Subgroups — C₃₁×#61259 #21518018

Order 7936.

1866 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3492	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3345	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3344	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3337	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3335	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3163	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3099	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3092	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3084	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	C₂²×C₃₁×#106	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×#554	—	2	2	no	—
20	C₃₁×#554	C₂²	1	4	yes	—
21	C₃₁×#553	—	2	2	no	—
22	C₃₁×#553	C₂²	1	4	yes	—
23	C₃₁×#552	—	2	2	no	—
24	C₃₁×#552	C₂²	1	4	yes	—
25	C₃₁×#547	—	2	2	no	—
26	C₃₁×#547	C₂²	1	4	yes	—
27	C₃₁×#544	C₂²	1	2	yes	—
28	C₃₁×#539	—	2	4	no	—
29	C₃₁×#537	C₂²	1	2	yes	—
30	C₃₁×#525	—	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	C₂²×C₄⋊C₄×C₃₁	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	complemented
40	C₃₁×#484	C₂²	1	4	yes	—
41	C₃₁×#483	C₂²	1	4	yes	—
42	C₃₁×#483	C₂²	1	4	yes	complemented
43	C₃₁×#482	C₂²	1	4	yes	complemented
44	C₃₁×#481	C₂²	1	4	yes	—
45	C₃₁×#481	C₂²	1	4	yes	complemented
46	C₃₁×#480	C₂²	1	4	yes	—
47	C₃₁×#479	C₂²	1	4	yes	complemented
48	C₃₁×#478	C₂²	1	4	yes	complemented
49	C₃₁×#476	C₂²	1	4	yes	complemented
50	C₃₁×#475	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	1	yes	characteristic
67	C₃₁×#427	C₂²	1	2	yes	complemented
68	C₂×C₃₁×#106	C₂²	1	4	yes	—
69	C₂×C₃₁×#106	C₂²	1	2	yes	—
70	C₂×C₃₁×#106	C₂²	1	4	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	—	2	4	no	—
82	C₂×#41×C₃₁	C₂³	1	2	yes	—
83	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₃₁	—	2	2	no	—
85	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×#135	—	2	4	no	—
87	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
88	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
89	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
90	C₃₁×#131	—	2	4	no	—
91	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
92	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
93	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
94	C₂²×C₈×C₃₁	—	2	4	no	—
95	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
96	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
97	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
98	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
99	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₃₁×C₄⋊Q₈	—	2	2	no	—
101	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
102	C₃₁×#126	—	2	4	no	—
103	C₃₁×#126	—	2	4	no	—
104	C₃₁×#125	—	2	4	no	—
105	C₃₁×#125	—	2	4	no	—
106	C₃₁×#125	—	2	2	no	—
107	C₃₁×#125	C₂³	1	4	yes	—
108	C₃₁×#123	—	2	4	no	—
109	C₃₁×#123	—	2	2	no	—
110	C₃₁×#123	—	2	4	no	—
