Subgroups — C₃₁×#61260 #21518019

Order 7936.

1782 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3493	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3349	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3347	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3340	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3337	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3163	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3098	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3093	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3083	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3060	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3050	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2952	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2929	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2912	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#556	—	2	2	no	—
20	C₃₁×#556	C₂²	1	4	yes	—
21	C₃₁×#555	—	2	2	no	—
22	C₃₁×#555	C₂²	1	4	yes	—
23	C₃₁×#553	—	2	2	no	—
24	C₃₁×#553	C₂²	1	4	yes	—
25	C₃₁×#552	—	2	2	no	—
26	C₃₁×#552	C₂²	1	4	yes	—
27	C₃₁×#544	C₂²	1	2	yes	—
28	C₃₁×#538	—	2	4	no	—
29	C₃₁×#537	C₂²	1	2	yes	—
30	C₃₁×#529	—	2	4	no	—
31	C₃₁×#525	C₂²	1	2	yes	—
32	C₃₁×#525	C₂²	1	1	yes	characteristic
33	C₃₁×#520	C₂²	1	2	yes	—
34	C₃₁×#520	C₂²	1	2	yes	—
35	C₂×C₃₁×#117	C₂²	1	2	yes	—
36	C₃₁×#501	C₂²	1	4	yes	complemented
37	C₃₁×#499	C₂²	1	2	yes	complemented
38	C₃₁×#498	C₂²	1	2	yes	complemented
39	C₃₁×#485	C₂²	1	4	yes	—
40	C₃₁×#485	C₂²	1	4	yes	—
41	C₃₁×#485	C₂²	1	4	yes	complemented
42	C₃₁×#484	C₂²	1	4	yes	complemented
43	C₃₁×#482	C₂²	1	4	yes	—
44	C₃₁×#482	C₂²	1	4	yes	—
45	C₃₁×#482	C₂²	1	4	yes	complemented
46	C₃₁×#480	C₂²	1	4	yes	complemented
47	C₃₁×#479	C₂²	1	4	yes	complemented
48	C₃₁×#478	C₂²	1	4	yes	complemented
49	C₃₁×#477	C₂²	1	4	yes	complemented
50	C₃₁×#474	C₂²	1	4	yes	complemented
51	C₃₁×#470	C₂²	1	2	yes	complemented
52	C₃₁×#469	C₂²	1	2	yes	complemented
53	C₃₁×#467	C₂²	1	2	yes	—
54	C₃₁×#466	C₂²	1	2	yes	—
55	C₃₁×#465	C₂²	1	2	yes	—
56	C₃₁×#465	C₂²	1	2	yes	—
57	C₃₁×#465	C₂²	1	4	yes	complemented
58	C₃₁×#442	C₂²	1	2	yes	complemented
59	C₃₁×#441	C₂²	1	2	yes	complemented
60	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
61	C₃₁×#435	C₂²	1	4	yes	—
62	C₈×D₄×C₃₁	C₂²	1	4	yes	—
63	C₃₁×#428	C₂²	1	2	yes	—
64	C₃₁×#428	C₂²	1	2	yes	—
65	C₃₁×#428	C₂²	1	4	yes	complemented
66	C₃₁×#427	C₂²	1	4	yes	—
67	C₃₁×#427	C₂²	1	4	yes	—
68	C₃₁×#427	C₂²	1	1	yes	characteristic
69	C₃₁×#427	C₂²	1	2	yes	complemented
70	C₂×C₃₁×#107	C₂²	1	2	yes	—
71	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
72	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
73	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
74	C₃₁×#424	C₂²	1	1	yes	characteristic
75	C₃₁×#417	C₂²	1	2	yes	—
76	C₃₁×#417	C₂²	1	2	yes	—
77	C₃₁×#416	C₂²	1	2	yes	—
78	C₃₁×#408	C₂²	1	1	yes	characteristic
79	C₃₁×#407	C₂²	1	1	yes	characteristic
80	C₃₁×#405	C₂²	1	1	yes	characteristic
81	C₂×#41×C₃₁	C₂³	1	2	yes	—
82	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#135	—	2	4	no	—
85	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
