yes	complemented
49	C₃₁×#463	C₂²	1	8	yes	complemented
50	C₃₁×#461	C₂²	1	2	yes	complemented
51	C₃₁×#460	C₂²	1	8	yes	complemented
52	C₃₁×#460	C₂²	1	4	yes	complemented
53	C₃₁×#459	C₂²	1	2	yes	complemented
54	C₃₁×#459	C₂²	1	4	yes	complemented
55	C₃₁×#459	C₂²	1	8	yes	complemented
56	C₃₁×#450	C₂²	1	2	yes	—
57	C₃₁×#450	C₂²	1	4	yes	—
58	C₃₁×#447	C₂²	1	2	yes	—
59	C₃₁×#447	C₂²	1	4	yes	—
60	C₃₁×#446	C₂²	1	4	yes	complemented
61	C₃₁×#446	C₂²	1	4	yes	complemented
62	C₃₁×#442	C₂²	1	8	yes	complemented
63	C₃₁×#440	C₂²	1	8	yes	complemented
64	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
65	C₄×D₈×C₃₁	C₂²	1	4	yes	complemented
66	C₃₁×#432	C₂²	1	4	yes	—
67	C₃₁×#431	C₂²	1	1	yes	characteristic
68	C₃₁×#431	C₂²	1	1	yes	characteristic
69	C₂×#41×C₃₁	—	2	4	no	—
70	C₂×#41×C₃₁	—	2	4	no	—
71	C₂×#41×C₃₁	—	2	1	no	—
72	C₂×#41×C₃₁	—	4	8	no	—
73	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
74	C₂²×D₄×C₃₁	—	2	2	no	—
75	C₂²×D₄×C₃₁	—	2	1	no	—
76	C₂²×D₄×C₃₁	—	2	2	no	—
77	C₂²×D₄×C₃₁	—	2	1	no	—
78	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
80	C₂³×C₄×C₃₁	—	2	2	no	—
81	C₂³×C₄×C₃₁	—	2	1	no	—
82	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
83	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
84	C₂×D₈×C₃₁	C₂³	1	4	yes	—
85	C₂×D₈×C₃₁	C₂³	1	8	yes	—
86	C₃₁×C₄⋊Q₈	—	2	2	no	—
87	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
90	C₃₁×#127	—	2	4	no	—
91	C₃₁×#127	—	2	1	no	—
92	C₃₁×#127	—	2	1	no	—
93	C₃₁×#127	C₂³	1	2	yes	—
94	C₃₁×#127	C₂³	1	1	yes	characteristic
95	C₃₁×#127	C₂³	1	8	yes	—
96	C₃₁×#127	C₂³	1	2	yes	—
97	C₃₁×#126	—	2	4	no	—
98	C₃₁×#126	—	2	4	no	—
99	C₃₁×#125	—	2	4	no	—
100	C₃₁×#125	—	2	1	no	—
101	C₃₁×#125	C₂³	1	4	yes	—
102	C₃₁×#124	—	2	2	no	—
103	C₃₁×#124	—	2	1	no	—
104	C₃₁×#124	C₂³	1	4	yes	—
105	C₃₁×#123	—	2	4	no	—
106	C₃₁×#123	—	2	4	no	—
107	C₃₁×C₂²⋊Q₈	—	2	2	no	—
108	C₃₁×C₂²⋊Q₈	—	2	2	no	—
109	C₃₁×C₂²⋊Q₈	—	2	1	no	—
110	C₃₁×C₂²⋊Q₈	—	2	4	no	—
111	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
112	C₃₁×#121	—	2	1	no	—
113	C₃₁×#121	—	2	4	no	—
114	C₃₁×#121	—	2	4	no	—
115	C₃₁×#121	—	2	8	no	—
116	C₃₁×#121	—	2	8	no	—
117	C₃₁×#121	C₂³	1	2	yes	—
118	C₄×Q₈×C₃₁	—	2	2	no	—
119	C₄×Q₈×C₃₁	—	2	1	no	—
120	C₄×Q₈×C₃₁	D₄	1	2	yes	—
121	C₄×Q₈×C₃₁	D₄	1	2	yes	—
122	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
123	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
124	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
125	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
126	C₄×D₄×C₃₁	—	2	4	no	—
127	C₄×D₄×C₃₁	—	2	2	no	—
128	C₄×D₄×C₃₁	—	2	4	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	C₂³	1	2	yes	—
133	C₄×D₄×C₃₁	C₂³	1	2	yes	—
134	C₄×D₄×C₃₁	C₂³	1	2	yes	—
135	C₄×D₄×C₃₁	C₂³	1	8	yes	—
136	C₄×D₄×C₃₁	C₂³	1	4	yes	—
137	C₃₁×#117	—	2	1	no	—
138	C₃₁×#117	—	2	2	no	—
139	C₃₁×#117	—	2	4	no	—
140	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