111	C₃₁×#123	—	2	4	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	—	2	4	no	—
117	C₃₁×C₂²⋊Q₈	—	2	4	no	—
118	C₃₁×C₂²⋊Q₈	—	2	4	no	—
119	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
120	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
121	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
122	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
123	C₃₁×#121	—	2	4	no	—
124	C₃₁×#121	—	2	4	no	—
125	C₃₁×#121	—	2	2	no	—
126	C₃₁×#121	D₄	1	2	yes	—
127	C₃₁×#121	C₂³	1	2	yes	—
128	C₃₁×#121	C₂³	1	2	yes	—
129	C₃₁×#121	C₂³	1	4	yes	—
130	C₃₁×#120	—	2	2	no	—
131	C₃₁×#120	D₄	1	2	yes	—
132	C₄×Q₈×C₃₁	—	2	2	no	—
133	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
134	C₄×D₄×C₃₁	—	2	4	no	—
135	C₄×D₄×C₃₁	—	2	2	no	—
136	C₄×D₄×C₃₁	—	2	4	no	—
137	C₄×D₄×C₃₁	—	2	2	no	—
138	C₄×D₄×C₃₁	—	2	2	no	—
139	C₄×D₄×C₃₁	—	2	4	no	—
140	C₄×D₄×C₃₁	—	2	4	no	—
141	C₄×D₄×C₃₁	C₂³	1	4	yes	—
142	C₄×D₄×C₃₁	C₂³	1	4	yes	—
143	C₃₁×#117	—	2	2	no	—
144	C₃₁×#117	—	2	4	no	—
145	C₃₁×#117	C₂³	1	2	yes	—
146	C₃₁×#117	C₂³	1	4	yes	—
147	C₃₁×#117	C₂³	1	1	yes	characteristic
148	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
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×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
156	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
157	C₃₁×#107	C₂³	1	2	yes	—
158	C₃₁×#107	C₂³	1	4	yes	—
159	C₃₁×#107	C₂³	1	4	yes	—
160	C₃₁×#106	C₂³	1	2	yes	—
161	C₃₁×#106	C₂³	1	2	yes	—
162	C₃₁×#106	C₂³	1	2	yes	—
163	C₃₁×#106	C₂³	1	2	yes	—
164	C₃₁×#106	C₂³	1	4	yes	—
165	C₃₁×#106	C₂³	1	4	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	#41×C₃₁	—	8	8	no	—
189	C₂×Q₈×C₃₁	—	2	4	no	—
190	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
191	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
192	C₂×D₄×C₃₁	—	2	2	no	—
193	C₂×D₄×C₃₁	—	2	2	no	—
194	C₂×D₄×C₃₁	—	2	2	no	—
195	C₂×D₄×C₃₁	—	2	4	no	—
196	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
197	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
198	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
199	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₃₁	—	2	2	no	—
201	C₂²×C₄×C₃₁	—	2	2	no	—
202	C₂²×C₄×C₃₁	—	2	4	no	—
203	C₂²×C₄×C₃₁	—	2	2	no	—
204	C₂²×C₄×C₃₁	—	2	4	no	—
205	C₂²×C₄×C₃₁	—	2	2	no	—
206	C₂²×C₄×C₃₁	—	2	4	no	—
207	C₂²×C₄×C₃₁	—	2	2	no	—
208	C₂²×C₄×C₃₁	—	2	4	no	—
209	C₂²×C₄×C₃₁	—	8	4	no	—
210	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
211	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
212	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
213	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
214	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
215	Q₁₆×C₃₁	—	2	2	no	—
216	SD₁₆×C₃₁	—	2	4	no	—
217	D₈×C₃₁	—	2	2	no	—
218	M₁₆×C₃₁	—	2	2	no	—
219	M₁₆×C₃₁	—	2	2	no	—
220	M₁₆×C₃₁	—	2	4	no	—
221	C₂×C₈×C₃₁	—	2	2	no	—
222	C₂×C₈×C₃₁	—	2	4	no	—
223	C₂×C₈×C₃₁	—	2	4	no	—
224	C₂×C₈×C₃₁	—	2	4	no	—
225	C₂×C₈×C₃₁	#41	1	4	yes	complemented
