Subgroups — C₃₁×#61338 #21518097

Order 7936.

1732 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3377	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3376	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3374	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3372	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3137	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#491	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3082	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3080	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3046	C₂	1	2	yes	maximal
12	C₃₁×#3039	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2961	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2947	C₂	1	2	yes	maximal, complemented
15	C₃₁×#2943	C₂	1	1	yes	maximal, characteristic
16	C₂×C₃₁×#137	C₂²	1	2	yes	—
17	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
18	C₃₁×#557	C₂²	1	4	yes	—
19	C₃₁×#557	C₂²	1	4	yes	—
20	C₃₁×#556	C₂²	1	4	yes	—
21	C₃₁×#554	—	2	4	no	—
22	C₃₁×#554	C₂²	1	4	yes	—
23	C₃₁×#552	—	2	4	no	—
24	C₃₁×#544	—	2	2	no	—
25	C₃₁×#544	C₂²	1	2	yes	—
26	C₃₁×#541	—	2	1	no	—
27	C₃₁×#541	—	2	1	no	—
28	C₃₁×#536	C₂²	1	2	yes	—
29	C₃₁×#533	—	2	2	no	—
30	C₃₁×#533	—	2	1	no	—
31	C₃₁×#533	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#125	—	2	1	no	—
34	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
35	C₃₁×#520	C₂²	1	2	yes	—
36	C₃₁×#519	C₂²	1	2	yes	—
37	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
38	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#491	C₂²	1	4	yes	—
40	C₃₁×#491	C₂²	1	4	yes	—
41	C₃₁×#491	C₂²	1	4	yes	—
42	C₃₁×#489	C₂²	1	4	yes	—
43	C₃₁×#488	C₂²	1	4	yes	—
44	C₃₁×#487	C₂²	1	4	yes	—
45	C₃₁×#479	C₂²	1	4	yes	—
46	C₃₁×#479	C₂²	1	4	yes	—
47	C₃₁×#479	C₂²	1	4	yes	—
48	C₃₁×#478	C₂²	1	4	yes	—
49	C₃₁×#477	C₂²	1	4	yes	—
50	C₃₁×#475	C₂²	1	4	yes	—
51	C₃₁×#464	C₂²	1	4	yes	—
52	C₃₁×#463	C₂²	1	4	yes	—
53	C₃₁×#462	C₂²	1	4	yes	—
54	C₃₁×#462	C₂²	1	4	yes	—
55	C₃₁×#461	C₂²	1	4	yes	—
56	C₃₁×#461	C₂²	1	4	yes	—
57	C₃₁×#446	C₂²	1	4	yes	—
58	C₃₁×#446	C₂²	1	4	yes	—
59	C₃₁×#442	C₂²	1	2	yes	—
60	C₃₁×#441	C₂²	1	2	yes	—
61	C₃₁×#441	C₂²	1	4	yes	—
62	C₃₁×#440	C₂²	1	2	yes	—
63	C₃₁×#440	C₂²	1	4	yes	—
64	C₃₁×#440	C₂²	1	2	yes	—
65	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
66	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
67	C₃₁×#428	C₂²	1	2	yes	—
68	C₃₁×#427	C₂²	1	1	yes	characteristic
69	C₃₁×#424	C₂²	1	2	yes	—
70	C₃₁×#423	C₂²	1	1	yes	characteristic
71	C₃₁×#419	C₂²	1	2	yes	—
72	C₃₁×#417	C₂²	1	2	yes	—
73	C₃₁×#416	C₂²	1	1	yes	characteristic
74	C₃₁×#416	C₂²	1	1	yes	characteristic
75	C₂×C₃₁×#103	C₂²	1	2	yes	—
76	C₂×C₃₁×#103	C₂²	1	2	yes	—
77	C₃₁×#405	C₂²	1	2	yes	—
78	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
79	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
81	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
