Subgroups — C₃₁×#24810 #21481569

Order 7936.

810 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3069	C₂	1	2	yes	maximal
2	C₂×C₃₁×#473	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#473	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2955	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2916	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2070	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2049	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1979	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1953	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1902	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1813	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#575	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#137	C₂²	1	2	yes	—
14	C₂×C₃₁×#137	C₂²	1	2	yes	—
15	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#568	C₂²	1	2	yes	—
17	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
19	C₃₁×#473	—	2	4	no	—
20	C₃₁×#473	—	2	2	no	—
21	C₃₁×#473	—	2	2	no	—
22	C₃₁×#473	—	2	2	no	—
23	C₃₁×#473	—	2	2	no	—
24	C₃₁×#471	—	2	4	no	—
25	C₃₁×#444	—	2	2	no	—
26	C₃₁×#443	—	2	2	no	—
27	C₃₁×#430	—	2	2	no	—
28	C₃₁×#430	C₂²	1	1	yes	characteristic
29	C₃₁×#430	C₂²	1	2	yes	—
30	C₃₁×#430	C₂²	1	2	yes	—
31	C₂×C₃₁×#108	—	2	2	no	—
32	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
35	C₃₁×#421	C₂²	1	1	yes	characteristic
36	C₃₁×#421	C₂²	1	1	yes	characteristic
37	C₃₁×#419	C₂²	1	2	yes	—
38	C₃₁×#413	C₂²	1	2	yes	—
39	C₂×C₃₁×#101	C₂²	1	2	yes	—
40	C₂×C₃₁×#101	C₂²	1	2	yes	—
41	C₂×C₃₁×#101	C₂²	1	2	yes	—
42	C₃₁×#405	C₂²	1	1	yes	characteristic
43	C₃₁×#344	C₂²	1	2	yes	—
44	C₃₁×#338	C₂²	1	2	yes	—
45	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
48	C₃₁×#137	—	2	1	no	—
49	C₃₁×#137	—	2	4	no	—
50	C₃₁×#137	—	2	4	no	—
51	C₃₁×#137	—	2	4	no	—
52	C₃₁×#137	—	4	2	no	—
53	C₃₁×#137	D₄	1	2	yes	—
54	C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
55	C₃₁×AGL(1,ℤ/8)	D₄	1	2	yes	—
56	C₃₁×#135	—	2	2	no	—
57	C₂×Q₁₆×C₃₁	—	2	2	no	—
58	C₂×Q₁₆×C₃₁	—	2	2	no	—
59	C₂×Q₁₆×C₃₁	—	2	1	no	—
60	C₂×Q₁₆×C₃₁	—	2	1	no	—
61	C₂×Q₁₆×C₃₁	—	2	1	no	—
62	C₂×Q₁₆×C₃₁	—	2	1	no	—
63	C₂×Q₁₆×C₃₁	—	2	1	no	—
64	C₂×Q₁₆×C₃₁	—	2	1	no	—
65	C₂×SD₁₆×C₃₁	—	2	2	no	—
66	C₂×SD₁₆×C₃₁	—	2	2	no	—
67	C₃₁×#131	—	2	1	no	—
68	C₃₁×#131	—	2	1	no	—
69	C₃₁×#131	—	2	4	no	—
70	C₃₁×#131	—	2	1	no	—
71	C₃₁×#131	—	2	1	no	—
72	C₃₁×#131	—	2	1	no	—
73	C₃₁×#131	—	2	1	no	—
74	C₃₁×#131	—	4	2	no	—
75	C₃₁×#131	D₄	1	2	yes	—
76	C₃₁×#131	D₄	1	2	yes	—
77	C₂×M₁₆×C₃₁	—	2	1	no	—
78	C₂×M₁₆×C₃₁	—	2	1	no	—
79	C₂×M₁₆×C₃₁	—	2	4	no	—
80	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
82	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
83	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
86	C₂²×C₈×C₃₁	—	2	1	no	—
87	C₂²×C₈×C₃₁	—	2	1	no	—
88	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₃₁×#117	C₂³	1	1	yes	characteristic
90	C₃₁×#108	—	2	4	no	—
91	C₃₁×#108	—	2	2	no	—
