Subgroups — C₃₁×#60880 #21517639

Order 7936.

1722 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3511	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3503	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3243	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3231	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3230	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3228	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3226	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3165	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3141	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3098	C₂	1	4	yes	maximal, complemented
13	C₃₁×#3083	C₂	1	4	yes	maximal, complemented
14	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic, complemented
15	Q₈²×C₃₁	—	2	1	no	—
16	C₃₁×#557	—	2	1	no	—
17	C₃₁×#556	—	2	1	no	—
18	C₃₁×#555	—	2	1	no	—
19	C₃₁×#554	—	2	1	no	—
20	C₃₁×#554	—	2	2	no	—
21	C₃₁×#554	—	2	1	no	—
22	C₃₁×#553	—	2	2	no	—
23	D₄×Q₈×C₃₁	—	2	1	no	—
24	D₄×Q₈×C₃₁	—	2	2	no	—
25	C₃₁×#548	—	2	2	no	—
26	C₃₁×#547	—	2	1	no	—
27	C₃₁×#544	C₂²	1	2	yes	—
28	C₃₁×#544	C₂²	1	4	yes	—
29	C₃₁×#542	C₂²	1	4	yes	—
30	C₃₁×#537	C₂²	1	2	yes	—
31	C₃₁×#533	C₂²	1	2	yes	—
32	C₃₁×#533	C₂²	1	1	yes	characteristic
33	C₃₁×#533	C₂²	1	2	yes	—
34	C₃₁×#532	C₂²	1	1	yes	characteristic
35	C₃₁×#532	C₂²	1	2	yes	—
36	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
39	C₃₁×#520	C₂²	1	2	yes	—
40	C₃₁×#520	C₂²	1	2	yes	—
41	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#501	C₂²	1	4	yes	—
43	C₃₁×#500	C₂²	1	2	yes	—
44	C₃₁×#500	C₂²	1	2	yes	—
45	C₃₁×#500	C₂²	1	4	yes	—
46	C₃₁×#500	C₂²	1	2	yes	—
47	C₃₁×#498	C₂²	1	2	yes	—
48	C₃₁×#489	C₂²	1	4	yes	—
49	C₃₁×#488	C₂²	1	4	yes	—
50	C₃₁×#487	C₂²	1	1	yes	characteristic
51	C₃₁×#487	C₂²	1	1	yes	characteristic
52	C₃₁×#487	C₂²	1	2	yes	—
53	C₃₁×#486	C₂²	1	1	yes	characteristic
54	C₃₁×#486	C₂²	1	1	yes	characteristic
55	C₃₁×#486	C₂²	1	2	yes	—
56	C₃₁×#485	C₂²	1	8	yes	—
57	C₃₁×#484	C₂²	1	8	yes	—
58	C₃₁×#482	C₂²	1	8	yes	—
59	C₃₁×#480	C₂²	1	8	yes	—
60	C₃₁×#479	C₂²	1	8	yes	—
61	C₃₁×#478	C₂²	1	8	yes	—
62	C₃₁×#477	C₂²	1	8	yes	—
63	C₃₁×#474	C₂²	1	8	yes	—
64	C₃₁×#433	C₂²	1	4	yes	—
65	C₃₁×#431	C₂²	1	2	yes	—
66	C₃₁×#431	C₂²	1	2	yes	—
67	C₃₁×#428	C₂²	1	4	yes	—
68	C₂×C₃₁×#107	C₂²	1	4	yes	—
69	C₃₁×#424	C₂²	1	2	yes	—
70	C₃₁×#417	C₂²	1	4	yes	—
71	C₃₁×#416	C₂²	1	4	yes	—
72	C₃₁×#408	C₂²	1	2	yes	—
73	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
74	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
75	C₂×#41×C₃₁	—	2	2	no	—
76	C₂×#41×C₃₁	C₂³	1	2	yes	—
77	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
78	C₂²×Q₈×C₃₁	—	2	1	no	—
79	C₂²×Q₈×C₃₁	—	2	1	no	—
80	C₂²×Q₈×C₃₁	—	2	2	no	—
81	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
82	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
83	C₃₁×C₄⋊Q₈	—	2	2	no	—
84	C₃₁×C₄⋊Q₈	—	2	4	no	—
85	C₃₁×C₄⋊Q₈	—	2	1	no	—
86	C₃₁×C₄⋊Q₈	—	2	1	no	—
87	C₃₁×C₄⋊Q₈	—	2	2	no	—
88	C₃₁×C₄⋊Q₈	—	2	1	no	—
