Subgroups — C₃₁×#61853 #21518612

Order 7936.

1648 conjugacy classes of subgroups.

The subgroup lattice has 702 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×#3502	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3387	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3386	C₂	1	2	yes	maximal, complemented
5	Q₈×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3371	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3369	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3364	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3244	C₂	1	4	yes	maximal, complemented
10	C₃₁×#3235	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#3231	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3121	C₂	1	4	yes	maximal, complemented
13	C₃₁×#3113	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#3103	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#2985	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#563	—	2	2	no	—
17	C₃₁×#559	C₂²	1	2	yes	—
18	Q₈²×C₃₁	—	2	1	no	—
19	Q₈²×C₃₁	C₂²	1	2	yes	—
20	Q₈²×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#557	—	2	1	no	—
22	C₃₁×#557	—	2	2	no	—
23	C₃₁×#557	C₂²	1	2	yes	—
24	C₃₁×#557	C₂²	1	4	yes	—
25	C₃₁×#556	—	2	1	no	—
26	C₃₁×#556	C₂²	1	2	yes	—
27	D₄×Q₈×C₃₁	—	2	1	no	—
28	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#544	—	2	2	no	—
30	C₃₁×#544	C₂²	1	2	yes	—
31	C₃₁×#542	—	2	2	no	—
32	C₃₁×#542	C₂²	1	4	yes	—
33	C₃₁×#536	—	2	2	no	—
34	C₃₁×#533	—	2	2	no	—
35	C₃₁×#533	C₂²	1	1	yes	characteristic
36	C₃₁×#533	C₂²	1	1	yes	characteristic
37	C₃₁×#533	C₂²	1	1	yes	characteristic
38	C₃₁×#532	C₂²	1	1	yes	characteristic
39	C₃₁×#519	C₂²	1	2	yes	—
40	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#501	C₂²	1	4	yes	—
42	C₃₁×#501	C₂²	1	2	yes	—
43	C₃₁×#500	C₂²	1	1	yes	characteristic
44	C₃₁×#498	C₂²	1	1	yes	characteristic
45	C₃₁×#498	C₂²	1	4	yes	—
46	C₃₁×#490	C₂²	1	2	yes	—
47	C₃₁×#489	C₂²	1	4	yes	—
48	C₃₁×#488	C₂²	1	4	yes	—
49	C₃₁×#487	C₂²	1	1	yes	characteristic
50	C₃₁×#487	C₂²	1	1	yes	characteristic
51	C₃₁×#485	C₂²	1	4	yes	—
52	C₃₁×#483	C₂²	1	2	yes	—
53	C₃₁×#482	C₂²	1	4	yes	—
54	C₃₁×#481	C₂²	1	2	yes	—
55	C₃₁×#479	C₂²	1	4	yes	—
56	C₃₁×#479	C₂²	1	4	yes	—
57	C₃₁×#478	C₂²	1	4	yes	—
58	C₃₁×#477	C₂²	1	4	yes	—
59	C₃₁×#476	C₂²	1	2	yes	—
60	C₃₁×#476	C₂²	1	4	yes	—
61	C₃₁×#475	C₂²	1	2	yes	—
62	C₃₁×#475	C₂²	1	2	yes	—
63	C₃₁×#475	C₂²	1	4	yes	—
64	C₃₁×#475	C₂²	1	4	yes	—
65	C₃₁×#474	C₂²	1	2	yes	—
66	C₃₁×#464	C₂²	1	4	yes	—
67	C₃₁×#464	C₂²	1	4	yes	—
68	C₃₁×#462	C₂²	1	2	yes	—
69	C₃₁×#460	C₂²	1	2	yes	—
70	C₃₁×#451	C₂²	1	2	yes	—
71	C₃₁×#448	C₂²	1	2	yes	—
72	C₃₁×#446	C₂²	1	2	yes	—
73	C₃₁×#446	C₂²	1	4	yes	—
74	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
75	C₃₁×#441	C₂²	1	4	yes	—
76	C₃₁×#441	C₂²	1	4	yes	—
77	C₃₁×#440	C₂²	1	4	yes	—
78	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
79	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
80	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
