Subgroups — C₃₁×#17691 #21474450

Order 7936.

900 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3510	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2971	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2970	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1537	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1535	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1532	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1529	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1527	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1527	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1526	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1524	C₂	1	2	yes	maximal, complemented
12	C₃₁×#559	—	2	1	no	—
13	C₃₁×#559	—	2	1	no	—
14	C₃₁×#556	—	2	1	no	—
15	C₃₁×#556	—	2	1	no	—
16	C₃₁×#551	—	2	2	no	—
17	C₃₁×#551	—	2	2	no	—
18	C₃₁×#542	C₄	1	2	yes	—
19	C₃₁×#542	C₄	1	2	yes	—
20	C₃₁×#536	C₄	1	1	yes	characteristic
21	C₃₁×#535	C₄	1	1	yes	characteristic
22	C₃₁×#533	C₂²	1	1	yes	characteristic
23	C₃₁×#529	C₂²	1	2	yes	—
24	C₃₁×#529	C₂²	1	2	yes	—
25	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
26	C₃₁×#519	C₄	1	1	yes	characteristic
27	C₃₁×#518	C₄	1	1	yes	characteristic
28	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#446	—	2	1	no	—
30	C₈×Q₈×C₃₁	—	2	1	no	—
31	C₃₁×#436	—	2	1	no	—
32	C₃₁×#436	—	2	2	no	—
33	C₃₁×#435	—	2	2	no	—
34	C₈×D₄×C₃₁	—	2	1	no	—
35	C₃₁×#432	C₂²	1	1	yes	characteristic
36	C₃₁×#432	C₂²	1	1	yes	characteristic
37	C₃₁×#432	C₂²	1	2	yes	complemented
38	C₃₁×#432	C₂²	1	2	yes	complemented
39	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
42	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
43	C₃₁×#330	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×#330	C₂²	1	2	yes	complemented
45	C₃₁×#330	C₂²	1	1	yes	characteristic, complemented
46	C₃₁×#330	C₂²	1	1	yes	characteristic, complemented
47	C₃₁×#330	C₂²	1	1	yes	characteristic, complemented
48	C₃₁×#330	C₂²	1	2	yes	complemented
49	C₃₁×#329	C₂²	1	2	yes	—
50	C₃₁×#329	C₂²	1	2	yes	—
51	C₃₁×#329	C₂²	1	2	yes	—
52	C₃₁×#329	C₂²	1	2	yes	—
53	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×M₁₆×C₃₁	—	2	1	no	—
57	C₂×M₁₆×C₃₁	—	2	1	no	—
58	C₂×M₁₆×C₃₁	—	2	2	no	—
59	C₂²×C₈×C₃₁	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	—	2	1	no	—
61	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
63	C₃₁×#127	—	2	1	no	—
64	C₃₁×#126	—	2	2	no	—
65	C₃₁×#126	—	2	2	no	—
66	C₃₁×#126	—	2	2	no	—
67	C₃₁×#126	—	2	2	no	—
68	C₃₁×#125	C₂³	1	1	yes	characteristic
69	C₃₁×#125	C₂³	1	1	yes	characteristic
70	C₃₁×#124	—	2	1	no	—
71	C₃₁×#124	—	2	1	no	—
72	C₃₁×#124	—	2	2	no	—
73	C₃₁×#124	—	2	2	no	—
74	C₃₁×#124	—	2	1	no	—
75	C₃₁×#124	—	2	1	no	—
76	C₃₁×#124	C₂×C₄	1	2	yes	—
77	C₃₁×#124	C₂³	1	2	yes	—
78	C₃₁×#123	—	2	2	no	—
79	C₃₁×#123	—	2	2	no	—
80	C₃₁×#123	—	2	2	no	—
81	C₃₁×#123	—	2	2	no	—
82	C₃₁×#123	—	2	2	no	—
83	C₃₁×C₂²⋊Q₈	—	2	1	no	—
84	C₃₁×C₂²⋊Q₈	—	2	1	no	—
85	C₃₁×C₂²⋊Q₈	—	2	2	no	—
86	C₃₁×C₂²⋊Q₈	—	2	2	no	—
87	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
88	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
89	C₃₁×#121	C₂×C₄	1	2	yes	—
90	C₃₁×#121	C₂×C₄	1	2	yes	—
91	C₃₁×#120	—	2	2	no	—
92	C₄×Q₈×C₃₁	—	2	1	no	—
93	C₄×Q₈×C₃₁	—	2	1	no	—
94	C₄×Q₈×C₃₁	—	2	1	no	—
95	C₄×Q₈×C₃₁	—	2	1	no	—
96	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
97	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