87	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₃₁×#131	—	2	4	no	—
89	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
90	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
91	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
92	C₂²×C₈×C₃₁	—	2	4	no	—
93	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
95	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
96	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
97	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
98	C₃₁×C₄⋊Q₈	—	2	2	no	—
99	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
100	C₃₁×#126	—	2	4	no	—
101	C₃₁×#126	—	2	4	no	—
102	C₃₁×#126	—	2	4	no	—
103	C₃₁×#126	—	2	4	no	—
104	C₃₁×#125	—	2	4	no	—
105	C₃₁×#125	—	2	2	no	—
106	C₃₁×#125	C₂³	1	4	yes	—
107	C₃₁×#124	—	2	4	no	—
108	C₃₁×#124	—	2	4	no	—
109	C₃₁×#124	—	2	4	no	—
110	C₃₁×#123	—	2	4	no	—
111	C₃₁×#123	—	2	2	no	—
112	C₃₁×#123	—	2	4	no	—
113	C₃₁×#123	D₄	1	2	yes	—
114	C₃₁×C₂²⋊Q₈	—	2	2	no	—
115	C₃₁×C₂²⋊Q₈	—	2	4	no	—
116	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
117	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
118	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
119	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
120	C₃₁×#121	—	2	4	no	—
121	C₃₁×#121	—	2	2	no	—
122	C₃₁×#121	—	2	4	no	—
123	C₃₁×#121	—	2	4	no	—
124	C₃₁×#121	D₄	1	2	yes	—
125	C₃₁×#121	C₂³	1	2	yes	—
126	C₃₁×#121	C₂³	1	2	yes	—
127	C₃₁×#121	C₂³	1	4	yes	—
128	C₃₁×#120	—	2	2	no	—
129	C₃₁×#120	D₄	1	2	yes	—
130	C₄×Q₈×C₃₁	—	2	2	no	—
131	C₄×Q₈×C₃₁	—	2	4	no	—
132	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
133	C₄×D₄×C₃₁	—	2	4	no	—
134	C₄×D₄×C₃₁	—	2	2	no	—
135	C₄×D₄×C₃₁	—	2	2	no	—
136	C₄×D₄×C₃₁	—	2	2	no	—
137	C₄×D₄×C₃₁	—	2	4	no	—
138	C₄×D₄×C₃₁	—	2	4	no	—
139	C₄×D₄×C₃₁	C₂³	1	4	yes	—
140	C₄×D₄×C₃₁	C₂³	1	4	yes	—
141	C₃₁×#117	—	2	2	no	—
142	C₃₁×#117	—	2	2	no	—
143	C₃₁×#117	—	2	2	no	—
144	C₃₁×#117	—	2	4	no	—
145	C₃₁×#117	—	2	4	no	—
146	C₃₁×#117	C₂³	1	2	yes	—
147	C₃₁×#117	C₂³	1	4	yes	—
148	C₃₁×#117	C₂³	1	4	yes	—
149	C₃₁×#117	C₂³	1	4	yes	—
150	C₃₁×#117	C₂³	1	1	yes	characteristic
151	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
153	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
154	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
155	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
156	C₂×C₄²×C₃₁	—	2	2	no	—
157	C₃₁×#107	C₂³	1	4	yes	—
158	C₃₁×#107	C₂³	1	4	yes	—
159	C₃₁×#107	C₂³	1	2	yes	—
160	C₃₁×#107	C₂³	1	4	yes	—
161	C₃₁×#107	C₂³	1	4	yes	—
162	C₃₁×#106	C₂³	1	2	yes	—
163	C₃₁×#106	C₂³	1	2	yes	—
164	C₃₁×#106	C₂³	1	2	yes	—
165	C₃₁×#106	C₂³	1	2	yes	—
166	C₃₁×#106	C₂³	1	1	yes	characteristic
167	C₃₁×#106	C₂³	1	4	yes	—
168	C₃₁×#106	C₂³	1	1	yes	characteristic
169	C₃₁×#106	C₂³	1	4	yes	—
170	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
171	C₃₁×#103	C₂³	1	2	yes	—
172	C₃₁×#103	C₂³	1	2	yes	—
173	C₃₁×#103	C₂³	1	4	yes	—
174	C₃₁×#103	C₂³	1	4	yes	—
175	C₃₁×#102	C₂³	1	2	yes	—
176	C₃₁×#102	C₂³	1	2	yes	—
177	C₃₁×#102	C₂³	1	4	yes	—
178	C₃₁×#102	C₂³	1	4	yes	—
179	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
180	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
181	C₃₁×#97	C₂³	1	2	yes	—
182	C₄×C₈×C₃₁	C₂³	1	2	yes	—