89	C₃₁×C₄⋊Q₈	—	2	2	no	—
90	C₃₁×C₄⋊Q₈	—	2	1	no	—
91	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
92	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
93	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
94	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
95	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
96	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
97	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
98	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
99	C₃₁×#127	C₂³	1	1	yes	characteristic
100	C₃₁×#126	—	2	4	no	—
101	C₃₁×#125	—	2	2	no	—
102	C₃₁×#125	—	2	4	no	—
103	C₃₁×#125	—	2	2	no	—
104	C₃₁×#125	C₂³	1	4	yes	—
105	C₃₁×#125	C₂³	1	2	yes	—
106	C₃₁×#124	—	2	4	no	—
107	C₃₁×#124	—	2	2	no	—
108	C₃₁×#124	C₂³	1	2	yes	—
109	C₃₁×#123	—	2	4	no	—
110	C₃₁×C₂²⋊Q₈	—	2	2	no	—
111	C₃₁×C₂²⋊Q₈	—	2	2	no	—
112	C₃₁×C₂²⋊Q₈	—	2	1	no	—
113	C₃₁×C₂²⋊Q₈	—	2	2	no	—
114	C₃₁×C₂²⋊Q₈	—	2	1	no	—
115	C₃₁×C₂²⋊Q₈	—	2	2	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	—	2	4	no	—
120	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
121	C₃₁×#121	—	2	2	no	—
122	C₃₁×#121	—	2	4	no	—
123	C₃₁×#121	—	2	1	no	—
124	C₃₁×#121	—	2	1	no	—
125	C₃₁×#121	—	2	2	no	—
126	C₃₁×#121	C₂³	1	4	yes	—
127	C₄×Q₈×C₃₁	—	2	2	no	—
128	C₄×Q₈×C₃₁	—	2	2	no	—
129	C₄×Q₈×C₃₁	—	2	1	no	—
130	C₄×Q₈×C₃₁	—	2	2	no	—
131	C₄×Q₈×C₃₁	—	2	2	no	—
132	C₄×Q₈×C₃₁	—	2	1	no	—
133	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
134	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
135	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
136	C₄×D₄×C₃₁	—	2	4	no	—
137	C₄×D₄×C₃₁	—	2	2	no	—
138	C₄×D₄×C₃₁	—	2	4	no	—
139	C₄×D₄×C₃₁	—	2	1	no	—
140	C₄×D₄×C₃₁	—	2	2	no	—
141	C₄×D₄×C₃₁	—	2	1	no	—
142	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
143	C₄×D₄×C₃₁	D₄	1	2	yes	—
144	C₄×D₄×C₃₁	D₄	1	2	yes	—
145	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
146	C₄×D₄×C₃₁	C₂³	1	4	yes	—
147	C₃₁×#117	—	2	2	no	—
148	C₃₁×#117	—	2	2	no	—
149	C₃₁×#117	C₂³	1	2	yes	—
150	C₃₁×#117	C₂³	1	4	yes	—
151	C₃₁×#117	C₂³	1	1	yes	characteristic
152	C₃₁×#117	C₂³	1	2	yes	—
153	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
154	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
155	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
158	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
159	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
160	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
161	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
162	C₃₁×#107	C₂³	1	8	yes	—
163	C₃₁×#107	C₂³	1	8	yes	—
164	C₃₁×#107	C₂³	1	8	yes	—
165	C₃₁×#106	C₂³	1	8	yes	—
166	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
167	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
168	C₃₁×#103	C₂³	1	4	yes	—
169	C₃₁×#103	C₂³	1	4	yes	—
170	C₃₁×#103	C₂³	1	8	yes	—
171	C₃₁×#102	C₂³	1	4	yes	—
172	C₃₁×#102	C₂³	1	4	yes	—
173	C₃₁×#102	C₂³	1	8	yes	—
174	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