92	C₃₁×#108	—	2	4	no	—
93	C₃₁×#108	—	2	1	no	—
94	C₃₁×#108	—	2	1	no	—
95	C₃₁×#108	—	2	1	no	—
96	C₃₁×#108	—	2	1	no	—
97	C₃₁×#108	D₄	1	2	yes	—
98	C₃₁×#108	D₄	1	2	yes	—
99	C₃₁×#104	—	2	2	no	—
100	C₃₁×#104	—	2	2	no	—
101	C₃₁×#103	—	4	2	no	—
102	C₃₁×#101	—	2	2	no	—
103	C₃₁×#101	—	2	4	no	—
104	C₃₁×#101	—	2	4	no	—
105	C₃₁×#101	—	2	4	no	—
106	C₃₁×#100	—	2	2	no	—
107	C₃₁×#97	—	2	1	no	—
108	C₄×C₈×C₃₁	—	2	1	no	—
109	#41×C₃₁	—	2	1	no	—
110	#41×C₃₁	—	2	1	no	—
111	#41×C₃₁	—	2	1	no	—
112	#41×C₃₁	—	2	1	no	—
113	#41×C₃₁	—	2	1	no	—
114	#41×C₃₁	—	2	1	no	—
115	#41×C₃₁	C₂×D₄	1	2	yes	—
116	#41×C₃₁	C₂×D₄	1	2	yes	—
117	C₂×Q₈×C₃₁	—	2	2	no	—
118	C₂×Q₈×C₃₁	—	2	2	no	—
119	C₂×Q₈×C₃₁	—	2	2	no	—
120	C₂×Q₈×C₃₁	—	2	2	no	—
121	C₂×Q₈×C₃₁	—	2	2	no	—
122	C₂×Q₈×C₃₁	—	2	2	no	—
123	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×Q₈×C₃₁	#41	1	1	yes	characteristic
125	C₂×D₄×C₃₁	—	2	1	no	—
126	C₂×D₄×C₃₁	—	2	1	no	—
127	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂×D₄×C₃₁	#41	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	—	2	1	no	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	2	2	no	—
132	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
133	Q₁₆×C₃₁	—	2	2	no	—
134	Q₁₆×C₃₁	—	4	4	no	—
135	Q₁₆×C₃₁	—	4	4	no	—
136	Q₁₆×C₃₁	—	4	1	no	—
137	Q₁₆×C₃₁	—	4	1	no	—
138	Q₁₆×C₃₁	—	4	1	no	—
139	Q₁₆×C₃₁	—	4	1	no	—
140	SD₁₆×C₃₁	—	2	2	no	—
141	SD₁₆×C₃₁	—	2	2	no	—
142	SD₁₆×C₃₁	—	4	4	no	—
143	SD₁₆×C₃₁	—	4	4	no	—
144	D₈×C₃₁	—	2	2	no	—
145	M₁₆×C₃₁	—	2	1	no	—
146	M₁₆×C₃₁	—	2	1	no	—
147	M₁₆×C₃₁	—	2	2	no	—
148	M₁₆×C₃₁	—	2	2	no	—
149	M₁₆×C₃₁	—	2	2	no	—
150	M₁₆×C₃₁	—	2	2	no	—
151	M₁₆×C₃₁	—	2	2	no	—
152	M₁₆×C₃₁	—	2	1	no	—
153	M₁₆×C₃₁	—	2	2	no	—
154	M₁₆×C₃₁	—	2	1	no	—
155	M₁₆×C₃₁	—	2	2	no	—
156	M₁₆×C₃₁	—	4	4	no	—
157	M₁₆×C₃₁	—	4	1	no	—
158	M₁₆×C₃₁	—	4	1	no	—
159	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
160	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
161	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
162	C₂×C₈×C₃₁	—	2	1	no	—
163	C₂×C₈×C₃₁	—	2	1	no	—
164	C₂×C₈×C₃₁	—	2	2	no	—
165	C₂×C₈×C₃₁	—	2	2	no	—
166	C₂×C₈×C₃₁	—	2	2	no	—
167	C₂×C₈×C₃₁	—	2	1	no	—
168	C₂×C₈×C₃₁	—	2	2	no	—
169	C₂×C₈×C₃₁	—	2	2	no	—
170	C₂×C₈×C₃₁	—	4	4	no	—
171	C₂×C₈×C₃₁	—	4	1	no	—
172	C₂×C₈×C₃₁	—	4	1	no	—
173	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
178	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₂²⋊C₄×C₃₁	—	2	1	no	—
181	C₄²×C₃₁	—	2	1	no	—
182	#24810	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂³×C₃₁	—	2	1	no	—
184	C₂³×C₃₁	—	2	1	no	—
185	C₂³×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
186	Q₈×C₃₁	—	2	1	no	—
187	Q₈×C₃₁	—	2	1	no	—
188	Q₈×C₃₁	—	2	1	no	—
189	Q₈×C₃₁	—	4	2	no	—
190	Q₈×C₃₁	—	4	2	no	—
191	Q₈×C₃₁	—	4	2	no	—
192	Q₈×C₃₁	—	4	2	no	—
193	Q₈×C₃₁	#121	1	2	yes	—
194	D₄×C₃₁	—	2	2	no	—
195	D₄×C₃₁	—	2	2	no	—
196	D₄×C₃₁	—	2	1	no	—
197	D₄×C₃₁	—	2	1	no	—
198	D₄×C₃₁	—	2	1	no	—
199	D₄×C₃₁	—	4	1	no	—
200	D₄×C₃₁	—	4	1	no	—
