Subgroups — C₃₁×#61915 #21518674

Order 7936.

1640 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3507	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3400	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3398	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3395	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3377	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3374	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3365	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3223	C₂	1	4	yes	maximal, complemented
10	C₃₁×#3216	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#3210	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#3120	C₂	1	4	yes	maximal, complemented
13	C₃₁×#3113	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#3104	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#2985	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#564	—	2	2	no	—
17	C₃₁×#559	C₂²	1	2	yes	—
18	C₃₁×#557	—	2	1	no	—
19	C₃₁×#557	—	2	1	no	—
20	C₃₁×#557	C₂²	1	2	yes	—
21	C₃₁×#557	C₂²	1	2	yes	—
22	C₃₁×#557	C₂²	1	2	yes	—
23	C₃₁×#557	C₂²	1	2	yes	—
24	C₃₁×#557	C₂²	1	2	yes	—
25	C₃₁×#556	—	2	1	no	—
26	C₃₁×#556	C₂²	1	4	yes	—
27	C₃₁×#554	—	2	1	no	—
28	C₃₁×#552	—	2	2	no	—
29	C₃₁×#541	—	2	2	no	—
30	C₃₁×#541	—	2	2	no	—
31	C₃₁×#541	C₂²	1	4	yes	—
32	C₃₁×#539	—	2	2	no	—
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×#497	C₂²	1	2	yes	—
42	C₃₁×#497	C₂²	1	4	yes	—
43	C₃₁×#494	C₂²	1	1	yes	characteristic
44	C₃₁×#494	C₂²	1	4	yes	—
45	C₃₁×#492	C₂²	1	1	yes	characteristic
46	C₃₁×#491	C₂²	1	4	yes	—
47	C₃₁×#489	C₂²	1	2	yes	—
48	C₃₁×#488	C₂²	1	4	yes	—
49	C₃₁×#487	C₂²	1	1	yes	characteristic
50	C₃₁×#486	C₂²	1	1	yes	characteristic
51	C₃₁×#484	C₂²	1	2	yes	—
52	C₃₁×#483	C₂²	1	2	yes	—
53	C₃₁×#479	C₂²	1	4	yes	—
54	C₃₁×#478	C₂²	1	4	yes	—
55	C₃₁×#477	C₂²	1	2	yes	—
56	C₃₁×#475	C₂²	1	2	yes	—
57	C₃₁×#464	C₂²	1	4	yes	—
58	C₃₁×#464	C₂²	1	4	yes	—
59	C₃₁×#464	C₂²	1	4	yes	—
60	C₃₁×#462	C₂²	1	2	yes	—
61	C₃₁×#462	C₂²	1	2	yes	—
62	C₃₁×#462	C₂²	1	2	yes	—
63	C₃₁×#462	C₂²	1	4	yes	—
64	C₃₁×#462	C₂²	1	4	yes	—
65	C₃₁×#462	C₂²	1	4	yes	—
66	C₃₁×#461	C₂²	1	4	yes	—
67	C₃₁×#460	C₂²	1	2	yes	—
68	C₃₁×#452	C₂²	1	4	yes	—
69	C₃₁×#452	C₂²	1	4	yes	—
70	C₃₁×#451	C₂²	1	2	yes	—
71	C₃₁×#451	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	2	yes	—
79	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
80	C₄×Q₁₆×C₃₁	C₂²	1	4	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	C₂³	1	2	yes	—
86	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
88	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
89	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
90	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
91	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
92	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
93	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
94	C₃₁×C₄⋊Q₈	—	2	2	no	—
95	C₃₁×C₄⋊Q₈	—	2	2	no	—
96	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
97	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
98	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
99	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