81	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
82	C₃₁×#433	C₂²	1	4	yes	—
83	C₃₁×#432	C₂²	1	1	yes	characteristic
84	C₃₁×#431	C₂²	1	1	yes	characteristic
85	C₂×#41×C₃₁	—	2	1	no	—
86	C₂²×Q₈×C₃₁	—	2	2	no	—
87	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
89	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
90	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
91	C₃₁×C₄⋊Q₈	—	2	2	no	—
92	C₃₁×C₄⋊Q₈	—	2	4	no	—
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	4	no	—
97	C₃₁×C₄⋊Q₈	—	2	2	no	—
98	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
99	C₃₁×C₄⋊Q₈	D₄	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×C₄⋊Q₈	C₂³	1	4	yes	—
105	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
106	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
107	C₃₁×#127	C₂³	1	1	yes	characteristic
108	C₃₁×#126	—	2	4	no	—
109	C₃₁×#126	—	2	4	no	—
110	C₃₁×#125	—	2	4	no	—
111	C₃₁×#125	—	2	4	no	—
112	C₃₁×#125	—	2	4	no	—
113	C₃₁×#125	—	2	2	no	—
114	C₃₁×#125	—	2	1	no	—
115	C₃₁×#125	D₄	1	2	yes	—
116	C₃₁×#125	C₂³	1	4	yes	—
117	C₃₁×#125	C₂³	1	2	yes	—
118	C₃₁×#124	—	2	2	no	—
119	C₃₁×#124	—	2	4	no	—
120	C₃₁×#124	—	2	1	no	—
121	C₃₁×#124	C₂³	1	4	yes	—
122	C₃₁×#123	—	2	4	no	—
123	C₃₁×#123	—	2	4	no	—
124	C₃₁×C₂²⋊Q₈	—	2	2	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	2	no	—
129	C₃₁×C₂²⋊Q₈	—	2	4	no	—
130	C₃₁×C₂²⋊Q₈	—	2	4	no	—
131	C₃₁×C₂²⋊Q₈	—	2	4	no	—
132	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
133	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
134	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
135	C₃₁×#121	—	2	1	no	—
136	C₃₁×#121	C₂³	1	2	yes	—
137	C₄×Q₈×C₃₁	—	2	2	no	—
138	C₄×Q₈×C₃₁	—	2	2	no	—
139	C₄×Q₈×C₃₁	—	2	2	no	—
140	C₄×Q₈×C₃₁	—	2	2	no	—
141	C₄×Q₈×C₃₁	—	2	4	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	C₂³	1	2	yes	—
149	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
150	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
151	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
152	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
153	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
154	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
155	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
156	C₄×D₄×C₃₁	—	2	2	no	—
157	C₄×D₄×C₃₁	—	2	1	no	—
158	C₄×D₄×C₃₁	C₂³	1	2	yes	—
159	C₄×D₄×C₃₁	C₂³	1	4	yes	—
160	C₃₁×#117	—	2	1	no	—
161	C₃₁×#117	—	2	2	no	—
162	C₃₁×#117	—	2	4	no	—
163	C₃₁×#117	C₂³	1	2	yes	—
164	C₃₁×#117	C₂³	1	2	yes	—
165	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
166	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
167	C₂×C₄²×C₃₁	—	2	1	no	—
168	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
169	C₃₁×#107	C₂³	1	4	yes	—
170	C₃₁×#107	C₂³	1	4	yes	—
171	C₃₁×#106	C₂³	1	2	yes	—
172	C₃₁×#106	C₂³	1	4	yes	—
173	C₃₁×#106	C₂³	1	2	yes	—
174	C₃₁×#106	C₂³	1	4	yes	—
175	C₃₁×#106	C₂³	1	4	yes	—
176	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