98	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
99	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
100	C₄×D₄×C₃₁	—	2	2	no	—
101	C₄×D₄×C₃₁	—	2	2	no	—
102	C₄×D₄×C₃₁	—	2	1	no	—
103	C₄×D₄×C₃₁	—	2	1	no	—
104	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
105	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
106	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
107	C₄×D₄×C₃₁	D₄	1	2	yes	—
108	C₄×D₄×C₃₁	D₄	1	2	yes	—
109	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
110	C₃₁×#117	—	2	2	no	—
111	C₃₁×#117	—	2	2	no	—
112	C₃₁×#117	—	2	2	no	—
113	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
114	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
115	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
116	C₃₁×#117	C₂×C₄	1	2	yes	—
117	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
118	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
119	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
120	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
121	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
122	C₃₁×C₄⋊C₈	—	2	1	no	—
123	C₃₁×C₄⋊C₈	—	2	1	no	—
124	C₃₁×C₄⋊C₈	—	2	2	no	—
125	C₃₁×C₄⋊C₈	—	2	2	no	—
126	C₃₁×C₂²⋊C₈	—	2	1	no	—
127	C₃₁×C₂²⋊C₈	—	2	1	no	—
128	C₃₁×C₂²⋊C₈	—	2	2	no	—
129	C₃₁×C₂²⋊C₈	—	2	2	no	—
130	C₃₁×#97	—	2	2	no	—
131	C₃₁×#97	C₂³	1	1	yes	characteristic
132	C₃₁×#97	C₂³	1	2	yes	—
133	C₃₁×#97	C₂³	1	1	yes	characteristic
134	C₃₁×#97	C₂³	1	1	yes	characteristic
135	C₃₁×#97	C₂³	1	1	yes	characteristic
136	C₃₁×#97	C₂³	1	2	yes	—
137	C₄×C₈×C₃₁	—	2	1	no	—
138	C₄×C₈×C₃₁	—	2	1	no	—
139	C₄×C₈×C₃₁	—	2	2	no	—
140	C₂⁴×C₃₁	—	2	1	no	—
141	#41×C₃₁	—	4	2	no	—
142	C₂×Q₈×C₃₁	—	2	1	no	—
143	C₂×Q₈×C₃₁	—	2	1	no	—
144	C₂×Q₈×C₃₁	—	2	1	no	—
145	C₂×Q₈×C₃₁	—	4	2	no	—
146	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
147	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
148	C₂×D₄×C₃₁	—	2	2	no	—
149	C₂×D₄×C₃₁	—	2	2	no	—
150	C₂×D₄×C₃₁	—	2	1	no	—
151	C₂×D₄×C₃₁	—	4	2	no	—
152	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
153	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
154	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
155	C₂²×C₄×C₃₁	—	2	1	no	—
156	C₂²×C₄×C₃₁	—	2	2	no	—
157	C₂²×C₄×C₃₁	—	2	2	no	—
158	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
164	M₁₆×C₃₁	—	4	2	no	—
165	M₁₆×C₃₁	—	4	2	no	—
166	M₁₆×C₃₁	—	4	4	no	—
167	C₂×C₈×C₃₁	—	2	1	no	—
168	C₂×C₈×C₃₁	—	2	1	no	—
169	C₂×C₈×C₃₁	—	2	1	no	—
170	C₂×C₈×C₃₁	—	2	1	no	—
171	C₂×C₈×C₃₁	—	2	2	no	—
172	C₂×C₈×C₃₁	—	2	2	no	—
173	C₂×C₈×C₃₁	—	4	4	no	—
174	C₄⋊C₄×C₃₁	—	2	2	no	—
175	C₄⋊C₄×C₃₁	—	2	2	no	—
176	C₄⋊C₄×C₃₁	—	2	2	no	—
177	C₄⋊C₄×C₃₁	—	2	2	no	—
178	C₄⋊C₄×C₃₁	—	2	1	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	—	2	1	no	—
181	C₄⋊C₄×C₃₁	—	2	1	no	—
182	C₄⋊C₄×C₃₁	—	2	1	no	—
183	C₄⋊C₄×C₃₁	—	2	1	no	—
184	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
185	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
186	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
187	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
188	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
189	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
190	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
191	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
192	C₂²⋊C₄×C₃₁	—	2	1	no	—
193	C₂²⋊C₄×C₃₁	—	2	1	no	—
194	C₂²⋊C₄×C₃₁	—	2	2	no	—