100	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
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	1	yes	characteristic
105	C₃₁×#127	C₂³	1	1	yes	characteristic
106	C₃₁×#126	—	2	4	no	—
107	C₃₁×#126	—	2	4	no	—
108	C₃₁×#126	—	2	4	no	—
109	C₃₁×#126	—	2	4	no	—
110	C₃₁×#125	—	2	2	no	—
111	C₃₁×#125	—	2	4	no	—
112	C₃₁×#125	—	2	2	no	—
113	C₃₁×#125	—	2	2	no	—
114	C₃₁×#125	—	2	4	no	—
115	C₃₁×#125	—	2	4	no	—
116	C₃₁×#125	—	2	4	no	—
117	C₃₁×#125	—	2	4	no	—
118	C₃₁×#125	—	2	4	no	—
119	C₃₁×#125	—	2	4	no	—
120	C₃₁×#125	—	2	1	no	—
121	C₃₁×#125	—	2	1	no	—
122	C₃₁×#125	D₄	1	2	yes	—
123	C₃₁×#125	C₂³	1	4	yes	—
124	C₃₁×#124	—	2	2	no	—
125	C₃₁×#124	C₂³	1	4	yes	—
126	C₃₁×#124	C₂³	1	2	yes	—
127	C₃₁×#123	—	2	4	no	—
128	C₃₁×#123	—	2	4	no	—
129	C₃₁×C₂²⋊Q₈	—	2	4	no	—
130	C₃₁×C₂²⋊Q₈	—	2	1	no	—
131	C₃₁×C₂²⋊Q₈	—	2	1	no	—
132	C₃₁×C₂²⋊Q₈	—	2	2	no	—
133	C₃₁×C₂²⋊Q₈	—	2	2	no	—
134	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
135	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
136	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
137	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
138	C₃₁×#121	—	2	1	no	—
139	C₃₁×#121	—	2	1	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	2	no	—
144	C₄×Q₈×C₃₁	—	2	2	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	4	yes	—
150	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
151	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
152	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
153	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
154	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
155	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
156	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
157	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
158	C₄×D₄×C₃₁	—	2	2	no	—
159	C₄×D₄×C₃₁	—	2	4	no	—
160	C₄×D₄×C₃₁	—	2	1	no	—
161	C₄×D₄×C₃₁	—	2	1	no	—
162	C₄×D₄×C₃₁	C₂³	1	4	yes	—
163	C₃₁×#117	—	2	1	no	—
164	C₃₁×#117	—	2	1	no	—
165	C₃₁×#117	—	2	2	no	—
166	C₃₁×#117	—	2	4	no	—
167	C₃₁×#117	C₂³	1	2	yes	—
168	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
169	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
170	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
171	C₂×C₄²×C₃₁	—	2	1	no	—
172	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
173	C₃₁×#107	C₂³	1	2	yes	—
174	C₃₁×#107	C₂³	1	2	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	2	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	4	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	—	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	—	4	2	no	—
202	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
203	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
204	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
205	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
206	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
207	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
208	C₂×D₄×C₃₁	—	2	2	no	—
209	C₂×D₄×C₃₁	—	2	1	no	—
210	C₂×D₄×C₃₁	—	2	1	no	—
211	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
212	C₂²×C₄×C₃₁	—	2	1	no	—
213	C₂²×C₄×C₃₁	—	2	1	no	—