141	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
142	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
143	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
144	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
145	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
146	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
147	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
148	C₃₁×#107	C₂³	1	4	yes	—
149	C₃₁×#106	C₂³	1	4	yes	—
150	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
151	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
152	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
153	C₃₁×#103	C₂³	1	4	yes	—
154	C₃₁×#102	C₂³	1	4	yes	—
155	C₃₁×#102	C₂³	1	4	yes	—
156	C₃₁×#102	C₂³	1	4	yes	—
157	C₃₁×#102	C₂³	1	8	yes	—
158	C₃₁×#102	C₂³	1	8	yes	—
159	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
160	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
161	C₃₁×#97	C₂³	1	8	yes	—
162	C₄×C₈×C₃₁	C₂³	1	2	yes	—
163	C₄×C₈×C₃₁	C₂³	1	2	yes	—
164	C₂⁴×C₃₁	—	2	2	no	—
165	C₂⁴×C₃₁	—	2	1	no	—
166	#41×C₃₁	—	4	16	no	—
167	#41×C₃₁	—	4	16	no	—
168	#41×C₃₁	—	4	4	no	—
169	C₂×Q₈×C₃₁	—	2	4	no	—
170	C₂×Q₈×C₃₁	—	2	1	no	—
171	C₂×Q₈×C₃₁	—	2	1	no	—
172	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
173	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
174	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
175	C₂×D₄×C₃₁	—	2	2	no	—
176	C₂×D₄×C₃₁	—	2	1	no	—
177	C₂×D₄×C₃₁	—	2	2	no	—
178	C₂×D₄×C₃₁	—	2	1	no	—
179	C₂×D₄×C₃₁	—	2	8	no	—
180	C₂×D₄×C₃₁	—	2	2	no	—
181	C₂×D₄×C₃₁	—	2	2	no	—
182	C₂×D₄×C₃₁	—	2	1	no	—
183	C₂×D₄×C₃₁	—	2	8	no	—
184	C₂×D₄×C₃₁	—	2	2	no	—
185	C₂×D₄×C₃₁	—	2	2	no	—
186	C₂×D₄×C₃₁	—	4	2	no	—
187	C₂×D₄×C₃₁	—	4	4	no	—
188	C₂×D₄×C₃₁	—	4	8	no	—
189	C₂×D₄×C₃₁	—	4	4	no	—
190	C₂×D₄×C₃₁	—	8	4	no	—
191	C₂×D₄×C₃₁	—	8	2	no	—
192	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
193	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
194	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
195	C₂²×C₄×C₃₁	—	2	1	no	—
196	C₂²×C₄×C₃₁	—	2	2	no	—
197	C₂²×C₄×C₃₁	—	2	8	no	—
198	C₂²×C₄×C₃₁	—	2	2	no	—
199	C₂²×C₄×C₃₁	—	2	2	no	—
200	C₂²×C₄×C₃₁	—	2	4	no	—
201	C₂²×C₄×C₃₁	—	4	8	no	—
202	C₂²×C₄×C₃₁	—	4	4	no	—
203	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
204	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
205	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
206	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
208	SD₁₆×C₃₁	—	4	4	no	—
209	SD₁₆×C₃₁	—	4	8	no	—
210	D₈×C₃₁	—	4	8	no	—
211	D₈×C₃₁	—	4	4	no	—
212	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
213	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
214	C₄⋊C₄×C₃₁	—	2	1	no	—
215	C₄⋊C₄×C₃₁	—	2	4	no	—
216	C₄⋊C₄×C₃₁	—	2	1	no	—
217	C₄⋊C₄×C₃₁	—	2	1	no	—
218	C₄⋊C₄×C₃₁	—	2	2	no	—
219	C₄⋊C₄×C₃₁	—	2	4	no	—
220	C₄⋊C₄×C₃₁	—	2	1	no	—
221	C₄⋊C₄×C₃₁	—	2	1	no	—
