Subgroups — C₃₁×#61806 #21518565

Order 7936.

1668 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3550	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3507	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3356	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3349	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3342	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3328	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3312	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3299	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3237	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3231	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3202	C₂	1	4	yes	maximal, complemented
12	C₃₁×#3113	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#3104	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#3097	C₂	1	4	yes	maximal, complemented
15	C₃₁×#2981	C₂	1	1	yes	maximal, characteristic
16	C₃₁×#563	—	2	2	no	—
17	C₃₁×#559	—	2	1	no	—
18	C₃₁×#559	—	2	1	no	—
19	Q₈²×C₃₁	—	2	1	no	—
20	C₃₁×#557	—	2	1	no	—
21	C₃₁×#556	—	2	2	no	—
22	C₃₁×#556	C₂²	1	2	yes	—
23	C₃₁×#556	C₂²	1	2	yes	—
24	C₃₁×#556	C₂²	1	2	yes	—
25	C₃₁×#554	C₂²	1	2	yes	—
26	C₃₁×#552	C₂²	1	4	yes	—
27	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
28	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#542	—	2	2	no	—
30	C₃₁×#542	—	2	2	no	—
31	C₃₁×#539	—	2	2	no	—
32	C₃₁×#536	C₂²	1	4	yes	—
33	C₃₁×#536	C₂²	1	2	yes	—
34	C₃₁×#533	C₂²	1	1	yes	characteristic
35	C₃₁×#533	C₂²	1	1	yes	characteristic
36	C₃₁×#533	C₂²	1	1	yes	characteristic
37	C₃₁×#532	C₂²	1	1	yes	characteristic
38	C₃₁×#529	—	2	2	no	—
39	C₃₁×#520	C₂²	1	2	yes	—
40	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#501	C₂²	1	2	yes	—
42	C₃₁×#500	C₂²	1	1	yes	characteristic
43	C₃₁×#498	C₂²	1	1	yes	characteristic
44	C₃₁×#491	C₂²	1	2	yes	—
45	C₃₁×#487	C₂²	1	1	yes	characteristic
46	C₃₁×#487	C₂²	1	1	yes	characteristic
47	C₃₁×#485	C₂²	1	4	yes	—
48	C₃₁×#485	C₂²	1	4	yes	—
49	C₃₁×#485	C₂²	1	4	yes	—
50	C₃₁×#485	C₂²	1	4	yes	—
51	C₃₁×#485	C₂²	1	4	yes	—
52	C₃₁×#485	C₂²	1	4	yes	—
53	C₃₁×#484	C₂²	1	4	yes	—
54	C₃₁×#484	C₂²	1	2	yes	—
55	C₃₁×#483	C₂²	1	2	yes	—
56	C₃₁×#483	C₂²	1	4	yes	—
57	C₃₁×#479	C₂²	1	4	yes	—
58	C₃₁×#477	C₂²	1	2	yes	—
59	C₃₁×#477	C₂²	1	2	yes	—
60	C₃₁×#475	C₂²	1	2	yes	—
61	C₃₁×#475	C₂²	1	2	yes	—
62	C₃₁×#470	C₂²	1	4	yes	—
63	C₃₁×#470	C₂²	1	4	yes	—
64	C₃₁×#467	C₂²	1	4	yes	—
65	C₃₁×#467	C₂²	1	4	yes	—
66	C₃₁×#462	C₂²	1	2	yes	—
67	C₃₁×#462	C₂²	1	2	yes	—
68	C₃₁×#451	C₂²	1	2	yes	—
69	C₃₁×#451	C₂²	1	2	yes	—
70	C₃₁×#451	C₂²	1	4	yes	—
71	C₃₁×#451	C₂²	1	4	yes	—
72	C₃₁×#441	C₂²	1	4	yes	—
73	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
74	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
75	C₃₁×#436	C₂²	1	2	yes	—
76	C₃₁×#435	C₂²	1	4	yes	—
77	C₈×D₄×C₃₁	C₂²	1	2	yes	—
78	C₃₁×#432	C₂²	1	1	yes	characteristic
79	C₃₁×#431	C₂²	1	1	yes	characteristic
80	C₃₁×#428	C₂²	1	4	yes	—
81	C₃₁×#427	C₂²	1	4	yes	—
82	C₃₁×#419	C₂²	1	4	yes	—
83	C₂×C₃₁×#103	C₂²	1	4	yes	—
84	C₃₁×#408	C₂²	1	4	yes	—
85	C₂×#41×C₃₁	C₂³	1	2	yes	—
86	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