214	C₂²×C₄×C₃₁	—	2	2	no	—
215	C₂²×C₄×C₃₁	—	2	4	no	—
216	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
217	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
218	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
219	Q₁₆×C₃₁	—	4	2	no	—
220	Q₁₆×C₃₁	—	4	4	no	—
221	Q₁₆×C₃₁	—	4	4	no	—
222	Q₁₆×C₃₁	—	4	2	no	—
223	Q₁₆×C₃₁	—	4	4	no	—
224	SD₁₆×C₃₁	—	4	4	no	—
225	SD₁₆×C₃₁	—	4	4	no	—
226	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
227	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
228	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
229	C₄⋊C₄×C₃₁	—	2	1	no	—
230	C₄⋊C₄×C₃₁	—	2	4	no	—
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	1	no	—
235	C₄⋊C₄×C₃₁	—	2	1	no	—
236	C₄⋊C₄×C₃₁	—	2	1	no	—
237	C₄⋊C₄×C₃₁	—	2	1	no	—
238	C₄⋊C₄×C₃₁	—	2	1	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	4	no	—
243	C₄⋊C₄×C₃₁	—	2	2	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	4	no	—
248	C₄⋊C₄×C₃₁	—	2	4	no	—
249	C₄⋊C₄×C₃₁	—	2	2	no	—
250	C₄⋊C₄×C₃₁	—	2	2	no	—
251	C₄⋊C₄×C₃₁	—	2	4	no	—
252	C₄⋊C₄×C₃₁	—	2	4	no	—
253	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
254	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
255	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
256	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
257	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
258	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
259	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
260	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
261	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
262	C₂²⋊C₄×C₃₁	—	2	2	no	—
263	C₂²⋊C₄×C₃₁	—	2	2	no	—
264	C₂²⋊C₄×C₃₁	—	2	2	no	—
265	C₂²⋊C₄×C₃₁	—	2	4	no	—
266	C₂²⋊C₄×C₃₁	—	2	4	no	—
267	C₂²⋊C₄×C₃₁	—	2	2	no	—
268	C₂²⋊C₄×C₃₁	—	2	4	no	—
269	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
270	C₄²×C₃₁	—	2	2	no	—
271	C₄²×C₃₁	—	2	2	no	—
272	C₄²×C₃₁	—	2	4	no	—
273	C₄²×C₃₁	—	2	2	no	—
274	C₄²×C₃₁	—	2	2	no	—
275	C₄²×C₃₁	—	2	4	no	—
276	C₄²×C₃₁	—	2	4	no	—
277	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
278	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
279	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
280	C₄²×C₃₁	C₂⁴	1	4	yes	—
281	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
282	#61915	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
283	C₂³×C₃₁	—	2	2	no	—
284	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
285	Q₈×C₃₁	—	2	4	no	—
286	Q₈×C₃₁	—	2	2	no	—
287	Q₈×C₃₁	—	2	4	no	—
288	Q₈×C₃₁	—	2	2	no	—
289	Q₈×C₃₁	—	2	2	no	—
290	Q₈×C₃₁	—	2	2	no	—
291	Q₈×C₃₁	—	2	4	no	—
292	Q₈×C₃₁	—	4	2	no	—
293	Q₈×C₃₁	—	4	2	no	—
294	Q₈×C₃₁	—	4	2	no	—
295	Q₈×C₃₁	—	4	2	no	—
296	D₄×C₃₁	—	2	4	no	—
297	D₄×C₃₁	—	4	4	no	—
298	D₄×C₃₁	—	4	2	no	—
299	D₄×C₃₁	—	8	1	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	—	4	2	no	—
312	C₂×C₄×C₃₁	—	8	4	no	—
313	C₂×C₄×C₃₁	—	8	1	no	—
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	C₂²×D₄	1	1	yes	characteristic
321	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
322	C₈×C₃₁	—	2	2	no	—
323	C₈×C₃₁	—	2	4	no	—
324	C₈×C₃₁	—	2	2	no	—
325	#3556	C₂×C₃₁	1	1	yes	characteristic, complemented