226	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
227	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
228	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
229	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
230	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
231	C₄⋊C₄×C₃₁	—	2	2	no	—
232	C₄⋊C₄×C₃₁	—	2	2	no	—
233	C₄⋊C₄×C₃₁	—	2	4	no	—
234	C₄⋊C₄×C₃₁	—	2	4	no	—
235	C₄⋊C₄×C₃₁	—	2	4	no	—
236	C₄⋊C₄×C₃₁	—	2	4	no	—
237	C₄⋊C₄×C₃₁	—	2	4	no	—
238	C₄⋊C₄×C₃₁	—	2	4	no	—
239	C₄⋊C₄×C₃₁	—	2	2	no	—
240	C₄⋊C₄×C₃₁	—	2	4	no	—
241	C₄⋊C₄×C₃₁	—	2	4	no	—
242	C₄⋊C₄×C₃₁	—	2	2	no	—
243	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
244	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
245	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
246	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
247	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
248	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
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	2	no	—
253	C₂²⋊C₄×C₃₁	—	2	4	no	—
254	C₂²⋊C₄×C₃₁	—	2	4	no	—
255	C₂²⋊C₄×C₃₁	—	2	4	no	—
256	C₂²⋊C₄×C₃₁	—	2	2	no	—
257	C₂²⋊C₄×C₃₁	—	2	2	no	—
258	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
259	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
260	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
261	C₄²×C₃₁	—	2	4	no	—
262	C₄²×C₃₁	—	2	4	no	—
263	C₄²×C₃₁	—	2	2	no	—
264	C₄²×C₃₁	C₂⁴	1	2	yes	—
265	#61259	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₂³×C₃₁	—	2	2	no	—
267	C₂³×C₃₁	—	2	4	no	—
268	C₂³×C₃₁	—	2	2	no	—
269	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
270	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
271	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
272	Q₈×C₃₁	—	2	2	no	—
273	Q₈×C₃₁	—	2	2	no	—
274	Q₈×C₃₁	—	2	2	no	—
275	Q₈×C₃₁	—	8	4	no	—
276	D₄×C₃₁	—	2	4	no	—
277	D₄×C₃₁	—	2	8	no	—
278	D₄×C₃₁	—	2	2	no	—
279	D₄×C₃₁	—	2	2	no	—
280	D₄×C₃₁	—	2	2	no	—
281	D₄×C₃₁	—	4	4	no	—
282	D₄×C₃₁	—	4	4	no	—
283	D₄×C₃₁	—	4	4	no	—
284	D₄×C₃₁	—	8	4	no	—
285	C₂×C₄×C₃₁	—	2	2	no	—
286	C₂×C₄×C₃₁	—	2	4	no	—
287	C₂×C₄×C₃₁	—	2	2	no	—
288	C₂×C₄×C₃₁	—	2	2	no	—
289	C₂×C₄×C₃₁	—	2	2	no	—
290	C₂×C₄×C₃₁	—	2	2	no	—
291	C₂×C₄×C₃₁	—	2	2	no	—
292	C₂×C₄×C₃₁	—	2	2	no	—
293	C₂×C₄×C₃₁	—	2	4	no	—
294	C₂×C₄×C₃₁	—	2	2	no	—
295	C₂×C₄×C₃₁	—	2	4	no	—
296	C₂×C₄×C₃₁	—	4	8	no	—
297	C₂×C₄×C₃₁	—	4	4	no	—
298	C₂×C₄×C₃₁	—	8	4	no	—
299	C₂×C₄×C₃₁	—	8	2	no	—
300	C₂×C₄×C₃₁	—	8	4	no	—
301	C₂×C₄×C₃₁	—	8	2	no	—
302	C₂×C₄×C₃₁	—	8	4	no	—
303	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
304	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
305	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
306	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
307	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
308	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
309	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
310	C₈×C₃₁	—	2	2	no	—
311	C₈×C₃₁	—	2	2	no	—