195	C₂²⋊C₄×C₃₁	—	2	2	no	—
196	C₂²⋊C₄×C₃₁	—	2	2	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	2	no	—
201	C₂²⋊C₄×C₃₁	—	2	2	no	—
202	C₂²⋊C₄×C₃₁	—	2	2	no	—
203	C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
205	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
206	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
207	C₄²×C₃₁	—	2	2	no	—
208	C₄²×C₃₁	—	2	2	no	—
209	C₄²×C₃₁	—	2	1	no	—
210	C₄²×C₃₁	—	2	1	no	—
211	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
212	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
213	C₄²×C₃₁	C₂×D₄	1	2	yes	—
214	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₃₁	C₂×D₄	1	2	yes	—
217	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
218	#17691	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂³×C₃₁	—	2	2	no	—
220	C₂³×C₃₁	—	4	2	no	—
221	C₂³×C₃₁	—	8	1	no	—
222	C₂³×C₃₁	—	8	1	no	—
223	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
224	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
225	Q₈×C₃₁	—	2	2	no	—
226	Q₈×C₃₁	—	4	2	no	—
227	Q₈×C₃₁	—	4	2	no	—
228	Q₈×C₃₁	—	8	1	no	—
229	Q₈×C₃₁	—	8	1	no	—
230	D₄×C₃₁	—	2	2	no	—
231	D₄×C₃₁	—	4	2	no	—
232	D₄×C₃₁	—	4	2	no	—
233	D₄×C₃₁	—	4	2	no	—
234	D₄×C₃₁	—	8	2	no	—
235	D₄×C₃₁	—	8	2	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	1	no	—
238	C₂×C₄×C₃₁	—	2	1	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	1	no	—
241	C₂×C₄×C₃₁	—	2	1	no	—
242	C₂×C₄×C₃₁	—	4	2	no	—
243	C₂×C₄×C₃₁	—	4	2	no	—
244	C₂×C₄×C₃₁	—	4	1	no	—
245	C₂×C₄×C₃₁	—	4	1	no	—
246	C₂×C₄×C₃₁	—	4	2	no	—
247	C₂×C₄×C₃₁	—	4	2	no	—
248	C₂×C₄×C₃₁	—	8	2	no	—
249	C₂×C₄×C₃₁	—	8	2	no	—
250	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
251	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
252	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
255	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
257	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
259	C₈×C₃₁	—	2	2	no	—
260	C₈×C₃₁	—	2	2	no	—
261	C₈×C₃₁	—	4	2	no	—
262	C₈×C₃₁	—	4	1	no	—
263	C₈×C₃₁	—	4	1	no	—
264	C₈×C₃₁	—	4	2	no	—
265	#3510	C₂×C₃₁	1	1	yes	characteristic
266	#2971	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#2970	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#1537	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#1535	C₂×C₃₁	1	2	yes	complemented
270	#1532	C₂×C₃₁	1	2	yes	complemented
271	#1529	C₂×C₃₁	1	2	yes	complemented
272	#1527	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#1527	C₂×C₃₁	1	1	yes	characteristic, complemented
274	#1526	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#1524	C₂×C₃₁	1	2	yes	complemented
276	C₂²×C₃₁	—	2	1	no	—
277	C₂²×C₃₁	—	2	2	no	—
278	C₂²×C₃₁	—	2	2	no	—
279	C₂²×C₃₁	—	2	2	no	—
280	C₂²×C₃₁	—	2	1	no	—
281	C₂²×C₃₁	—	2	1	no	—
282	C₂²×C₃₁	—	4	2	no	—
283	C₂²×C₃₁	—	4	2	no	—
284	C₂²×C₃₁	—	4	2	no	—
285	C₂²×C₃₁	—	16	2	no	—
286	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
287	C₄×C₃₁	—	2	1	no	—
288	C₄×C₃₁	—	2	2	no	—
289	C₄×C₃₁	—	2	1	no	—
290	C₄×C₃₁	—	2	1	no	—
291	C₄×C₃₁	—	2	2	no	—
292	C₄×C₃₁	—	2	2	no	—
293	C₄×C₃₁	—	2	1	no	—
294	C₄×C₃₁	—	2	1	no	—
295	C₄×C₃₁	—	4	2	no	—
296	C₄×C₃₁	—	4	1	no	—
297	C₄×C₃₁	—	4	1	no	—
298	C₄×C₃₁	—	4	1	no	—
299	C₄×C₃₁	—	4	1	no	—
300	C₄×C₃₁	#413	1	2	yes	—
301	#559	—	2	1	no	—
302	#559	—	2	1	no	—
303	#556	—	2	1	no	—
304	#556	—	2	1	no	—
305	#551	—	2	2	no	—
306	#551	—	2	2	no	—
307	#542	C₄×C₃₁	1	2	yes	—
308	#542	C₄×C₃₁	1	2	yes	—