326	#3507	C₂×C₃₁	1	1	yes	characteristic
327	#3400	C₂×C₃₁	1	2	yes	complemented
328	#3398	C₂×C₃₁	1	4	yes	complemented
329	#3395	C₂×C₃₁	1	2	yes	complemented
330	#3377	C₂×C₃₁	1	4	yes	complemented
331	#3374	C₂×C₃₁	1	2	yes	complemented
332	#3365	C₂×C₃₁	1	2	yes	complemented
333	#3223	C₂×C₃₁	1	4	yes	complemented
334	#3216	C₂×C₃₁	1	1	yes	characteristic
335	#3210	C₂×C₃₁	1	1	yes	characteristic
336	#3120	C₂×C₃₁	1	4	yes	complemented
337	#3113	C₂×C₃₁	1	1	yes	characteristic, complemented
338	#3104	C₂×C₃₁	1	1	yes	characteristic, complemented
339	#2985	C₂×C₃₁	1	1	yes	characteristic, complemented
340	C₂²×C₃₁	—	2	1	no	—
341	C₂²×C₃₁	—	2	1	no	—
342	C₂²×C₃₁	—	2	1	no	—
343	C₂²×C₃₁	—	4	2	no	—
344	C₂²×C₃₁	—	4	2	no	—
345	C₂²×C₃₁	—	4	2	no	—
346	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
347	C₄×C₃₁	—	2	1	no	—
348	C₄×C₃₁	—	2	1	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	—	4	2	no	—
360	C₄×C₃₁	#578	1	2	yes	—
361	C₄×C₃₁	#585	1	2	yes	—
362	#564	—	2	2	no	—
363	#559	C₂²×C₃₁	1	2	yes	—
364	#557	—	2	1	no	—
365	#557	—	2	1	no	—
366	#557	C₂²×C₃₁	1	2	yes	—
367	#557	C₂²×C₃₁	1	2	yes	—
368	#557	C₂²×C₃₁	1	2	yes	—
369	#557	C₂²×C₃₁	1	2	yes	—
370	#557	C₂²×C₃₁	1	2	yes	—
371	#556	—	2	1	no	—
372	#556	C₂²×C₃₁	1	4	yes	—
373	#554	—	2	1	no	—
374	#552	—	2	2	no	—
375	#541	—	2	2	no	—
376	#541	—	2	2	no	—
377	#541	C₂²×C₃₁	1	4	yes	—
378	#539	—	2	2	no	—
379	#536	C₂²×C₃₁	1	2	yes	—
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	#529	—	2	2	no	—
385	#520	C₂²×C₃₁	1	2	yes	—
386	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
387	#497	C₂²×C₃₁	1	2	yes	—
388	#497	C₂²×C₃₁	1	4	yes	—
389	#494	C₂²×C₃₁	1	1	yes	characteristic
390	#494	C₂²×C₃₁	1	4	yes	—
391	#492	C₂²×C₃₁	1	1	yes	characteristic
392	#491	C₂²×C₃₁	1	4	yes	—
393	#489	C₂²×C₃₁	1	2	yes	—
394	#488	C₂²×C₃₁	1	4	yes	—
395	#487	C₂²×C₃₁	1	1	yes	characteristic
396	#486	C₂²×C₃₁	1	1	yes	characteristic
397	#484	C₂²×C₃₁	1	2	yes	—
398	#483	C₂²×C₃₁	1	2	yes	—
399	#479	C₂²×C₃₁	1	4	yes	—
400	#478	C₂²×C₃₁	1	4	yes	—
401	#477	C₂²×C₃₁	1	2	yes	—
402	#475	C₂²×C₃₁	1	2	yes	—
403	#464	C₂²×C₃₁	1	4	yes	—
404	#464	C₂²×C₃₁	1	4	yes	—
405	#464	C₂²×C₃₁	1	4	yes	—
406	#462	C₂²×C₃₁	1	2	yes	—
407	#462	C₂²×C₃₁	1	2	yes	—
408	#462	C₂²×C₃₁	1	2	yes	—
409	#462	C₂²×C₃₁	1	4	yes	—
410	#462	C₂²×C₃₁	1	4	yes	—
411	#462	C₂²×C₃₁	1	4	yes	—
412	#461	C₂²×C₃₁	1	4	yes	—
413	#460	C₂²×C₃₁	1	2	yes	—
414	#452	C₂²×C₃₁	1	4	yes	—
415	#452	C₂²×C₃₁	1	4	yes	—
416	#451	C₂²×C₃₁	1	2	yes	—
417	#451	C₂²×C₃₁	1	2	yes	—
418	#446	C₂²×C₃₁	1	2	yes	—
419	#446	C₂²×C₃₁	1	4	yes	—
420	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
421	#441	C₂²×C₃₁	1	4	yes	—
422	#441	C₂²×C₃₁	1	4	yes	—
423	#440	C₂²×C₃₁	1	4	yes	—
424	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
425	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
426	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
427	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
428	#433	C₂²×C₃₁	1	4	yes	—
429	#432	C₂²×C₃₁	1	1	yes	characteristic
430	#431	C₂²×C₃₁	1	1	yes	characteristic
431	C₂×C₃₁	—	4	1	no	—
432	C₂×C₃₁	—	8	2	no	—
433	C₂×C₃₁	#3487	1	1	yes	characteristic
434	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