177	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
178	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
179	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
180	C₃₁×#103	C₂³	1	2	yes	—
181	C₃₁×#103	C₂³	1	4	yes	—
182	C₃₁×#103	C₂³	1	2	yes	—
183	C₃₁×#103	C₂³	1	4	yes	—
184	C₃₁×#103	C₂³	1	4	yes	—
185	C₃₁×#103	C₂³	1	4	yes	—
186	C₃₁×#102	C₂³	1	2	yes	—
187	C₃₁×#102	C₂³	1	2	yes	—
188	C₃₁×#102	C₂³	1	4	yes	—
189	C₃₁×#102	C₂³	1	4	yes	—
190	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
191	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
192	C₃₁×#97	C₂³	1	2	yes	—
193	C₃₁×#97	C₂³	1	4	yes	—
194	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
195	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
196	C₄×C₈×C₃₁	C₂³	1	4	yes	—
197	#41×C₃₁	—	8	2	no	—
198	C₂×Q₈×C₃₁	—	2	4	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	—	2	4	no	—
205	C₂×Q₈×C₃₁	—	2	1	no	—
206	C₂×Q₈×C₃₁	—	8	4	no	—
207	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
208	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
209	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
210	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
211	C₂×D₄×C₃₁	—	2	2	no	—
212	C₂×D₄×C₃₁	—	2	1	no	—
213	C₂×D₄×C₃₁	—	2	1	no	—
214	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
215	C₂²×C₄×C₃₁	—	2	1	no	—
216	C₂²×C₄×C₃₁	—	2	1	no	—
217	C₂²×C₄×C₃₁	—	2	2	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	Q₁₆×C₃₁	—	2	8	no	—
223	SD₁₆×C₃₁	—	2	4	no	—
224	SD₁₆×C₃₁	—	2	4	no	—
225	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
226	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
227	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
228	C₄⋊C₄×C₃₁	—	2	2	no	—
229	C₄⋊C₄×C₃₁	—	2	2	no	—
230	C₄⋊C₄×C₃₁	—	2	1	no	—
231	C₄⋊C₄×C₃₁	—	2	1	no	—
232	C₄⋊C₄×C₃₁	—	2	4	no	—
233	C₄⋊C₄×C₃₁	—	2	4	no	—
234	C₄⋊C₄×C₃₁	—	2	1	no	—
235	C₄⋊C₄×C₃₁	—	2	1	no	—
236	C₄⋊C₄×C₃₁	—	2	1	no	—
237	C₄⋊C₄×C₃₁	—	2	2	no	—
238	C₄⋊C₄×C₃₁	—	2	1	no	—
239	C₄⋊C₄×C₃₁	—	2	2	no	—
240	C₄⋊C₄×C₃₁	—	2	4	no	—
241	C₄⋊C₄×C₃₁	—	2	1	no	—
242	C₄⋊C₄×C₃₁	—	2	4	no	—
243	C₄⋊C₄×C₃₁	—	2	4	no	—
244	C₄⋊C₄×C₃₁	—	2	2	no	—
245	C₄⋊C₄×C₃₁	—	2	4	no	—
246	C₄⋊C₄×C₃₁	—	2	4	no	—
247	C₄⋊C₄×C₃₁	—	2	2	no	—
248	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
249	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
250	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
251	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
252	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
253	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
254	C₄⋊C₄×C₃₁	C₂⁴	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	C₂²⋊C₄×C₃₁	—	2	2	no	—
260	C₂²⋊C₄×C₃₁	—	2	2	no	—
261	C₂²⋊C₄×C₃₁	—	2	4	no	—
262	C₂²⋊C₄×C₃₁	—	2	2	no	—
263	C₂²⋊C₄×C₃₁	—	2	4	no	—
264	C₂²⋊C₄×C₃₁	—	2	2	no	—
265	C₂²⋊C₄×C₃₁	—	2	4	no	—
266	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
267	C₄²×C₃₁	—	2	2	no	—
268	C₄²×C₃₁	—	2	2	no	—
269	C₄²×C₃₁	—	2	2	no	—
270	C₄²×C₃₁	—	2	4	no	—