82	C₃₁×#137	—	2	8	no	—
83	C₃₁×#135	—	2	4	no	—
84	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
85	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
87	C₂×SD₁₆×C₃₁	C₂³	1	1	yes	characteristic
88	C₂×SD₁₆×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
90	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
91	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
92	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₃₁×C₄⋊Q₈	—	2	4	no	—
94	C₃₁×C₄⋊Q₈	—	2	2	no	—
95	C₃₁×C₄⋊Q₈	—	2	1	no	—
96	C₃₁×C₄⋊Q₈	—	2	1	no	—
97	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
98	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
99	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
100	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
101	C₃₁×#126	—	2	4	no	—
102	C₃₁×#126	—	2	4	no	—
103	C₃₁×#125	—	2	2	no	—
104	C₃₁×#125	—	2	4	no	—
105	C₃₁×#125	—	2	4	no	—
106	C₃₁×#125	—	2	4	no	—
107	C₃₁×#125	—	2	4	no	—
108	C₃₁×#125	—	2	4	no	—
109	C₃₁×#125	—	2	1	no	—
110	C₃₁×#125	—	2	1	no	—
111	C₃₁×#125	D₄	1	1	yes	characteristic
112	C₃₁×#125	D₄	1	1	yes	characteristic
113	C₃₁×#125	C₂³	1	4	yes	—
114	C₃₁×#125	C₂³	1	4	yes	—
115	C₃₁×#124	C₂³	1	4	yes	—
116	C₃₁×#123	—	2	4	no	—
117	C₃₁×C₂²⋊Q₈	—	2	4	no	—
118	C₃₁×C₂²⋊Q₈	—	2	4	no	—
119	C₃₁×C₂²⋊Q₈	—	2	2	no	—
120	C₃₁×C₂²⋊Q₈	—	2	2	no	—
121	C₃₁×C₂²⋊Q₈	—	2	4	no	—
122	C₃₁×C₂²⋊Q₈	—	2	4	no	—
123	C₃₁×C₂²⋊Q₈	—	2	4	no	—
124	C₃₁×C₂²⋊Q₈	D₄	1	4	yes	—
125	C₄×Q₈×C₃₁	—	2	4	no	—
126	C₄×Q₈×C₃₁	—	2	2	no	—
127	C₄×Q₈×C₃₁	—	2	2	no	—
128	C₄×Q₈×C₃₁	—	2	4	no	—
129	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
130	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
131	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
132	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
133	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
134	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
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	C₂³	1	4	yes	—
139	C₄×D₄×C₃₁	C₂³	1	2	yes	—
140	C₄×D₄×C₃₁	C₂³	1	2	yes	—
141	C₃₁×#117	—	2	2	no	—
142	C₃₁×#117	—	2	4	no	—
143	C₃₁×#117	—	2	4	no	—
144	C₃₁×#117	—	2	1	no	—
145	C₃₁×#117	—	2	1	no	—
146	C₃₁×#117	C₂³	1	1	yes	characteristic
147	C₃₁×#117	C₂³	1	1	yes	characteristic
148	C₃₁×#117	C₂³	1	2	yes	—
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	—	2	4	no	—
153	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
154	C₂×C₄²×C₃₁	—	2	1	no	—
155	C₂×C₄²×C₃₁	—	2	1	no	—
156	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
157	C₃₁×#107	C₂³	1	2	yes	—
158	C₃₁×#107	C₂³	1	4	yes	—
159	C₃₁×#106	C₂³	1	2	yes	—
160	C₃₁×#106	C₂³	1	4	yes	—
161	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
162	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
163	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
164	C₃₁×#103	C₂³	1	4	yes	—
165	C₃₁×#103	C₂³	1	4	yes	—