435	C₂×C₃₁	#3586	1	1	yes	characteristic
436	C₂×#41	C₂³×C₃₁	1	2	yes	—
437	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
438	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
439	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
440	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
441	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
442	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
443	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
444	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
445	C₄⋊Q₈	—	2	2	no	—
446	C₄⋊Q₈	—	2	2	no	—
447	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
448	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
449	C₄⋊Q₈	C₂³×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	4	yes	—
453	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
454	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
455	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
456	#127	C₂³×C₃₁	1	1	yes	characteristic
457	#126	—	2	4	no	—
458	#126	—	2	4	no	—
459	#126	—	2	4	no	—
460	#126	—	2	4	no	—
461	#125	—	2	2	no	—
462	#125	—	2	4	no	—
463	#125	—	2	2	no	—
464	#125	—	2	2	no	—
465	#125	—	2	4	no	—
466	#125	—	2	4	no	—
467	#125	—	2	4	no	—
468	#125	—	2	4	no	—
469	#125	—	2	4	no	—
470	#125	—	2	4	no	—
471	#125	—	2	1	no	—
472	#125	—	2	1	no	—
473	#125	D₄×C₃₁	1	2	yes	—
474	#125	C₂³×C₃₁	1	4	yes	—
475	#124	—	2	2	no	—
476	#124	C₂³×C₃₁	1	2	yes	—
477	#124	C₂³×C₃₁	1	4	yes	—
478	#123	—	2	4	no	—
479	#123	—	2	4	no	—
480	C₂²⋊Q₈	—	2	4	no	—
481	C₂²⋊Q₈	—	2	1	no	—
482	C₂²⋊Q₈	—	2	1	no	—
483	C₂²⋊Q₈	—	2	2	no	—
484	C₂²⋊Q₈	—	2	2	no	—
485	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
486	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
487	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
488	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
489	#121	—	2	1	no	—
490	#121	—	2	1	no	—
491	C₄×Q₈	—	2	2	no	—
492	C₄×Q₈	—	2	2	no	—
493	C₄×Q₈	—	2	2	no	—
494	C₄×Q₈	—	2	2	no	—
495	C₄×Q₈	—	2	2	no	—
496	C₄×Q₈	—	2	1	no	—
497	C₄×Q₈	—	2	1	no	—
498	C₄×Q₈	—	2	1	no	—
499	C₄×Q₈	—	2	1	no	—
500	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
501	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
502	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
503	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
504	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
505	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
506	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
507	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
508	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
509	C₄×D₄	—	2	2	no	—
510	C₄×D₄	—	2	4	no	—
511	C₄×D₄	—	2	1	no	—
512	C₄×D₄	—	2	1	no	—
513	C₄×D₄	C₂³×C₃₁	1	4	yes	—
514	#117	—	2	1	no	—
515	#117	—	2	1	no	—
516	#117	—	2	2	no	—
517	#117	—	2	4	no	—
518	#117	C₂³×C₃₁	1	2	yes	—
519	C₂×C₄⋊C₄	—	2	1	no	—
520	C₂×C₄⋊C₄	—	2	2	no	—
521	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
522	C₂×C₄²	—	2	1	no	—
523	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
524	#107	C₂³×C₃₁	1	2	yes	—
525	#107	C₂³×C₃₁	1	2	yes	—
526	#106	C₂³×C₃₁	1	4	yes	—