222	C₄⋊C₄×C₃₁	—	2	2	no	—
223	C₄⋊C₄×C₃₁	—	2	4	no	—
224	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
225	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
226	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
227	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
228	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
229	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
230	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
231	C₂²⋊C₄×C₃₁	—	2	2	no	—
232	C₂²⋊C₄×C₃₁	—	2	8	no	—
233	C₂²⋊C₄×C₃₁	—	2	2	no	—
234	C₂²⋊C₄×C₃₁	—	2	2	no	—
235	C₂²⋊C₄×C₃₁	—	2	2	no	—
236	C₂²⋊C₄×C₃₁	—	2	4	no	—
237	C₂²⋊C₄×C₃₁	—	2	4	no	—
238	C₂²⋊C₄×C₃₁	—	2	2	no	—
239	C₂²⋊C₄×C₃₁	—	2	4	no	—
240	C₄²×C₃₁	—	2	4	no	—
241	C₄²×C₃₁	—	2	2	no	—
242	C₄²×C₃₁	C₂×D₄	1	4	yes	—
243	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
244	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
245	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
246	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
247	C₄²×C₃₁	C₂⁴	1	4	yes	—
248	#61890	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
249	C₂³×C₃₁	—	2	2	no	—
250	C₂³×C₃₁	—	2	2	no	—
251	C₂³×C₃₁	—	2	4	no	—
252	C₂³×C₃₁	—	4	2	no	—
253	C₂³×C₃₁	—	4	4	no	—
254	C₂³×C₃₁	—	4	2	no	—
255	C₂³×C₃₁	—	4	4	no	—
256	C₂³×C₃₁	—	4	4	no	—
257	C₂³×C₃₁	—	4	2	no	—
258	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
259	Q₈×C₃₁	—	2	4	no	—
260	Q₈×C₃₁	—	2	4	no	—
261	Q₈×C₃₁	—	2	4	no	—
262	Q₈×C₃₁	—	4	2	no	—
263	Q₈×C₃₁	—	8	1	no	—
264	D₄×C₃₁	—	2	4	no	—
265	D₄×C₃₁	—	2	2	no	—
266	D₄×C₃₁	—	2	8	no	—
267	D₄×C₃₁	—	2	2	no	—
268	D₄×C₃₁	—	4	2	no	—
269	D₄×C₃₁	—	4	4	no	—
270	D₄×C₃₁	—	4	16	no	—
271	D₄×C₃₁	—	4	4	no	—
272	D₄×C₃₁	—	4	4	no	—
273	D₄×C₃₁	—	4	4	no	—
274	D₄×C₃₁	—	4	2	no	—
275	D₄×C₃₁	—	8	1	no	—
276	D₄×C₃₁	—	8	8	no	—
277	D₄×C₃₁	—	8	2	no	—
278	D₄×C₃₁	—	8	2	no	—
279	C₂×C₄×C₃₁	—	2	4	no	—
280	C₂×C₄×C₃₁	—	2	2	no	—
281	C₂×C₄×C₃₁	—	2	2	no	—
282	C₂×C₄×C₃₁	—	2	2	no	—
283	C₂×C₄×C₃₁	—	2	2	no	—
284	C₂×C₄×C₃₁	—	2	2	no	—
285	C₂×C₄×C₃₁	—	4	2	no	—
286	C₂×C₄×C₃₁	—	4	1	no	—
287	C₂×C₄×C₃₁	—	4	16	no	—
288	C₂×C₄×C₃₁	—	4	2	no	—
289	C₂×C₄×C₃₁	—	4	4	no	—
290	C₂×C₄×C₃₁	—	4	4	no	—
291	C₂×C₄×C₃₁	—	4	4	no	—
292	C₂×C₄×C₃₁	—	8	4	no	—
293	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
294	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
295	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
296	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
297	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
298	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
299	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
300	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
301	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
302	C₈×C₃₁	—	2	4	no	—
303	C₈×C₃₁	—	2	4	no	—
304	#3507	C₂×C₃₁	1	1	yes	characteristic, complemented
305	C₂×#550	C₂×C₃₁	1	1	yes	characteristic, complemented
