Subgroups — C₃₁×#24711 #21481470

Order 7936.

828 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#432	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2857	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2075	C₂	1	2	yes	maximal
6	C₃₁×#2035	C₂	1	2	yes	maximal
7	C₃₁×#1987	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1944	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1935	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
14	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#477	—	2	4	no	—
18	C₃₁×#432	—	2	4	no	—
19	C₂×C₃₁×#107	C₂²	1	2	yes	—
20	C₂×C₃₁×#106	C₂²	1	2	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#103	C₂²	1	2	yes	—
24	C₂×C₃₁×#103	C₂²	1	2	yes	—
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
26	C₂×C₃₁×#97	C₂²	1	2	yes	—
27	C₃₁×#391	—	2	1	no	—
28	C₃₁×#391	C₂²	1	1	yes	characteristic
29	C₃₁×#389	—	2	1	no	—
30	C₃₁×#389	C₂²	1	2	yes	—
31	C₃₁×#389	C₂²	1	1	yes	characteristic
32	C₃₁×#387	—	2	1	no	—
33	C₃₁×#385	—	2	1	no	—
34	C₃₁×#384	C₂²	1	1	yes	characteristic
35	C₃₁×#380	—	2	1	no	—
36	C₃₁×#380	C₂²	1	1	yes	characteristic
37	C₃₁×#380	C₂²	1	2	yes	—
38	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
39	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#340	C₂²	1	2	yes	—
44	C₃₁×#340	C₂²	1	2	yes	—
45	C₂²×Q₈×C₃₁	—	2	1	no	—
46	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₃₁×C₄⋊Q₈	—	2	2	no	—
53	C₃₁×C₄⋊Q₈	—	2	2	no	—
54	C₃₁×C₂²⋊Q₈	—	2	2	no	—
55	C₃₁×C₂²⋊Q₈	—	2	2	no	—
56	C₃₁×C₂²⋊Q₈	—	4	4	no	—
57	C₄×Q₈×C₃₁	—	2	2	no	—
58	C₄×Q₈×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	—	2	2	no	—
75	C₂×C₄²×C₃₁	—	2	2	no	—
76	C₂×C₄²×C₃₁	—	2	1	no	—
77	C₂×C₄²×C₃₁	—	2	1	no	—
78	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₃₁×#107	—	2	4	no	—
83	C₃₁×#106	—	2	4	no	—
84	C₃₁×C₄⋊C₈	—	2	2	no	—
85	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
86	C₃₁×#103	—	2	4	no	—
87	C₃₁×#103	—	2	4	no	—
88	C₃₁×C₂²⋊C₈	—	2	4	no	—
89	C₃₁×#97	—	2	4	no	—
90	C₃₁×#95	—	2	1	no	—
91	C₃₁×#95	—	2	1	no	—
92	C₃₁×#95	—	2	1	no	—
93	C₃₁×#95	—	2	1	no	—
94	C₃₁×#95	—	2	1	no	—
95	C₃₁×#95	—	2	2	no	—
96	C₃₁×#95	—	2	2	no	—
97	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×Q₈×C₃₁	—	2	1	no	—
99	C₂×Q₈×C₃₁	—	2	1	no	—
100	C₂×Q₈×C₃₁	—	2	1	no	—
101	C₂×Q₈×C₃₁	—	2	1	no	—
102	C₂×Q₈×C₃₁	—	2	1	no	—
103	C₂×Q₈×C₃₁	—	4	2	no	—
104	C₂×Q₈×C₃₁	—	4	2	no	—
105	C₂×Q₈×C₃₁	—	4	2	no	—
106	C₂×Q₈×C₃₁	#41	1	2	yes	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	—	4	1	no	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
137	C₄⋊C₄×C₃₁	—	2	4	no	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₄⋊C₄×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	2	1	no	—
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	4	2	no	—
148	C₄⋊C₄×C₃₁	—	4	2	no	—
149	C₄⋊C₄×C₃₁	—	4	4	no	—
150	C₄⋊C₄×C₃₁	—	4	2	no	—
151	C₄⋊C₄×C₃₁	—	4	2	no	—
152	C₄⋊C₄×C₃₁	—	4	2	no	—
153	C₄⋊C₄×C₃₁	#41	1	2	yes	—
154	C₂²⋊C₄×C₃₁	—	2	4	no	—
155	C₂²⋊C₄×C₃₁	—	4	2	no	—
156	C₂²⋊C₄×C₃₁	—	4	4	no	—
157	C₄²×C₃₁	—	2	2	no	—
158	C₄²×C₃₁	—	2	2	no	—
159	C₄²×C₃₁	—	2	1	no	—
160	C₄²×C₃₁	—	2	1	no	—
161	C₄²×C₃₁	—	4	2	no	—
162	C₄²×C₃₁	—	4	2	no	—
163	C₄²×C₃₁	C₂×D₄	1	2	yes	—
164	C₄²×C₃₁	#41	1	2	yes	—
