Subgroups — C₃₁×#24649 #21481408

Order 7936.

884 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#460	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#441	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2548	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2356	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2055	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2015	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1983	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1961	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1926	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1805	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#124	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#460	—	2	4	no	—
19	C₃₁×#441	—	2	4	no	—
20	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#103	C₂²	1	2	yes	—
26	C₂×C₃₁×#102	C₂²	1	2	yes	—
27	C₂×C₃₁×#102	C₂²	1	2	yes	—
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
30	C₃₁×#394	—	2	1	no	—
31	C₃₁×#394	—	2	1	no	—
32	C₃₁×#391	C₂²	1	2	yes	—
33	C₃₁×#390	—	2	2	no	—
34	C₃₁×#390	C₂²	1	2	yes	—
35	C₃₁×#387	—	2	2	no	—
36	C₃₁×#383	—	2	1	no	—
37	C₃₁×#382	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
39	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#376	—	2	1	no	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₂²×D₄×C₃₁	—	2	1	no	—
47	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
48	C₂×Q₁₆×C₃₁	—	2	2	no	—
49	C₂×Q₁₆×C₃₁	—	2	2	no	—
50	C₂×Q₁₆×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	—	2	4	no	—
52	C₂×SD₁₆×C₃₁	—	4	2	no	—
53	C₂×SD₁₆×C₃₁	—	4	2	no	—
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₃₁×#127	—	2	2	no	—
58	C₃₁×#127	—	2	2	no	—
59	C₃₁×#124	—	2	4	no	—
60	C₃₁×#124	—	4	2	no	—
61	C₄×Q₈×C₃₁	—	2	2	no	—
62	C₄×Q₈×C₃₁	—	2	2	no	—
63	C₄×Q₈×C₃₁	—	2	2	no	—
64	C₄×Q₈×C₃₁	D₄	1	4	yes	—
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
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	2	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₄²×C₃₁	—	2	1	no	—
76	C₂×C₄²×C₃₁	—	2	1	no	—
77	C₂×C₄²×C₃₁	—	2	2	no	—
78	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×#106	—	2	2	no	—
81	C₃₁×C₄⋊C₈	—	2	2	no	—
82	C₃₁×C₄⋊C₈	—	4	2	no	—
83	C₃₁×#103	—	2	2	no	—
84	C₃₁×#103	—	2	2	no	—
85	C₃₁×#103	—	4	2	no	—
86	C₃₁×#102	—	2	4	no	—
87	C₃₁×#102	—	2	4	no	—
88	C₃₁×#102	—	4	2	no	—
89	C₃₁×#97	—	2	2	no	—
90	C₃₁×#95	—	2	1	no	—
91	C₃₁×#95	—	2	2	no	—
92	C₃₁×#95	—	2	2	no	—
93	C₃₁×#95	—	2	1	no	—
94	C₃₁×#95	—	2	2	no	—
95	C₃₁×#95	D₄	1	1	yes	characteristic
96	C₃₁×#95	D₄	1	1	yes	characteristic
97	C₂⁴×C₃₁	—	2	2	no	—
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	—	2	1	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	2	1	no	—
106	C₂×Q₈×C₃₁	—	2	1	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
109	C₂×Q₈×C₃₁	#41	1	2	yes	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	4	4	no	—
117	C₂×D₄×C₃₁	—	4	2	no	—
118	C₂×D₄×C₃₁	—	4	2	no	—
119	C₂×D₄×C₃₁	—	8	2	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
130	Q₁₆×C₃₁	—	2	8	no	—
131	SD₁₆×C₃₁	—	4	8	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	#41	1	2	yes	—
141	C₄⋊C₄×C₃₁	—	2	4	no	—
