Subgroups — C₃₁×#33218 #21489977

Order 7936.

870 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#501	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2857	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2851	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1992	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1942	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1941	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1847	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1809	C₂	1	2	yes	maximal
11	C₃₁×#1809	C₂	1	2	yes	maximal
12	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#501	—	2	4	no	—
18	C₂×C₃₁×#107	C₂²	1	2	yes	—
19	C₂×C₃₁×#107	C₂²	1	2	yes	—
20	C₂×C₃₁×#106	C₂²	1	2	yes	—
21	C₂×C₃₁×#106	C₂²	1	2	yes	—
22	C₄×M₁₆×C₃₁	—	2	4	no	—
23	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#391	—	2	1	no	—
27	C₃₁×#389	—	2	1	no	—
28	C₃₁×#389	C₂²	1	1	yes	characteristic
29	C₃₁×#389	C₂²	1	1	yes	characteristic
30	C₃₁×#389	C₂²	1	2	yes	—
31	C₃₁×#387	—	2	1	no	—
32	C₃₁×#387	—	2	1	no	—
33	C₃₁×#385	—	2	1	no	—
34	C₃₁×#385	—	2	1	no	—
35	C₃₁×#384	C₂²	1	1	yes	characteristic
36	C₃₁×#382	C₂²	1	1	yes	characteristic
37	C₃₁×#380	—	2	1	no	—
38	C₃₁×#380	C₂²	1	1	yes	characteristic
39	C₃₁×#380	C₂²	1	1	yes	characteristic
40	C₃₁×#380	C₂²	1	2	yes	—
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	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₃₁	—	2	4	no	—
50	C₂×M₁₆×C₃₁	—	2	4	no	—
51	C₂×M₁₆×C₃₁	Q₈	1	8	yes	complemented
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₃₁×C₄⋊Q₈	—	2	2	no	—
55	C₃₁×C₄⋊Q₈	—	2	2	no	—
56	C₃₁×#125	—	2	2	no	—
57	C₃₁×#125	—	2	2	no	—
58	C₃₁×C₂²⋊Q₈	—	4	4	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	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	—	2	1	no	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₄²×C₃₁	—	2	2	no	—
77	C₂×C₄²×C₃₁	—	2	2	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×#107	—	2	4	no	—
84	C₃₁×#106	—	2	4	no	—
85	C₃₁×#106	—	2	4	no	—
86	C₃₁×#97	—	2	2	no	—
87	C₃₁×#97	D₄	1	4	yes	—
88	C₄×C₈×C₃₁	—	2	4	no	—
89	C₃₁×#95	—	2	2	no	—
90	C₃₁×#95	—	2	1	no	—
91	C₃₁×#95	—	2	1	no	—
92	C₃₁×#95	—	2	2	no	—
93	C₃₁×#95	—	2	1	no	—
94	C₃₁×#95	—	2	1	no	—
95	C₃₁×#95	—	2	1	no	—
96	C₃₁×#95	—	2	1	no	—
97	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×Q₈×C₃₁	—	4	2	no	—
99	C₂×Q₈×C₃₁	—	4	2	no	—
100	C₂×Q₈×C₃₁	—	4	2	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
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	1	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	4	1	no	—
115	C₂²×C₄×C₃₁	—	4	1	no	—
116	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
122	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
123	M₁₆×C₃₁	—	2	16	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
127	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
128	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
129	C₂×C₈×C₃₁	#41	1	4	yes	—
130	C₄⋊C₄×C₃₁	—	2	1	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	1	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	1	no	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	4	2	no	—
141	C₄⋊C₄×C₃₁	—	4	2	no	—
142	C₄⋊C₄×C₃₁	—	4	2	no	—
143	C₄⋊C₄×C₃₁	—	4	2	no	—
144	C₄⋊C₄×C₃₁	—	4	2	no	—
145	C₄⋊C₄×C₃₁	—	4	2	no	—
146	C₄⋊C₄×C₃₁	—	4	4	no	—
147	C₄⋊C₄×C₃₁	—	4	2	no	—