306	#3399	C₂×C₃₁	1	4	yes	complemented
307	#3397	C₂×C₃₁	1	4	yes	complemented
308	#3368	C₂×C₃₁	1	4	yes	complemented
309	#3366	C₂×C₃₁	1	4	yes	complemented
310	#3217	C₂×C₃₁	1	4	yes	complemented
311	#3208	C₂×C₃₁	1	1	yes	characteristic, complemented
312	#3205	C₂×C₃₁	1	1	yes	characteristic, complemented
313	#3114	C₂×C₃₁	1	4	yes	complemented
314	#3101	C₂×C₃₁	1	1	yes	characteristic, complemented
315	#3100	C₂×C₃₁	1	1	yes	characteristic, complemented
316	#2983	C₂×C₃₁	1	1	yes	characteristic, complemented
317	C₂²×C₃₁	—	2	1	no	—
318	C₂²×C₃₁	—	2	1	no	—
319	C₂²×C₃₁	—	4	2	no	—
320	C₂²×C₃₁	—	4	2	no	—
321	C₂²×C₃₁	—	4	4	no	—
322	C₂²×C₃₁	—	4	2	no	—
323	C₂²×C₃₁	—	4	4	no	—
324	C₂²×C₃₁	—	4	2	no	—
325	C₂²×C₃₁	—	4	4	no	—
326	C₂²×C₃₁	—	4	2	no	—
327	C₂²×C₃₁	—	4	2	no	—
328	C₂²×C₃₁	—	8	8	no	—
329	C₂²×C₃₁	—	8	4	no	—
330	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
331	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
332	C₄×C₃₁	—	2	4	no	—
333	C₄×C₃₁	—	2	1	no	—
334	C₄×C₃₁	—	2	1	no	—
335	C₄×C₃₁	—	2	1	no	—
336	C₄×C₃₁	—	2	4	no	—
337	C₄×C₃₁	—	4	2	no	—
338	C₄×C₃₁	—	4	2	no	—
339	C₄×C₃₁	—	4	2	no	—
340	C₄×C₃₁	—	4	2	no	—
341	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
342	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
343	C₄×C₃₁	#585	1	2	yes	—
344	C₂²×#41	—	2	1	no	—
345	#559	C₂²×C₃₁	1	2	yes	complemented
346	#557	C₂²×C₃₁	1	2	yes	complemented
347	#556	—	2	2	no	—
348	#556	C₂²×C₃₁	1	2	yes	complemented
349	#554	C₂²×C₃₁	1	2	yes	complemented
350	#552	—	2	2	no	—
351	#550	—	2	2	no	—
352	#550	—	2	2	no	—
353	#550	C₂²×C₃₁	1	4	yes	—
354	#550	C₂²×C₃₁	1	4	yes	—
355	#539	—	2	2	no	—
356	#536	—	2	1	no	—
357	#533	C₂²×C₃₁	1	1	yes	characteristic
358	#532	C₂²×C₃₁	1	1	yes	characteristic
359	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
360	C₂×#127	C₂²×C₃₁	1	2	yes	—
361	#529	C₂²×C₃₁	1	4	yes	—
362	C₂×#121	—	2	1	no	—
363	C₂×#121	—	2	2	no	—
364	#520	—	2	1	no	—
365	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
366	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
367	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
368	#496	C₂²×C₃₁	1	8	yes	complemented
369	#493	C₂²×C₃₁	1	2	yes	complemented
370	#492	C₂²×C₃₁	1	2	yes	complemented
371	#490	C₂²×C₃₁	1	8	yes	complemented
372	#486	C₂²×C₃₁	1	2	yes	complemented
373	#486	C₂²×C₃₁	1	2	yes	complemented
374	#483	C₂²×C₃₁	1	2	yes	—
375	#480	C₂²×C₃₁	1	2	yes	—
376	#478	C₂²×C₃₁	1	8	yes	complemented
377	#476	C₂²×C₃₁	1	2	yes	complemented
378	#474	C₂²×C₃₁	1	2	yes	complemented
379	#463	C₂²×C₃₁	1	8	yes	complemented
380	#461	C₂²×C₃₁	1	2	yes	complemented
381	#460	C₂²×C₃₁	1	8	yes	complemented
382	#460	C₂²×C₃₁	1	4	yes	complemented
383	#459	C₂²×C₃₁	1	2	yes	complemented
384	#459	C₂²×C₃₁	1	4	yes	complemented
385	#459	C₂²×C₃₁	1	8	yes	complemented
386	#450	C₂²×C₃₁	1	2	yes	—
387	#450	C₂²×C₃₁	1	4	yes	—
388	#447	C₂²×C₃₁	1	2	yes	—
389	#447	C₂²×C₃₁	1	4	yes	—