309	#536	C₄×C₃₁	1	1	yes	characteristic
310	#535	C₄×C₃₁	1	1	yes	characteristic
311	#533	C₂²×C₃₁	1	1	yes	characteristic
312	#529	C₂²×C₃₁	1	2	yes	—
313	#529	C₂²×C₃₁	1	2	yes	—
314	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
315	#519	C₄×C₃₁	1	1	yes	characteristic
316	#518	C₄×C₃₁	1	1	yes	characteristic
317	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
318	#446	—	2	1	no	—
319	C₈×Q₈	—	2	1	no	—
320	#436	—	2	1	no	—
321	#436	—	2	2	no	—
322	#435	—	2	2	no	—
323	C₈×D₄	—	2	1	no	—
324	#432	C₂²×C₃₁	1	1	yes	characteristic
325	#432	C₂²×C₃₁	1	1	yes	characteristic
326	#432	C₂²×C₃₁	1	2	yes	complemented
327	#432	C₂²×C₃₁	1	2	yes	complemented
328	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
329	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
330	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
331	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
332	#330	C₂²×C₃₁	1	2	yes	complemented
333	#330	C₂²×C₃₁	1	2	yes	complemented
334	#330	C₂²×C₃₁	1	1	yes	characteristic, complemented
335	#330	C₂²×C₃₁	1	1	yes	characteristic, complemented
336	#330	C₂²×C₃₁	1	1	yes	characteristic, complemented
337	#330	C₂²×C₃₁	1	1	yes	characteristic, complemented
338	#329	C₂²×C₃₁	1	2	yes	—
339	#329	C₂²×C₃₁	1	2	yes	—
340	#329	C₂²×C₃₁	1	2	yes	—
341	#329	C₂²×C₃₁	1	2	yes	—
342	C₂×C₃₁	—	4	2	no	—
343	C₂×C₃₁	—	4	1	no	—
344	C₂×C₃₁	—	8	2	no	—
345	C₂×C₃₁	C₂×#413	1	1	yes	characteristic
346	C₂×C₃₁	#2901	1	1	yes	characteristic
347	C₂×C₃₁	#2901	1	1	yes	characteristic
348	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
349	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
350	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
351	C₂×M₁₆	—	2	1	no	—
352	C₂×M₁₆	—	2	1	no	—
353	C₂×M₁₆	—	2	2	no	—
354	C₂²×C₈	—	2	2	no	—
355	C₄⋊Q₈	—	2	1	no	—
356	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
357	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
358	#127	—	2	1	no	—
359	#126	—	2	2	no	—
360	#126	—	2	2	no	—
361	#126	—	2	2	no	—
362	#126	—	2	2	no	—
363	#125	C₂³×C₃₁	1	1	yes	characteristic
364	#125	C₂³×C₃₁	1	1	yes	characteristic
365	#124	—	2	1	no	—
366	#124	—	2	1	no	—
367	#124	—	2	2	no	—
368	#124	—	2	2	no	—
369	#124	—	2	1	no	—
370	#124	—	2	1	no	—
371	#124	C₂×C₄×C₃₁	1	2	yes	—
372	#124	C₂³×C₃₁	1	2	yes	—
373	#123	—	2	2	no	—
374	#123	—	2	2	no	—
375	#123	—	2	2	no	—
376	#123	—	2	2	no	—
377	#123	—	2	2	no	—
378	C₂²⋊Q₈	—	2	1	no	—
379	C₂²⋊Q₈	—	2	1	no	—
380	C₂²⋊Q₈	—	2	2	no	—
381	C₂²⋊Q₈	—	2	2	no	—
382	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
383	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
384	#121	C₂×C₄×C₃₁	1	2	yes	—
385	#121	C₂×C₄×C₃₁	1	2	yes	—
386	#120	—	2	2	no	—
387	C₄×Q₈	—	2	1	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	—	2	1	no	—
390	C₄×Q₈	—	2	1	no	—
391	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
392	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
393	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
394	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	1	no	—
398	C₄×D₄	—	2	1	no	—
399	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
400	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
401	C₄×D₄	D₄×C₃₁	1	2	yes	—
402	C₄×D₄	D₄×C₃₁	1	2	yes	—
403	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
404	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
405	#117	—	2	2	no	—
406	#117	—	2	2	no	—
407	#117	—	2	2	no	—