87	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
89	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
90	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
91	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
92	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
93	C₃₁×C₄⋊Q₈	—	2	2	no	—
94	C₃₁×C₄⋊Q₈	—	2	4	no	—
95	C₃₁×C₄⋊Q₈	—	2	2	no	—
96	C₃₁×C₄⋊Q₈	—	2	2	no	—
97	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
98	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
99	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
100	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
101	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
102	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
103	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
104	C₃₁×#127	D₄	1	1	yes	characteristic
105	C₃₁×#126	—	2	4	no	—
106	C₃₁×#126	—	2	4	no	—
107	C₃₁×#126	—	2	4	no	—
108	C₃₁×#126	—	2	4	no	—
109	C₃₁×#125	—	2	2	no	—
110	C₃₁×#125	—	2	4	no	—
111	C₃₁×#125	—	2	1	no	—
112	C₃₁×#125	—	2	1	no	—
113	C₃₁×#125	C₂³	1	2	yes	—
114	C₃₁×#124	—	2	2	no	—
115	C₃₁×#124	—	2	4	no	—
116	C₃₁×#124	—	2	2	no	—
117	C₃₁×#124	—	2	2	no	—
118	C₃₁×#124	—	2	2	no	—
119	C₃₁×#124	D₄	1	2	yes	—
120	C₃₁×#123	—	2	4	no	—
121	C₃₁×#123	—	2	4	no	—
122	C₃₁×#123	—	2	4	no	—
123	C₃₁×#123	—	2	4	no	—
124	C₃₁×C₂²⋊Q₈	—	2	1	no	—
125	C₃₁×C₂²⋊Q₈	—	2	1	no	—
126	C₃₁×C₂²⋊Q₈	—	2	1	no	—
127	C₃₁×C₂²⋊Q₈	—	2	1	no	—
128	C₃₁×C₂²⋊Q₈	—	2	4	no	—
129	C₃₁×C₂²⋊Q₈	—	2	4	no	—
130	C₃₁×C₂²⋊Q₈	—	2	4	no	—
131	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
132	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
133	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
134	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
135	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
136	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
137	C₃₁×#121	D₄	1	2	yes	—
138	C₃₁×#121	D₄	1	2	yes	—
139	C₄×Q₈×C₃₁	—	2	2	no	—
140	C₄×Q₈×C₃₁	—	2	2	no	—
141	C₄×Q₈×C₃₁	—	2	2	no	—
142	C₄×Q₈×C₃₁	—	2	2	no	—
143	C₄×Q₈×C₃₁	—	2	1	no	—
144	C₄×Q₈×C₃₁	—	2	1	no	—
145	C₄×Q₈×C₃₁	—	2	1	no	—
146	C₄×Q₈×C₃₁	—	2	1	no	—
147	C₄×Q₈×C₃₁	—	2	1	no	—
148	C₄×Q₈×C₃₁	—	2	1	no	—
149	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
150	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
151	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
152	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
153	C₄×D₄×C₃₁	—	2	4	no	—
154	C₄×D₄×C₃₁	—	2	2	no	—
155	C₄×D₄×C₃₁	—	2	2	no	—
156	C₄×D₄×C₃₁	—	2	2	no	—
157	C₄×D₄×C₃₁	—	2	2	no	—
158	C₄×D₄×C₃₁	C₂³	1	2	yes	—
159	C₄×D₄×C₃₁	C₂³	1	2	yes	—
160	C₃₁×#117	—	2	1	no	—
161	C₃₁×#117	—	2	1	no	—
162	C₃₁×#117	—	2	4	no	—
163	C₃₁×#117	C₂³	1	4	yes	—
164	C₃₁×#117	C₂³	1	4	yes	—
165	C₃₁×#117	C₂³	1	2	yes	—
166	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
168	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
169	C₂×C₄²×C₃₁	—	2	1	no	—
170	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
171	C₃₁×#107	C₂³	1	2	yes	—
172	C₃₁×#107	C₂³	1	4	yes	—
173	C₃₁×#107	C₂³	1	2	yes	—
174	C₃₁×#107	C₂³	1	4	yes	—
175	C₃₁×#106	C₂³	1	4	yes	—