166	C₃₁×#103	C₂³	1	4	yes	—
167	C₃₁×#103	C₂³	1	4	yes	—
168	C₃₁×#103	C₂³	1	2	yes	—
169	C₃₁×#103	C₂³	1	2	yes	—
170	C₃₁×#103	C₂³	1	4	yes	—
171	C₃₁×#103	C₂³	1	4	yes	—
172	C₃₁×#103	C₂³	1	4	yes	—
173	C₃₁×#102	C₂³	1	2	yes	—
174	C₃₁×#102	C₂³	1	2	yes	—
175	C₃₁×#102	C₂³	1	4	yes	—
176	C₃₁×#97	C₂³	1	2	yes	—
177	C₃₁×#97	C₂³	1	2	yes	—
178	C₃₁×#97	C₂³	1	4	yes	—
179	C₄×C₈×C₃₁	C₂³	1	4	yes	—
180	#41×C₃₁	—	2	2	no	—
181	#41×C₃₁	—	2	2	no	—
182	#41×C₃₁	—	2	2	no	—
183	#41×C₃₁	#41	1	4	yes	—
184	C₂×Q₈×C₃₁	—	2	2	no	—
185	C₂×Q₈×C₃₁	—	2	4	no	—
186	C₂×Q₈×C₃₁	—	2	4	no	—
187	C₂×Q₈×C₃₁	—	2	4	no	—
188	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
189	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
190	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
191	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
192	C₂×D₄×C₃₁	—	2	1	no	—
193	C₂×D₄×C₃₁	—	2	1	no	—
194	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
195	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
196	C₂²×C₄×C₃₁	—	2	1	no	—
197	C₂²×C₄×C₃₁	—	2	2	no	—
198	C₂²×C₄×C₃₁	—	2	2	no	—
199	C₂²×C₄×C₃₁	—	2	2	no	—
200	C₂²×C₄×C₃₁	—	2	1	no	—
201	C₂²×C₄×C₃₁	—	2	2	no	—
202	C₂²×C₄×C₃₁	—	2	4	no	—
203	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
204	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
205	Q₁₆×C₃₁	—	2	2	no	—
206	Q₁₆×C₃₁	—	2	8	no	—
207	SD₁₆×C₃₁	—	2	2	no	—
208	SD₁₆×C₃₁	—	2	2	no	—
209	SD₁₆×C₃₁	—	2	8	no	—
210	D₈×C₃₁	—	2	2	no	—
211	M₁₆×C₃₁	—	2	4	no	—
212	C₂×C₈×C₃₁	—	2	2	no	—
213	C₂×C₈×C₃₁	—	2	2	no	—
214	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
215	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
216	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
217	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
218	C₄⋊C₄×C₃₁	—	2	4	no	—
219	C₄⋊C₄×C₃₁	—	2	2	no	—
220	C₄⋊C₄×C₃₁	—	2	4	no	—
221	C₄⋊C₄×C₃₁	—	2	4	no	—
222	C₄⋊C₄×C₃₁	—	2	4	no	—
223	C₄⋊C₄×C₃₁	—	2	4	no	—
224	C₄⋊C₄×C₃₁	—	2	4	no	—
225	C₄⋊C₄×C₃₁	—	2	4	no	—
226	C₄⋊C₄×C₃₁	—	2	2	no	—
227	C₄⋊C₄×C₃₁	—	2	2	no	—
228	C₄⋊C₄×C₃₁	—	2	4	no	—
229	C₄⋊C₄×C₃₁	—	2	4	no	—
230	C₄⋊C₄×C₃₁	—	2	4	no	—
231	C₄⋊C₄×C₃₁	—	2	4	no	—
232	C₄⋊C₄×C₃₁	—	2	4	no	—
233	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
234	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
235	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
236	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
237	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
238	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
239	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
240	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
241	C₂²⋊C₄×C₃₁	—	2	1	no	—
242	C₂²⋊C₄×C₃₁	—	2	1	no	—
243	C₂²⋊C₄×C₃₁	—	2	2	no	—
244	C₂²⋊C₄×C₃₁	—	2	4	no	—
245	C₂²⋊C₄×C₃₁	—	2	2	no	—
246	C₂²⋊C₄×C₃₁	—	2	4	no	—
247	C₂²⋊C₄×C₃₁	—	2	4	no	—
248	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