390	#446	C₂²×C₃₁	1	4	yes	complemented
391	#446	C₂²×C₃₁	1	4	yes	complemented
392	#442	C₂²×C₃₁	1	8	yes	complemented
393	#440	C₂²×C₃₁	1	8	yes	complemented
394	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
395	C₄×D₈	C₂²×C₃₁	1	4	yes	complemented
396	#432	C₂²×C₃₁	1	4	yes	—
397	#431	C₂²×C₃₁	1	1	yes	characteristic
398	#431	C₂²×C₃₁	1	1	yes	characteristic
399	C₂×C₃₁	—	2	2	no	—
400	C₂×C₃₁	—	8	2	no	—
401	C₂×C₃₁	—	8	2	no	—
402	C₂×C₃₁	—	8	4	no	—
403	C₂×C₃₁	#3487	1	1	yes	characteristic
404	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
405	C₂×C₃₁	#3585	1	1	yes	characteristic
406	C₂×#41	—	2	4	no	—
407	C₂×#41	—	2	4	no	—
408	C₂×#41	—	2	1	no	—
409	C₂×#41	—	4	8	no	—
410	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
411	C₂²×D₄	—	2	2	no	—
412	C₂²×D₄	—	2	1	no	—
413	C₂²×D₄	—	2	2	no	—
414	C₂²×D₄	—	2	1	no	—
415	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
416	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
417	C₂³×C₄	—	2	2	no	—
418	C₂³×C₄	—	2	1	no	—
419	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
420	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
421	C₂×D₈	C₂³×C₃₁	1	4	yes	—
422	C₂×D₈	C₂³×C₃₁	1	8	yes	—
423	C₄⋊Q₈	—	2	2	no	—
424	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
425	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
426	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
427	#127	—	2	4	no	—
428	#127	—	2	1	no	—
429	#127	—	2	1	no	—
430	#127	C₂³×C₃₁	1	2	yes	—
431	#127	C₂³×C₃₁	1	8	yes	—
432	#127	C₂³×C₃₁	1	2	yes	—
433	#127	C₂³×C₃₁	1	1	yes	characteristic
434	#126	—	2	4	no	—
435	#126	—	2	4	no	—
436	#125	—	2	4	no	—
437	#125	—	2	1	no	—
438	#125	C₂³×C₃₁	1	4	yes	—
439	#124	—	2	2	no	—
440	#124	—	2	1	no	—
441	#124	C₂³×C₃₁	1	4	yes	—
442	#123	—	2	4	no	—
443	#123	—	2	4	no	—
444	C₂²⋊Q₈	—	2	2	no	—
445	C₂²⋊Q₈	—	2	2	no	—
446	C₂²⋊Q₈	—	2	1	no	—
447	C₂²⋊Q₈	—	2	4	no	—
448	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
449	#121	—	2	1	no	—
450	#121	—	2	4	no	—
451	#121	—	2	4	no	—
452	#121	—	2	8	no	—
453	#121	—	2	8	no	—
454	#121	C₂³×C₃₁	1	2	yes	—
455	C₄×Q₈	—	2	2	no	—
456	C₄×Q₈	—	2	1	no	—
457	C₄×Q₈	D₄×C₃₁	1	2	yes	—
458	C₄×Q₈	D₄×C₃₁	1	2	yes	—
459	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
460	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
461	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
462	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
463	C₄×D₄	—	2	4	no	—
464	C₄×D₄	—	2	2	no	—
465	C₄×D₄	—	2	4	no	—
466	C₄×D₄	—	2	1	no	—
467	C₄×D₄	—	2	1	no	—
468	C₄×D₄	—	2	1	no	—
469	C₄×D₄	C₂³×C₃₁	1	2	yes	—
470	C₄×D₄	C₂³×C₃₁	1	2	yes	—
471	C₄×D₄	C₂³×C₃₁	1	8	yes	—
472	C₄×D₄	C₂³×C₃₁	1	4	yes	—
473	C₄×D₄	C₂³×C₃₁	1	2	yes	—
474	#117	—	2	1	no	—
475	#117	—	2	2	no	—
476	#117	—	2	4	no	—
477	C₂×C₄⋊C₄	—	2	2	no	—
478	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
479	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