142	C₄⋊C₄×C₃₁	—	2	1	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	2	no	—
147	C₄⋊C₄×C₃₁	—	4	2	no	—
148	C₄⋊C₄×C₃₁	—	4	2	no	—
149	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
150	C₂²⋊C₄×C₃₁	—	4	4	no	—
151	C₂²⋊C₄×C₃₁	—	4	4	no	—
152	C₂²⋊C₄×C₃₁	—	8	2	no	—
153	C₂²⋊C₄×C₃₁	—	8	2	no	—
154	C₄²×C₃₁	—	2	4	no	—
155	C₄²×C₃₁	—	2	2	no	—
156	C₄²×C₃₁	—	2	2	no	—
157	C₄²×C₃₁	—	4	4	no	—
158	C₄²×C₃₁	—	4	2	no	—
159	C₄²×C₃₁	C₂×D₄	1	2	yes	—
160	C₄²×C₃₁	#41	1	2	yes	—
161	#24649	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂³×C₃₁	—	4	2	no	—
163	C₂³×C₃₁	—	4	2	no	—
164	C₂³×C₃₁	—	4	2	no	—
165	C₂³×C₃₁	—	4	2	no	—
166	C₂³×C₃₁	—	4	2	no	—
167	C₂³×C₃₁	—	4	2	no	—
168	C₂³×C₃₁	—	4	2	no	—
169	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
170	Q₈×C₃₁	—	2	2	no	—
171	Q₈×C₃₁	—	2	2	no	—
172	Q₈×C₃₁	—	2	2	no	—
173	Q₈×C₃₁	—	2	2	no	—
174	Q₈×C₃₁	—	4	1	no	—
175	Q₈×C₃₁	—	4	1	no	—
176	Q₈×C₃₁	—	4	1	no	—
177	Q₈×C₃₁	—	4	1	no	—
178	D₄×C₃₁	—	4	2	no	—
179	D₄×C₃₁	—	4	2	no	—
180	D₄×C₃₁	—	4	2	no	—
181	D₄×C₃₁	—	4	2	no	—
182	D₄×C₃₁	—	8	4	no	—
183	D₄×C₃₁	—	8	4	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	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	2	no	—
200	C₂×C₄×C₃₁	—	4	1	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	#121	1	2	yes	—
206	C₂×C₄×C₃₁	#121	1	2	yes	—
207	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
209	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
213	C₈×C₃₁	—	2	2	no	—
214	C₈×C₃₁	—	4	1	no	—
215	C₈×C₃₁	—	4	2	no	—
216	C₂×#460	C₂×C₃₁	1	1	yes	characteristic
217	C₂×#441	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#2548	C₂×C₃₁	1	1	yes	characteristic
219	#2356	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#2055	C₂×C₃₁	1	2	yes	complemented
221	#2015	C₂×C₃₁	1	2	yes	complemented
222	#1983	C₂×C₃₁	1	2	yes	complemented
223	#1961	C₂×C₃₁	1	1	yes	characteristic
224	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#1926	C₂×C₃₁	1	2	yes	complemented
226	#1805	C₂×C₃₁	1	1	yes	characteristic, complemented
227	C₂²×C₃₁	—	8	2	no	—
228	C₂²×C₃₁	—	8	2	no	—
229	C₂²×C₃₁	—	8	2	no	—
230	C₂²×C₃₁	—	8	2	no	—
231	C₂²×C₃₁	—	8	2	no	—
232	C₂²×C₃₁	—	8	2	no	—
233	C₂²×C₃₁	—	8	2	no	—
234	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
235	C₂²×C₃₁	#546	1	1	yes	characteristic
236	C₂²×C₃₁	#550	1	1	yes	characteristic
237	C₂²×C₃₁	#575	1	2	yes	—
238	C₂²×C₃₁	#577	1	2	yes	—
239	C₄×C₃₁	—	2	2	no	—
240	C₄×C₃₁	—	2	2	no	—
241	C₄×C₃₁	—	4	2	no	—
242	C₄×C₃₁	—	4	2	no	—
243	C₄×C₃₁	—	4	2	no	—
244	C₄×C₃₁	—	4	2	no	—
245	C₄×C₃₁	—	4	2	no	—
246	C₄×C₃₁	—	8	2	no	—
247	C₄×C₃₁	—	8	1	no	—
248	C₄×C₃₁	#525	1	1	yes	characteristic
249	C₄×C₃₁	#535	1	1	yes	characteristic
250	C₄×C₃₁	#553	1	1	yes	characteristic
251	C₄×C₃₁	#555	1	1	yes	characteristic
252	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
253	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
254	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×#124	C₂²×C₃₁	1	2	yes	—
256	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
258	#460	—	2	4	no	—
259	#441	—	2	4	no	—
260	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
263	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×#103	C₂²×C₃₁	1	2	yes	—
266	C₂×#102	C₂²×C₃₁	1	2	yes	—