249	C₄²×C₃₁	—	2	1	no	—
250	C₄²×C₃₁	—	2	1	no	—
251	C₄²×C₃₁	—	2	2	no	—
252	C₄²×C₃₁	—	2	2	no	—
253	C₄²×C₃₁	—	2	4	no	—
254	C₄²×C₃₁	—	2	4	no	—
255	C₄²×C₃₁	—	2	4	no	—
256	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
257	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
258	C₄²×C₃₁	C₂⁴	1	4	yes	—
259	C₄²×C₃₁	C₂⁴	1	2	yes	—
260	#61338	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
261	C₂³×C₃₁	—	2	1	no	—
262	C₂³×C₃₁	—	2	1	no	—
263	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
264	Q₈×C₃₁	—	2	1	no	—
265	Q₈×C₃₁	—	2	4	no	—
266	Q₈×C₃₁	—	2	1	no	—
267	Q₈×C₃₁	—	2	4	no	—
268	Q₈×C₃₁	—	2	1	no	—
269	Q₈×C₃₁	—	4	4	no	—
270	Q₈×C₃₁	—	4	4	no	—
271	Q₈×C₃₁	—	8	4	no	—
272	Q₈×C₃₁	C₂×#41	1	2	yes	—
273	D₄×C₃₁	—	2	4	no	—
274	D₄×C₃₁	—	2	1	no	—
275	D₄×C₃₁	—	2	1	no	—
276	D₄×C₃₁	—	2	1	no	—
277	D₄×C₃₁	—	4	2	no	—
278	D₄×C₃₁	C₂×#41	1	2	yes	—
279	C₂×C₄×C₃₁	—	2	1	no	—
280	C₂×C₄×C₃₁	—	2	4	no	—
281	C₂×C₄×C₃₁	—	2	2	no	—
282	C₂×C₄×C₃₁	—	2	4	no	—
283	C₂×C₄×C₃₁	—	2	2	no	—
284	C₂×C₄×C₃₁	—	2	1	no	—
285	C₂×C₄×C₃₁	—	2	4	no	—
286	C₂×C₄×C₃₁	—	4	2	no	—
287	C₂×C₄×C₃₁	—	4	4	no	—
288	C₂×C₄×C₃₁	—	4	4	no	—
289	C₂×C₄×C₃₁	—	4	2	no	—
290	C₂×C₄×C₃₁	—	8	2	no	—
291	C₂×C₄×C₃₁	—	8	4	no	—
292	C₂×C₄×C₃₁	—	8	2	no	—
293	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
294	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
295	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
296	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
297	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
298	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
299	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
300	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
301	C₈×C₃₁	—	2	1	no	—
302	C₈×C₃₁	—	2	1	no	—
303	C₈×C₃₁	—	2	4	no	—
304	C₈×C₃₁	—	2	2	no	—
305	#3509	C₂×C₃₁	1	1	yes	characteristic, complemented
306	#3504	C₂×C₃₁	1	1	yes	characteristic, complemented
307	#3377	C₂×C₃₁	1	4	yes	complemented
308	#3376	C₂×C₃₁	1	4	yes	complemented
309	#3374	C₂×C₃₁	1	4	yes	complemented
310	#3372	C₂×C₃₁	1	4	yes	complemented
311	#3137	C₂×C₃₁	1	2	yes	complemented
312	C₂×#491	C₂×C₃₁	1	1	yes	characteristic, complemented
313	#3082	C₂×C₃₁	1	2	yes	complemented
314	#3080	C₂×C₃₁	1	1	yes	characteristic, complemented
315	#3046	C₂×C₃₁	1	2	yes	—
316	#3039	C₂×C₃₁	1	1	yes	characteristic, complemented
317	#2961	C₂×C₃₁	1	1	yes	characteristic, complemented
318	#2947	C₂×C₃₁	1	2	yes	complemented
319	#2943	C₂×C₃₁	1	1	yes	characteristic
320	C₂²×C₃₁	—	2	2	no	—
321	C₂²×C₃₁	—	2	2	no	—
322	C₂²×C₃₁	—	4	2	no	—
323	C₂²×C₃₁	—	4	2	no	—
324	C₂²×C₃₁	—	4	1	no	—
325	C₂²×C₃₁	#548	1	2	yes	—
326	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
327	C₄×C₃₁	—	2	2	no	—
328	C₄×C₃₁	—	2	2	no	—
329	C₄×C₃₁	—	2	4	no	—
330	C₄×C₃₁	—	2	4	no	—
331	C₄×C₃₁	—	2	2	no	—
332	C₄×C₃₁	—	4	2	no	—
333	C₄×C₃₁	—	4	4	no	—
334	C₄×C₃₁	—	4	2	no	—