148	C₂²⋊C₄×C₃₁	—	4	2	no	—
149	C₂²⋊C₄×C₃₁	—	4	2	no	—
150	C₂²⋊C₄×C₃₁	—	4	4	no	—
151	C₄²×C₃₁	—	2	2	no	—
152	C₄²×C₃₁	—	2	2	no	—
153	C₄²×C₃₁	—	2	1	no	—
154	C₄²×C₃₁	—	2	1	no	—
155	C₄²×C₃₁	C₂×D₄	1	2	yes	—
156	C₄²×C₃₁	#41	1	2	yes	—
157	#33218	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂³×C₃₁	—	2	1	no	—
159	C₂³×C₃₁	—	2	1	no	—
160	C₂³×C₃₁	—	2	1	no	—
161	C₂³×C₃₁	—	2	1	no	—
162	C₂³×C₃₁	—	2	1	no	—
163	C₂³×C₃₁	—	2	1	no	—
164	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
165	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
166	Q₈×C₃₁	—	8	4	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	4	no	—
178	C₂×C₄×C₃₁	—	2	4	no	—
179	C₂×C₄×C₃₁	—	4	4	no	—
180	C₂×C₄×C₃₁	—	4	4	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	—	4	1	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	1	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	1	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	2	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	#121	1	2	yes	—
200	C₂×C₄×C₃₁	#121	1	2	yes	—
201	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
202	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
203	C₂×C₄×C₃₁	#137	1	2	yes	—
204	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
206	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
207	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
208	C₈×C₃₁	—	2	4	no	—
209	C₈×C₃₁	C₂²⋊Q₈	1	8	yes	complemented
210	C₂×#501	C₂×C₃₁	1	1	yes	characteristic, complemented
211	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
212	#2857	C₂×C₃₁	1	1	yes	characteristic
213	#2851	C₂×C₃₁	1	1	yes	characteristic
214	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#1992	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#1942	C₂×C₃₁	1	2	yes	complemented
217	#1941	C₂×C₃₁	1	2	yes	complemented
218	#1847	C₂×C₃₁	1	1	yes	characteristic, complemented
219	#1809	C₂×C₃₁	1	2	yes	—
220	#1809	C₂×C₃₁	1	2	yes	—
221	C₂²×C₃₁	—	2	2	no	—
222	C₂²×C₃₁	—	2	2	no	—
223	C₂²×C₃₁	—	2	2	no	—
224	C₂²×C₃₁	—	4	1	no	—
225	C₂²×C₃₁	—	4	1	no	—
226	C₂²×C₃₁	—	4	1	no	—
227	C₂²×C₃₁	—	4	1	no	—
228	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
229	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
230	C₂²×C₃₁	#532	1	1	yes	characteristic
231	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
232	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
233	C₂²×C₃₁	#575	1	1	yes	characteristic
234	C₂²×C₃₁	#575	1	1	yes	characteristic
235	C₄×C₃₁	—	2	2	no	—
236	C₄×C₃₁	—	2	2	no	—
237	C₄×C₃₁	—	2	2	no	—
238	C₄×C₃₁	—	2	2	no	—
239	C₄×C₃₁	—	2	2	no	—
240	C₄×C₃₁	—	8	2	no	—
241	C₄×C₃₁	—	8	1	no	—
242	C₄×C₃₁	—	8	1	no	—
243	C₄×C₃₁	—	8	1	no	—
244	C₄×C₃₁	—	8	2	no	—
245	C₄×C₃₁	—	8	1	no	—
246	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
247	C₄×C₃₁	#533	1	2	yes	—
248	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
249	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×C₂²⋊Q₈	—	2	1	no	—
252	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
253	#501	—	2	4	no	—
254	C₂×#107	C₂²×C₃₁	1	2	yes	—
255	C₂×#107	C₂²×C₃₁	1	2	yes	—
256	C₂×#106	C₂²×C₃₁	1	2	yes	—
257	C₂×#106	C₂²×C₃₁	1	2	yes	—
258	C₄×M₁₆	—	2	4	no	—
259	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
262	#391	—	2	1	no	—
263	#389	—	2	1	no	—
264	#389	C₂²×C₃₁	1	1	yes	characteristic