312	C₈×C₃₁	—	2	2	no	—
313	C₈×C₃₁	C₂×#41	1	2	yes	—
314	C₈×C₃₁	C₂×#41	1	2	yes	—
315	#3492	C₂×C₃₁	1	2	yes	complemented
316	#3345	C₂×C₃₁	1	4	yes	complemented
317	#3344	C₂×C₃₁	1	4	yes	complemented
318	#3337	C₂×C₃₁	1	4	yes	complemented
319	#3335	C₂×C₃₁	1	4	yes	complemented
320	#3163	C₂×C₃₁	1	1	yes	characteristic, complemented
321	#3099	C₂×C₃₁	1	2	yes	complemented
322	#3092	C₂×C₃₁	1	2	yes	complemented
323	#3084	C₂×C₃₁	1	2	yes	complemented
324	#3060	C₂×C₃₁	1	1	yes	characteristic, complemented
325	#3050	C₂×C₃₁	1	1	yes	characteristic, complemented
326	#2952	C₂×C₃₁	1	1	yes	characteristic, complemented
327	#2929	C₂×C₃₁	1	1	yes	characteristic, complemented
328	#2912	C₂×C₃₁	1	1	yes	characteristic, complemented
329	C₂²×#106	C₂×C₃₁	1	1	yes	characteristic, complemented
330	C₂²×C₃₁	—	2	8	no	—
331	C₂²×C₃₁	—	2	2	no	—
332	C₂²×C₃₁	—	2	2	no	—
333	C₂²×C₃₁	—	2	2	no	—
334	C₂²×C₃₁	—	2	2	no	—
335	C₂²×C₃₁	—	2	4	no	—
336	C₂²×C₃₁	—	4	2	no	—
337	C₂²×C₃₁	—	4	2	no	—
338	C₂²×C₃₁	—	4	2	no	—
339	C₂²×C₃₁	#547	1	2	yes	—
340	C₂²×C₃₁	#577	1	4	yes	—
341	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
342	C₄×C₃₁	—	2	2	no	—
343	C₄×C₃₁	—	2	2	no	—
344	C₄×C₃₁	—	2	4	no	—
345	C₄×C₃₁	—	4	2	no	—
346	C₄×C₃₁	—	4	4	no	—
347	C₄×C₃₁	—	4	4	no	—
348	C₄×C₃₁	—	4	2	no	—
349	C₄×C₃₁	—	4	2	no	—
350	C₄×C₃₁	#547	1	2	yes	—
351	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
352	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
353	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
354	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
355	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
356	#554	—	2	2	no	—
357	#554	C₂²×C₃₁	1	4	yes	—
358	#553	—	2	2	no	—
359	#553	C₂²×C₃₁	1	4	yes	—
360	#552	—	2	2	no	—
361	#552	C₂²×C₃₁	1	4	yes	—
362	#547	—	2	2	no	—
363	#547	C₂²×C₃₁	1	4	yes	—
364	#544	C₂²×C₃₁	1	2	yes	—
365	#539	—	2	4	no	—
366	#537	C₂²×C₃₁	1	2	yes	—
367	#525	—	2	4	no	—
368	#525	C₂²×C₃₁	1	2	yes	—
369	#525	C₂²×C₃₁	1	1	yes	characteristic
370	#520	C₂²×C₃₁	1	2	yes	—
371	#520	C₂²×C₃₁	1	2	yes	—
372	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
373	#501	C₂²×C₃₁	1	4	yes	complemented
374	#499	C₂²×C₃₁	1	2	yes	complemented
375	#498	C₂²×C₃₁	1	2	yes	complemented
376	#485	C₂²×C₃₁	1	4	yes	complemented
377	#484	C₂²×C₃₁	1	4	yes	—
378	#483	C₂²×C₃₁	1	4	yes	—
379	#483	C₂²×C₃₁	1	4	yes	complemented
380	#482	C₂²×C₃₁	1	4	yes	complemented
381	#481	C₂²×C₃₁	1	4	yes	—
382	#481	C₂²×C₃₁	1	4	yes	complemented
383	#480	C₂²×C₃₁	1	4	yes	—
384	#479	C₂²×C₃₁	1	4	yes	complemented
385	#478	C₂²×C₃₁	1	4	yes	complemented
386	#476	C₂²×C₃₁	1	4	yes	complemented
387	#475	C₂²×C₃₁	1	4	yes	complemented
388	#470	C₂²×C₃₁	1	2	yes	complemented
389	#469	C₂²×C₃₁	1	2	yes	complemented
390	#467	C₂²×C₃₁	1	2	yes	—
391	#466	C₂²×C₃₁	1	2	yes	—
392	#465	C₂²×C₃₁	1	2	yes	—
393	#465	C₂²×C₃₁	1	2	yes	—
394	#465	C₂²×C₃₁	1	4	yes	complemented
395	#442	C₂²×C₃₁	1	2	yes	complemented
396	#441	C₂²×C₃₁	1	2	yes	complemented