335	C₄×C₃₁	—	4	1	no	—
336	C₄×C₃₁	—	4	4	no	—
337	C₄×C₃₁	#548	1	2	yes	—
338	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
339	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
340	C₂×#137	C₂²×C₃₁	1	2	yes	—
341	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
342	#557	C₂²×C₃₁	1	4	yes	—
343	#557	C₂²×C₃₁	1	4	yes	—
344	#556	C₂²×C₃₁	1	4	yes	—
345	#554	—	2	4	no	—
346	#554	C₂²×C₃₁	1	4	yes	—
347	#552	—	2	4	no	—
348	#544	—	2	2	no	—
349	#544	C₂²×C₃₁	1	2	yes	—
350	#541	—	2	1	no	—
351	#541	—	2	1	no	—
352	#536	C₂²×C₃₁	1	2	yes	—
353	#533	—	2	2	no	—
354	#533	—	2	1	no	—
355	#533	C₂²×C₃₁	1	1	yes	characteristic
356	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
357	C₂×#125	—	2	1	no	—
358	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
359	#520	C₂²×C₃₁	1	2	yes	—
360	#519	C₂²×C₃₁	1	2	yes	—
361	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
362	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
363	#491	C₂²×C₃₁	1	4	yes	—
364	#491	C₂²×C₃₁	1	4	yes	—
365	#491	C₂²×C₃₁	1	4	yes	—
366	#489	C₂²×C₃₁	1	4	yes	—
367	#488	C₂²×C₃₁	1	4	yes	—
368	#487	C₂²×C₃₁	1	4	yes	—
369	#479	C₂²×C₃₁	1	4	yes	—
370	#479	C₂²×C₃₁	1	4	yes	—
371	#479	C₂²×C₃₁	1	4	yes	—
372	#478	C₂²×C₃₁	1	4	yes	—
373	#477	C₂²×C₃₁	1	4	yes	—
374	#475	C₂²×C₃₁	1	4	yes	—
375	#464	C₂²×C₃₁	1	4	yes	—
376	#463	C₂²×C₃₁	1	4	yes	—
377	#462	C₂²×C₃₁	1	4	yes	—
378	#462	C₂²×C₃₁	1	4	yes	—
379	#461	C₂²×C₃₁	1	4	yes	—
380	#461	C₂²×C₃₁	1	4	yes	—
381	#446	C₂²×C₃₁	1	4	yes	—
382	#446	C₂²×C₃₁	1	4	yes	—
383	#442	C₂²×C₃₁	1	2	yes	—
384	#441	C₂²×C₃₁	1	2	yes	—
385	#441	C₂²×C₃₁	1	4	yes	—
386	#440	C₂²×C₃₁	1	2	yes	—
387	#440	C₂²×C₃₁	1	4	yes	—
388	#440	C₂²×C₃₁	1	2	yes	—
389	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
390	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
391	#428	C₂²×C₃₁	1	2	yes	—
392	#427	C₂²×C₃₁	1	1	yes	characteristic
393	#424	C₂²×C₃₁	1	2	yes	—
394	#423	C₂²×C₃₁	1	1	yes	characteristic
395	#419	C₂²×C₃₁	1	2	yes	—
396	#417	C₂²×C₃₁	1	2	yes	—
397	#416	C₂²×C₃₁	1	1	yes	characteristic
398	#416	C₂²×C₃₁	1	1	yes	characteristic
399	C₂×#103	C₂²×C₃₁	1	2	yes	—
400	C₂×#103	C₂²×C₃₁	1	2	yes	—
401	#405	C₂²×C₃₁	1	2	yes	—
402	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
403	C₂×C₃₁	—	2	2	no	—
404	C₂×C₃₁	—	4	2	no	—
405	C₂×C₃₁	—	8	1	no	—
406	C₂×C₃₁	C₂×#548	1	1	yes	characteristic
407	C₂×C₃₁	#3586	1	2	yes	—
408	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
409	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
410	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
411	#137	—	2	8	no	—
412	#135	—	2	4	no	—
413	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
414	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
415	C₂×SD₁₆	C₂³×C₃₁	1	1	yes	characteristic
416	C₂×SD₁₆	C₂³×C₃₁	1	1	yes	characteristic
417	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
418	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