480	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
481	C₂×C₂²⋊C₄	—	2	2	no	—
482	C₂×C₂²⋊C₄	—	2	1	no	—
483	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
484	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
485	#107	C₂³×C₃₁	1	4	yes	—
486	#106	C₂³×C₃₁	1	4	yes	—
487	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
488	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
489	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
490	#103	C₂³×C₃₁	1	4	yes	—
491	#102	C₂³×C₃₁	1	4	yes	—
492	#102	C₂³×C₃₁	1	4	yes	—
493	#102	C₂³×C₃₁	1	4	yes	—
494	#102	C₂³×C₃₁	1	8	yes	—
495	#102	C₂³×C₃₁	1	8	yes	—
496	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
497	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
498	#97	C₂³×C₃₁	1	8	yes	—
499	C₄×C₈	C₂³×C₃₁	1	2	yes	—
500	C₄×C₈	C₂³×C₃₁	1	2	yes	—
501	C₃₁	#61890	1	1	yes	characteristic, direct product factor, Sylow
502	C₂⁴	—	2	2	no	—
503	C₂⁴	—	2	1	no	—
504	#41	—	4	16	no	—
505	#41	—	4	16	no	—
506	#41	—	4	4	no	—
507	C₂×Q₈	—	2	4	no	—
508	C₂×Q₈	—	2	1	no	—
509	C₂×Q₈	—	2	1	no	—
510	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
511	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
512	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
513	C₂×D₄	—	2	2	no	—
514	C₂×D₄	—	2	1	no	—
515	C₂×D₄	—	2	2	no	—
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	2	8	no	—
518	C₂×D₄	—	2	2	no	—
519	C₂×D₄	—	2	2	no	—
520	C₂×D₄	—	2	1	no	—
521	C₂×D₄	—	2	8	no	—
522	C₂×D₄	—	2	2	no	—
523	C₂×D₄	—	2	2	no	—
524	C₂×D₄	—	4	2	no	—
525	C₂×D₄	—	4	4	no	—
526	C₂×D₄	—	4	8	no	—
527	C₂×D₄	—	4	4	no	—
528	C₂×D₄	—	8	4	no	—
529	C₂×D₄	—	8	2	no	—
530	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
531	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
532	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
533	C₂²×C₄	—	2	1	no	—
534	C₂²×C₄	—	2	2	no	—
535	C₂²×C₄	—	2	8	no	—
536	C₂²×C₄	—	2	2	no	—
537	C₂²×C₄	—	2	2	no	—
538	C₂²×C₄	—	2	4	no	—
539	C₂²×C₄	—	4	8	no	—
540	C₂²×C₄	—	4	4	no	—
541	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
542	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
543	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
544	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
545	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
546	SD₁₆	—	4	4	no	—
547	SD₁₆	—	4	8	no	—
548	D₈	—	4	8	no	—
549	D₈	—	4	4	no	—
550	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
551	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
552	C₄⋊C₄	—	2	1	no	—
553	C₄⋊C₄	—	2	4	no	—
554	C₄⋊C₄	—	2	1	no	—
555	C₄⋊C₄	—	2	1	no	—
556	C₄⋊C₄	—	2	2	no	—
557	C₄⋊C₄	—	2	4	no	—
558	C₄⋊C₄	—	2	1	no	—
559	C₄⋊C₄	—	2	1	no	—
560	C₄⋊C₄	—	2	2	no	—
561	C₄⋊C₄	—	2	4	no	—
562	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
563	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
564	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