183	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
184	#41×C₃₁	—	2	2	no	—
185	#41×C₃₁	—	2	2	no	—
186	#41×C₃₁	—	2	4	no	—
187	#41×C₃₁	—	2	4	no	—
188	C₂×Q₈×C₃₁	—	2	4	no	—
189	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
190	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
191	C₂×D₄×C₃₁	—	2	2	no	—
192	C₂×D₄×C₃₁	—	2	2	no	—
193	C₂×D₄×C₃₁	—	2	2	no	—
194	C₂×D₄×C₃₁	—	2	4	no	—
195	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
196	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
197	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
198	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₃₁	—	2	2	no	—
200	C₂²×C₄×C₃₁	—	2	2	no	—
201	C₂²×C₄×C₃₁	—	2	2	no	—
202	C₂²×C₄×C₃₁	—	2	2	no	—
203	C₂²×C₄×C₃₁	—	2	4	no	—
204	C₂²×C₄×C₃₁	—	2	2	no	—
205	C₂²×C₄×C₃₁	—	2	4	no	—
206	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
208	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
209	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
210	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
211	Q₁₆×C₃₁	—	2	2	no	—
212	SD₁₆×C₃₁	—	2	4	no	—
213	D₈×C₃₁	—	2	2	no	—
214	M₁₆×C₃₁	—	2	2	no	—
215	M₁₆×C₃₁	—	2	2	no	—
216	M₁₆×C₃₁	—	2	4	no	—
217	C₂×C₈×C₃₁	—	2	2	no	—
218	C₂×C₈×C₃₁	—	2	4	no	—
219	C₂×C₈×C₃₁	—	2	4	no	—
220	C₂×C₈×C₃₁	—	2	4	no	—
221	C₂×C₈×C₃₁	#41	1	4	yes	—
222	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
223	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
224	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
225	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
226	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
227	C₄⋊C₄×C₃₁	—	2	2	no	—
228	C₄⋊C₄×C₃₁	—	2	4	no	—
229	C₄⋊C₄×C₃₁	—	2	2	no	—
230	C₄⋊C₄×C₃₁	—	2	4	no	—
231	C₄⋊C₄×C₃₁	—	2	2	no	—
232	C₄⋊C₄×C₃₁	—	2	4	no	—
233	C₄⋊C₄×C₃₁	—	2	4	no	—
234	C₄⋊C₄×C₃₁	—	2	2	no	—
235	C₄⋊C₄×C₃₁	—	2	4	no	—
236	C₄⋊C₄×C₃₁	—	2	4	no	—
237	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
238	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
239	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
240	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
241	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
242	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
243	C₂²⋊C₄×C₃₁	—	2	4	no	—
244	C₂²⋊C₄×C₃₁	—	2	4	no	—
245	C₂²⋊C₄×C₃₁	—	2	4	no	—
246	C₂²⋊C₄×C₃₁	—	2	2	no	—
247	C₂²⋊C₄×C₃₁	—	2	4	no	—
248	C₂²⋊C₄×C₃₁	—	2	2	no	—
249	C₂²⋊C₄×C₃₁	—	2	2	no	—
250	C₂²⋊C₄×C₃₁	—	2	2	no	—
251	C₂²⋊C₄×C₃₁	—	2	4	no	—
252	C₂²⋊C₄×C₃₁	—	2	4	no	—
253	C₂²⋊C₄×C₃₁	—	2	2	no	—
254	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
255	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
256	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
257	C₄²×C₃₁	—	2	4	no	—
258	C₄²×C₃₁	—	2	4	no	—
259	C₄²×C₃₁	—	2	4	no	—
260	C₄²×C₃₁	—	2	4	no	—
261	C₄²×C₃₁	—	2	2	no	—
262	C₄²×C₃₁	C₂⁴	1	2	yes	—
263	#61260	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
264	C₂³×C₃₁	—	2	2	no	—
265	C₂³×C₃₁	—	2	2	no	—
266	C₂³×C₃₁	—	4	2	no	—
267	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
268	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
269	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