271	C₄²×C₃₁	—	2	4	no	—
272	C₄²×C₃₁	—	2	4	no	—
273	C₄²×C₃₁	—	2	2	no	—
274	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
275	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
276	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
277	C₄²×C₃₁	C₂⁴	1	4	yes	—
278	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
279	#61853	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂³×C₃₁	—	2	2	no	—
281	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
282	Q₈×C₃₁	—	2	4	no	—
283	Q₈×C₃₁	—	2	4	no	—
284	Q₈×C₃₁	—	2	2	no	—
285	Q₈×C₃₁	—	2	4	no	—
286	Q₈×C₃₁	—	4	2	no	—
287	Q₈×C₃₁	—	4	2	no	—
288	Q₈×C₃₁	—	4	2	no	—
289	Q₈×C₃₁	—	8	1	no	—
290	Q₈×C₃₁	—	8	4	no	—
291	Q₈×C₃₁	—	8	4	no	—
292	Q₈×C₃₁	—	8	1	no	—
293	Q₈×C₃₁	2₋¹⁺⁴	1	4	yes	—
294	D₄×C₃₁	—	2	2	no	—
295	D₄×C₃₁	—	2	2	no	—
296	D₄×C₃₁	—	4	2	no	—
297	D₄×C₃₁	—	8	1	no	—
298	D₄×C₃₁	—	8	2	no	—
299	C₂×C₄×C₃₁	—	2	4	no	—
300	C₂×C₄×C₃₁	—	2	2	no	—
301	C₂×C₄×C₃₁	—	2	2	no	—
302	C₂×C₄×C₃₁	—	2	2	no	—
303	C₂×C₄×C₃₁	—	2	2	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	—	4	1	no	—
310	C₂×C₄×C₃₁	—	4	4	no	—
311	C₂×C₄×C₃₁	—	8	4	no	—
312	C₂×C₄×C₃₁	—	8	1	no	—
313	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
314	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
315	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
316	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
317	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
318	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
319	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
320	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
321	C₈×C₃₁	—	2	2	no	—
322	C₈×C₃₁	—	2	2	no	—
323	C₈×C₃₁	—	2	4	no	—
324	#3550	C₂×C₃₁	1	1	yes	characteristic
325	#3502	C₂×C₃₁	1	1	yes	characteristic, complemented
326	#3387	C₂×C₃₁	1	4	yes	complemented
327	#3386	C₂×C₃₁	1	2	yes	complemented
328	Q₈×SD₁₆	C₂×C₃₁	1	2	yes	complemented
329	#3371	C₂×C₃₁	1	4	yes	complemented
330	#3369	C₂×C₃₁	1	2	yes	complemented
331	#3364	C₂×C₃₁	1	2	yes	complemented
332	#3244	C₂×C₃₁	1	4	yes	complemented
333	#3235	C₂×C₃₁	1	1	yes	characteristic
334	#3231	C₂×C₃₁	1	1	yes	characteristic, complemented
335	#3121	C₂×C₃₁	1	4	yes	complemented
336	#3113	C₂×C₃₁	1	1	yes	characteristic, complemented
337	#3103	C₂×C₃₁	1	1	yes	characteristic
338	#2985	C₂×C₃₁	1	1	yes	characteristic, complemented
339	C₂²×C₃₁	—	2	1	no	—
340	C₂²×C₃₁	—	2	1	no	—
341	C₂²×C₃₁	—	2	1	no	—
342	C₂²×C₃₁	—	4	2	no	—
343	C₂²×C₃₁	—	4	2	no	—
344	C₂²×C₃₁	—	4	2	no	—
345	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
346	C₄×C₃₁	—	2	1	no	—
347	C₄×C₃₁	—	2	1	no	—
348	C₄×C₃₁	—	2	4	no	—
349	C₄×C₃₁	—	2	4	no	—
350	C₄×C₃₁	—	2	1	no	—
351	C₄×C₃₁	—	2	1	no	—
352	C₄×C₃₁	—	2	4	no	—
353	C₄×C₃₁	—	4	2	no	—
354	C₄×C₃₁	—	4	2	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	#577	1	2	yes	—
360	C₄×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
361	#563	—	2	2	no	—
362	#559	C₂²×C₃₁	1	2	yes	—
363	Q₈²	—	2	1	no	—
364	Q₈²	C₂²×C₃₁	1	2	yes	—