201	D₄×C₃₁	#121	1	2	yes	—
202	C₂×C₄×C₃₁	—	2	1	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	4	1	no	—
207	C₂×C₄×C₃₁	—	4	1	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
218	C₈×C₃₁	—	2	2	no	—
219	C₈×C₃₁	—	2	2	no	—
220	C₈×C₃₁	—	2	4	no	—
221	C₈×C₃₁	—	4	1	no	—
222	C₈×C₃₁	—	4	2	no	—
223	C₈×C₃₁	—	4	2	no	—
224	C₈×C₃₁	#121	1	1	yes	characteristic
225	C₈×C₃₁	#121	1	1	yes	characteristic
226	C₈×C₃₁	#121	1	1	yes	characteristic
227	C₈×C₃₁	#121	1	1	yes	characteristic
228	#3069	C₂×C₃₁	1	2	yes	—
229	C₂×#473	C₂×C₃₁	1	1	yes	characteristic, complemented
230	C₂×#473	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#2955	C₂×C₃₁	1	1	yes	characteristic
232	#2916	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#2070	C₂×C₃₁	1	2	yes	complemented
234	#2049	C₂×C₃₁	1	2	yes	complemented
235	#1979	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#1953	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#1902	C₂×C₃₁	1	2	yes	complemented
238	#1813	C₂×C₃₁	1	1	yes	characteristic, complemented
239	C₂²×C₃₁	—	2	2	no	—
240	C₂²×C₃₁	—	4	1	no	—
241	C₂²×C₃₁	—	4	1	no	—
242	C₂²×C₃₁	—	8	1	no	—
243	C₂²×C₃₁	—	8	1	no	—
244	C₂²×C₃₁	#537	1	1	yes	characteristic
245	C₂²×C₃₁	#547	1	1	yes	characteristic
246	C₂²×C₃₁	#548	1	1	yes	characteristic
247	C₄×C₃₁	—	2	2	no	—
248	C₄×C₃₁	—	4	1	no	—
249	C₄×C₃₁	—	8	1	no	—
250	C₄×C₃₁	—	8	2	no	—
251	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
252	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
253	C₄×C₃₁	D₄²	1	1	yes	characteristic
254	C₄×C₃₁	#553	1	1	yes	characteristic
255	#575	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×#137	C₂²×C₃₁	1	2	yes	—
257	C₂×#137	C₂²×C₃₁	1	2	yes	—
258	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
259	#568	C₂²×C₃₁	1	2	yes	—
260	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
262	#473	—	2	4	no	—
263	#473	—	2	2	no	—
264	#473	—	2	2	no	—
265	#473	—	2	2	no	—
266	#473	—	2	2	no	—
267	#471	—	2	4	no	—
268	#444	—	2	2	no	—
269	#443	—	2	2	no	—
270	#430	—	2	2	no	—
271	#430	C₂²×C₃₁	1	2	yes	—
272	#430	C₂²×C₃₁	1	1	yes	characteristic
273	#430	C₂²×C₃₁	1	2	yes	—
274	C₂×#108	—	2	2	no	—
275	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
278	#421	C₂²×C₃₁	1	1	yes	characteristic
279	#421	C₂²×C₃₁	1	1	yes	characteristic
280	#419	C₂²×C₃₁	1	2	yes	—
281	#413	C₂²×C₃₁	1	2	yes	—
282	C₂×#101	C₂²×C₃₁	1	2	yes	—
283	C₂×#101	C₂²×C₃₁	1	2	yes	—
284	C₂×#101	C₂²×C₃₁	1	2	yes	—
285	#405	C₂²×C₃₁	1	1	yes	characteristic
286	#344	C₂²×C₃₁	1	2	yes	—
287	#338	C₂²×C₃₁	1	2	yes	—
288	C₂×C₃₁	—	2	1	no	—
289	C₂×C₃₁	—	2	1	no	—
290	C₂×C₃₁	—	2	1	no	—
291	C₂×C₃₁	—	4	2	no	—
292	C₂×C₃₁	—	8	1	no	—
293	C₂×C₃₁	#2416	1	1	yes	characteristic
294	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
295	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
296	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
297	#137	—	2	1	no	—
298	#137	—	2	4	no	—
299	#137	—	2	4	no	—
300	#137	—	2	4	no	—
301	#137	—	4	2	no	—