175	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
176	C₃₁×#97	C₂³	1	2	yes	—
177	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
178	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
179	C₄×C₈×C₃₁	C₂³	1	2	yes	—
180	C₄×C₈×C₃₁	C₂³	1	2	yes	—
181	#41×C₃₁	—	2	4	no	—
182	#41×C₃₁	—	2	8	no	—
183	#41×C₃₁	—	8	4	no	—
184	C₂×Q₈×C₃₁	—	2	1	no	—
185	C₂×Q₈×C₃₁	—	2	2	no	—
186	C₂×Q₈×C₃₁	—	2	1	no	—
187	C₂×Q₈×C₃₁	—	2	2	no	—
188	C₂×Q₈×C₃₁	—	2	2	no	—
189	C₂×Q₈×C₃₁	—	2	2	no	—
190	C₂×Q₈×C₃₁	—	2	4	no	—
191	C₂×Q₈×C₃₁	—	8	2	no	—
192	C₂×Q₈×C₃₁	—	8	2	no	—
193	C₂×Q₈×C₃₁	—	8	4	no	—
194	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
196	C₂×D₄×C₃₁	—	2	2	no	—
197	C₂×D₄×C₃₁	—	2	4	no	—
198	C₂×D₄×C₃₁	—	2	1	no	—
199	C₂×D₄×C₃₁	—	2	1	no	—
200	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
201	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
202	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
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	2	no	—
207	C₂²×C₄×C₃₁	—	2	4	no	—
208	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
209	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
210	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
211	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
212	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
213	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
214	M₁₆×C₃₁	—	2	4	no	—
215	C₂×C₈×C₃₁	—	2	4	no	—
216	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
217	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
218	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
219	C₄⋊C₄×C₃₁	—	2	2	no	—
220	C₄⋊C₄×C₃₁	—	2	1	no	—
221	C₄⋊C₄×C₃₁	—	2	2	no	—
222	C₄⋊C₄×C₃₁	—	2	2	no	—
223	C₄⋊C₄×C₃₁	—	2	4	no	—
224	C₄⋊C₄×C₃₁	—	2	4	no	—
225	C₄⋊C₄×C₃₁	—	2	2	no	—
226	C₄⋊C₄×C₃₁	—	2	2	no	—
227	C₄⋊C₄×C₃₁	—	2	4	no	—
228	C₄⋊C₄×C₃₁	—	2	2	no	—
229	C₄⋊C₄×C₃₁	—	2	1	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	2	no	—
235	C₄⋊C₄×C₃₁	—	2	2	no	—
236	C₄⋊C₄×C₃₁	—	2	4	no	—
237	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
238	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
239	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
240	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
241	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
242	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
243	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
244	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
245	C₂²⋊C₄×C₃₁	—	2	4	no	—
246	C₂²⋊C₄×C₃₁	—	2	4	no	—
247	C₂²⋊C₄×C₃₁	—	2	2	no	—
248	C₂²⋊C₄×C₃₁	—	2	4	no	—
249	C₂²⋊C₄×C₃₁	—	2	4	no	—
250	C₂²⋊C₄×C₃₁	—	2	2	no	—
251	C₂²⋊C₄×C₃₁	—	2	2	no	—
252	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
253	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
254	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
255	C₄²×C₃₁	—	2	4	no	—
256	C₄²×C₃₁	—	2	4	no	—
257	C₄²×C₃₁	—	2	2	no	—
258	C₄²×C₃₁	C₂×D₄	1	2	yes	—
259	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