419	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
420	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
421	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
422	C₄⋊Q₈	—	2	4	no	—
423	C₄⋊Q₈	—	2	2	no	—
424	C₄⋊Q₈	—	2	1	no	—
425	C₄⋊Q₈	—	2	1	no	—
426	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
427	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
428	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
429	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
430	#126	—	2	4	no	—
431	#126	—	2	4	no	—
432	#125	—	2	2	no	—
433	#125	—	2	4	no	—
434	#125	—	2	4	no	—
435	#125	—	2	4	no	—
436	#125	—	2	4	no	—
437	#125	—	2	4	no	—
438	#125	—	2	1	no	—
439	#125	—	2	1	no	—
440	#125	D₄×C₃₁	1	1	yes	characteristic
441	#125	D₄×C₃₁	1	1	yes	characteristic
442	#125	C₂³×C₃₁	1	4	yes	—
443	#125	C₂³×C₃₁	1	4	yes	—
444	#124	C₂³×C₃₁	1	4	yes	—
445	#123	—	2	4	no	—
446	C₂²⋊Q₈	—	2	4	no	—
447	C₂²⋊Q₈	—	2	4	no	—
448	C₂²⋊Q₈	—	2	2	no	—
449	C₂²⋊Q₈	—	2	2	no	—
450	C₂²⋊Q₈	—	2	4	no	—
451	C₂²⋊Q₈	—	2	4	no	—
452	C₂²⋊Q₈	—	2	4	no	—
453	C₂²⋊Q₈	D₄×C₃₁	1	4	yes	—
454	C₄×Q₈	—	2	4	no	—
455	C₄×Q₈	—	2	2	no	—
456	C₄×Q₈	—	2	2	no	—
457	C₄×Q₈	—	2	4	no	—
458	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
459	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
460	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
461	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
462	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
463	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
464	C₄×D₄	—	2	2	no	—
465	C₄×D₄	—	2	2	no	—
466	C₄×D₄	—	2	4	no	—
467	C₄×D₄	C₂³×C₃₁	1	2	yes	—
468	C₄×D₄	C₂³×C₃₁	1	2	yes	—
469	C₄×D₄	C₂³×C₃₁	1	4	yes	—
470	#117	—	2	2	no	—
471	#117	—	2	4	no	—
472	#117	—	2	4	no	—
473	#117	—	2	1	no	—
474	#117	—	2	1	no	—
475	#117	C₂³×C₃₁	1	1	yes	characteristic
476	#117	C₂³×C₃₁	1	1	yes	characteristic
477	#117	C₂³×C₃₁	1	2	yes	—
478	C₂×C₄⋊C₄	—	2	2	no	—
479	C₂×C₄⋊C₄	—	2	2	no	—
480	C₂×C₄⋊C₄	—	2	2	no	—
481	C₂×C₄⋊C₄	—	2	4	no	—
482	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
483	C₂×C₄²	—	2	1	no	—
484	C₂×C₄²	—	2	1	no	—
485	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
486	#107	C₂³×C₃₁	1	2	yes	—
487	#107	C₂³×C₃₁	1	4	yes	—
488	#106	C₂³×C₃₁	1	2	yes	—
489	#106	C₂³×C₃₁	1	4	yes	—
490	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
491	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
492	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
493	#103	C₂³×C₃₁	1	4	yes	—
494	#103	C₂³×C₃₁	1	4	yes	—
495	#103	C₂³×C₃₁	1	4	yes	—
496	#103	C₂³×C₃₁	1	4	yes	—
497	#103	C₂³×C₃₁	1	2	yes	—
498	#103	C₂³×C₃₁	1	2	yes	—
499	#103	C₂³×C₃₁	1	4	yes	—
500	#103	C₂³×C₃₁	1	4	yes	—
501	#103	C₂³×C₃₁	1	4	yes	—
502	#102	C₂³×C₃₁	1	2	yes	—
503	#102	C₂³×C₃₁	1	2	yes	—
504	#102	C₂³×C₃₁	1	4	yes	—
505	#97	C₂³×C₃₁	1	2	yes	—
506	#97	C₂³×C₃₁	1	2	yes	—