270	Q₈×C₃₁	—	2	2	no	—
271	Q₈×C₃₁	—	2	2	no	—
272	Q₈×C₃₁	—	2	2	no	—
273	Q₈×C₃₁	—	8	4	no	—
274	D₄×C₃₁	—	2	4	no	—
275	D₄×C₃₁	—	2	2	no	—
276	D₄×C₃₁	—	2	2	no	—
277	D₄×C₃₁	—	2	2	no	—
278	D₄×C₃₁	—	4	4	no	—
279	D₄×C₃₁	—	4	4	no	—
280	D₄×C₃₁	—	8	2	no	—
281	D₄×C₃₁	—	8	4	no	—
282	D₄×C₃₁	—	8	2	no	—
283	C₂×C₄×C₃₁	—	2	2	no	—
284	C₂×C₄×C₃₁	—	2	4	no	—
285	C₂×C₄×C₃₁	—	2	2	no	—
286	C₂×C₄×C₃₁	—	2	2	no	—
287	C₂×C₄×C₃₁	—	2	2	no	—
288	C₂×C₄×C₃₁	—	2	2	no	—
289	C₂×C₄×C₃₁	—	2	4	no	—
290	C₂×C₄×C₃₁	—	2	2	no	—
291	C₂×C₄×C₃₁	—	2	2	no	—
292	C₂×C₄×C₃₁	—	2	4	no	—
293	C₂×C₄×C₃₁	—	2	2	no	—
294	C₂×C₄×C₃₁	—	4	4	no	—
295	C₂×C₄×C₃₁	—	8	4	no	—
296	C₂×C₄×C₃₁	—	8	2	no	—
297	C₂×C₄×C₃₁	—	8	4	no	—
298	C₂×C₄×C₃₁	—	8	2	no	—
299	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
300	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
301	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
302	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
303	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
304	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
305	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
306	C₈×C₃₁	—	2	2	no	—
307	C₈×C₃₁	—	2	2	no	—
308	C₈×C₃₁	—	2	2	no	—
309	C₈×C₃₁	C₂×#41	1	2	yes	—
310	C₈×C₃₁	C₂×#41	1	2	yes	—
311	#3493	C₂×C₃₁	1	2	yes	complemented
312	#3349	C₂×C₃₁	1	4	yes	complemented
313	#3347	C₂×C₃₁	1	4	yes	complemented
314	#3340	C₂×C₃₁	1	4	yes	complemented
315	#3337	C₂×C₃₁	1	4	yes	complemented
316	#3163	C₂×C₃₁	1	1	yes	characteristic, complemented
317	#3098	C₂×C₃₁	1	2	yes	complemented
318	#3093	C₂×C₃₁	1	2	yes	complemented
319	#3083	C₂×C₃₁	1	2	yes	complemented
320	#3060	C₂×C₃₁	1	1	yes	characteristic, complemented
321	#3050	C₂×C₃₁	1	1	yes	characteristic, complemented
322	#2952	C₂×C₃₁	1	1	yes	characteristic, complemented
323	#2929	C₂×C₃₁	1	1	yes	characteristic, complemented
324	#2912	C₂×C₃₁	1	1	yes	characteristic, complemented
325	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
326	C₂²×C₃₁	—	2	2	no	—
327	C₂²×C₃₁	—	2	4	no	—
328	C₂²×C₃₁	—	2	2	no	—
329	C₂²×C₃₁	—	2	2	no	—
330	C₂²×C₃₁	—	2	2	no	—
331	C₂²×C₃₁	—	2	2	no	—
332	C₂²×C₃₁	—	4	4	no	—
333	C₂²×C₃₁	—	4	2	no	—
334	C₂²×C₃₁	—	4	2	no	—
335	C₂²×C₃₁	—	4	2	no	—
336	C₂²×C₃₁	#547	1	2	yes	—
337	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
338	C₄×C₃₁	—	2	4	no	—
339	C₄×C₃₁	—	2	2	no	—
340	C₄×C₃₁	—	2	2	no	—
341	C₄×C₃₁	—	4	4	no	—
342	C₄×C₃₁	—	4	2	no	—
343	C₄×C₃₁	—	4	4	no	—
344	C₄×C₃₁	—	4	2	no	—
345	C₄×C₃₁	—	4	2	no	—
346	C₄×C₃₁	#547	1	2	yes	—
347	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
348	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
349	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
350	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
351	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
352	#556	—	2	2	no	—
353	#556	C₂²×C₃₁	1	4	yes	—
354	#555	—	2	2	no	—
355	#555	C₂²×C₃₁	1	4	yes	—
356	#553	—	2	2	no	—
357	#553	C₂²×C₃₁	1	4	yes	—
358	#552	—	2	2	no	—
359	#552	C₂²×C₃₁	1	4	yes	—
360	#544	C₂²×C₃₁	1	2	yes	—
361	#538	—	2	4	no	—
362	#537	C₂²×C₃₁	1	2	yes	—
363	#529	—	2	4	no	—
364	#525	C₂²×C₃₁	1	2	yes	—
365	#525	C₂²×C₃₁	1	1	yes	characteristic
366	#520	C₂²×C₃₁	1	2	yes	—