527	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
528	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
529	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
530	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
531	#103	C₂³×C₃₁	1	2	yes	—
532	#103	C₂³×C₃₁	1	4	yes	—
533	#103	C₂³×C₃₁	1	2	yes	—
534	#103	C₂³×C₃₁	1	2	yes	—
535	#103	C₂³×C₃₁	1	2	yes	—
536	#103	C₂³×C₃₁	1	4	yes	—
537	#103	C₂³×C₃₁	1	4	yes	—
538	#103	C₂³×C₃₁	1	4	yes	—
539	#102	C₂³×C₃₁	1	4	yes	—
540	#102	C₂³×C₃₁	1	4	yes	—
541	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
542	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
543	#97	C₂³×C₃₁	1	2	yes	—
544	#97	C₂³×C₃₁	1	4	yes	—
545	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
546	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
547	C₄×C₈	C₂³×C₃₁	1	4	yes	—
548	C₃₁	#61915	1	1	yes	characteristic, direct product factor, Sylow
549	#41	—	4	4	no	—
550	C₂×Q₈	—	2	1	no	—
551	C₂×Q₈	—	2	1	no	—
552	C₂×Q₈	—	2	1	no	—
553	C₂×Q₈	—	4	2	no	—
554	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
555	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
556	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
557	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
558	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
559	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
560	C₂×D₄	—	2	2	no	—
561	C₂×D₄	—	2	1	no	—
562	C₂×D₄	—	2	1	no	—
563	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
564	C₂²×C₄	—	2	1	no	—
565	C₂²×C₄	—	2	1	no	—
566	C₂²×C₄	—	2	2	no	—
567	C₂²×C₄	—	2	4	no	—
568	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
569	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
570	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
571	Q₁₆	—	4	2	no	—
572	Q₁₆	—	4	4	no	—
573	Q₁₆	—	4	4	no	—
574	Q₁₆	—	4	2	no	—
575	Q₁₆	—	4	4	no	—
576	SD₁₆	—	4	4	no	—
577	SD₁₆	—	4	4	no	—
578	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
579	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
580	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	2	4	no	—
583	C₄⋊C₄	—	2	2	no	—
584	C₄⋊C₄	—	2	2	no	—
585	C₄⋊C₄	—	2	4	no	—
586	C₄⋊C₄	—	2	1	no	—
587	C₄⋊C₄	—	2	1	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	2	1	no	—
590	C₄⋊C₄	—	2	1	no	—
591	C₄⋊C₄	—	2	1	no	—
592	C₄⋊C₄	—	2	1	no	—
593	C₄⋊C₄	—	2	1	no	—
594	C₄⋊C₄	—	2	4	no	—
595	C₄⋊C₄	—	2	2	no	—
596	C₄⋊C₄	—	2	2	no	—
597	C₄⋊C₄	—	2	4	no	—
598	C₄⋊C₄	—	2	4	no	—
599	C₄⋊C₄	—	2	4	no	—
600	C₄⋊C₄	—	2	4	no	—
601	C₄⋊C₄	—	2	2	no	—
602	C₄⋊C₄	—	2	2	no	—
603	C₄⋊C₄	—	2	4	no	—
604	C₄⋊C₄	—	2	4	no	—
605	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
606	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
607	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
608	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
609	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
610	C₄⋊C₄	C₂×D₄×C₃₁	1	4	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	C₂²⋊C₄	—	2	2	no	—
615	C₂²⋊C₄	—	2	2	no	—
616	C₂²⋊C₄	—	2	2	no	—
617	C₂²⋊C₄	—	2	4	no	—
618	C₂²⋊C₄	—	2	4	no	—
619	C₂²⋊C₄	—	2	2	no	—
620	C₂²⋊C₄	—	2	4	no	—
621	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