365	Q₈²	C₂²×C₃₁	1	2	yes	—
366	#557	—	2	1	no	—
367	#557	—	2	2	no	—
368	#557	C₂²×C₃₁	1	2	yes	—
369	#557	C₂²×C₃₁	1	4	yes	—
370	#556	—	2	1	no	—
371	#556	C₂²×C₃₁	1	2	yes	—
372	D₄×Q₈	—	2	1	no	—
373	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
374	#544	—	2	2	no	—
375	#544	C₂²×C₃₁	1	2	yes	—
376	#542	—	2	2	no	—
377	#542	C₂²×C₃₁	1	4	yes	—
378	#536	—	2	2	no	—
379	#533	—	2	2	no	—
380	#533	C₂²×C₃₁	1	1	yes	characteristic
381	#533	C₂²×C₃₁	1	1	yes	characteristic
382	#533	C₂²×C₃₁	1	1	yes	characteristic
383	#532	C₂²×C₃₁	1	1	yes	characteristic
384	#519	C₂²×C₃₁	1	2	yes	—
385	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
386	#501	C₂²×C₃₁	1	4	yes	—
387	#501	C₂²×C₃₁	1	2	yes	—
388	#500	C₂²×C₃₁	1	1	yes	characteristic
389	#498	C₂²×C₃₁	1	1	yes	characteristic
390	#498	C₂²×C₃₁	1	4	yes	—
391	#490	C₂²×C₃₁	1	2	yes	—
392	#489	C₂²×C₃₁	1	4	yes	—
393	#488	C₂²×C₃₁	1	4	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	#483	C₂²×C₃₁	1	2	yes	—
398	#482	C₂²×C₃₁	1	4	yes	—
399	#481	C₂²×C₃₁	1	2	yes	—
400	#479	C₂²×C₃₁	1	4	yes	—
401	#479	C₂²×C₃₁	1	4	yes	—
402	#478	C₂²×C₃₁	1	4	yes	—
403	#477	C₂²×C₃₁	1	4	yes	—
404	#476	C₂²×C₃₁	1	2	yes	—
405	#476	C₂²×C₃₁	1	4	yes	—
406	#475	C₂²×C₃₁	1	2	yes	—
407	#475	C₂²×C₃₁	1	2	yes	—
408	#475	C₂²×C₃₁	1	4	yes	—
409	#475	C₂²×C₃₁	1	4	yes	—
410	#474	C₂²×C₃₁	1	2	yes	—
411	#464	C₂²×C₃₁	1	4	yes	—
412	#464	C₂²×C₃₁	1	4	yes	—
413	#462	C₂²×C₃₁	1	2	yes	—
414	#460	C₂²×C₃₁	1	2	yes	—
415	#451	C₂²×C₃₁	1	2	yes	—
416	#448	C₂²×C₃₁	1	2	yes	—
417	#446	C₂²×C₃₁	1	2	yes	—
418	#446	C₂²×C₃₁	1	4	yes	—
419	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
420	#441	C₂²×C₃₁	1	4	yes	—
421	#441	C₂²×C₃₁	1	4	yes	—
422	#440	C₂²×C₃₁	1	4	yes	—
423	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
424	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
425	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
426	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
427	#433	C₂²×C₃₁	1	4	yes	—
428	#432	C₂²×C₃₁	1	1	yes	characteristic
429	#431	C₂²×C₃₁	1	1	yes	characteristic
430	C₂×C₃₁	—	4	1	no	—
431	C₂×C₃₁	—	8	2	no	—
432	C₂×C₃₁	#3486	1	1	yes	characteristic
433	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
434	C₂×C₃₁	#3586	1	1	yes	characteristic
435	C₂×#41	—	2	1	no	—
436	C₂²×Q₈	—	2	2	no	—
437	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
438	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
439	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
440	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
441	C₄⋊Q₈	—	2	2	no	—
442	C₄⋊Q₈	—	2	4	no	—
443	C₄⋊Q₈	—	2	2	no	—
444	C₄⋊Q₈	—	2	4	no	—
445	C₄⋊Q₈	—	2	2	no	—
446	C₄⋊Q₈	—	2	4	no	—
447	C₄⋊Q₈	—	2	2	no	—
448	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
449	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
450	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
451	C₄⋊Q₈	C₂³×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	4	yes	—
456	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