260	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
261	C₄²×C₃₁	C₂×D₄	1	2	yes	—
262	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
263	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
264	C₄²×C₃₁	C₂⁴	1	2	yes	—
265	#60880	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₂³×C₃₁	—	2	2	no	—
267	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
268	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
269	Q₈×C₃₁	—	2	2	no	—
270	Q₈×C₃₁	—	2	4	no	—
271	Q₈×C₃₁	—	4	4	no	—
272	Q₈×C₃₁	—	4	2	no	—
273	Q₈×C₃₁	—	4	4	no	—
274	Q₈×C₃₁	—	4	2	no	—
275	Q₈×C₃₁	—	8	4	no	—
276	Q₈×C₃₁	—	8	2	no	—
277	Q₈×C₃₁	—	8	2	no	—
278	D₄×C₃₁	—	2	4	no	—
279	D₄×C₃₁	—	2	2	no	—
280	D₄×C₃₁	—	2	4	no	—
281	D₄×C₃₁	—	4	4	no	—
282	D₄×C₃₁	—	4	2	no	—
283	D₄×C₃₁	—	4	2	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	4	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	4	no	—
294	C₂×C₄×C₃₁	—	2	2	no	—
295	C₂×C₄×C₃₁	—	4	4	no	—
296	C₂×C₄×C₃₁	—	8	4	no	—
297	C₂×C₄×C₃₁	—	8	2	no	—
298	C₂×C₄×C₃₁	—	8	2	no	—
299	C₂×C₄×C₃₁	—	8	4	no	—
300	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
301	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
302	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
303	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
304	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
305	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
306	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
307	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
308	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	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	4	no	—
313	#3511	C₂×C₃₁	1	1	yes	characteristic
314	#3503	C₂×C₃₁	1	1	yes	characteristic, complemented
315	#3243	C₂×C₃₁	1	8	yes	complemented
316	#3231	C₂×C₃₁	1	2	yes	complemented
317	#3230	C₂×C₃₁	1	2	yes	complemented
318	#3228	C₂×C₃₁	1	2	yes	complemented
319	#3226	C₂×C₃₁	1	2	yes	complemented
320	#3165	C₂×C₃₁	1	1	yes	characteristic, complemented
321	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
322	#3141	C₂×C₃₁	1	1	yes	characteristic, complemented
323	#3131	C₂×C₃₁	1	1	yes	characteristic, complemented
324	#3098	C₂×C₃₁	1	4	yes	complemented
325	#3083	C₂×C₃₁	1	4	yes	complemented
326	#2922	C₂×C₃₁	1	1	yes	characteristic, complemented
327	C₂²×C₃₁	—	2	1	no	—
328	C₂²×C₃₁	—	2	1	no	—
329	C₂²×C₃₁	—	2	2	no	—
330	C₂²×C₃₁	—	2	2	no	—
331	C₂²×C₃₁	—	2	2	no	—
332	C₂²×C₃₁	—	4	2	no	—
333	C₂²×C₃₁	—	4	2	no	—
334	C₂²×C₃₁	—	4	2	no	—
335	C₂²×C₃₁	#536	1	2	yes	—
336	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
337	C₄×C₃₁	—	2	2	no	—
338	C₄×C₃₁	—	2	2	no	—
339	C₄×C₃₁	—	2	1	no	—
340	C₄×C₃₁	—	2	2	no	—
341	C₄×C₃₁	—	2	1	no	—
342	C₄×C₃₁	—	4	2	no	—
343	C₄×C₃₁	—	4	4	no	—
344	C₄×C₃₁	—	4	4	no	—
345	C₄×C₃₁	—	4	2	no	—
346	C₄×C₃₁	—	4	2	no	—
347	C₄×C₃₁	#536	1	2	yes	—
348	C₄×C₃₁	#576	1	2	yes	—
349	C₄×C₃₁	#578	1	2	yes	—
350	C₄×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
351	Q₈²	—	2	1	no	—
352	#557	—	2	1	no	—