176	C₃₁×#106	C₂³	1	4	yes	—
177	C₃₁×#106	C₂³	1	4	yes	—
178	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
179	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
180	C₃₁×#103	C₂³	1	2	yes	—
181	C₃₁×#103	C₂³	1	4	yes	—
182	C₃₁×#103	C₂³	1	4	yes	—
183	C₃₁×#103	C₂³	1	4	yes	—
184	C₃₁×#103	C₂³	1	2	yes	—
185	C₃₁×#103	C₂³	1	4	yes	—
186	C₃₁×#103	C₂³	1	4	yes	—
187	C₃₁×#103	C₂³	1	2	yes	—
188	C₃₁×#103	C₂³	1	2	yes	—
189	C₃₁×#103	C₂³	1	4	yes	—
190	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
191	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
192	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
193	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
194	C₃₁×#97	C₂³	1	2	yes	—
195	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
196	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
197	#41×C₃₁	—	4	4	no	—
198	C₂×Q₈×C₃₁	—	2	1	no	—
199	C₂×Q₈×C₃₁	—	2	1	no	—
200	C₂×Q₈×C₃₁	—	2	1	no	—
201	C₂×Q₈×C₃₁	—	2	1	no	—
202	C₂×Q₈×C₃₁	—	2	1	no	—
203	C₂×Q₈×C₃₁	—	2	4	no	—
204	C₂×Q₈×C₃₁	—	4	2	no	—
205	C₂×Q₈×C₃₁	—	4	4	no	—
206	C₂×Q₈×C₃₁	—	4	4	no	—
207	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
208	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
209	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
210	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
211	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
212	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
213	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
214	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
215	C₂²×C₄×C₃₁	—	2	1	no	—
216	C₂²×C₄×C₃₁	—	2	1	no	—
217	C₂²×C₄×C₃₁	—	2	4	no	—
218	C₂²×C₄×C₃₁	—	2	4	no	—
219	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
220	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
221	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
222	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
223	Q₁₆×C₃₁	—	4	2	no	—
224	Q₁₆×C₃₁	—	4	4	no	—
225	Q₁₆×C₃₁	—	4	2	no	—
226	M₁₆×C₃₁	—	4	4	no	—
227	C₂×C₈×C₃₁	—	4	4	no	—
228	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
229	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
230	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
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	4	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	4	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	—	2	1	no	—
244	C₄⋊C₄×C₃₁	—	2	4	no	—
245	C₄⋊C₄×C₃₁	—	2	4	no	—
246	C₄⋊C₄×C₃₁	—	2	2	no	—
247	C₄⋊C₄×C₃₁	—	2	1	no	—
248	C₄⋊C₄×C₃₁	—	2	2	no	—
249	C₄⋊C₄×C₃₁	—	2	2	no	—
250	C₄⋊C₄×C₃₁	—	2	4	no	—
251	C₄⋊C₄×C₃₁	—	2	2	no	—
252	C₄⋊C₄×C₃₁	—	2	2	no	—
253	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
254	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
255	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
256	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
257	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
258	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
259	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
260	C₂²⋊C₄×C₃₁	—	2	2	no	—
261	C₂²⋊C₄×C₃₁	—	2	2	no	—
262	C₂²⋊C₄×C₃₁	—	2	4	no	—