397	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
398	#435	C₂²×C₃₁	1	4	yes	—
399	C₈×D₄	C₂²×C₃₁	1	4	yes	—
400	#428	C₂²×C₃₁	1	2	yes	—
401	#428	C₂²×C₃₁	1	2	yes	—
402	#428	C₂²×C₃₁	1	4	yes	complemented
403	#427	C₂²×C₃₁	1	1	yes	characteristic
404	#427	C₂²×C₃₁	1	2	yes	complemented
405	C₂×#106	C₂²×C₃₁	1	4	yes	—
406	C₂×#106	C₂²×C₃₁	1	2	yes	—
407	C₂×#106	C₂²×C₃₁	1	4	yes	—
408	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
409	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
410	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
411	#424	C₂²×C₃₁	1	1	yes	characteristic
412	#417	C₂²×C₃₁	1	2	yes	—
413	#417	C₂²×C₃₁	1	2	yes	—
414	#416	C₂²×C₃₁	1	2	yes	—
415	#408	C₂²×C₃₁	1	1	yes	characteristic
416	#407	C₂²×C₃₁	1	1	yes	characteristic
417	#405	C₂²×C₃₁	1	1	yes	characteristic
418	C₂×C₃₁	—	2	2	no	—
419	C₂×C₃₁	—	2	4	no	—
420	C₂×C₃₁	—	4	2	no	—
421	C₂×C₃₁	—	8	2	no	—
422	C₂×C₃₁	C₂×#547	1	1	yes	characteristic
423	C₂×C₃₁	C₂×#577	1	2	yes	—
424	C₂×#41	—	2	4	no	—
425	C₂×#41	C₂³×C₃₁	1	2	yes	—
426	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
427	C₂³×C₄	—	2	2	no	—
428	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
429	#135	—	2	4	no	—
430	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
431	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
432	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
433	#131	—	2	4	no	—
434	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
435	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
436	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
437	C₂²×C₈	—	2	4	no	—
438	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
439	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
440	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
441	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
442	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
443	C₄⋊Q₈	—	2	2	no	—
444	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
445	#126	—	2	4	no	—
446	#126	—	2	4	no	—
447	#125	—	2	4	no	—
448	#125	—	2	4	no	—
449	#125	—	2	2	no	—
450	#125	C₂³×C₃₁	1	4	yes	—
451	#123	—	2	4	no	—
452	#123	—	2	2	no	—
453	#123	—	2	4	no	—
454	#123	—	2	4	no	—
455	#123	—	2	4	no	—
456	#123	D₄×C₃₁	1	2	yes	—
457	C₂²⋊Q₈	—	2	2	no	—
458	C₂²⋊Q₈	—	2	4	no	—
459	C₂²⋊Q₈	—	2	4	no	—
460	C₂²⋊Q₈	—	2	4	no	—
461	C₂²⋊Q₈	—	2	4	no	—
462	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
463	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
464	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
465	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
466	#121	—	2	4	no	—
467	#121	—	2	4	no	—
468	#121	—	2	2	no	—
469	#121	D₄×C₃₁	1	2	yes	—
470	#121	C₂³×C₃₁	1	2	yes	—
471	#121	C₂³×C₃₁	1	2	yes	—
472	#121	C₂³×C₃₁	1	4	yes	—
473	#120	—	2	2	no	—
474	#120	D₄×C₃₁	1	2	yes	—
475	C₄×Q₈	—	2	2	no	—
476	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
477	C₄×D₄	—	2	4	no	—
478	C₄×D₄	—	2	2	no	—
479	C₄×D₄	—	2	4	no	—
480	C₄×D₄	—	2	2	no	—