408	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
409	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
410	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
411	#117	C₂×C₄×C₃₁	1	2	yes	—
412	C₂×C₂²⋊C₄	—	2	1	no	—
413	C₂×C₂²⋊C₄	—	2	1	no	—
414	C₂×C₂²⋊C₄	—	2	2	no	—
415	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
416	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
417	C₄⋊C₈	—	2	1	no	—
418	C₄⋊C₈	—	2	1	no	—
419	C₄⋊C₈	—	2	2	no	—
420	C₄⋊C₈	—	2	2	no	—
421	C₂²⋊C₈	—	2	1	no	—
422	C₂²⋊C₈	—	2	1	no	—
423	C₂²⋊C₈	—	2	2	no	—
424	C₂²⋊C₈	—	2	2	no	—
425	#97	—	2	2	no	—
426	#97	C₂³×C₃₁	1	1	yes	characteristic
427	#97	C₂³×C₃₁	1	2	yes	—
428	#97	C₂³×C₃₁	1	1	yes	characteristic
429	#97	C₂³×C₃₁	1	1	yes	characteristic
430	#97	C₂³×C₃₁	1	1	yes	characteristic
431	#97	C₂³×C₃₁	1	2	yes	—
432	C₄×C₈	—	2	1	no	—
433	C₄×C₈	—	2	1	no	—
434	C₄×C₈	—	2	2	no	—
435	C₃₁	#17691	1	1	yes	characteristic, direct product factor, Sylow
436	C₂⁴	—	2	1	no	—
437	#41	—	4	2	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	—	4	2	no	—
442	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
443	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	—	2	1	no	—
447	C₂×D₄	—	4	2	no	—
448	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
449	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
450	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	2	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
455	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
456	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
457	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
458	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
459	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
460	M₁₆	—	4	2	no	—
461	M₁₆	—	4	2	no	—
462	M₁₆	—	4	4	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	2	no	—
468	C₂×C₈	—	2	2	no	—
469	C₂×C₈	—	4	4	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
481	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
482	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
483	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
484	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
485	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
486	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
487	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
488	C₂²⋊C₄	—	2	1	no	—
489	C₂²⋊C₄	—	2	1	no	—
490	C₂²⋊C₄	—	2	2	no	—
491	C₂²⋊C₄	—	2	2	no	—
492	C₂²⋊C₄	—	2	2	no	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	2	2	no	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₂²⋊C₄	—	2	2	no	—
499	C₂²⋊C₄	—	2	2	no	—
500	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
501	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
502	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
503	C₄²	—	2	2	no	—
504	C₄²	—	2	2	no	—
505	C₄²	—	2	1	no	—
506	C₄²	—	2	1	no	—
507	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
508	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
509	C₄²	C₂×D₄×C₃₁	1	2	yes	—
510	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
511	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
512	C₄²	C₂×D₄×C₃₁	1	2	yes	—
513	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
514	C₂³	—	2	2	no	—
515	C₂³	—	4	2	no	—
516	C₂³	—	8	1	no	—
517	C₂³	—	8	1	no	—
518	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