353	#556	—	2	1	no	—
354	#555	—	2	1	no	—
355	#554	—	2	1	no	—
356	#554	—	2	2	no	—
357	#554	—	2	1	no	—
358	#553	—	2	2	no	—
359	D₄×Q₈	—	2	1	no	—
360	D₄×Q₈	—	2	2	no	—
361	#548	—	2	2	no	—
362	#547	—	2	1	no	—
363	#544	C₂²×C₃₁	1	2	yes	—
364	#544	C₂²×C₃₁	1	4	yes	—
365	#542	C₂²×C₃₁	1	4	yes	—
366	#537	C₂²×C₃₁	1	2	yes	—
367	#533	C₂²×C₃₁	1	2	yes	—
368	#533	C₂²×C₃₁	1	1	yes	characteristic
369	#533	C₂²×C₃₁	1	2	yes	—
370	#532	C₂²×C₃₁	1	1	yes	characteristic
371	#532	C₂²×C₃₁	1	2	yes	—
372	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
373	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
374	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
375	#520	C₂²×C₃₁	1	2	yes	—
376	#520	C₂²×C₃₁	1	2	yes	—
377	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
378	#501	C₂²×C₃₁	1	4	yes	—
379	#500	C₂²×C₃₁	1	2	yes	—
380	#500	C₂²×C₃₁	1	2	yes	—
381	#500	C₂²×C₃₁	1	4	yes	—
382	#500	C₂²×C₃₁	1	2	yes	—
383	#498	C₂²×C₃₁	1	2	yes	—
384	#489	C₂²×C₃₁	1	4	yes	—
385	#488	C₂²×C₃₁	1	4	yes	—
386	#487	C₂²×C₃₁	1	1	yes	characteristic
387	#487	C₂²×C₃₁	1	1	yes	characteristic
388	#487	C₂²×C₃₁	1	2	yes	—
389	#486	C₂²×C₃₁	1	1	yes	characteristic
390	#486	C₂²×C₃₁	1	1	yes	characteristic
391	#486	C₂²×C₃₁	1	2	yes	—
392	#485	C₂²×C₃₁	1	8	yes	—
393	#484	C₂²×C₃₁	1	8	yes	—
394	#482	C₂²×C₃₁	1	8	yes	—
395	#480	C₂²×C₃₁	1	8	yes	—
396	#479	C₂²×C₃₁	1	8	yes	—
397	#478	C₂²×C₃₁	1	8	yes	—
398	#477	C₂²×C₃₁	1	8	yes	—
399	#474	C₂²×C₃₁	1	8	yes	—
400	#433	C₂²×C₃₁	1	4	yes	—
401	#431	C₂²×C₃₁	1	2	yes	—
402	#431	C₂²×C₃₁	1	2	yes	—
403	#428	C₂²×C₃₁	1	4	yes	—
404	C₂×#107	C₂²×C₃₁	1	4	yes	—
405	#424	C₂²×C₃₁	1	2	yes	—
406	#417	C₂²×C₃₁	1	4	yes	—
407	#416	C₂²×C₃₁	1	4	yes	—
408	#408	C₂²×C₃₁	1	2	yes	—
409	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
410	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
411	C₂×C₃₁	—	2	2	no	—
412	C₂×C₃₁	—	4	2	no	—
413	C₂×C₃₁	—	8	2	no	—
414	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
415	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
416	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
417	C₂×#41	—	2	2	no	—
418	C₂×#41	C₂³×C₃₁	1	2	yes	—
419	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
420	C₂²×Q₈	—	2	1	no	—
421	C₂²×Q₈	—	2	1	no	—
422	C₂²×Q₈	—	2	2	no	—
423	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
424	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
425	C₄⋊Q₈	—	2	2	no	—
426	C₄⋊Q₈	—	2	4	no	—
427	C₄⋊Q₈	—	2	1	no	—
428	C₄⋊Q₈	—	2	1	no	—
429	C₄⋊Q₈	—	2	2	no	—
430	C₄⋊Q₈	—	2	1	no	—
431	C₄⋊Q₈	—	2	2	no	—
432	C₄⋊Q₈	—	2	1	no	—
433	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
434	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
435	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
436	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
437	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
438	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
439	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
440	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