267	C₂×#102	C₂²×C₃₁	1	2	yes	—
268	C₂×#102	C₂²×C₃₁	1	2	yes	—
269	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
270	#394	—	2	1	no	—
271	#394	—	2	1	no	—
272	#391	C₂²×C₃₁	1	2	yes	—
273	#390	—	2	2	no	—
274	#390	C₂²×C₃₁	1	2	yes	—
275	#387	—	2	2	no	—
276	#383	—	2	1	no	—
277	#382	C₂²×C₃₁	1	1	yes	characteristic
278	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
279	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
280	#376	—	2	1	no	—
281	#340	C₂²×C₃₁	1	2	yes	—
282	#336	C₂²×C₃₁	1	2	yes	—
283	C₂×C₃₁	—	16	2	no	—
284	C₂×C₃₁	#2414	1	1	yes	characteristic
285	C₂×C₃₁	#3040	1	1	yes	characteristic
286	C₂×C₃₁	#3043	1	1	yes	characteristic
287	C₂×C₃₁	#3317	1	2	yes	—
288	C₂×C₃₁	#3401	1	2	yes	—
289	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
290	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₂²×D₄	—	2	1	no	—
292	C₂²×D₄	—	2	1	no	—
293	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
294	C₂×Q₁₆	—	2	2	no	—
295	C₂×Q₁₆	—	2	2	no	—
296	C₂×Q₁₆	—	2	2	no	—
297	C₂×SD₁₆	—	2	4	no	—
298	C₂×SD₁₆	—	4	2	no	—
299	C₂×SD₁₆	—	4	2	no	—
300	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
301	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
302	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
303	#127	—	2	2	no	—
304	#127	—	2	2	no	—
305	#124	—	2	4	no	—
306	#124	—	4	2	no	—
307	C₄×Q₈	—	2	2	no	—
308	C₄×Q₈	—	2	2	no	—
309	C₄×Q₈	—	2	2	no	—
310	C₄×Q₈	D₄×C₃₁	1	4	yes	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
314	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
315	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
316	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
317	C₂×C₂²⋊C₄	—	2	2	no	—
318	C₂×C₂²⋊C₄	—	2	2	no	—
319	C₂×C₂²⋊C₄	—	2	2	no	—
320	C₂×C₂²⋊C₄	—	2	2	no	—
321	C₂×C₄²	—	2	1	no	—
322	C₂×C₄²	—	2	1	no	—
323	C₂×C₄²	—	2	2	no	—
324	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
325	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
326	#106	—	2	2	no	—
327	C₄⋊C₈	—	2	2	no	—
328	C₄⋊C₈	—	4	2	no	—
329	#103	—	2	2	no	—
330	#103	—	2	2	no	—
331	#103	—	4	2	no	—
332	#102	—	2	4	no	—
333	#102	—	2	4	no	—
334	#102	—	4	2	no	—
335	#97	—	2	2	no	—
336	#95	—	2	1	no	—
337	#95	—	2	2	no	—
338	#95	—	2	2	no	—
339	#95	—	2	1	no	—
340	#95	—	2	2	no	—
341	#95	D₄×C₃₁	1	1	yes	characteristic
342	#95	D₄×C₃₁	1	1	yes	characteristic
343	C₃₁	#24649	1	1	yes	characteristic, direct product factor, Sylow
344	C₂⁴	—	2	2	no	—
345	C₂×Q₈	—	2	1	no	—
346	C₂×Q₈	—	2	1	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	1	no	—
350	C₂×Q₈	—	2	1	no	—
351	C₂×Q₈	—	2	1	no	—
352	C₂×Q₈	—	2	1	no	—
353	C₂×Q₈	—	2	1	no	—
354	C₂×Q₈	—	2	1	no	—
355	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
356	C₂×Q₈	#41×C₃₁	1	2	yes	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	4	4	no	—
364	C₂×D₄	—	4	2	no	—
365	C₂×D₄	—	4	2	no	—
366	C₂×D₄	—	8	2	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
373	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
374	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
375	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
376	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
377	Q₁₆	—	2	8	no	—