457	#127	C₂³×C₃₁	1	1	yes	characteristic
458	#126	—	2	4	no	—
459	#126	—	2	4	no	—
460	#125	—	2	4	no	—
461	#125	—	2	4	no	—
462	#125	—	2	4	no	—
463	#125	—	2	2	no	—
464	#125	—	2	1	no	—
465	#125	D₄×C₃₁	1	2	yes	—
466	#125	C₂³×C₃₁	1	2	yes	—
467	#125	C₂³×C₃₁	1	4	yes	—
468	#124	—	2	2	no	—
469	#124	—	2	4	no	—
470	#124	—	2	1	no	—
471	#124	C₂³×C₃₁	1	4	yes	—
472	#123	—	2	4	no	—
473	#123	—	2	4	no	—
474	C₂²⋊Q₈	—	2	2	no	—
475	C₂²⋊Q₈	—	2	1	no	—
476	C₂²⋊Q₈	—	2	1	no	—
477	C₂²⋊Q₈	—	2	1	no	—
478	C₂²⋊Q₈	—	2	2	no	—
479	C₂²⋊Q₈	—	2	4	no	—
480	C₂²⋊Q₈	—	2	4	no	—
481	C₂²⋊Q₈	—	2	4	no	—
482	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
483	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
484	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
485	#121	—	2	1	no	—
486	#121	C₂³×C₃₁	1	2	yes	—
487	C₄×Q₈	—	2	2	no	—
488	C₄×Q₈	—	2	2	no	—
489	C₄×Q₈	—	2	2	no	—
490	C₄×Q₈	—	2	2	no	—
491	C₄×Q₈	—	2	4	no	—
492	C₄×Q₈	—	2	2	no	—
493	C₄×Q₈	—	2	1	no	—
494	C₄×Q₈	—	2	1	no	—
495	C₄×Q₈	—	2	1	no	—
496	C₄×Q₈	—	2	1	no	—
497	C₄×Q₈	—	2	1	no	—
498	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
499	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
500	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
501	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
502	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
503	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
504	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
505	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
506	C₄×D₄	—	2	2	no	—
507	C₄×D₄	—	2	1	no	—
508	C₄×D₄	C₂³×C₃₁	1	2	yes	—
509	C₄×D₄	C₂³×C₃₁	1	4	yes	—
510	#117	—	2	1	no	—
511	#117	—	2	2	no	—
512	#117	—	2	4	no	—
513	#117	C₂³×C₃₁	1	2	yes	—
514	#117	C₂³×C₃₁	1	2	yes	—
515	C₂×C₄⋊C₄	—	2	1	no	—
516	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
517	C₂×C₄²	—	2	1	no	—
518	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
519	#107	C₂³×C₃₁	1	4	yes	—
520	#107	C₂³×C₃₁	1	4	yes	—
521	#106	C₂³×C₃₁	1	2	yes	—
522	#106	C₂³×C₃₁	1	4	yes	—
523	#106	C₂³×C₃₁	1	2	yes	—
524	#106	C₂³×C₃₁	1	4	yes	—
525	#106	C₂³×C₃₁	1	4	yes	—
526	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
527	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
528	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
529	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
530	#103	C₂³×C₃₁	1	2	yes	—
531	#103	C₂³×C₃₁	1	4	yes	—
532	#103	C₂³×C₃₁	1	2	yes	—
533	#103	C₂³×C₃₁	1	4	yes	—
534	#103	C₂³×C₃₁	1	4	yes	—
535	#103	C₂³×C₃₁	1	4	yes	—
536	#102	C₂³×C₃₁	1	2	yes	—
537	#102	C₂³×C₃₁	1	2	yes	—
538	#102	C₂³×C₃₁	1	4	yes	—
539	#102	C₂³×C₃₁	1	4	yes	—
540	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
541	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
542	#97	C₂³×C₃₁	1	2	yes	—
543	#97	C₂³×C₃₁	1	4	yes	—
544	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
545	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
546	C₄×C₈	C₂³×C₃₁	1	4	yes	—