441	#127	C₂³×C₃₁	1	1	yes	characteristic
442	#126	—	2	4	no	—
443	#125	—	2	2	no	—
444	#125	—	2	4	no	—
445	#125	—	2	2	no	—
446	#125	C₂³×C₃₁	1	4	yes	—
447	#125	C₂³×C₃₁	1	2	yes	—
448	#124	—	2	4	no	—
449	#124	—	2	2	no	—
450	#124	C₂³×C₃₁	1	2	yes	—
451	#123	—	2	4	no	—
452	C₂²⋊Q₈	—	2	2	no	—
453	C₂²⋊Q₈	—	2	2	no	—
454	C₂²⋊Q₈	—	2	1	no	—
455	C₂²⋊Q₈	—	2	2	no	—
456	C₂²⋊Q₈	—	2	1	no	—
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	C₂³×C₃₁	1	4	yes	—
463	#121	—	2	2	no	—
464	#121	—	2	4	no	—
465	#121	—	2	1	no	—
466	#121	—	2	1	no	—
467	#121	—	2	2	no	—
468	#121	C₂³×C₃₁	1	4	yes	—
469	C₄×Q₈	—	2	2	no	—
470	C₄×Q₈	—	2	2	no	—
471	C₄×Q₈	—	2	1	no	—
472	C₄×Q₈	—	2	2	no	—
473	C₄×Q₈	—	2	2	no	—
474	C₄×Q₈	—	2	1	no	—
475	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
476	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
477	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
478	C₄×D₄	—	2	4	no	—
479	C₄×D₄	—	2	2	no	—
480	C₄×D₄	—	2	4	no	—
481	C₄×D₄	—	2	1	no	—
482	C₄×D₄	—	2	2	no	—
483	C₄×D₄	—	2	1	no	—
484	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
485	C₄×D₄	D₄×C₃₁	1	2	yes	—
486	C₄×D₄	D₄×C₃₁	1	2	yes	—
487	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
488	C₄×D₄	C₂³×C₃₁	1	4	yes	—
489	#117	—	2	2	no	—
490	#117	—	2	2	no	—
491	#117	C₂³×C₃₁	1	2	yes	—
492	#117	C₂³×C₃₁	1	4	yes	—
493	#117	C₂³×C₃₁	1	1	yes	characteristic
494	#117	C₂³×C₃₁	1	2	yes	—
495	C₂×C₄⋊C₄	—	2	1	no	—
496	C₂×C₄⋊C₄	—	2	1	no	—
497	C₂×C₄⋊C₄	—	2	2	no	—
498	C₂×C₄⋊C₄	—	2	2	no	—
499	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
500	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
501	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
502	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
503	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
504	#107	C₂³×C₃₁	1	8	yes	—
505	#107	C₂³×C₃₁	1	8	yes	—
506	#107	C₂³×C₃₁	1	8	yes	—
507	#106	C₂³×C₃₁	1	8	yes	—
508	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
509	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
510	#103	C₂³×C₃₁	1	4	yes	—
511	#103	C₂³×C₃₁	1	4	yes	—
512	#103	C₂³×C₃₁	1	8	yes	—
513	#102	C₂³×C₃₁	1	4	yes	—
514	#102	C₂³×C₃₁	1	4	yes	—
515	#102	C₂³×C₃₁	1	8	yes	—
516	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
517	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
518	#97	C₂³×C₃₁	1	2	yes	—
519	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
520	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
521	C₄×C₈	C₂³×C₃₁	1	2	yes	—
522	C₄×C₈	C₂³×C₃₁	1	2	yes	—
523	C₃₁	#60880	1	1	yes	characteristic, direct product factor, Sylow
524	#41	—	2	4	no	—
525	#41	—	2	8	no	—
526	#41	—	8	4	no	—
527	C₂×Q₈	—	2	1	no	—
528	C₂×Q₈	—	2	2	no	—
529	C₂×Q₈	—	2	1	no	—
530	C₂×Q₈	—	2	2	no	—
531	C₂×Q₈	—	2	2	no	—
532	C₂×Q₈	—	2	2	no	—
533	C₂×Q₈	—	2	4	no	—
534	C₂×Q₈	—	8	2	no	—
535	C₂×Q₈	—	8	2	no	—
536	C₂×Q₈	—	8	4	no	—
537	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
538	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