622	C₄²	—	2	2	no	—
623	C₄²	—	2	2	no	—
624	C₄²	—	2	4	no	—
625	C₄²	—	2	2	no	—
626	C₄²	—	2	2	no	—
627	C₄²	—	2	4	no	—
628	C₄²	—	2	4	no	—
629	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
630	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
631	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
632	C₄²	C₂⁴×C₃₁	1	4	yes	—
633	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
634	C₂³	—	2	2	no	—
635	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
636	Q₈	—	2	4	no	—
637	Q₈	—	2	2	no	—
638	Q₈	—	2	4	no	—
639	Q₈	—	2	2	no	—
640	Q₈	—	2	2	no	—
641	Q₈	—	2	2	no	—
642	Q₈	—	2	4	no	—
643	Q₈	—	4	2	no	—
644	Q₈	—	4	2	no	—
645	Q₈	—	4	2	no	—
646	Q₈	—	4	2	no	—
647	D₄	—	2	4	no	—
648	D₄	—	4	4	no	—
649	D₄	—	4	2	no	—
650	D₄	—	8	1	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	—	2	2	no	—
660	C₂×C₄	—	4	1	no	—
661	C₂×C₄	—	4	4	no	—
662	C₂×C₄	—	4	2	no	—
663	C₂×C₄	—	8	4	no	—
664	C₂×C₄	—	8	1	no	—
665	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
666	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
667	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
668	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
669	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
670	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
671	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
672	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
673	C₈	—	2	2	no	—
674	C₈	—	2	4	no	—
675	C₈	—	2	2	no	—
676	C₂²	—	2	1	no	—
677	C₂²	—	2	1	no	—
678	C₂²	—	2	1	no	—
679	C₂²	—	4	2	no	—
680	C₂²	—	4	2	no	—
681	C₂²	—	4	2	no	—
682	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
683	C₄	—	2	1	no	—
684	C₄	—	2	1	no	—
685	C₄	—	2	4	no	—
686	C₄	—	2	1	no	—
687	C₄	—	2	1	no	—
688	C₄	—	2	4	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	—	4	2	no	—
695	C₄	—	4	2	no	—
696	C₄	C₃₁×#578	1	2	yes	—
697	C₄	C₃₁×#585	1	2	yes	—
698	C₂	—	4	1	no	—
699	C₂	—	8	2	no	—
700	C₂	C₃₁×#3487	1	1	yes	characteristic
701	C₂	C₂×C₃₁×#578	1	1	yes	characteristic
702	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
1. C₃₁×#3556	C₂	semidirect
3. C₃₁×#3400	C₂	semidirect
4. C₃₁×#3398	C₂	semidirect
5. C₃₁×#3395	C₂	semidirect
6. C₃₁×#3377	C₂	semidirect
7. C₃₁×#3374	C₂	semidirect
8. C₃₁×#3365	C₂	semidirect
9. C₃₁×#3223	C₂	semidirect
12. C₃₁×#3120	C₂	semidirect
13. C₃₁×#3113	C₂	semidirect
14. C₃₁×#3104	C₂	semidirect
15. C₃₁×#2985	C₂	semidirect
282. #61915	C₃₁	direct (N × H)
325. #3556	C₂×C₃₁	semidirect
327. #3400	C₂×C₃₁	semidirect
328. #3398	C₂×C₃₁	semidirect
329. #3395	C₂×C₃₁	semidirect
330. #3377	C₂×C₃₁	semidirect
331. #3374	C₂×C₃₁	semidirect
332. #3365	C₂×C₃₁	semidirect
333. #3223	C₂×C₃₁	semidirect
336. #3120	C₂×C₃₁	semidirect
337. #3113	C₂×C₃₁	semidirect
338. #3104	C₂×C₃₁	semidirect
339. #2985	C₂×C₃₁	semidirect
548. C₃₁	#61915	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×#3556	2
C₃₁×#3507	2
C₃₁×#3400	2
C₃₁×#3398	2
C₃₁×#3395	2
C₃₁×#3377	2
C₃₁×#3374	2
C₃₁×#3365	2
C₃₁×#3223	2
C₃₁×#3216	2
C₃₁×#3210	2
C₃₁×#3120	2
C₃₁×#3113	2
C₃₁×#3104	2
C₃₁×#2985	2
#61915	31

Subgroups — C31×#61915 #21518674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61915 #21518674