165	#24711	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂³×C₃₁	—	2	1	no	—
167	C₂³×C₃₁	—	2	1	no	—
168	C₂³×C₃₁	—	2	1	no	—
169	C₂³×C₃₁	—	2	1	no	—
170	C₂³×C₃₁	—	2	1	no	—
171	C₂³×C₃₁	—	2	1	no	—
172	C₂³×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
173	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
174	Q₈×C₃₁	—	2	2	no	—
175	Q₈×C₃₁	—	2	2	no	—
176	Q₈×C₃₁	—	4	1	no	—
177	Q₈×C₃₁	—	4	1	no	—
178	Q₈×C₃₁	—	8	4	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	4	4	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	1	no	—
199	C₂×C₄×C₃₁	—	4	1	no	—
200	C₂×C₄×C₃₁	—	4	1	no	—
201	C₂×C₄×C₃₁	—	4	4	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	4	1	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	4	1	no	—
210	C₂×C₄×C₃₁	—	4	1	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	#121	1	2	yes	—
215	C₂×C₄×C₃₁	#121	1	2	yes	—
216	C₂×C₄×C₃₁	#137	1	2	yes	—
217	C₂×C₄×C₃₁	#137	1	2	yes	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
222	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
223	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
224	C₈×C₃₁	—	2	4	no	—
225	C₈×C₃₁	—	4	2	no	—
226	C₂×#477	C₂×C₃₁	1	1	yes	characteristic, complemented
227	C₂×#432	C₂×C₃₁	1	1	yes	characteristic
228	#2857	C₂×C₃₁	1	1	yes	characteristic
229	C₄×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
230	#2075	C₂×C₃₁	1	2	yes	—
231	#2035	C₂×C₃₁	1	2	yes	—
232	#1987	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#1944	C₂×C₃₁	1	2	yes	complemented
235	#1935	C₂×C₃₁	1	2	yes	complemented
236	#1803	C₂×C₃₁	1	1	yes	characteristic, complemented
237	C₂²×C₃₁	—	2	2	no	—
238	C₂²×C₃₁	—	2	2	no	—
239	C₂²×C₃₁	—	2	2	no	—
240	C₂²×C₃₁	—	4	1	no	—
241	C₂²×C₃₁	—	4	1	no	—
242	C₂²×C₃₁	—	4	1	no	—
243	C₂²×C₃₁	—	4	1	no	—
244	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
245	C₂²×C₃₁	#525	1	1	yes	characteristic
246	C₂²×C₃₁	#536	1	1	yes	characteristic
247	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
249	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
250	C₂²×C₃₁	#575	1	1	yes	characteristic
251	C₄×C₃₁	—	2	2	no	—
252	C₄×C₃₁	—	4	2	no	—
253	C₄×C₃₁	—	4	1	no	—
254	C₄×C₃₁	—	4	2	no	—
255	C₄×C₃₁	—	4	1	no	—
256	C₄×C₃₁	—	4	1	no	—
257	C₄×C₃₁	—	8	2	no	—
258	C₄×C₃₁	—	8	1	no	—
259	C₄×C₃₁	—	8	1	no	—
260	C₄×C₃₁	—	8	2	no	—
261	C₄×C₃₁	#470	1	4	yes	—
262	C₄×C₃₁	#547	1	1	yes	characteristic
263	C₄×C₃₁	#548	1	1	yes	characteristic
264	C₄×C₃₁	#552	1	1	yes	characteristic
265	C₄×C₃₁	#556	1	1	yes	characteristic
266	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×C₂²⋊Q₈	—	2	1	no	—
268	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
270	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
271	#477	—	2	4	no	—
272	#432	—	2	4	no	—
273	C₂×#107	C₂²×C₃₁	1	2	yes	—
274	C₂×#106	C₂²×C₃₁	1	2	yes	—
275	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×#103	C₂²×C₃₁	1	2	yes	—
278	C₂×#103	C₂²×C₃₁	1	2	yes	—
279	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
280	C₂×#97	C₂²×C₃₁	1	2	yes	—
281	#391	—	2	1	no	—
282	#391	C₂²×C₃₁	1	1	yes	characteristic
283	#389	—	2	1	no	—
284	#389	C₂²×C₃₁	1	2	yes	—
285	#389	C₂²×C₃₁	1	1	yes	characteristic
286	#387	—	2	1	no	—
287	#385	—	2	1	no	—