378	SD₁₆	—	4	8	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	#41×C₃₁	1	2	yes	—
388	C₄⋊C₄	—	2	4	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	4	2	no	—
395	C₄⋊C₄	—	4	2	no	—
396	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
397	C₂²⋊C₄	—	4	4	no	—
398	C₂²⋊C₄	—	4	4	no	—
399	C₂²⋊C₄	—	8	2	no	—
400	C₂²⋊C₄	—	8	2	no	—
401	C₄²	—	2	4	no	—
402	C₄²	—	2	2	no	—
403	C₄²	—	2	2	no	—
404	C₄²	—	4	4	no	—
405	C₄²	—	4	2	no	—
406	C₄²	C₂×D₄×C₃₁	1	2	yes	—
407	C₄²	#41×C₃₁	1	2	yes	—
408	C₂³	—	4	2	no	—
409	C₂³	—	4	2	no	—
410	C₂³	—	4	2	no	—
411	C₂³	—	4	2	no	—
412	C₂³	—	4	2	no	—
413	C₂³	—	4	2	no	—
414	C₂³	—	4	2	no	—
415	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
416	Q₈	—	2	2	no	—
417	Q₈	—	2	2	no	—
418	Q₈	—	2	2	no	—
419	Q₈	—	2	2	no	—
420	Q₈	—	4	1	no	—
421	Q₈	—	4	1	no	—
422	Q₈	—	4	1	no	—
423	Q₈	—	4	1	no	—
424	D₄	—	4	2	no	—
425	D₄	—	4	2	no	—
426	D₄	—	4	2	no	—
427	D₄	—	4	2	no	—
428	D₄	—	8	4	no	—
429	D₄	—	8	4	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	C₃₁×#121	1	2	yes	—
452	C₂×C₄	C₃₁×#121	1	2	yes	—
453	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
454	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
455	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
456	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
457	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
458	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
459	C₈	—	2	2	no	—
460	C₈	—	4	1	no	—
461	C₈	—	4	2	no	—
462	C₂²	—	8	2	no	—
463	C₂²	—	8	2	no	—
464	C₂²	—	8	2	no	—
465	C₂²	—	8	2	no	—
466	C₂²	—	8	2	no	—
467	C₂²	—	8	2	no	—
468	C₂²	—	8	2	no	—
469	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
470	C₂²	C₃₁×#546	1	1	yes	characteristic
471	C₂²	C₃₁×#550	1	1	yes	characteristic
472	C₂²	C₃₁×#575	1	2	yes	—
473	C₂²	C₃₁×#577	1	2	yes	—
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	4	2	no	—
477	C₄	—	4	2	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	2	no	—
480	C₄	—	4	2	no	—
481	C₄	—	8	2	no	—
482	C₄	—	8	1	no	—
483	C₄	C₃₁×#525	1	1	yes	characteristic
484	C₄	C₃₁×#535	1	1	yes	characteristic
485	C₄	C₃₁×#553	1	1	yes	characteristic
486	C₄	C₃₁×#555	1	1	yes	characteristic
487	C₂	—	16	2	no	—
488	C₂	C₃₁×#2414	1	1	yes	characteristic
489	C₂	C₃₁×#3040	1	1	yes	characteristic
490	C₂	C₃₁×#3043	1	1	yes	characteristic
491	C₂	C₃₁×#3317	1	2	yes	—
492	C₂	C₃₁×#3401	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#441	C₂	semidirect
4. C₃₁×#2356	C₂	semidirect
5. C₃₁×#2055	C₂	semidirect
6. C₃₁×#2015	C₂	semidirect
7. C₃₁×#1983	C₂	semidirect
9. C₃₁×#1940	C₂	semidirect
10. C₃₁×#1926	C₂	semidirect
11. C₃₁×#1805	C₂	semidirect
161. #24649	C₃₁	direct (N × H)
217. C₂×#441	C₂×C₃₁	semidirect
219. #2356	C₂×C₃₁	semidirect
220. #2055	C₂×C₃₁	semidirect
221. #2015	C₂×C₃₁	semidirect
222. #1983	C₂×C₃₁	semidirect
224. #1940	C₂×C₃₁	semidirect
225. #1926	C₂×C₃₁	semidirect
226. #1805	C₂×C₃₁	semidirect
343. C₃₁	#24649	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×#460	2
C₂×C₃₁×#441	2
C₃₁×#2548	2
C₃₁×#2356	2
C₃₁×#2055	2
C₃₁×#2015	2
C₃₁×#1983	2
C₃₁×#1961	2
C₃₁×#1940	2
C₃₁×#1926	2
C₃₁×#1805	2
#24649	31

Subgroups — C31×#24649 #21481408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24649 #21481408