547	C₃₁	#61853	1	1	yes	characteristic, direct product factor, Sylow
548	#41	—	8	2	no	—
549	C₂×Q₈	—	2	4	no	—
550	C₂×Q₈	—	2	1	no	—
551	C₂×Q₈	—	2	1	no	—
552	C₂×Q₈	—	2	1	no	—
553	C₂×Q₈	—	2	1	no	—
554	C₂×Q₈	—	2	4	no	—
555	C₂×Q₈	—	2	4	no	—
556	C₂×Q₈	—	2	1	no	—
557	C₂×Q₈	—	8	4	no	—
558	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
559	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
560	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
561	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
562	C₂×D₄	—	2	2	no	—
563	C₂×D₄	—	2	1	no	—
564	C₂×D₄	—	2	1	no	—
565	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
566	C₂²×C₄	—	2	1	no	—
567	C₂²×C₄	—	2	1	no	—
568	C₂²×C₄	—	2	2	no	—
569	C₂²×C₄	—	2	4	no	—
570	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
571	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
572	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
573	Q₁₆	—	2	8	no	—
574	SD₁₆	—	2	4	no	—
575	SD₁₆	—	2	4	no	—
576	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
577	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
578	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
579	C₄⋊C₄	—	2	2	no	—
580	C₄⋊C₄	—	2	2	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	2	1	no	—
583	C₄⋊C₄	—	2	4	no	—
584	C₄⋊C₄	—	2	4	no	—
585	C₄⋊C₄	—	2	1	no	—
586	C₄⋊C₄	—	2	1	no	—
587	C₄⋊C₄	—	2	1	no	—
588	C₄⋊C₄	—	2	2	no	—
589	C₄⋊C₄	—	2	1	no	—
590	C₄⋊C₄	—	2	2	no	—
591	C₄⋊C₄	—	2	4	no	—
592	C₄⋊C₄	—	2	1	no	—
593	C₄⋊C₄	—	2	4	no	—
594	C₄⋊C₄	—	2	4	no	—
595	C₄⋊C₄	—	2	2	no	—
596	C₄⋊C₄	—	2	4	no	—
597	C₄⋊C₄	—	2	4	no	—
598	C₄⋊C₄	—	2	2	no	—
599	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
600	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
601	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
602	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
603	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
604	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
605	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
606	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
607	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
608	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
609	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
610	C₂²⋊C₄	—	2	2	no	—
611	C₂²⋊C₄	—	2	2	no	—
612	C₂²⋊C₄	—	2	4	no	—
613	C₂²⋊C₄	—	2	2	no	—
614	C₂²⋊C₄	—	2	4	no	—
615	C₂²⋊C₄	—	2	2	no	—
616	C₂²⋊C₄	—	2	4	no	—
617	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
618	C₄²	—	2	2	no	—
619	C₄²	—	2	2	no	—
620	C₄²	—	2	2	no	—
621	C₄²	—	2	4	no	—
622	C₄²	—	2	4	no	—
623	C₄²	—	2	4	no	—
624	C₄²	—	2	2	no	—
625	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
626	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
627	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
628	C₄²	C₂⁴×C₃₁	1	4	yes	—
629	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
630	C₂³	—	2	2	no	—
631	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
632	Q₈	—	2	4	no	—
633	Q₈	—	2	4	no	—