288	#384	C₂²×C₃₁	1	1	yes	characteristic
289	#380	—	2	1	no	—
290	#380	C₂²×C₃₁	1	1	yes	characteristic
291	#380	C₂²×C₃₁	1	2	yes	—
292	C₄×C₄⋊C₄	—	2	1	no	—
293	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
294	C₄×C₂²⋊C₄	—	2	1	no	—
295	#340	C₂²×C₃₁	1	2	yes	—
296	#340	C₂²×C₃₁	1	2	yes	—
297	#340	C₂²×C₃₁	1	2	yes	—
298	#340	C₂²×C₃₁	1	2	yes	—
299	C₂×C₃₁	—	4	2	no	—
300	C₂×C₃₁	#2415	1	1	yes	characteristic
301	C₂×C₃₁	C₂×#470	1	1	yes	characteristic
302	C₂×C₃₁	#3055	1	1	yes	characteristic
303	C₂×C₃₁	#3112	1	1	yes	characteristic
304	C₂×C₃₁	#3116	1	1	yes	characteristic
305	C₂×C₃₁	#3237	1	1	yes	characteristic
306	C₂×C₃₁	#3238	1	1	yes	characteristic
307	C₂²×Q₈	—	2	1	no	—
308	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
309	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
310	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
311	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
313	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
314	C₄⋊Q₈	—	2	2	no	—
315	C₄⋊Q₈	—	2	2	no	—
316	C₂²⋊Q₈	—	2	2	no	—
317	C₂²⋊Q₈	—	2	2	no	—
318	C₂²⋊Q₈	—	4	4	no	—
319	C₄×Q₈	—	2	2	no	—
320	C₄×Q₈	—	2	2	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	—	2	2	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	—	2	1	no	—
327	C₂×C₄⋊C₄	—	2	1	no	—
328	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
329	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
330	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
331	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
332	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
333	C₂×C₂²⋊C₄	—	2	1	no	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	—	2	1	no	—
336	C₂×C₄²	—	2	2	no	—
337	C₂×C₄²	—	2	2	no	—
338	C₂×C₄²	—	2	1	no	—
339	C₂×C₄²	—	2	1	no	—
340	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
341	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
342	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
343	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
344	#107	—	2	4	no	—
345	#106	—	2	4	no	—
346	C₄⋊C₈	—	2	2	no	—
347	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
348	#103	—	2	4	no	—
349	#103	—	2	4	no	—
350	C₂²⋊C₈	—	2	4	no	—
351	#97	—	2	4	no	—
352	#95	—	2	1	no	—
353	#95	—	2	1	no	—
354	#95	—	2	1	no	—
355	#95	—	2	1	no	—
356	#95	—	2	1	no	—
357	#95	—	2	2	no	—
358	#95	—	2	2	no	—
359	C₃₁	#24711	1	1	yes	characteristic, direct product factor, Sylow
360	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
361	C₂×Q₈	—	2	1	no	—
362	C₂×Q₈	—	2	1	no	—
363	C₂×Q₈	—	2	1	no	—
364	C₂×Q₈	—	2	1	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	—	4	2	no	—
367	C₂×Q₈	—	4	2	no	—
368	C₂×Q₈	—	4	2	no	—
369	C₂×Q₈	#41×C₃₁	1	2	yes	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	4	1	no	—
388	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
389	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
390	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
393	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	4	2	no	—
411	C₄⋊C₄	—	4	2	no	—
412	C₄⋊C₄	—	4	4	no	—
413	C₄⋊C₄	—	4	2	no	—
414	C₄⋊C₄	—	4	2	no	—
415	C₄⋊C₄	—	4	2	no	—
416	C₄⋊C₄	#41×C₃₁	1	2	yes	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	—	4	2	no	—