481	C₄×D₄	—	2	2	no	—
482	C₄×D₄	—	2	4	no	—
483	C₄×D₄	—	2	4	no	—
484	C₄×D₄	C₂³×C₃₁	1	4	yes	—
485	C₄×D₄	C₂³×C₃₁	1	4	yes	—
486	#117	—	2	2	no	—
487	#117	—	2	4	no	—
488	#117	C₂³×C₃₁	1	2	yes	—
489	#117	C₂³×C₃₁	1	4	yes	—
490	#117	C₂³×C₃₁	1	1	yes	characteristic
491	C₂×C₄⋊C₄	—	2	2	no	—
492	C₂×C₄⋊C₄	—	2	2	no	—
493	C₂×C₄⋊C₄	—	2	2	no	—
494	C₂×C₄⋊C₄	—	2	2	no	—
495	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
496	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
497	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
498	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
499	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
500	#107	C₂³×C₃₁	1	2	yes	—
501	#107	C₂³×C₃₁	1	4	yes	—
502	#107	C₂³×C₃₁	1	4	yes	—
503	#106	C₂³×C₃₁	1	2	yes	—
504	#106	C₂³×C₃₁	1	2	yes	—
505	#106	C₂³×C₃₁	1	2	yes	—
506	#106	C₂³×C₃₁	1	2	yes	—
507	#106	C₂³×C₃₁	1	4	yes	—
508	#106	C₂³×C₃₁	1	4	yes	—
509	#106	C₂³×C₃₁	1	1	yes	characteristic
510	#106	C₂³×C₃₁	1	4	yes	—
511	#106	C₂³×C₃₁	1	1	yes	characteristic
512	#106	C₂³×C₃₁	1	4	yes	—
513	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
514	#103	C₂³×C₃₁	1	2	yes	—
515	#103	C₂³×C₃₁	1	2	yes	—
516	#103	C₂³×C₃₁	1	4	yes	—
517	#103	C₂³×C₃₁	1	4	yes	—
518	#102	C₂³×C₃₁	1	2	yes	—
519	#102	C₂³×C₃₁	1	2	yes	—
520	#102	C₂³×C₃₁	1	4	yes	—
521	#102	C₂³×C₃₁	1	4	yes	—
522	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
523	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
524	#97	C₂³×C₃₁	1	2	yes	—
525	C₄×C₈	C₂³×C₃₁	1	2	yes	—
526	C₃₁	#61259	1	1	yes	characteristic, direct product factor, Sylow
527	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
528	#41	—	2	2	no	—
529	#41	—	2	2	no	—
530	#41	—	2	4	no	—
531	#41	—	2	4	no	—
532	#41	—	8	8	no	—
533	C₂×Q₈	—	2	4	no	—
534	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
535	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
536	C₂×D₄	—	2	2	no	—
537	C₂×D₄	—	2	2	no	—
538	C₂×D₄	—	2	2	no	—
539	C₂×D₄	—	2	4	no	—
540	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
541	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
542	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
543	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
544	C₂²×C₄	—	2	2	no	—
545	C₂²×C₄	—	2	2	no	—
546	C₂²×C₄	—	2	4	no	—
547	C₂²×C₄	—	2	2	no	—
548	C₂²×C₄	—	2	4	no	—
549	C₂²×C₄	—	2	2	no	—
550	C₂²×C₄	—	2	4	no	—
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	4	no	—
553	C₂²×C₄	—	8	4	no	—
554	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
555	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
556	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
557	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
558	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
559	Q₁₆	—	2	2	no	—
560	SD₁₆	—	2	4	no	—
561	D₈	—	2	2	no	—
562	M₁₆	—	2	2	no	—
563	M₁₆	—	2	2	no	—
564	M₁₆	—	2	4	no	—
565	C₂×C₈	—	2	2	no	—
566	C₂×C₈	—	2	4	no	—
567	C₂×C₈	—	2	4	no	—
568	C₂×C₈	—	2	4	no	—