367	#520	C₂²×C₃₁	1	2	yes	—
368	C₂×#117	C₂²×C₃₁	1	2	yes	—
369	#501	C₂²×C₃₁	1	4	yes	complemented
370	#499	C₂²×C₃₁	1	2	yes	complemented
371	#498	C₂²×C₃₁	1	2	yes	complemented
372	#485	C₂²×C₃₁	1	4	yes	—
373	#485	C₂²×C₃₁	1	4	yes	—
374	#485	C₂²×C₃₁	1	4	yes	complemented
375	#484	C₂²×C₃₁	1	4	yes	complemented
376	#482	C₂²×C₃₁	1	4	yes	—
377	#482	C₂²×C₃₁	1	4	yes	—
378	#482	C₂²×C₃₁	1	4	yes	complemented
379	#480	C₂²×C₃₁	1	4	yes	complemented
380	#479	C₂²×C₃₁	1	4	yes	complemented
381	#478	C₂²×C₃₁	1	4	yes	complemented
382	#477	C₂²×C₃₁	1	4	yes	complemented
383	#474	C₂²×C₃₁	1	4	yes	complemented
384	#470	C₂²×C₃₁	1	2	yes	complemented
385	#469	C₂²×C₃₁	1	2	yes	complemented
386	#467	C₂²×C₃₁	1	2	yes	—
387	#466	C₂²×C₃₁	1	2	yes	—
388	#465	C₂²×C₃₁	1	2	yes	—
389	#465	C₂²×C₃₁	1	2	yes	—
390	#465	C₂²×C₃₁	1	4	yes	complemented
391	#442	C₂²×C₃₁	1	2	yes	complemented
392	#441	C₂²×C₃₁	1	2	yes	complemented
393	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
394	#435	C₂²×C₃₁	1	4	yes	—
395	C₈×D₄	C₂²×C₃₁	1	4	yes	—
396	#428	C₂²×C₃₁	1	2	yes	—
397	#428	C₂²×C₃₁	1	2	yes	—
398	#428	C₂²×C₃₁	1	4	yes	complemented
399	#427	C₂²×C₃₁	1	4	yes	—
400	#427	C₂²×C₃₁	1	4	yes	—
401	#427	C₂²×C₃₁	1	1	yes	characteristic
402	#427	C₂²×C₃₁	1	2	yes	complemented
403	C₂×#107	C₂²×C₃₁	1	2	yes	—
404	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
405	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
406	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
407	#424	C₂²×C₃₁	1	1	yes	characteristic
408	#417	C₂²×C₃₁	1	2	yes	—
409	#417	C₂²×C₃₁	1	2	yes	—
410	#416	C₂²×C₃₁	1	2	yes	—
411	#408	C₂²×C₃₁	1	1	yes	characteristic
412	#407	C₂²×C₃₁	1	1	yes	characteristic
413	#405	C₂²×C₃₁	1	1	yes	characteristic
414	C₂×C₃₁	—	2	2	no	—
415	C₂×C₃₁	—	4	2	no	—
416	C₂×C₃₁	—	4	2	no	—
417	C₂×C₃₁	—	8	2	no	—
418	C₂×C₃₁	C₂×#547	1	1	yes	characteristic
419	C₂×C₃₁	#3584	1	2	yes	—
420	C₂×#41	C₂³×C₃₁	1	2	yes	—
421	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
422	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
423	#135	—	2	4	no	—
424	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
425	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
426	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
427	#131	—	2	4	no	—
428	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
429	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
430	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
431	C₂²×C₈	—	2	4	no	—
432	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
433	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
434	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
435	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
436	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
437	C₄⋊Q₈	—	2	2	no	—
438	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
439	#126	—	2	4	no	—
440	#126	—	2	4	no	—
441	#126	—	2	4	no	—
442	#126	—	2	4	no	—
443	#125	—	2	4	no	—
444	#125	—	2	2	no	—
445	#125	C₂³×C₃₁	1	4	yes	—
446	#124	—	2	4	no	—
447	#124	—	2	4	no	—
448	#124	—	2	4	no	—
449	#123	—	2	4	no	—
450	#123	—	2	2	no	—
451	#123	—	2	4	no	—
452	#123	D₄×C₃₁	1	2	yes	—
453	C₂²⋊Q₈	—	2	2	no	—
454	C₂²⋊Q₈	—	2	4	no	—