263	C₂²⋊C₄×C₃₁	—	2	2	no	—
264	C₂²⋊C₄×C₃₁	—	2	4	no	—
265	C₂²⋊C₄×C₃₁	—	2	4	no	—
266	C₂²⋊C₄×C₃₁	—	2	4	no	—
267	C₂²⋊C₄×C₃₁	—	2	4	no	—
268	C₂²⋊C₄×C₃₁	—	2	4	no	—
269	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
270	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
271	C₄²×C₃₁	—	2	2	no	—
272	C₄²×C₃₁	—	2	2	no	—
273	C₄²×C₃₁	—	2	4	no	—
274	C₄²×C₃₁	—	2	2	no	—
275	C₄²×C₃₁	—	2	2	no	—
276	C₄²×C₃₁	—	2	2	no	—
277	C₄²×C₃₁	—	2	4	no	—
278	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
279	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
280	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
281	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
282	#61806	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
283	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
284	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
285	Q₈×C₃₁	—	2	2	no	—
286	Q₈×C₃₁	—	2	2	no	—
287	Q₈×C₃₁	—	2	2	no	—
288	Q₈×C₃₁	—	2	4	no	—
289	Q₈×C₃₁	—	2	2	no	—
290	Q₈×C₃₁	—	4	2	no	—
291	Q₈×C₃₁	—	4	2	no	—
292	Q₈×C₃₁	—	4	2	no	—
293	Q₈×C₃₁	—	4	2	no	—
294	Q₈×C₃₁	—	4	2	no	—
295	Q₈×C₃₁	—	8	1	no	—
296	Q₈×C₃₁	—	8	4	no	—
297	D₄×C₃₁	—	2	4	no	—
298	D₄×C₃₁	—	4	2	no	—
299	D₄×C₃₁	—	4	2	no	—
300	D₄×C₃₁	—	4	4	no	—
301	C₂×C₄×C₃₁	—	2	2	no	—
302	C₂×C₄×C₃₁	—	2	4	no	—
303	C₂×C₄×C₃₁	—	2	4	no	—
304	C₂×C₄×C₃₁	—	2	2	no	—
305	C₂×C₄×C₃₁	—	2	2	no	—
306	C₂×C₄×C₃₁	—	2	2	no	—
307	C₂×C₄×C₃₁	—	2	2	no	—
308	C₂×C₄×C₃₁	—	2	2	no	—
309	C₂×C₄×C₃₁	—	2	2	no	—
310	C₂×C₄×C₃₁	—	2	2	no	—
311	C₂×C₄×C₃₁	—	2	2	no	—
312	C₂×C₄×C₃₁	—	2	2	no	—
313	C₂×C₄×C₃₁	—	4	1	no	—
314	C₂×C₄×C₃₁	—	4	2	no	—
315	C₂×C₄×C₃₁	—	8	1	no	—
316	C₂×C₄×C₃₁	—	8	4	no	—
317	C₂×C₄×C₃₁	—	8	4	no	—
318	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
319	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
320	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
321	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
322	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
323	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
324	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
325	C₈×C₃₁	—	2	2	no	—
326	C₈×C₃₁	—	2	4	no	—
327	C₈×C₃₁	—	2	2	no	—
328	#3550	C₂×C₃₁	1	1	yes	characteristic
329	#3507	C₂×C₃₁	1	1	yes	characteristic
330	#3356	C₂×C₃₁	1	2	yes	complemented
331	#3349	C₂×C₃₁	1	4	yes	complemented
332	#3342	C₂×C₃₁	1	2	yes	complemented
333	#3328	C₂×C₃₁	1	2	yes	complemented
334	#3312	C₂×C₃₁	1	4	yes	complemented
335	#3299	C₂×C₃₁	1	2	yes	complemented
336	#3237	C₂×C₃₁	1	1	yes	characteristic, complemented
337	#3231	C₂×C₃₁	1	1	yes	characteristic, complemented
338	#3202	C₂×C₃₁	1	4	yes	complemented
339	#3113	C₂×C₃₁	1	1	yes	characteristic, complemented
340	#3104	C₂×C₃₁	1	1	yes	characteristic, complemented
341	#3097	C₂×C₃₁	1	4	yes	complemented
342	#2981	C₂×C₃₁	1	1	yes	characteristic
343	C₂²×C₃₁	—	2	1	no	—
344	C₂²×C₃₁	—	2	4	no	—
345	C₂²×C₃₁	—	2	1	no	—
346	C₂²×C₃₁	—	2	4	no	—
347	C₂²×C₃₁	—	2	1	no	—
348	C₂²×C₃₁	—	2	4	no	—
349	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
350	C₄×C₃₁	—	2	1	no	—
351	C₄×C₃₁	—	2	1	no	—
352	C₄×C₃₁	—	2	1	no	—
353	C₄×C₃₁	—	2	4	no	—
354	C₄×C₃₁	—	2	1	no	—