539	C₂×D₄	—	2	2	no	—
540	C₂×D₄	—	2	4	no	—
541	C₂×D₄	—	2	1	no	—
542	C₂×D₄	—	2	1	no	—
543	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
544	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
545	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
546	C₂²×C₄	—	2	2	no	—
547	C₂²×C₄	—	2	4	no	—
548	C₂²×C₄	—	2	2	no	—
549	C₂²×C₄	—	2	2	no	—
550	C₂²×C₄	—	2	4	no	—
551	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
552	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
553	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
554	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
555	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
556	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
557	M₁₆	—	2	4	no	—
558	C₂×C₈	—	2	4	no	—
559	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
560	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
561	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
562	C₄⋊C₄	—	2	2	no	—
563	C₄⋊C₄	—	2	1	no	—
564	C₄⋊C₄	—	2	2	no	—
565	C₄⋊C₄	—	2	2	no	—
566	C₄⋊C₄	—	2	4	no	—
567	C₄⋊C₄	—	2	4	no	—
568	C₄⋊C₄	—	2	2	no	—
569	C₄⋊C₄	—	2	2	no	—
570	C₄⋊C₄	—	2	4	no	—
571	C₄⋊C₄	—	2	2	no	—
572	C₄⋊C₄	—	2	1	no	—
573	C₄⋊C₄	—	2	1	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	4	no	—
576	C₄⋊C₄	—	2	4	no	—
577	C₄⋊C₄	—	2	2	no	—
578	C₄⋊C₄	—	2	2	no	—
579	C₄⋊C₄	—	2	4	no	—
580	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
581	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
582	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
583	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
584	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
585	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
586	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
587	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
588	C₂²⋊C₄	—	2	4	no	—
589	C₂²⋊C₄	—	2	4	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	—	2	2	no	—
595	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
596	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
597	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
598	C₄²	—	2	4	no	—
599	C₄²	—	2	4	no	—
600	C₄²	—	2	2	no	—
601	C₄²	C₂×D₄×C₃₁	1	2	yes	—
602	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
603	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
604	C₄²	C₂×D₄×C₃₁	1	2	yes	—
605	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
606	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
607	C₄²	C₂⁴×C₃₁	1	2	yes	—
608	C₂³	—	2	2	no	—
609	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
610	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
611	Q₈	—	2	2	no	—
612	Q₈	—	2	4	no	—
613	Q₈	—	4	4	no	—
614	Q₈	—	4	2	no	—
615	Q₈	—	4	4	no	—
616	Q₈	—	4	2	no	—
617	Q₈	—	8	4	no	—
618	Q₈	—	8	2	no	—
619	Q₈	—	8	2	no	—
620	D₄	—	2	4	no	—
621	D₄	—	2	2	no	—
622	D₄	—	2	4	no	—
623	D₄	—	4	4	no	—
624	D₄	—	4	2	no	—
625	D₄	—	4	2	no	—
