Subgroups — C₃₁×#24770 #21481529

Order 7936.

924 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#472	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3038	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#459	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₃₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2906	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2063	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2045	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1965	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1907	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1800	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
14	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×#131	C₂²	1	2	yes	—
16	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
17	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#472	—	2	8	no	—
20	C₃₁×#460	—	2	1	no	—
21	C₃₁×#460	—	2	1	no	—
22	C₃₁×#459	—	2	1	no	—
23	C₃₁×#459	—	2	1	no	—
24	C₃₁×#459	—	2	1	no	—
25	C₃₁×#459	—	2	1	no	—
26	C₃₁×#459	—	2	2	no	—
27	C₃₁×#442	—	2	1	no	—
28	C₃₁×#440	—	2	1	no	—
29	C₄×SD₁₆×C₃₁	—	2	1	no	—
30	C₄×D₈×C₃₁	—	2	1	no	—
31	C₂×C₃₁×#108	C₂²	1	2	yes	—
32	C₃₁×#428	C₂²	1	1	yes	characteristic
33	C₃₁×#424	—	2	2	no	—
34	C₃₁×#424	C₂²	1	2	yes	—
35	C₃₁×#423	—	2	1	no	—
36	C₃₁×#423	C₂²	1	1	yes	characteristic
37	C₃₁×#423	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₃₁×#418	C₂²	1	2	yes	—
41	C₃₁×#418	C₂²	1	1	yes	characteristic
42	C₃₁×#417	C₂²	1	1	yes	characteristic
43	C₂×C₃₁×#102	C₂²	1	2	yes	—
44	C₂×C₃₁×#100	C₂²	1	2	yes	—
45	C₂×C₃₁×#100	C₂²	1	2	yes	—
46	C₂×C₃₁×#100	C₂²	1	2	yes	—
47	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
48	C₃₁×#343	C₂²	1	2	yes	—
49	C₃₁×#341	C₂²	1	2	yes	—
50	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	—	2	1	no	—
53	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
54	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₃₁	—	2	1	no	—
56	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
57	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
58	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
59	C₃₁×AGL(1,ℤ/8)	D₄	1	4	yes	complemented
60	C₂×SD₁₆×C₃₁	—	2	1	no	—
61	C₂×SD₁₆×C₃₁	—	2	2	no	—
62	C₂×D₈×C₃₁	—	2	1	no	—
63	C₂×D₈×C₃₁	—	2	2	no	—
64	C₂×D₈×C₃₁	—	2	4	no	—
65	C₂×D₈×C₃₁	—	2	2	no	—
66	C₂×D₈×C₃₁	—	2	2	no	—
67	C₂×D₈×C₃₁	—	4	2	no	—
68	C₃₁×#131	—	2	4	no	—
69	C₃₁×#131	—	4	2	no	—
70	C₂×M₁₆×C₃₁	—	2	2	no	—
71	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
73	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
74	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₃₁	—	2	2	no	—
76	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
77	C₃₁×#127	—	2	1	no	—
78	C₃₁×#127	—	2	1	no	—
79	C₃₁×#127	—	2	1	no	—
80	C₃₁×#127	—	2	1	no	—
81	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
82	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
83	C₄×D₄×C₃₁	—	2	1	no	—
84	C₄×D₄×C₃₁	—	2	1	no	—
85	C₄×D₄×C₃₁	—	2	1	no	—
86	C₄×D₄×C₃₁	—	2	1	no	—