355	C₄×C₃₁	—	4	2	no	—
356	C₄×C₃₁	—	4	2	no	—
357	C₄×C₃₁	—	4	2	no	—
358	C₄×C₃₁	—	4	2	no	—
359	C₄×C₃₁	—	4	2	no	—
360	C₄×C₃₁	—	4	2	no	—
361	C₄×C₃₁	—	4	2	no	—
362	C₄×C₃₁	—	4	2	no	—
363	C₄×C₃₁	#578	1	2	yes	—
364	C₄×C₃₁	#585	1	2	yes	—
365	#563	—	2	2	no	—
366	#559	—	2	1	no	—
367	#559	—	2	1	no	—
368	Q₈²	—	2	1	no	—
369	#557	—	2	1	no	—
370	#556	—	2	2	no	—
371	#556	C₂²×C₃₁	1	2	yes	—
372	#556	C₂²×C₃₁	1	2	yes	—
373	#556	C₂²×C₃₁	1	2	yes	—
374	#554	C₂²×C₃₁	1	2	yes	—
375	#552	C₂²×C₃₁	1	4	yes	—
376	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
377	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
378	#542	—	2	2	no	—
379	#542	—	2	2	no	—
380	#539	—	2	2	no	—
381	#536	C₂²×C₃₁	1	4	yes	—
382	#536	C₂²×C₃₁	1	2	yes	—
383	#533	C₂²×C₃₁	1	1	yes	characteristic
384	#533	C₂²×C₃₁	1	1	yes	characteristic
385	#533	C₂²×C₃₁	1	1	yes	characteristic
386	#532	C₂²×C₃₁	1	1	yes	characteristic
387	#529	—	2	2	no	—
388	#520	C₂²×C₃₁	1	2	yes	—
389	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
390	#501	C₂²×C₃₁	1	2	yes	—
391	#500	C₂²×C₃₁	1	1	yes	characteristic
392	#498	C₂²×C₃₁	1	1	yes	characteristic
393	#491	C₂²×C₃₁	1	2	yes	—
394	#487	C₂²×C₃₁	1	1	yes	characteristic
395	#487	C₂²×C₃₁	1	1	yes	characteristic
396	#485	C₂²×C₃₁	1	4	yes	—
397	#485	C₂²×C₃₁	1	4	yes	—
398	#485	C₂²×C₃₁	1	4	yes	—
399	#485	C₂²×C₃₁	1	4	yes	—
400	#485	C₂²×C₃₁	1	4	yes	—
401	#485	C₂²×C₃₁	1	4	yes	—
402	#484	C₂²×C₃₁	1	4	yes	—
403	#484	C₂²×C₃₁	1	2	yes	—
404	#483	C₂²×C₃₁	1	2	yes	—
405	#483	C₂²×C₃₁	1	4	yes	—
406	#479	C₂²×C₃₁	1	4	yes	—
407	#477	C₂²×C₃₁	1	2	yes	—
408	#477	C₂²×C₃₁	1	2	yes	—
409	#475	C₂²×C₃₁	1	2	yes	—
410	#475	C₂²×C₃₁	1	2	yes	—
411	#470	C₂²×C₃₁	1	4	yes	—
412	#470	C₂²×C₃₁	1	4	yes	—
413	#467	C₂²×C₃₁	1	4	yes	—
414	#467	C₂²×C₃₁	1	4	yes	—
415	#462	C₂²×C₃₁	1	2	yes	—
416	#462	C₂²×C₃₁	1	2	yes	—
417	#451	C₂²×C₃₁	1	4	yes	—
418	#451	C₂²×C₃₁	1	2	yes	—
419	#451	C₂²×C₃₁	1	2	yes	—
420	#451	C₂²×C₃₁	1	4	yes	—
421	#441	C₂²×C₃₁	1	4	yes	—
422	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
423	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
424	#436	C₂²×C₃₁	1	2	yes	—
425	#435	C₂²×C₃₁	1	4	yes	—
426	C₈×D₄	C₂²×C₃₁	1	2	yes	—
427	#432	C₂²×C₃₁	1	1	yes	characteristic
428	#431	C₂²×C₃₁	1	1	yes	characteristic
429	#428	C₂²×C₃₁	1	4	yes	—
430	#427	C₂²×C₃₁	1	4	yes	—
431	#419	C₂²×C₃₁	1	4	yes	—
432	C₂×#103	C₂²×C₃₁	1	4	yes	—
433	#408	C₂²×C₃₁	1	4	yes	—
434	C₂×C₃₁	—	4	1	no	—
435	C₂×C₃₁	—	4	4	no	—
436	C₂×C₃₁	#3485	1	1	yes	characteristic
437	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
438	C₂×C₃₁	#3586	1	1	yes	characteristic
439	C₂×#41	C₂³×C₃₁	1	2	yes	—
440	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
441	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
442	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
443	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
444	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
445	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
446	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
447	C₄⋊Q₈	—	2	2	no	—
448	C₄⋊Q₈	—	2	4	no	—
449	C₄⋊Q₈	—	2	2	no	—
450	C₄⋊Q₈	—	2	2	no	—
451	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