265	#389	C₂²×C₃₁	1	1	yes	characteristic
266	#389	C₂²×C₃₁	1	2	yes	—
267	#387	—	2	1	no	—
268	#387	—	2	1	no	—
269	#385	—	2	1	no	—
270	#385	—	2	1	no	—
271	#384	C₂²×C₃₁	1	1	yes	characteristic
272	#382	C₂²×C₃₁	1	1	yes	characteristic
273	#380	—	2	1	no	—
274	#380	C₂²×C₃₁	1	1	yes	characteristic
275	#380	C₂²×C₃₁	1	1	yes	characteristic
276	#380	C₂²×C₃₁	1	2	yes	—
277	#340	C₂²×C₃₁	1	2	yes	—
278	#340	C₂²×C₃₁	1	2	yes	—
279	#340	C₂²×C₃₁	1	2	yes	—
280	#340	C₂²×C₃₁	1	2	yes	—
281	C₂×C₃₁	—	4	2	no	—
282	C₂×C₃₁	#2857	1	1	yes	characteristic
283	C₂×C₃₁	#3055	1	1	yes	characteristic
284	C₂×C₃₁	#3056	1	1	yes	characteristic
285	C₂×C₃₁	#3165	1	1	yes	characteristic
286	C₂×C₃₁	#3166	1	1	yes	characteristic
287	C₂×C₃₁	#3171	1	1	yes	characteristic
288	C₂×C₃₁	#3172	1	1	yes	characteristic
289	C₂²×Q₈	—	2	1	no	—
290	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
291	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
292	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
293	C₂×M₁₆	—	2	4	no	—
294	C₂×M₁₆	—	2	4	no	—
295	C₂×M₁₆	Q₈×C₃₁	1	8	yes	complemented
296	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
297	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
298	C₄⋊Q₈	—	2	2	no	—
299	C₄⋊Q₈	—	2	2	no	—
300	#125	—	2	2	no	—
301	#125	—	2	2	no	—
302	C₂²⋊Q₈	—	4	4	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	2	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	—	2	1	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
313	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
314	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
315	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
316	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
317	C₂×C₂²⋊C₄	—	2	2	no	—
318	C₂×C₂²⋊C₄	—	2	1	no	—
319	C₂×C₂²⋊C₄	—	2	1	no	—
320	C₂×C₄²	—	2	2	no	—
321	C₂×C₄²	—	2	2	no	—
322	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
323	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
324	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
325	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
326	#107	—	2	4	no	—
327	#107	—	2	4	no	—
328	#106	—	2	4	no	—
329	#106	—	2	4	no	—
330	#97	—	2	2	no	—
331	#97	D₄×C₃₁	1	4	yes	—
332	C₄×C₈	—	2	4	no	—
333	#95	—	2	2	no	—
334	#95	—	2	1	no	—
335	#95	—	2	1	no	—
336	#95	—	2	2	no	—
337	#95	—	2	1	no	—
338	#95	—	2	1	no	—
339	#95	—	2	1	no	—
340	#95	—	2	1	no	—
341	C₃₁	#33218	1	1	yes	characteristic, direct product factor, Sylow
342	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
343	C₂×Q₈	—	4	2	no	—
344	C₂×Q₈	—	4	2	no	—
345	C₂×Q₈	—	4	2	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	4	1	no	—
360	C₂²×C₄	—	4	1	no	—
361	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
363	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
364	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
365	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
366	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
367	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
368	M₁₆	—	2	16	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
372	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
373	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
374	C₂×C₈	#41×C₃₁	1	4	yes	—
375	C₄⋊C₄	—	2	1	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	4	2	no	—