87	C₄×D₄×C₃₁	—	2	1	no	—
88	C₄×D₄×C₃₁	—	2	1	no	—
89	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
90	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
91	C₃₁×#117	—	2	1	no	—
92	C₃₁×#117	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
94	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
95	C₂×C₄²×C₃₁	—	2	1	no	—
96	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
97	C₃₁×#108	—	2	4	no	—
98	C₃₁×#107	—	2	1	no	—
99	C₃₁×#106	—	2	1	no	—
100	C₃₁×C₄⋊C₈	—	2	2	no	—
101	C₃₁×C₄⋊C₈	—	2	2	no	—
102	C₃₁×C₄⋊C₈	—	2	1	no	—
103	C₃₁×C₄⋊C₈	—	2	1	no	—
104	C₃₁×C₄⋊C₈	—	2	1	no	—
105	C₃₁×C₄⋊C₈	—	2	1	no	—
106	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
107	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
108	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
109	C₃₁×#103	—	2	1	no	—
110	C₃₁×#102	—	2	1	no	—
111	C₃₁×#102	—	2	2	no	—
112	C₃₁×#102	—	2	2	no	—
113	C₃₁×#102	—	2	1	no	—
114	C₃₁×#102	—	2	1	no	—
115	C₃₁×#102	—	2	2	no	—
116	C₃₁×#102	—	2	2	no	—
117	C₃₁×#100	—	2	4	no	—
118	C₃₁×#100	—	2	4	no	—
119	C₃₁×#100	—	4	2	no	—
120	C₃₁×#97	—	2	1	no	—
121	C₄×C₈×C₃₁	—	2	1	no	—
122	C₂⁴×C₃₁	—	2	1	no	—
123	C₂⁴×C₃₁	—	2	2	no	—
124	#41×C₃₁	—	2	1	no	—
125	#41×C₃₁	—	2	1	no	—
126	#41×C₃₁	C₂×D₄	1	2	yes	—
127	#41×C₃₁	#41	1	2	yes	—
128	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂×D₄×C₃₁	—	2	1	no	—
130	C₂×D₄×C₃₁	—	2	1	no	—
131	C₂×D₄×C₃₁	—	2	1	no	—
132	C₂×D₄×C₃₁	—	2	2	no	—
133	C₂×D₄×C₃₁	—	2	2	no	—
134	C₂×D₄×C₃₁	—	2	2	no	—
135	C₂×D₄×C₃₁	—	2	1	no	—
136	C₂×D₄×C₃₁	—	2	2	no	—
137	C₂×D₄×C₃₁	—	2	1	no	—
138	C₂×D₄×C₃₁	—	2	2	no	—
139	C₂×D₄×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	—	2	2	no	—
141	C₂×D₄×C₃₁	—	4	1	no	—
142	C₂×D₄×C₃₁	—	4	1	no	—
143	C₂×D₄×C₃₁	—	4	1	no	—
144	C₂×D₄×C₃₁	—	4	1	no	—
145	C₂×D₄×C₃₁	—	8	2	no	—
146	C₂×D₄×C₃₁	—	8	2	no	—
147	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
148	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	—	2	1	no	—
150	C₂²×C₄×C₃₁	—	2	1	no	—
151	C₂²×C₄×C₃₁	—	2	1	no	—
152	C₂²×C₄×C₃₁	—	2	1	no	—
153	C₂²×C₄×C₃₁	—	4	1	no	—
154	C₂²×C₄×C₃₁	—	4	1	no	—
155	C₂²×C₄×C₃₁	—	8	2	no	—
156	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
159	SD₁₆×C₃₁	—	2	4	no	—
160	SD₁₆×C₃₁	—	4	4	no	—
161	D₈×C₃₁	—	2	4	no	—
162	D₈×C₃₁	—	4	4	no	—
163	D₈×C₃₁	—	4	4	no	—
164	D₈×C₃₁	—	4	4	no	—
165	M₁₆×C₃₁	—	2	1	no	—
166	M₁₆×C₃₁	—	2	2	no	—
167	M₁₆×C₃₁	—	2	2	no	—
168	M₁₆×C₃₁	—	2	2	no	—
169	M₁₆×C₃₁	—	2	1	no	—
170	M₁₆×C₃₁	—	2	2	no	—
171	M₁₆×C₃₁	—	2	1	no	—
172	M₁₆×C₃₁	—	4	4	no	—
173	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
174	C₂×C₈×C₃₁	—	2	1	no	—
175	C₂×C₈×C₃₁	—	2	2	no	—
176	C₂×C₈×C₃₁	—	2	2	no	—
177	C₂×C₈×C₃₁	—	2	1	no	—
178	C₂×C₈×C₃₁	—	2	2	no	—
179	C₂×C₈×C₃₁	—	2	2	no	—
180	C₂×C₈×C₃₁	—	4	4	no	—
181	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
182	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
183	C₄⋊C₄×C₃₁	—	2	1	no	—
184	C₄⋊C₄×C₃₁	—	2	1	no	—
185	C₄⋊C₄×C₃₁	—	2	1	no	—
186	C₄⋊C₄×C₃₁	—	2	1	no	—
187	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