452	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
453	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
454	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
455	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
456	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
457	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
458	#127	D₄×C₃₁	1	1	yes	characteristic
459	#126	—	2	4	no	—
460	#126	—	2	4	no	—
461	#126	—	2	4	no	—
462	#126	—	2	4	no	—
463	#125	—	2	2	no	—
464	#125	—	2	4	no	—
465	#125	—	2	1	no	—
466	#125	—	2	1	no	—
467	#125	C₂³×C₃₁	1	2	yes	—
468	#124	—	2	2	no	—
469	#124	—	2	4	no	—
470	#124	—	2	2	no	—
471	#124	—	2	2	no	—
472	#124	—	2	2	no	—
473	#124	D₄×C₃₁	1	2	yes	—
474	#123	—	2	4	no	—
475	#123	—	2	4	no	—
476	#123	—	2	4	no	—
477	#123	—	2	4	no	—
478	C₂²⋊Q₈	—	2	1	no	—
479	C₂²⋊Q₈	—	2	1	no	—
480	C₂²⋊Q₈	—	2	1	no	—
481	C₂²⋊Q₈	—	2	1	no	—
482	C₂²⋊Q₈	—	2	4	no	—
483	C₂²⋊Q₈	—	2	4	no	—
484	C₂²⋊Q₈	—	2	4	no	—
485	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
486	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
487	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
488	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
489	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
490	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
491	#121	D₄×C₃₁	1	2	yes	—
492	#121	D₄×C₃₁	1	2	yes	—
493	C₄×Q₈	—	2	2	no	—
494	C₄×Q₈	—	2	2	no	—
495	C₄×Q₈	—	2	2	no	—
496	C₄×Q₈	—	2	2	no	—
497	C₄×Q₈	—	2	1	no	—
498	C₄×Q₈	—	2	1	no	—
499	C₄×Q₈	—	2	1	no	—
500	C₄×Q₈	—	2	1	no	—
501	C₄×Q₈	—	2	1	no	—
502	C₄×Q₈	—	2	1	no	—
503	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
504	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
505	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
506	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
507	C₄×D₄	—	2	4	no	—
508	C₄×D₄	—	2	2	no	—
509	C₄×D₄	—	2	2	no	—
510	C₄×D₄	—	2	2	no	—
511	C₄×D₄	—	2	2	no	—
512	C₄×D₄	C₂³×C₃₁	1	2	yes	—
513	C₄×D₄	C₂³×C₃₁	1	2	yes	—
514	#117	—	2	1	no	—
515	#117	—	2	1	no	—
516	#117	—	2	4	no	—
517	#117	C₂³×C₃₁	1	2	yes	—
518	#117	C₂³×C₃₁	1	4	yes	—
519	#117	C₂³×C₃₁	1	4	yes	—
520	C₂×C₄⋊C₄	—	2	1	no	—
521	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
522	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
523	C₂×C₄²	—	2	1	no	—
524	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
525	#107	C₂³×C₃₁	1	2	yes	—
526	#107	C₂³×C₃₁	1	4	yes	—
527	#107	C₂³×C₃₁	1	2	yes	—
528	#107	C₂³×C₃₁	1	4	yes	—
529	#106	C₂³×C₃₁	1	4	yes	—
530	#106	C₂³×C₃₁	1	4	yes	—
531	#106	C₂³×C₃₁	1	4	yes	—
532	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
533	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
534	#103	C₂³×C₃₁	1	2	yes	—
535	#103	C₂³×C₃₁	1	4	yes	—
536	#103	C₂³×C₃₁	1	4	yes	—
537	#103	C₂³×C₃₁	1	4	yes	—
538	#103	C₂³×C₃₁	1	2	yes	—
539	#103	C₂³×C₃₁	1	4	yes	—
540	#103	C₂³×C₃₁	1	4	yes	—
541	#103	C₂³×C₃₁	1	2	yes	—
542	#103	C₂³×C₃₁	1	2	yes	—
543	#103	C₂³×C₃₁	1	4	yes	—
544	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
545	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
546	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