386	C₄⋊C₄	—	4	2	no	—
387	C₄⋊C₄	—	4	2	no	—
388	C₄⋊C₄	—	4	2	no	—
389	C₄⋊C₄	—	4	2	no	—
390	C₄⋊C₄	—	4	2	no	—
391	C₄⋊C₄	—	4	4	no	—
392	C₄⋊C₄	—	4	2	no	—
393	C₂²⋊C₄	—	4	2	no	—
394	C₂²⋊C₄	—	4	2	no	—
395	C₂²⋊C₄	—	4	4	no	—
396	C₄²	—	2	2	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	2	1	no	—
399	C₄²	—	2	1	no	—
400	C₄²	C₂×D₄×C₃₁	1	2	yes	—
401	C₄²	#41×C₃₁	1	2	yes	—
402	C₂³	—	2	1	no	—
403	C₂³	—	2	1	no	—
404	C₂³	—	2	1	no	—
405	C₂³	—	2	1	no	—
406	C₂³	—	2	1	no	—
407	C₂³	—	2	1	no	—
408	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
409	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
410	Q₈	—	8	4	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	4	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	4	4	no	—
424	C₂×C₄	—	4	4	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	C₃₁×#121	1	2	yes	—
444	C₂×C₄	C₃₁×#121	1	2	yes	—
445	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
446	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
447	C₂×C₄	C₃₁×#137	1	2	yes	—
448	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
449	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
450	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
451	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
452	C₈	—	2	4	no	—
453	C₈	C₃₁×C₂²⋊Q₈	1	8	yes	complemented
454	C₂²	—	2	2	no	—
455	C₂²	—	2	2	no	—
456	C₂²	—	2	2	no	—
457	C₂²	—	4	1	no	—
458	C₂²	—	4	1	no	—
459	C₂²	—	4	1	no	—
460	C₂²	—	4	1	no	—
461	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
462	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
463	C₂²	C₃₁×#532	1	1	yes	characteristic
464	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
465	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
466	C₂²	C₃₁×#575	1	1	yes	characteristic
467	C₂²	C₃₁×#575	1	1	yes	characteristic
468	C₄	—	2	2	no	—
469	C₄	—	2	2	no	—
470	C₄	—	2	2	no	—
471	C₄	—	2	2	no	—
472	C₄	—	2	2	no	—
473	C₄	—	8	2	no	—
474	C₄	—	8	1	no	—
475	C₄	—	8	1	no	—
476	C₄	—	8	2	no	—
477	C₄	—	8	1	no	—
478	C₄	—	8	1	no	—
479	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
480	C₄	C₃₁×#533	1	2	yes	—
481	C₂	—	4	2	no	—
482	C₂	C₃₁×#2857	1	1	yes	characteristic
483	C₂	C₃₁×#3055	1	1	yes	characteristic
484	C₂	C₃₁×#3056	1	1	yes	characteristic
485	C₂	C₃₁×#3165	1	1	yes	characteristic
486	C₂	C₃₁×#3166	1	1	yes	characteristic
487	C₂	C₃₁×#3171	1	1	yes	characteristic
488	C₂	C₃₁×#3172	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×#501	C₂	semidirect
5. C₃₁×#1993	C₂	semidirect
6. C₃₁×#1992	C₂	semidirect
7. C₃₁×#1942	C₂	semidirect
8. C₃₁×#1941	C₂	semidirect
9. C₃₁×#1847	C₂	semidirect
51. C₂×M₁₆×C₃₁	Q₈	semidirect
157. #33218	C₃₁	direct (N × H)
209. C₈×C₃₁	C₂²⋊Q₈	semidirect
210. C₂×#501	C₂×C₃₁	semidirect
214. #1993	C₂×C₃₁	semidirect
215. #1992	C₂×C₃₁	semidirect
216. #1942	C₂×C₃₁	semidirect
217. #1941	C₂×C₃₁	semidirect
218. #1847	C₂×C₃₁	semidirect
295. C₂×M₁₆	Q₈×C₃₁	semidirect
341. C₃₁	#33218	direct (N × H)
453. C₈	C₃₁×C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#501	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#2857	2
C₃₁×#2851	2
C₃₁×#1993	2
C₃₁×#1992	2
C₃₁×#1942	2
C₃₁×#1941	2
C₃₁×#1847	2
C₃₁×#1809	2
#33218	31

Subgroups — C31×#33218 #21489977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33218 #21489977