626	D₄	—	8	4	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	2	4	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	4	no	—
631	C₂×C₄	—	2	4	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	2	no	—
634	C₂×C₄	—	2	4	no	—
635	C₂×C₄	—	2	4	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	4	4	no	—
638	C₂×C₄	—	8	4	no	—
639	C₂×C₄	—	8	2	no	—
640	C₂×C₄	—	8	2	no	—
641	C₂×C₄	—	8	4	no	—
642	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
643	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
644	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
645	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
646	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
647	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
648	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
649	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
650	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
651	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
652	C₈	—	2	2	no	—
653	C₈	—	2	2	no	—
654	C₈	—	2	4	no	—
655	C₂²	—	2	1	no	—
656	C₂²	—	2	1	no	—
657	C₂²	—	2	2	no	—
658	C₂²	—	2	2	no	—
659	C₂²	—	2	2	no	—
660	C₂²	—	4	2	no	—
661	C₂²	—	4	2	no	—
662	C₂²	—	4	2	no	—
663	C₂²	C₃₁×#536	1	2	yes	—
664	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
665	C₄	—	2	2	no	—
666	C₄	—	2	2	no	—
667	C₄	—	2	1	no	—
668	C₄	—	2	2	no	—
669	C₄	—	2	1	no	—
670	C₄	—	4	2	no	—
671	C₄	—	4	4	no	—
672	C₄	—	4	4	no	—
673	C₄	—	4	2	no	—
674	C₄	—	4	2	no	—
675	C₄	C₃₁×#536	1	2	yes	—
676	C₄	C₃₁×#576	1	2	yes	—
677	C₄	C₃₁×#578	1	2	yes	—
678	C₄	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
679	C₂	—	2	2	no	—
680	C₂	—	4	2	no	—
681	C₂	—	8	2	no	—
682	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
683	C₂	C₂×C₃₁×#576	1	1	yes	characteristic
684	C₂	C₂×C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3511	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3503	C₂	semidirect
3. C₃₁×#3243	C₂	semidirect
4. C₃₁×#3231	C₂	semidirect
5. C₃₁×#3230	C₂	semidirect
6. C₃₁×#3228	C₂	semidirect
7. C₃₁×#3226	C₂	semidirect
8. C₃₁×#3165	C₂	semidirect
9. C₂×C₃₁×#500	C₂	semidirect
10. C₃₁×#3141	C₂	semidirect
11. C₃₁×#3131	C₂	semidirect
12. C₃₁×#3098	C₂	semidirect
13. C₃₁×#3083	C₂	semidirect
14. C₃₁×#2922	C₂	semidirect
265. #60880	C₃₁	direct (N × H)
314. #3503	C₂×C₃₁	semidirect
315. #3243	C₂×C₃₁	semidirect
316. #3231	C₂×C₃₁	semidirect
317. #3230	C₂×C₃₁	semidirect
318. #3228	C₂×C₃₁	semidirect
319. #3226	C₂×C₃₁	semidirect
320. #3165	C₂×C₃₁	semidirect
321. C₂×#500	C₂×C₃₁	semidirect
322. #3141	C₂×C₃₁	semidirect
323. #3131	C₂×C₃₁	semidirect
324. #3098	C₂×C₃₁	semidirect
325. #3083	C₂×C₃₁	semidirect
326. #2922	C₂×C₃₁	semidirect
523. C₃₁	#60880	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×#3511	2
C₃₁×#3503	2
C₃₁×#3243	2
C₃₁×#3231	2
C₃₁×#3230	2
C₃₁×#3228	2
C₃₁×#3226	2
C₃₁×#3165	2
C₂×C₃₁×#500	2
C₃₁×#3141	2
C₃₁×#3131	2
C₃₁×#3098	2
C₃₁×#3083	2
C₃₁×#2922	2
#60880	31

Subgroups — C31×#60880 #21517639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60880 #21517639