519	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
520	Q₈	—	2	2	no	—
521	Q₈	—	4	2	no	—
522	Q₈	—	4	2	no	—
523	Q₈	—	8	1	no	—
524	Q₈	—	8	1	no	—
525	D₄	—	2	2	no	—
526	D₄	—	4	2	no	—
527	D₄	—	4	2	no	—
528	D₄	—	4	2	no	—
529	D₄	—	8	2	no	—
530	D₄	—	8	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	4	2	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	—	4	2	no	—
543	C₂×C₄	—	8	2	no	—
544	C₂×C₄	—	8	2	no	—
545	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
546	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
547	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
548	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
549	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
550	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
551	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
552	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
553	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
554	C₈	—	2	2	no	—
555	C₈	—	2	2	no	—
556	C₈	—	4	2	no	—
557	C₈	—	4	1	no	—
558	C₈	—	4	1	no	—
559	C₈	—	4	2	no	—
560	C₂²	—	2	1	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	2	2	no	—
564	C₂²	—	2	1	no	—
565	C₂²	—	2	1	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	4	2	no	—
569	C₂²	—	16	2	no	—
570	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
571	C₄	—	2	1	no	—
572	C₄	—	2	2	no	—
573	C₄	—	2	1	no	—
574	C₄	—	2	1	no	—
575	C₄	—	2	2	no	—
576	C₄	—	2	2	no	—
577	C₄	—	2	1	no	—
578	C₄	—	2	1	no	—
579	C₄	—	4	2	no	—
580	C₄	—	4	1	no	—
581	C₄	—	4	1	no	—
582	C₄	—	4	1	no	—
583	C₄	—	4	1	no	—
584	C₄	C₃₁×#413	1	2	yes	—
585	C₂	—	4	1	no	—
586	C₂	—	4	2	no	—
587	C₂	—	8	2	no	—
588	C₂	C₂×C₃₁×#413	1	1	yes	characteristic
589	C₂	C₃₁×#2901	1	1	yes	characteristic
590	C₂	C₃₁×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#535	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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×#2971	C₂	semidirect
3. C₃₁×#2970	C₂	semidirect
4. C₃₁×#1537	C₂	semidirect
5. C₃₁×#1535	C₂	semidirect
6. C₃₁×#1532	C₂	semidirect
7. C₃₁×#1529	C₂	semidirect
8. C₃₁×#1527	C₂	semidirect
9. C₃₁×#1527	C₂	semidirect
10. C₃₁×#1526	C₂	semidirect
11. C₃₁×#1524	C₂	semidirect
37. C₃₁×#432	C₂²	semidirect
38. C₃₁×#432	C₂²	semidirect
41. C₄×M₁₆×C₃₁	C₂²	semidirect
42. C₂×C₃₁×#97	C₂²	semidirect
43. C₃₁×#330	C₂²	semidirect
44. C₃₁×#330	C₂²	semidirect
45. C₃₁×#330	C₂²	semidirect
46. C₃₁×#330	C₂²	semidirect
47. C₃₁×#330	C₂²	semidirect
48. C₃₁×#330	C₂²	semidirect
218. #17691	C₃₁	direct (N × H)
266. #2971	C₂×C₃₁	semidirect
267. #2970	C₂×C₃₁	semidirect
268. #1537	C₂×C₃₁	semidirect
269. #1535	C₂×C₃₁	semidirect
270. #1532	C₂×C₃₁	semidirect
271. #1529	C₂×C₃₁	semidirect
272. #1527	C₂×C₃₁	semidirect
273. #1527	C₂×C₃₁	semidirect
274. #1526	C₂×C₃₁	semidirect
275. #1524	C₂×C₃₁	semidirect
326. #432	C₂²×C₃₁	semidirect
327. #432	C₂²×C₃₁	semidirect
330. C₄×M₁₆	C₂²×C₃₁	semidirect
331. C₂×#97	C₂²×C₃₁	semidirect
332. #330	C₂²×C₃₁	semidirect
333. #330	C₂²×C₃₁	semidirect
334. #330	C₂²×C₃₁	semidirect
335. #330	C₂²×C₃₁	semidirect
336. #330	C₂²×C₃₁	semidirect
337. #330	C₂²×C₃₁	semidirect
435. C₃₁	#17691	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×#3510	2
C₃₁×#2971	2
C₃₁×#2970	2
C₃₁×#1537	2
C₃₁×#1535	2
C₃₁×#1532	2
C₃₁×#1529	2
C₃₁×#1527	2
C₃₁×#1526	2
C₃₁×#1524	2
#17691	31

Subgroups — C31×#17691 #21474450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17691 #21474450