302	#137	D₄×C₃₁	1	2	yes	—
303	AGL(1,ℤ/8)	—	2	1	no	—
304	AGL(1,ℤ/8)	D₄×C₃₁	1	2	yes	—
305	#135	—	2	2	no	—
306	C₂×Q₁₆	—	2	2	no	—
307	C₂×Q₁₆	—	2	2	no	—
308	C₂×Q₁₆	—	2	1	no	—
309	C₂×Q₁₆	—	2	1	no	—
310	C₂×Q₁₆	—	2	1	no	—
311	C₂×Q₁₆	—	2	1	no	—
312	C₂×Q₁₆	—	2	1	no	—
313	C₂×Q₁₆	—	2	1	no	—
314	C₂×SD₁₆	—	2	2	no	—
315	C₂×SD₁₆	—	2	2	no	—
316	#131	—	2	1	no	—
317	#131	—	2	1	no	—
318	#131	—	2	4	no	—
319	#131	—	2	1	no	—
320	#131	—	2	1	no	—
321	#131	—	2	1	no	—
322	#131	—	2	1	no	—
323	#131	—	4	2	no	—
324	#131	D₄×C₃₁	1	2	yes	—
325	#131	D₄×C₃₁	1	2	yes	—
326	C₂×M₁₆	—	2	1	no	—
327	C₂×M₁₆	—	2	1	no	—
328	C₂×M₁₆	—	2	4	no	—
329	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
330	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
331	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
332	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
333	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
334	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
335	C₂²×C₈	—	2	1	no	—
336	C₂²×C₈	—	2	1	no	—
337	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
338	#117	C₂³×C₃₁	1	1	yes	characteristic
339	#108	—	2	4	no	—
340	#108	—	2	2	no	—
341	#108	—	2	4	no	—
342	#108	—	2	1	no	—
343	#108	—	2	1	no	—
344	#108	—	2	1	no	—
345	#108	—	2	1	no	—
346	#108	D₄×C₃₁	1	2	yes	—
347	#108	D₄×C₃₁	1	2	yes	—
348	#104	—	2	2	no	—
349	#104	—	2	2	no	—
350	#103	—	4	2	no	—
351	#101	—	2	2	no	—
352	#101	—	2	4	no	—
353	#101	—	2	4	no	—
354	#101	—	2	4	no	—
355	#100	—	2	2	no	—
356	#97	—	2	1	no	—
357	C₄×C₈	—	2	1	no	—
358	C₃₁	#24810	1	1	yes	characteristic, direct product factor, Sylow
359	#41	—	2	1	no	—
360	#41	—	2	1	no	—
361	#41	—	2	1	no	—
362	#41	—	2	1	no	—
363	#41	—	2	1	no	—
364	#41	—	2	1	no	—
365	#41	C₂×D₄×C₃₁	1	2	yes	—
366	#41	C₂×D₄×C₃₁	1	2	yes	—
367	C₂×Q₈	—	2	2	no	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	2	2	no	—
370	C₂×Q₈	—	2	2	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₂×Q₈	#41×C₃₁	1	1	yes	characteristic
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂×D₄	#41×C₃₁	1	1	yes	characteristic
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
383	Q₁₆	—	2	2	no	—
384	Q₁₆	—	4	4	no	—
385	Q₁₆	—	4	4	no	—
386	Q₁₆	—	4	1	no	—
387	Q₁₆	—	4	1	no	—
388	Q₁₆	—	4	1	no	—
389	Q₁₆	—	4	1	no	—
390	SD₁₆	—	2	2	no	—
391	SD₁₆	—	2	2	no	—
392	SD₁₆	—	4	4	no	—
393	SD₁₆	—	4	4	no	—
394	D₈	—	2	2	no	—
395	M₁₆	—	2	1	no	—
396	M₁₆	—	2	1	no	—
397	M₁₆	—	2	2	no	—
398	M₁₆	—	2	2	no	—
399	M₁₆	—	2	2	no	—
400	M₁₆	—	2	2	no	—
401	M₁₆	—	2	2	no	—
402	M₁₆	—	2	1	no	—
403	M₁₆	—	2	2	no	—
404	M₁₆	—	2	1	no	—
405	M₁₆	—	2	2	no	—
406	M₁₆	—	4	4	no	—
407	M₁₆	—	4	1	no	—
408	M₁₆	—	4	1	no	—
409	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
410	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
411	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	4	4	no	—
421	C₂×C₈	—	4	1	no	—
422	C₂×C₈	—	4	1	no	—
423	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