455	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
456	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
457	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
458	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
459	#121	—	2	4	no	—
460	#121	—	2	2	no	—
461	#121	—	2	4	no	—
462	#121	—	2	4	no	—
463	#121	D₄×C₃₁	1	2	yes	—
464	#121	C₂³×C₃₁	1	2	yes	—
465	#121	C₂³×C₃₁	1	2	yes	—
466	#121	C₂³×C₃₁	1	4	yes	—
467	#120	—	2	2	no	—
468	#120	D₄×C₃₁	1	2	yes	—
469	C₄×Q₈	—	2	2	no	—
470	C₄×Q₈	—	2	4	no	—
471	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
472	C₄×D₄	—	2	4	no	—
473	C₄×D₄	—	2	2	no	—
474	C₄×D₄	—	2	2	no	—
475	C₄×D₄	—	2	2	no	—
476	C₄×D₄	—	2	4	no	—
477	C₄×D₄	—	2	4	no	—
478	C₄×D₄	C₂³×C₃₁	1	4	yes	—
479	C₄×D₄	C₂³×C₃₁	1	4	yes	—
480	#117	—	2	2	no	—
481	#117	—	2	2	no	—
482	#117	—	2	2	no	—
483	#117	—	2	4	no	—
484	#117	—	2	4	no	—
485	#117	C₂³×C₃₁	1	4	yes	—
486	#117	C₂³×C₃₁	1	4	yes	—
487	#117	C₂³×C₃₁	1	2	yes	—
488	#117	C₂³×C₃₁	1	4	yes	—
489	#117	C₂³×C₃₁	1	1	yes	characteristic
490	C₂×C₄⋊C₄	—	2	2	no	—
491	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
492	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
493	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
494	C₂×C₂²⋊C₄	—	2	2	no	—
495	C₂×C₄²	—	2	2	no	—
496	#107	C₂³×C₃₁	1	4	yes	—
497	#107	C₂³×C₃₁	1	4	yes	—
498	#107	C₂³×C₃₁	1	2	yes	—
499	#107	C₂³×C₃₁	1	4	yes	—
500	#107	C₂³×C₃₁	1	4	yes	—
501	#106	C₂³×C₃₁	1	2	yes	—
502	#106	C₂³×C₃₁	1	2	yes	—
503	#106	C₂³×C₃₁	1	2	yes	—
504	#106	C₂³×C₃₁	1	2	yes	—
505	#106	C₂³×C₃₁	1	1	yes	characteristic
506	#106	C₂³×C₃₁	1	4	yes	—
507	#106	C₂³×C₃₁	1	1	yes	characteristic
508	#106	C₂³×C₃₁	1	4	yes	—
509	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
510	#103	C₂³×C₃₁	1	2	yes	—
511	#103	C₂³×C₃₁	1	2	yes	—
512	#103	C₂³×C₃₁	1	4	yes	—
513	#103	C₂³×C₃₁	1	4	yes	—
514	#102	C₂³×C₃₁	1	2	yes	—
515	#102	C₂³×C₃₁	1	2	yes	—
516	#102	C₂³×C₃₁	1	4	yes	—
517	#102	C₂³×C₃₁	1	4	yes	—
518	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
519	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
520	#97	C₂³×C₃₁	1	2	yes	—
521	C₄×C₈	C₂³×C₃₁	1	2	yes	—
522	C₃₁	#61260	1	1	yes	characteristic, direct product factor, Sylow
523	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
524	#41	—	2	2	no	—
525	#41	—	2	2	no	—
526	#41	—	2	4	no	—
527	#41	—	2	4	no	—
528	C₂×Q₈	—	2	4	no	—
529	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
530	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
531	C₂×D₄	—	2	2	no	—
532	C₂×D₄	—	2	2	no	—
533	C₂×D₄	—	2	2	no	—
534	C₂×D₄	—	2	4	no	—
535	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
536	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
537	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
538	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
539	C₂²×C₄	—	2	2	no	—
540	C₂²×C₄	—	2	2	no	—
541	C₂²×C₄	—	2	2	no	—
542	C₂²×C₄	—	2	2	no	—
543	C₂²×C₄	—	2	4	no	—
544	C₂²×C₄	—	2	2	no	—
545	C₂²×C₄	—	2	4	no	—
546	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
547	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
548	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
549	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