569	C₂×C₈	#41×C₃₁	1	4	yes	complemented
570	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
571	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
572	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
573	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
574	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
575	C₄⋊C₄	—	2	2	no	—
576	C₄⋊C₄	—	2	2	no	—
577	C₄⋊C₄	—	2	4	no	—
578	C₄⋊C₄	—	2	4	no	—
579	C₄⋊C₄	—	2	4	no	—
580	C₄⋊C₄	—	2	4	no	—
581	C₄⋊C₄	—	2	4	no	—
582	C₄⋊C₄	—	2	4	no	—
583	C₄⋊C₄	—	2	2	no	—
584	C₄⋊C₄	—	2	4	no	—
585	C₄⋊C₄	—	2	4	no	—
586	C₄⋊C₄	—	2	2	no	—
587	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
588	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
589	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
590	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
591	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
592	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
593	C₂²⋊C₄	—	2	2	no	—
594	C₂²⋊C₄	—	2	2	no	—
595	C₂²⋊C₄	—	2	4	no	—
596	C₂²⋊C₄	—	2	2	no	—
597	C₂²⋊C₄	—	2	4	no	—
598	C₂²⋊C₄	—	2	4	no	—
599	C₂²⋊C₄	—	2	4	no	—
600	C₂²⋊C₄	—	2	2	no	—
601	C₂²⋊C₄	—	2	2	no	—
602	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
603	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
604	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
605	C₄²	—	2	4	no	—
606	C₄²	—	2	4	no	—
607	C₄²	—	2	2	no	—
608	C₄²	C₂⁴×C₃₁	1	2	yes	—
609	C₂³	—	2	2	no	—
610	C₂³	—	2	4	no	—
611	C₂³	—	2	2	no	—
612	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
613	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
614	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
615	Q₈	—	2	2	no	—
616	Q₈	—	2	2	no	—
617	Q₈	—	2	2	no	—
618	Q₈	—	8	4	no	—
619	D₄	—	2	4	no	—
620	D₄	—	2	8	no	—
621	D₄	—	2	2	no	—
622	D₄	—	2	2	no	—
623	D₄	—	2	2	no	—
624	D₄	—	4	4	no	—
625	D₄	—	4	4	no	—
626	D₄	—	4	4	no	—
627	D₄	—	8	4	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	2	4	no	—
630	C₂×C₄	—	2	2	no	—
631	C₂×C₄	—	2	2	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	2	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	2	no	—
636	C₂×C₄	—	2	4	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	4	no	—
639	C₂×C₄	—	4	8	no	—
640	C₂×C₄	—	4	4	no	—
641	C₂×C₄	—	8	4	no	—
642	C₂×C₄	—	8	2	no	—
643	C₂×C₄	—	8	4	no	—
644	C₂×C₄	—	8	2	no	—
645	C₂×C₄	—	8	4	no	—
646	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
647	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
648	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
649	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
650	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
651	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
652	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
653	C₈	—	2	2	no	—
654	C₈	—	2	2	no	—
655	C₈	—	2	2	no	—
656	C₈	C₂×#41×C₃₁	1	2	yes	—
657	C₈	C₂×#41×C₃₁	1	2	yes	—
658	C₂²	—	2	8	no	—
659	C₂²	—	2	2	no	—
660	C₂²	—	2	2	no	—
661	C₂²	—	2	2	no	—