507	#97	C₂³×C₃₁	1	4	yes	—
508	C₄×C₈	C₂³×C₃₁	1	4	yes	—
509	C₃₁	#61338	1	1	yes	characteristic, direct product factor, Sylow
510	#41	—	2	2	no	—
511	#41	—	2	2	no	—
512	#41	—	2	2	no	—
513	#41	#41×C₃₁	1	4	yes	—
514	C₂×Q₈	—	2	2	no	—
515	C₂×Q₈	—	2	4	no	—
516	C₂×Q₈	—	2	4	no	—
517	C₂×Q₈	—	2	4	no	—
518	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
519	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
520	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
521	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
522	C₂×D₄	—	2	1	no	—
523	C₂×D₄	—	2	1	no	—
524	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
525	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
526	C₂²×C₄	—	2	1	no	—
527	C₂²×C₄	—	2	2	no	—
528	C₂²×C₄	—	2	2	no	—
529	C₂²×C₄	—	2	2	no	—
530	C₂²×C₄	—	2	1	no	—
531	C₂²×C₄	—	2	2	no	—
532	C₂²×C₄	—	2	4	no	—
533	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
534	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
535	Q₁₆	—	2	2	no	—
536	Q₁₆	—	2	8	no	—
537	SD₁₆	—	2	2	no	—
538	SD₁₆	—	2	2	no	—
539	SD₁₆	—	2	8	no	—
540	D₈	—	2	2	no	—
541	M₁₆	—	2	4	no	—
542	C₂×C₈	—	2	2	no	—
543	C₂×C₈	—	2	2	no	—
544	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
545	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
546	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
547	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
548	C₄⋊C₄	—	2	4	no	—
549	C₄⋊C₄	—	2	2	no	—
550	C₄⋊C₄	—	2	4	no	—
551	C₄⋊C₄	—	2	4	no	—
552	C₄⋊C₄	—	2	4	no	—
553	C₄⋊C₄	—	2	4	no	—
554	C₄⋊C₄	—	2	4	no	—
555	C₄⋊C₄	—	2	4	no	—
556	C₄⋊C₄	—	2	2	no	—
557	C₄⋊C₄	—	2	2	no	—
558	C₄⋊C₄	—	2	4	no	—
559	C₄⋊C₄	—	2	4	no	—
560	C₄⋊C₄	—	2	4	no	—
561	C₄⋊C₄	—	2	4	no	—
562	C₄⋊C₄	—	2	4	no	—
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	2	yes	—
566	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
567	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
568	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
569	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
570	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
571	C₂²⋊C₄	—	2	1	no	—
572	C₂²⋊C₄	—	2	1	no	—
573	C₂²⋊C₄	—	2	2	no	—
574	C₂²⋊C₄	—	2	4	no	—
575	C₂²⋊C₄	—	2	2	no	—
576	C₂²⋊C₄	—	2	4	no	—
577	C₂²⋊C₄	—	2	4	no	—
578	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
579	C₄²	—	2	1	no	—
580	C₄²	—	2	1	no	—
581	C₄²	—	2	2	no	—
582	C₄²	—	2	2	no	—
583	C₄²	—	2	4	no	—
584	C₄²	—	2	4	no	—
585	C₄²	—	2	4	no	—
586	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
587	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
588	C₄²	C₂⁴×C₃₁	1	4	yes	—
589	C₄²	C₂⁴×C₃₁	1	2	yes	—
590	C₂³	—	2	1	no	—
591	C₂³	—	2	1	no	—
592	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
593	Q₈	—	2	1	no	—
594	Q₈	—	2	4	no	—
595	Q₈	—	2	1	no	—
596	Q₈	—	2	4	no	—
597	Q₈	—	2	1	no	—
598	Q₈	—	4	4	no	—
599	Q₈	—	4	4	no	—
600	Q₈	—	8	4	no	—
601	Q₈	C₂×#41×C₃₁	1	2	yes	—