550	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
551	Q₁₆	—	2	2	no	—
552	SD₁₆	—	2	4	no	—
553	D₈	—	2	2	no	—
554	M₁₆	—	2	2	no	—
555	M₁₆	—	2	2	no	—
556	M₁₆	—	2	4	no	—
557	C₂×C₈	—	2	2	no	—
558	C₂×C₈	—	2	4	no	—
559	C₂×C₈	—	2	4	no	—
560	C₂×C₈	—	2	4	no	—
561	C₂×C₈	#41×C₃₁	1	4	yes	—
562	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
563	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
564	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
565	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
566	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
567	C₄⋊C₄	—	2	2	no	—
568	C₄⋊C₄	—	2	4	no	—
569	C₄⋊C₄	—	2	2	no	—
570	C₄⋊C₄	—	2	4	no	—
571	C₄⋊C₄	—	2	2	no	—
572	C₄⋊C₄	—	2	4	no	—
573	C₄⋊C₄	—	2	4	no	—
574	C₄⋊C₄	—	2	2	no	—
575	C₄⋊C₄	—	2	4	no	—
576	C₄⋊C₄	—	2	4	no	—
577	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
578	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
579	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
580	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
581	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
582	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
583	C₂²⋊C₄	—	2	4	no	—
584	C₂²⋊C₄	—	2	4	no	—
585	C₂²⋊C₄	—	2	4	no	—
586	C₂²⋊C₄	—	2	2	no	—
587	C₂²⋊C₄	—	2	4	no	—
588	C₂²⋊C₄	—	2	2	no	—
589	C₂²⋊C₄	—	2	2	no	—
590	C₂²⋊C₄	—	2	2	no	—
591	C₂²⋊C₄	—	2	4	no	—
592	C₂²⋊C₄	—	2	4	no	—
593	C₂²⋊C₄	—	2	2	no	—
594	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
595	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
596	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
597	C₄²	—	2	4	no	—
598	C₄²	—	2	4	no	—
599	C₄²	—	2	4	no	—
600	C₄²	—	2	4	no	—
601	C₄²	—	2	2	no	—
602	C₄²	C₂⁴×C₃₁	1	2	yes	—
603	C₂³	—	2	2	no	—
604	C₂³	—	2	2	no	—
605	C₂³	—	4	2	no	—
606	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
607	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
608	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
609	Q₈	—	2	2	no	—
610	Q₈	—	2	2	no	—
611	Q₈	—	2	2	no	—
612	Q₈	—	8	4	no	—
613	D₄	—	2	4	no	—
614	D₄	—	2	2	no	—
615	D₄	—	2	2	no	—
616	D₄	—	2	2	no	—
617	D₄	—	4	4	no	—
618	D₄	—	4	4	no	—
619	D₄	—	8	2	no	—
620	D₄	—	8	4	no	—
621	D₄	—	8	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	4	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	2	4	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	2	no	—
631	C₂×C₄	—	2	4	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	4	4	no	—
634	C₂×C₄	—	8	4	no	—
635	C₂×C₄	—	8	2	no	—
636	C₂×C₄	—	8	4	no	—
637	C₂×C₄	—	8	2	no	—
638	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
639	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
640	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
641	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
642	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
643	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
644	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
645	C₈	—	2	2	no	—
646	C₈	—	2	2	no	—
647	C₈	—	2	2	no	—
648	C₈	C₂×#41×C₃₁	1	2	yes	—
649	C₈	C₂×#41×C₃₁	1	2	yes	—
650	C₂²	—	2	2	no	—
651	C₂²	—	2	2	no	—
652	C₂²	—	2	2	no	—
653	C₂²	—	2	4	no	—
654	C₂²	—	2	2	no	—
655	C₂²	—	2	2	no	—