424	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
425	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
426	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
427	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
428	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
429	C₄⋊C₄	—	2	1	no	—
430	C₂²⋊C₄	—	2	1	no	—
431	C₄²	—	2	1	no	—
432	C₂³	—	2	1	no	—
433	C₂³	—	2	1	no	—
434	C₂³	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
435	Q₈	—	2	1	no	—
436	Q₈	—	2	1	no	—
437	Q₈	—	2	1	no	—
438	Q₈	—	4	2	no	—
439	Q₈	—	4	2	no	—
440	Q₈	—	4	2	no	—
441	Q₈	—	4	2	no	—
442	Q₈	C₃₁×#121	1	2	yes	—
443	D₄	—	2	2	no	—
444	D₄	—	2	2	no	—
445	D₄	—	2	1	no	—
446	D₄	—	2	1	no	—
447	D₄	—	2	1	no	—
448	D₄	—	4	1	no	—
449	D₄	—	4	1	no	—
450	D₄	C₃₁×#121	1	2	yes	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
462	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
463	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
464	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
465	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
466	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
467	C₈	—	2	2	no	—
468	C₈	—	2	2	no	—
469	C₈	—	2	4	no	—
470	C₈	—	4	1	no	—
471	C₈	—	4	2	no	—
472	C₈	—	4	2	no	—
473	C₈	C₃₁×#121	1	1	yes	characteristic
474	C₈	C₃₁×#121	1	1	yes	characteristic
475	C₈	C₃₁×#121	1	1	yes	characteristic
476	C₈	C₃₁×#121	1	1	yes	characteristic
477	C₂²	—	2	2	no	—
478	C₂²	—	4	1	no	—
479	C₂²	—	4	1	no	—
480	C₂²	—	8	1	no	—
481	C₂²	—	8	1	no	—
482	C₂²	C₃₁×#537	1	1	yes	characteristic
483	C₂²	C₃₁×#547	1	1	yes	characteristic
484	C₂²	C₃₁×#548	1	1	yes	characteristic
485	C₄	—	2	2	no	—
486	C₄	—	4	1	no	—
487	C₄	—	8	1	no	—
488	C₄	—	8	2	no	—
489	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
490	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
491	C₄	D₄²×C₃₁	1	1	yes	characteristic
492	C₄	C₃₁×#553	1	1	yes	characteristic
493	C₂	—	2	1	no	—
494	C₂	—	2	1	no	—
495	C₂	—	2	1	no	—
496	C₂	—	4	2	no	—
497	C₂	—	8	1	no	—
498	C₂	C₃₁×#2416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#2416
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#473	C₂	semidirect
3. C₂×C₃₁×#473	C₂	semidirect
5. C₃₁×#2916	C₂	semidirect
6. C₃₁×#2070	C₂	semidirect
7. C₃₁×#2049	C₂	semidirect
8. C₃₁×#1979	C₂	semidirect
9. C₃₁×#1953	C₂	semidirect
10. C₃₁×#1902	C₂	semidirect
11. C₃₁×#1813	C₂	semidirect
182. #24810	C₃₁	direct (N × H)
229. C₂×#473	C₂×C₃₁	semidirect
230. C₂×#473	C₂×C₃₁	semidirect
232. #2916	C₂×C₃₁	semidirect
233. #2070	C₂×C₃₁	semidirect
234. #2049	C₂×C₃₁	semidirect
235. #1979	C₂×C₃₁	semidirect
236. #1953	C₂×C₃₁	semidirect
237. #1902	C₂×C₃₁	semidirect
238. #1813	C₂×C₃₁	semidirect
358. C₃₁	#24810	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×#3069	2
C₂×C₃₁×#473	2
C₃₁×#2955	2
C₃₁×#2916	2
C₃₁×#2070	2
C₃₁×#2049	2
C₃₁×#1979	2
C₃₁×#1953	2
C₃₁×#1902	2
C₃₁×#1813	2
#24810	31

Subgroups — C31×#24810 #21481569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24810 #21481569