547	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
548	#97	C₂³×C₃₁	1	2	yes	—
549	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
550	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
551	C₃₁	#61806	1	1	yes	characteristic, direct product factor, Sylow
552	#41	—	4	4	no	—
553	C₂×Q₈	—	2	1	no	—
554	C₂×Q₈	—	2	1	no	—
555	C₂×Q₈	—	2	1	no	—
556	C₂×Q₈	—	2	1	no	—
557	C₂×Q₈	—	2	1	no	—
558	C₂×Q₈	—	2	4	no	—
559	C₂×Q₈	—	4	2	no	—
560	C₂×Q₈	—	4	4	no	—
561	C₂×Q₈	—	4	4	no	—
562	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
563	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
564	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
565	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
566	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
567	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
568	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
569	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
570	C₂²×C₄	—	2	1	no	—
571	C₂²×C₄	—	2	1	no	—
572	C₂²×C₄	—	2	4	no	—
573	C₂²×C₄	—	2	4	no	—
574	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
575	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
576	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
577	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
578	Q₁₆	—	4	2	no	—
579	Q₁₆	—	4	4	no	—
580	Q₁₆	—	4	2	no	—
581	M₁₆	—	4	4	no	—
582	C₂×C₈	—	4	4	no	—
583	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
584	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
585	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
586	C₄⋊C₄	—	2	2	no	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	2	4	no	—
590	C₄⋊C₄	—	2	2	no	—
591	C₄⋊C₄	—	2	1	no	—
592	C₄⋊C₄	—	2	1	no	—
593	C₄⋊C₄	—	2	4	no	—
594	C₄⋊C₄	—	2	1	no	—
595	C₄⋊C₄	—	2	1	no	—
596	C₄⋊C₄	—	2	1	no	—
597	C₄⋊C₄	—	2	1	no	—
598	C₄⋊C₄	—	2	1	no	—
599	C₄⋊C₄	—	2	4	no	—
600	C₄⋊C₄	—	2	4	no	—
601	C₄⋊C₄	—	2	2	no	—
602	C₄⋊C₄	—	2	1	no	—
603	C₄⋊C₄	—	2	2	no	—
604	C₄⋊C₄	—	2	2	no	—
605	C₄⋊C₄	—	2	4	no	—
606	C₄⋊C₄	—	2	2	no	—
607	C₄⋊C₄	—	2	2	no	—
608	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
609	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
610	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
611	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
612	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
613	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
614	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
615	C₂²⋊C₄	—	2	2	no	—
616	C₂²⋊C₄	—	2	2	no	—
617	C₂²⋊C₄	—	2	4	no	—
618	C₂²⋊C₄	—	2	2	no	—
619	C₂²⋊C₄	—	2	4	no	—
620	C₂²⋊C₄	—	2	4	no	—
621	C₂²⋊C₄	—	2	4	no	—
622	C₂²⋊C₄	—	2	4	no	—
623	C₂²⋊C₄	—	2	4	no	—
624	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
625	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
626	C₄²	—	2	2	no	—
627	C₄²	—	2	2	no	—
628	C₄²	—	2	4	no	—
629	C₄²	—	2	2	no	—
630	C₄²	—	2	2	no	—
631	C₄²	—	2	2	no	—
632	C₄²	—	2	4	no	—
633	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
634	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
635	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