656	C₂²	—	4	4	no	—
657	C₂²	—	4	2	no	—
658	C₂²	—	4	2	no	—
659	C₂²	—	4	2	no	—
660	C₂²	C₃₁×#547	1	2	yes	—
661	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
662	C₄	—	2	2	no	—
663	C₄	—	2	2	no	—
664	C₄	—	2	4	no	—
665	C₄	—	4	2	no	—
666	C₄	—	4	4	no	—
667	C₄	—	4	2	no	—
668	C₄	—	4	4	no	—
669	C₄	—	4	2	no	—
670	C₄	C₃₁×#547	1	2	yes	—
671	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
672	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
673	C₂	—	2	2	no	—
674	C₂	—	4	2	no	—
675	C₂	—	4	2	no	—
676	C₂	—	8	2	no	—
677	C₂	C₂×C₃₁×#547	1	1	yes	characteristic
678	C₂	C₃₁×#3584	1	2	yes	—

	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 ◁ C₃₁×#525 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3493	C₂	semidirect
2. C₃₁×#3349	C₂	semidirect
3. C₃₁×#3347	C₂	semidirect
4. C₃₁×#3340	C₂	semidirect
5. C₃₁×#3337	C₂	semidirect
6. C₃₁×#3163	C₂	semidirect
7. C₃₁×#3098	C₂	semidirect
8. C₃₁×#3093	C₂	semidirect
9. C₃₁×#3083	C₂	semidirect
10. C₃₁×#3060	C₂	semidirect
11. C₃₁×#3050	C₂	semidirect
12. C₃₁×#2952	C₂	semidirect
13. C₃₁×#2929	C₂	semidirect
14. C₃₁×#2912	C₂	semidirect
15. C₂×C₃₁×#427	C₂	semidirect
36. C₃₁×#501	C₂²	semidirect
37. C₃₁×#499	C₂²	semidirect
38. C₃₁×#498	C₂²	semidirect
41. C₃₁×#485	C₂²	semidirect
42. C₃₁×#484	C₂²	semidirect
45. C₃₁×#482	C₂²	semidirect
46. C₃₁×#480	C₂²	semidirect
47. C₃₁×#479	C₂²	semidirect
48. C₃₁×#478	C₂²	semidirect
49. C₃₁×#477	C₂²	semidirect
50. C₃₁×#474	C₂²	semidirect
51. C₃₁×#470	C₂²	semidirect
52. C₃₁×#469	C₂²	semidirect
57. C₃₁×#465	C₂²	semidirect
58. C₃₁×#442	C₂²	semidirect
59. C₃₁×#441	C₂²	semidirect
60. C₄×SD₁₆×C₃₁	C₂²	semidirect
65. C₃₁×#428	C₂²	semidirect
69. C₃₁×#427	C₂²	semidirect
73. C₂×C₃₁×#106	C₂²	semidirect
263. #61260	C₃₁	direct (N × H)
311. #3493	C₂×C₃₁	semidirect
312. #3349	C₂×C₃₁	semidirect
313. #3347	C₂×C₃₁	semidirect
314. #3340	C₂×C₃₁	semidirect
315. #3337	C₂×C₃₁	semidirect
316. #3163	C₂×C₃₁	semidirect
317. #3098	C₂×C₃₁	semidirect
318. #3093	C₂×C₃₁	semidirect
319. #3083	C₂×C₃₁	semidirect
320. #3060	C₂×C₃₁	semidirect
321. #3050	C₂×C₃₁	semidirect
322. #2952	C₂×C₃₁	semidirect
323. #2929	C₂×C₃₁	semidirect
324. #2912	C₂×C₃₁	semidirect
325. C₂×#427	C₂×C₃₁	semidirect
369. #501	C₂²×C₃₁	semidirect
370. #499	C₂²×C₃₁	semidirect
371. #498	C₂²×C₃₁	semidirect
374. #485	C₂²×C₃₁	semidirect
375. #484	C₂²×C₃₁	semidirect
378. #482	C₂²×C₃₁	semidirect
379. #480	C₂²×C₃₁	semidirect
380. #479	C₂²×C₃₁	semidirect
381. #478	C₂²×C₃₁	semidirect
382. #477	C₂²×C₃₁	semidirect
383. #474	C₂²×C₃₁	semidirect
384. #470	C₂²×C₃₁	semidirect
385. #469	C₂²×C₃₁	semidirect
390. #465	C₂²×C₃₁	semidirect
391. #442	C₂²×C₃₁	semidirect
392. #441	C₂²×C₃₁	semidirect
393. C₄×SD₁₆	C₂²×C₃₁	semidirect
398. #428	C₂²×C₃₁	semidirect
402. #427	C₂²×C₃₁	semidirect
406. C₂×#106	C₂²×C₃₁	semidirect
522. C₃₁	#61260	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×#3493	2
C₃₁×#3349	2
C₃₁×#3347	2
C₃₁×#3340	2
C₃₁×#3337	2
C₃₁×#3163	2
C₃₁×#3098	2
C₃₁×#3093	2
C₃₁×#3083	2
C₃₁×#3060	2
C₃₁×#3050	2
C₃₁×#2952	2
C₃₁×#2929	2
C₃₁×#2912	2
C₂×C₃₁×#427	2
#61260	31

Subgroups — C31×#61260 #21518019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61260 #21518019