634	Q₈	—	2	2	no	—
635	Q₈	—	2	4	no	—
636	Q₈	—	4	2	no	—
637	Q₈	—	4	2	no	—
638	Q₈	—	4	2	no	—
639	Q₈	—	8	1	no	—
640	Q₈	—	8	4	no	—
641	Q₈	—	8	4	no	—
642	Q₈	—	8	1	no	—
643	Q₈	C₃₁×2₋¹⁺⁴	1	4	yes	—
644	D₄	—	2	2	no	—
645	D₄	—	2	2	no	—
646	D₄	—	4	2	no	—
647	D₄	—	8	1	no	—
648	D₄	—	8	2	no	—
649	C₂×C₄	—	2	4	no	—
650	C₂×C₄	—	2	2	no	—
651	C₂×C₄	—	2	2	no	—
652	C₂×C₄	—	2	2	no	—
653	C₂×C₄	—	2	2	no	—
654	C₂×C₄	—	2	2	no	—
655	C₂×C₄	—	2	2	no	—
656	C₂×C₄	—	2	2	no	—
657	C₂×C₄	—	2	2	no	—
658	C₂×C₄	—	2	2	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	—	4	4	no	—
661	C₂×C₄	—	8	4	no	—
662	C₂×C₄	—	8	1	no	—
663	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
664	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
665	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
666	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
667	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
668	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
669	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
670	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
671	C₈	—	2	2	no	—
672	C₈	—	2	2	no	—
673	C₈	—	2	4	no	—
674	C₂²	—	2	1	no	—
675	C₂²	—	2	1	no	—
676	C₂²	—	2	1	no	—
677	C₂²	—	4	2	no	—
678	C₂²	—	4	2	no	—
679	C₂²	—	4	2	no	—
680	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
681	C₄	—	2	4	no	—
682	C₄	—	2	1	no	—
683	C₄	—	2	1	no	—
684	C₄	—	2	4	no	—
685	C₄	—	2	1	no	—
686	C₄	—	2	1	no	—
687	C₄	—	2	4	no	—
688	C₄	—	4	2	no	—
689	C₄	—	4	2	no	—
690	C₄	—	4	2	no	—
691	C₄	—	4	2	no	—
692	C₄	—	4	2	no	—
693	C₄	—	4	2	no	—
694	C₄	C₃₁×#577	1	2	yes	—
695	C₄	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
696	C₂	—	4	1	no	—
697	C₂	—	8	2	no	—
698	C₂	C₃₁×#3486	1	1	yes	characteristic
699	C₂	C₂×C₃₁×#577	1	1	yes	characteristic
700	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×#533 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3502	C₂	semidirect
3. C₃₁×#3387	C₂	semidirect
4. C₃₁×#3386	C₂	semidirect
5. Q₈×SD₁₆×C₃₁	C₂	semidirect
6. C₃₁×#3371	C₂	semidirect
7. C₃₁×#3369	C₂	semidirect
8. C₃₁×#3364	C₂	semidirect
9. C₃₁×#3244	C₂	semidirect
11. C₃₁×#3231	C₂	semidirect
12. C₃₁×#3121	C₂	semidirect
13. C₃₁×#3113	C₂	semidirect
15. C₃₁×#2985	C₂	semidirect
279. #61853	C₃₁	direct (N × H)
325. #3502	C₂×C₃₁	semidirect
326. #3387	C₂×C₃₁	semidirect
327. #3386	C₂×C₃₁	semidirect
328. Q₈×SD₁₆	C₂×C₃₁	semidirect
329. #3371	C₂×C₃₁	semidirect
330. #3369	C₂×C₃₁	semidirect
331. #3364	C₂×C₃₁	semidirect
332. #3244	C₂×C₃₁	semidirect
334. #3231	C₂×C₃₁	semidirect
335. #3121	C₂×C₃₁	semidirect
336. #3113	C₂×C₃₁	semidirect
338. #2985	C₂×C₃₁	semidirect
547. C₃₁	#61853	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×#3502	2
C₃₁×#3387	2
C₃₁×#3386	2
Q₈×SD₁₆×C₃₁	2
C₃₁×#3371	2
C₃₁×#3369	2
C₃₁×#3364	2
C₃₁×#3244	2
C₃₁×#3235	2
C₃₁×#3231	2
C₃₁×#3121	2
C₃₁×#3113	2
C₃₁×#3103	2
C₃₁×#2985	2
#61853	31

Subgroups — C31×#61853 #21518612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61853 #21518612