636	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
637	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
638	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
639	Q₈	—	2	2	no	—
640	Q₈	—	2	2	no	—
641	Q₈	—	2	2	no	—
642	Q₈	—	2	4	no	—
643	Q₈	—	2	2	no	—
644	Q₈	—	4	2	no	—
645	Q₈	—	4	2	no	—
646	Q₈	—	4	2	no	—
647	Q₈	—	4	2	no	—
648	Q₈	—	4	2	no	—
649	Q₈	—	8	1	no	—
650	Q₈	—	8	4	no	—
651	D₄	—	2	4	no	—
652	D₄	—	4	2	no	—
653	D₄	—	4	2	no	—
654	D₄	—	4	4	no	—
655	C₂×C₄	—	2	2	no	—
656	C₂×C₄	—	2	4	no	—
657	C₂×C₄	—	2	4	no	—
658	C₂×C₄	—	2	2	no	—
659	C₂×C₄	—	2	2	no	—
660	C₂×C₄	—	2	2	no	—
661	C₂×C₄	—	2	2	no	—
662	C₂×C₄	—	2	2	no	—
663	C₂×C₄	—	2	2	no	—
664	C₂×C₄	—	2	2	no	—
665	C₂×C₄	—	2	2	no	—
666	C₂×C₄	—	2	2	no	—
667	C₂×C₄	—	4	1	no	—
668	C₂×C₄	—	4	2	no	—
669	C₂×C₄	—	8	1	no	—
670	C₂×C₄	—	8	4	no	—
671	C₂×C₄	—	8	4	no	—
672	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
673	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
674	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
675	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
676	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
677	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
678	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
679	C₈	—	2	2	no	—
680	C₈	—	2	4	no	—
681	C₈	—	2	2	no	—
682	C₂²	—	2	1	no	—
683	C₂²	—	2	4	no	—
684	C₂²	—	2	1	no	—
685	C₂²	—	2	4	no	—
686	C₂²	—	2	1	no	—
687	C₂²	—	2	4	no	—
688	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
689	C₄	—	2	1	no	—
690	C₄	—	2	4	no	—
691	C₄	—	2	1	no	—
692	C₄	—	2	1	no	—
693	C₄	—	2	1	no	—
694	C₄	—	4	2	no	—
695	C₄	—	4	2	no	—
696	C₄	—	4	2	no	—
697	C₄	—	4	2	no	—
698	C₄	—	4	2	no	—
699	C₄	—	4	2	no	—
700	C₄	—	4	2	no	—
701	C₄	—	4	2	no	—
702	C₄	C₃₁×#578	1	2	yes	—
703	C₄	C₃₁×#585	1	2	yes	—
704	C₂	—	4	1	no	—
705	C₂	—	4	4	no	—
706	C₂	C₃₁×#3485	1	1	yes	characteristic
707	C₂	C₂×C₃₁×#578	1	1	yes	characteristic
708	C₂	C₃₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#532 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#3356	C₂	semidirect
4. C₃₁×#3349	C₂	semidirect
5. C₃₁×#3342	C₂	semidirect
6. C₃₁×#3328	C₂	semidirect
7. C₃₁×#3312	C₂	semidirect
8. C₃₁×#3299	C₂	semidirect
9. C₃₁×#3237	C₂	semidirect
10. C₃₁×#3231	C₂	semidirect
11. C₃₁×#3202	C₂	semidirect
12. C₃₁×#3113	C₂	semidirect
13. C₃₁×#3104	C₂	semidirect
14. C₃₁×#3097	C₂	semidirect
282. #61806	C₃₁	direct (N × H)
330. #3356	C₂×C₃₁	semidirect
331. #3349	C₂×C₃₁	semidirect
332. #3342	C₂×C₃₁	semidirect
333. #3328	C₂×C₃₁	semidirect
334. #3312	C₂×C₃₁	semidirect
335. #3299	C₂×C₃₁	semidirect
336. #3237	C₂×C₃₁	semidirect
337. #3231	C₂×C₃₁	semidirect
338. #3202	C₂×C₃₁	semidirect
339. #3113	C₂×C₃₁	semidirect
340. #3104	C₂×C₃₁	semidirect
341. #3097	C₂×C₃₁	semidirect
551. C₃₁	#61806	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×#3550	2
C₃₁×#3507	2
C₃₁×#3356	2
C₃₁×#3349	2
C₃₁×#3342	2
C₃₁×#3328	2
C₃₁×#3312	2
C₃₁×#3299	2
C₃₁×#3237	2
C₃₁×#3231	2
C₃₁×#3202	2
C₃₁×#3113	2
C₃₁×#3104	2
C₃₁×#3097	2
C₃₁×#2981	2
#61806	31

Subgroups — C31×#61806 #21518565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61806 #21518565