565	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
566	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
567	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
568	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
569	C₂²⋊C₄	—	2	2	no	—
570	C₂²⋊C₄	—	2	8	no	—
571	C₂²⋊C₄	—	2	2	no	—
572	C₂²⋊C₄	—	2	2	no	—
573	C₂²⋊C₄	—	2	2	no	—
574	C₂²⋊C₄	—	2	4	no	—
575	C₂²⋊C₄	—	2	4	no	—
576	C₂²⋊C₄	—	2	2	no	—
577	C₂²⋊C₄	—	2	4	no	—
578	C₄²	—	2	4	no	—
579	C₄²	—	2	2	no	—
580	C₄²	C₂×D₄×C₃₁	1	4	yes	—
581	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
582	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
583	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
584	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
585	C₄²	C₂⁴×C₃₁	1	4	yes	—
586	C₂³	—	2	2	no	—
587	C₂³	—	2	2	no	—
588	C₂³	—	2	4	no	—
589	C₂³	—	4	2	no	—
590	C₂³	—	4	4	no	—
591	C₂³	—	4	2	no	—
592	C₂³	—	4	4	no	—
593	C₂³	—	4	4	no	—
594	C₂³	—	4	2	no	—
595	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
596	Q₈	—	2	4	no	—
597	Q₈	—	2	4	no	—
598	Q₈	—	2	4	no	—
599	Q₈	—	4	2	no	—
600	Q₈	—	8	1	no	—
601	D₄	—	2	4	no	—
602	D₄	—	2	2	no	—
603	D₄	—	2	8	no	—
604	D₄	—	2	2	no	—
605	D₄	—	4	2	no	—
606	D₄	—	4	4	no	—
607	D₄	—	4	16	no	—
608	D₄	—	4	4	no	—
609	D₄	—	4	4	no	—
610	D₄	—	4	4	no	—
611	D₄	—	4	2	no	—
612	D₄	—	8	1	no	—
613	D₄	—	8	8	no	—
614	D₄	—	8	2	no	—
615	D₄	—	8	2	no	—
616	C₂×C₄	—	2	4	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	4	2	no	—
623	C₂×C₄	—	4	1	no	—
624	C₂×C₄	—	4	16	no	—
625	C₂×C₄	—	4	2	no	—
626	C₂×C₄	—	4	4	no	—
627	C₂×C₄	—	4	4	no	—
628	C₂×C₄	—	4	4	no	—
629	C₂×C₄	—	8	4	no	—
630	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
631	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
632	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
633	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
634	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
635	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
636	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
637	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
638	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
639	C₈	—	2	4	no	—
640	C₈	—	2	4	no	—
641	C₂²	—	2	1	no	—
642	C₂²	—	2	1	no	—
643	C₂²	—	4	2	no	—
644	C₂²	—	4	4	no	—
645	C₂²	—	4	2	no	—
646	C₂²	—	4	4	no	—
647	C₂²	—	4	2	no	—
648	C₂²	—	4	2	no	—
649	C₂²	—	4	2	no	—
650	C₂²	—	4	2	no	—
651	C₂²	—	4	4	no	—
652	C₂²	—	8	8	no	—
653	C₂²	—	8	4	no	—
654	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
655	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
656	C₄	—	2	1	no	—
657	C₄	—	2	4	no	—
658	C₄	—	2	4	no	—
659	C₄	—	2	1	no	—
660	C₄	—	2	1	no	—
661	C₄	—	4	2	no	—
662	C₄	—	4	2	no	—
663	C₄	—	4	2	no	—
664	C₄	—	4	2	no	—
665	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
666	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
667	C₄	C₃₁×#585	1	2	yes	—
668	C₂	—	2	2	no	—