602	D₄	—	2	4	no	—
603	D₄	—	2	1	no	—
604	D₄	—	2	1	no	—
605	D₄	—	2	1	no	—
606	D₄	—	4	2	no	—
607	D₄	C₂×#41×C₃₁	1	2	yes	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	2	4	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	2	4	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	4	no	—
615	C₂×C₄	—	4	2	no	—
616	C₂×C₄	—	4	4	no	—
617	C₂×C₄	—	4	4	no	—
618	C₂×C₄	—	4	2	no	—
619	C₂×C₄	—	8	2	no	—
620	C₂×C₄	—	8	4	no	—
621	C₂×C₄	—	8	2	no	—
622	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
623	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
624	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
625	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
626	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
627	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
628	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
629	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
630	C₈	—	2	1	no	—
631	C₈	—	2	1	no	—
632	C₈	—	2	4	no	—
633	C₈	—	2	2	no	—
634	C₂²	—	2	2	no	—
635	C₂²	—	2	2	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	1	no	—
638	C₂²	—	4	2	no	—
639	C₂²	C₃₁×#548	1	2	yes	—
640	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
641	C₄	—	2	2	no	—
642	C₄	—	2	2	no	—
643	C₄	—	2	2	no	—
644	C₄	—	2	4	no	—
645	C₄	—	2	4	no	—
646	C₄	—	4	1	no	—
647	C₄	—	4	4	no	—
648	C₄	—	4	2	no	—
649	C₄	—	4	4	no	—
650	C₄	—	4	2	no	—
651	C₄	C₃₁×#548	1	2	yes	—
652	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
653	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
654	C₂	—	2	2	no	—
655	C₂	—	4	2	no	—
656	C₂	—	8	1	no	—
657	C₂	C₂×C₃₁×#548	1	1	yes	characteristic
658	C₂	C₃₁×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3509	C₂	semidirect
2. C₃₁×#3504	C₂	semidirect
3. C₃₁×#3377	C₂	semidirect
4. C₃₁×#3376	C₂	semidirect
5. C₃₁×#3374	C₂	semidirect
6. C₃₁×#3372	C₂	semidirect
7. C₃₁×#3137	C₂	semidirect
8. C₂×C₃₁×#491	C₂	semidirect
9. C₃₁×#3082	C₂	semidirect
10. C₃₁×#3080	C₂	semidirect
12. C₃₁×#3039	C₂	semidirect
13. C₃₁×#2961	C₂	semidirect
14. C₃₁×#2947	C₂	semidirect
260. #61338	C₃₁	direct (N × H)
305. #3509	C₂×C₃₁	semidirect
306. #3504	C₂×C₃₁	semidirect
307. #3377	C₂×C₃₁	semidirect
308. #3376	C₂×C₃₁	semidirect
309. #3374	C₂×C₃₁	semidirect
310. #3372	C₂×C₃₁	semidirect
311. #3137	C₂×C₃₁	semidirect
312. C₂×#491	C₂×C₃₁	semidirect
313. #3082	C₂×C₃₁	semidirect
314. #3080	C₂×C₃₁	semidirect
316. #3039	C₂×C₃₁	semidirect
317. #2961	C₂×C₃₁	semidirect
318. #2947	C₂×C₃₁	semidirect
509. C₃₁	#61338	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×#3509	2
C₃₁×#3504	2
C₃₁×#3377	2
C₃₁×#3376	2
C₃₁×#3374	2
C₃₁×#3372	2
C₃₁×#3137	2
C₂×C₃₁×#491	2
C₃₁×#3082	2
C₃₁×#3080	2
C₃₁×#3046	2
C₃₁×#3039	2
C₃₁×#2961	2
C₃₁×#2947	2
C₃₁×#2943	2
#61338	31

Subgroups — C31×#61338 #21518097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61338 #21518097