419	C₂²⋊C₄	—	4	4	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	2	2	no	—
422	C₄²	—	2	1	no	—
423	C₄²	—	2	1	no	—
424	C₄²	—	4	2	no	—
425	C₄²	—	4	2	no	—
426	C₄²	C₂×D₄×C₃₁	1	2	yes	—
427	C₄²	#41×C₃₁	1	2	yes	—
428	C₂³	—	2	1	no	—
429	C₂³	—	2	1	no	—
430	C₂³	—	2	1	no	—
431	C₂³	—	2	1	no	—
432	C₂³	—	2	1	no	—
433	C₂³	—	2	1	no	—
434	C₂³	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
435	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
436	Q₈	—	2	2	no	—
437	Q₈	—	2	2	no	—
438	Q₈	—	4	1	no	—
439	Q₈	—	4	1	no	—
440	Q₈	—	8	4	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	4	4	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	4	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	1	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	C₃₁×#121	1	2	yes	—
477	C₂×C₄	C₃₁×#121	1	2	yes	—
478	C₂×C₄	C₃₁×#137	1	2	yes	—
479	C₂×C₄	C₃₁×#137	1	2	yes	—
480	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
481	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
482	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
483	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
484	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
485	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
486	C₈	—	2	4	no	—
487	C₈	—	4	2	no	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	4	1	no	—
492	C₂²	—	4	1	no	—
493	C₂²	—	4	1	no	—
494	C₂²	—	4	1	no	—
495	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
496	C₂²	C₃₁×#525	1	1	yes	characteristic
497	C₂²	C₃₁×#536	1	1	yes	characteristic
498	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
499	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
500	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
501	C₂²	C₃₁×#575	1	1	yes	characteristic
502	C₄	—	2	2	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	1	no	—
505	C₄	—	4	2	no	—
506	C₄	—	4	1	no	—
507	C₄	—	4	1	no	—
508	C₄	—	8	2	no	—
509	C₄	—	8	1	no	—
510	C₄	—	8	2	no	—
511	C₄	—	8	1	no	—
512	C₄	C₃₁×#470	1	4	yes	—
513	C₄	C₃₁×#547	1	1	yes	characteristic
514	C₄	C₃₁×#548	1	1	yes	characteristic
515	C₄	C₃₁×#552	1	1	yes	characteristic
516	C₄	C₃₁×#556	1	1	yes	characteristic
517	C₂	—	4	2	no	—
518	C₂	C₃₁×#2415	1	1	yes	characteristic
519	C₂	C₂×C₃₁×#470	1	1	yes	characteristic
520	C₂	C₃₁×#3055	1	1	yes	characteristic
521	C₂	C₃₁×#3112	1	1	yes	characteristic
522	C₂	C₃₁×#3116	1	1	yes	characteristic
523	C₂	C₃₁×#3237	1	1	yes	characteristic
524	C₂	C₃₁×#3238	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#477	C₂	semidirect
7. C₃₁×#1987	C₂	semidirect
8. C₃₁×#1948	C₂	semidirect
9. C₃₁×#1944	C₂	semidirect
10. C₃₁×#1935	C₂	semidirect
11. C₃₁×#1803	C₂	semidirect
165. #24711	C₃₁	direct (N × H)
226. C₂×#477	C₂×C₃₁	semidirect
232. #1987	C₂×C₃₁	semidirect
233. #1948	C₂×C₃₁	semidirect
234. #1944	C₂×C₃₁	semidirect
235. #1935	C₂×C₃₁	semidirect
236. #1803	C₂×C₃₁	semidirect
359. C₃₁	#24711	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#477	2
C₂×C₃₁×#432	2
C₃₁×#2857	2
C₄×C₃₁×C₂²⋊Q₈	2
C₃₁×#2075	2
C₃₁×#2035	2
C₃₁×#1987	2
C₃₁×#1948	2
C₃₁×#1944	2
C₃₁×#1935	2
C₃₁×#1803	2
#24711	31

Subgroups — C31×#24711 #21481470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24711 #21481470