662	C₂²	—	2	2	no	—
663	C₂²	—	2	4	no	—
664	C₂²	—	4	2	no	—
665	C₂²	—	4	2	no	—
666	C₂²	—	4	2	no	—
667	C₂²	C₃₁×#547	1	2	yes	—
668	C₂²	C₃₁×#577	1	4	yes	—
669	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
670	C₄	—	2	2	no	—
671	C₄	—	2	2	no	—
672	C₄	—	2	4	no	—
673	C₄	—	4	2	no	—
674	C₄	—	4	4	no	—
675	C₄	—	4	4	no	—
676	C₄	—	4	2	no	—
677	C₄	—	4	2	no	—
678	C₄	C₃₁×#547	1	2	yes	—
679	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
680	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
681	C₂	—	2	2	no	—
682	C₂	—	2	4	no	—
683	C₂	—	4	2	no	—
684	C₂	—	8	2	no	—
685	C₂	C₂×C₃₁×#547	1	1	yes	characteristic
686	C₂	C₂×C₃₁×#577	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×#3492	C₂	semidirect
2. C₃₁×#3345	C₂	semidirect
3. C₃₁×#3344	C₂	semidirect
4. C₃₁×#3337	C₂	semidirect
5. C₃₁×#3335	C₂	semidirect
6. C₃₁×#3163	C₂	semidirect
7. C₃₁×#3099	C₂	semidirect
8. C₃₁×#3092	C₂	semidirect
9. C₃₁×#3084	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. C₂²×C₃₁×#106	C₂	semidirect
36. C₃₁×#501	C₂²	semidirect
37. C₃₁×#499	C₂²	semidirect
38. C₃₁×#498	C₂²	semidirect
39. C₃₁×#485	C₂²	semidirect
42. C₃₁×#483	C₂²	semidirect
43. C₃₁×#482	C₂²	semidirect
45. C₃₁×#481	C₂²	semidirect
47. C₃₁×#479	C₂²	semidirect
48. C₃₁×#478	C₂²	semidirect
49. C₃₁×#476	C₂²	semidirect
50. C₃₁×#475	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
67. C₃₁×#427	C₂²	semidirect
73. C₂×C₃₁×#106	C₂²	semidirect
225. C₂×C₈×C₃₁	#41	semidirect
265. #61259	C₃₁	direct (N × H)
315. #3492	C₂×C₃₁	semidirect
316. #3345	C₂×C₃₁	semidirect
317. #3344	C₂×C₃₁	semidirect
318. #3337	C₂×C₃₁	semidirect
319. #3335	C₂×C₃₁	semidirect
320. #3163	C₂×C₃₁	semidirect
321. #3099	C₂×C₃₁	semidirect
322. #3092	C₂×C₃₁	semidirect
323. #3084	C₂×C₃₁	semidirect
324. #3060	C₂×C₃₁	semidirect
325. #3050	C₂×C₃₁	semidirect
326. #2952	C₂×C₃₁	semidirect
327. #2929	C₂×C₃₁	semidirect
328. #2912	C₂×C₃₁	semidirect
329. C₂²×#106	C₂×C₃₁	semidirect
373. #501	C₂²×C₃₁	semidirect
374. #499	C₂²×C₃₁	semidirect
375. #498	C₂²×C₃₁	semidirect
376. #485	C₂²×C₃₁	semidirect
379. #483	C₂²×C₃₁	semidirect
380. #482	C₂²×C₃₁	semidirect
382. #481	C₂²×C₃₁	semidirect
384. #479	C₂²×C₃₁	semidirect
385. #478	C₂²×C₃₁	semidirect
386. #476	C₂²×C₃₁	semidirect
387. #475	C₂²×C₃₁	semidirect
388. #470	C₂²×C₃₁	semidirect
389. #469	C₂²×C₃₁	semidirect
394. #465	C₂²×C₃₁	semidirect
395. #442	C₂²×C₃₁	semidirect
396. #441	C₂²×C₃₁	semidirect
397. C₄×SD₁₆	C₂²×C₃₁	semidirect
402. #428	C₂²×C₃₁	semidirect
404. #427	C₂²×C₃₁	semidirect
410. C₂×#106	C₂²×C₃₁	semidirect
526. C₃₁	#61259	direct (N × H)
569. C₂×C₈	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3492	2
C₃₁×#3345	2
C₃₁×#3344	2
C₃₁×#3337	2
C₃₁×#3335	2
C₃₁×#3163	2
C₃₁×#3099	2
C₃₁×#3092	2
C₃₁×#3084	2
C₃₁×#3060	2
C₃₁×#3050	2
C₃₁×#2952	2
C₃₁×#2929	2
C₃₁×#2912	2
C₂²×C₃₁×#106	2
#61259	31

Subgroups — C31×#61259 #21518018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61259 #21518018