669	C₂	—	8	2	no	—
670	C₂	—	8	4	no	—
671	C₂	—	8	2	no	—
672	C₂	C₃₁×#3487	1	1	yes	characteristic
673	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
674	C₂	C₃₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3507	C₂	semidirect
2. C₂×C₃₁×#550	C₂	semidirect
3. C₃₁×#3399	C₂	semidirect
4. C₃₁×#3397	C₂	semidirect
5. C₃₁×#3368	C₂	semidirect
6. C₃₁×#3366	C₂	semidirect
7. C₃₁×#3217	C₂	semidirect
8. C₃₁×#3208	C₂	semidirect
9. C₃₁×#3205	C₂	semidirect
10. C₃₁×#3114	C₂	semidirect
11. C₃₁×#3101	C₂	semidirect
12. C₃₁×#3100	C₂	semidirect
13. C₃₁×#2983	C₂	semidirect
15. C₃₁×#559	C₂²	semidirect
16. C₃₁×#557	C₂²	semidirect
18. C₃₁×#556	C₂²	semidirect
19. C₃₁×#554	C₂²	semidirect
38. C₃₁×#496	C₂²	semidirect
39. C₃₁×#493	C₂²	semidirect
40. C₃₁×#492	C₂²	semidirect
41. C₃₁×#490	C₂²	semidirect
42. C₃₁×#486	C₂²	semidirect
43. C₃₁×#486	C₂²	semidirect
46. C₃₁×#478	C₂²	semidirect
47. C₃₁×#476	C₂²	semidirect
48. C₃₁×#474	C₂²	semidirect
49. C₃₁×#463	C₂²	semidirect
50. C₃₁×#461	C₂²	semidirect
51. C₃₁×#460	C₂²	semidirect
52. C₃₁×#460	C₂²	semidirect
53. C₃₁×#459	C₂²	semidirect
54. C₃₁×#459	C₂²	semidirect
55. C₃₁×#459	C₂²	semidirect
60. C₃₁×#446	C₂²	semidirect
61. C₃₁×#446	C₂²	semidirect
62. C₃₁×#442	C₂²	semidirect
63. C₃₁×#440	C₂²	semidirect
64. C₄×SD₁₆×C₃₁	C₂²	semidirect
65. C₄×D₈×C₃₁	C₂²	semidirect
248. #61890	C₃₁	direct (N × H)
304. #3507	C₂×C₃₁	semidirect
305. C₂×#550	C₂×C₃₁	semidirect
306. #3399	C₂×C₃₁	semidirect
307. #3397	C₂×C₃₁	semidirect
308. #3368	C₂×C₃₁	semidirect
309. #3366	C₂×C₃₁	semidirect
310. #3217	C₂×C₃₁	semidirect
311. #3208	C₂×C₃₁	semidirect
312. #3205	C₂×C₃₁	semidirect
313. #3114	C₂×C₃₁	semidirect
314. #3101	C₂×C₃₁	semidirect
315. #3100	C₂×C₃₁	semidirect
316. #2983	C₂×C₃₁	semidirect
345. #559	C₂²×C₃₁	semidirect
346. #557	C₂²×C₃₁	semidirect
348. #556	C₂²×C₃₁	semidirect
349. #554	C₂²×C₃₁	semidirect
368. #496	C₂²×C₃₁	semidirect
369. #493	C₂²×C₃₁	semidirect
370. #492	C₂²×C₃₁	semidirect
371. #490	C₂²×C₃₁	semidirect
372. #486	C₂²×C₃₁	semidirect
373. #486	C₂²×C₃₁	semidirect
376. #478	C₂²×C₃₁	semidirect
377. #476	C₂²×C₃₁	semidirect
378. #474	C₂²×C₃₁	semidirect
379. #463	C₂²×C₃₁	semidirect
380. #461	C₂²×C₃₁	semidirect
381. #460	C₂²×C₃₁	semidirect
382. #460	C₂²×C₃₁	semidirect
383. #459	C₂²×C₃₁	semidirect
384. #459	C₂²×C₃₁	semidirect
385. #459	C₂²×C₃₁	semidirect
390. #446	C₂²×C₃₁	semidirect
391. #446	C₂²×C₃₁	semidirect
392. #442	C₂²×C₃₁	semidirect
393. #440	C₂²×C₃₁	semidirect
394. C₄×SD₁₆	C₂²×C₃₁	semidirect
395. C₄×D₈	C₂²×C₃₁	semidirect
501. C₃₁	#61890	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×#3507	2
C₂×C₃₁×#550	2
C₃₁×#3399	2
C₃₁×#3397	2
C₃₁×#3368	2
C₃₁×#3366	2
C₃₁×#3217	2
C₃₁×#3208	2
C₃₁×#3205	2
C₃₁×#3114	2
C₃₁×#3101	2
C₃₁×#3100	2
C₃₁×#2983	2
#61890	31

Subgroups — C31×#61890 #21518649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61890 #21518649