189	C₂²⋊C₄×C₃₁	—	2	1	no	—
190	C₂²⋊C₄×C₃₁	—	2	1	no	—
191	C₂²⋊C₄×C₃₁	—	2	1	no	—
192	C₂²⋊C₄×C₃₁	—	4	1	no	—
193	C₂²⋊C₄×C₃₁	—	4	1	no	—
194	C₄²×C₃₁	—	2	1	no	—
195	C₄²×C₃₁	—	2	1	no	—
196	C₄²×C₃₁	—	2	1	no	—
197	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
200	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
201	#24770	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂³×C₃₁	—	2	1	no	—
203	C₂³×C₃₁	—	2	2	no	—
204	C₂³×C₃₁	—	4	1	no	—
205	C₂³×C₃₁	—	4	1	no	—
206	C₂³×C₃₁	—	4	2	no	—
207	C₂³×C₃₁	—	4	2	no	—
208	C₂³×C₃₁	—	4	2	no	—
209	C₂³×C₃₁	—	8	1	no	—
210	C₂³×C₃₁	—	8	1	no	—
211	C₂³×C₃₁	—	8	2	no	—
212	C₂³×C₃₁	—	8	2	no	—
213	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
214	Q₈×C₃₁	—	2	1	no	—
215	Q₈×C₃₁	#121	1	2	yes	—
216	D₄×C₃₁	—	2	2	no	—
217	D₄×C₃₁	—	2	2	no	—
218	D₄×C₃₁	—	2	2	no	—
219	D₄×C₃₁	—	2	1	no	—
220	D₄×C₃₁	—	2	2	no	—
221	D₄×C₃₁	—	4	2	no	—
222	D₄×C₃₁	—	4	1	no	—
223	D₄×C₃₁	—	4	2	no	—
224	D₄×C₃₁	—	4	2	no	—
225	D₄×C₃₁	—	4	1	no	—
226	D₄×C₃₁	—	4	2	no	—
227	D₄×C₃₁	—	8	2	no	—
228	D₄×C₃₁	—	8	2	no	—
229	D₄×C₃₁	—	8	2	no	—
230	D₄×C₃₁	—	8	2	no	—
231	D₄×C₃₁	#121	1	2	yes	—
232	C₂×C₄×C₃₁	—	2	1	no	—
233	C₂×C₄×C₃₁	—	2	2	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	1	no	—
237	C₂×C₄×C₃₁	—	2	1	no	—
238	C₂×C₄×C₃₁	—	4	1	no	—
239	C₂×C₄×C₃₁	—	4	2	no	—
240	C₂×C₄×C₃₁	—	4	1	no	—
241	C₂×C₄×C₃₁	—	8	1	no	—
242	C₂×C₄×C₃₁	—	8	2	no	—
243	C₂×C₄×C₃₁	—	8	2	no	—
244	C₂×C₄×C₃₁	—	8	1	no	—
245	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
246	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
247	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
248	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
249	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
253	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
254	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
255	C₈×C₃₁	—	2	2	no	—
256	C₈×C₃₁	—	2	2	no	—
257	C₈×C₃₁	—	2	2	no	—
258	C₈×C₃₁	—	4	2	no	—
259	C₈×C₃₁	—	4	1	no	—
260	C₈×C₃₁	—	4	2	no	—
261	C₂×#472	C₂×C₃₁	1	2	yes	complemented
262	#3038	C₂×C₃₁	1	1	yes	characteristic, complemented
263	C₂×#459	C₂×C₃₁	1	1	yes	characteristic, complemented
264	C₄×AGL(1,ℤ/8)	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#2906	C₂×C₃₁	1	1	yes	characteristic, complemented
266	#2063	C₂×C₃₁	1	2	yes	complemented
267	#2045	C₂×C₃₁	1	2	yes	complemented
268	#1965	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#1907	C₂×C₃₁	1	2	yes	complemented
271	#1800	C₂×C₃₁	1	1	yes	characteristic, complemented
272	C₂²×C₃₁	—	2	1	no	—
273	C₂²×C₃₁	—	2	1	no	—
274	C₂²×C₃₁	—	2	2	no	—
275	C₂²×C₃₁	—	4	1	no	—
276	C₂²×C₃₁	—	4	1	no	—
277	C₂²×C₃₁	—	4	2	no	—
278	C₂²×C₃₁	—	8	2	no	—
279	C₂²×C₃₁	—	8	2	no	—
280	C₂²×C₃₁	—	8	1	no	—
281	C₂²×C₃₁	—	8	1	no	—
282	C₂²×C₃₁	—	8	2	no	—
283	C₂²×C₃₁	—	8	2	no	—
284	C₂²×C₃₁	—	8	2	no	—
285	C₂²×C₃₁	—	16	2	no	—
286	C₂²×C₃₁	—	16	2	no	—
287	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
288	C₂²×C₃₁	#547	1	1	yes	characteristic
289	C₂²×C₃₁	#553	1	1	yes	characteristic
290	C₄×C₃₁	—	2	2	no	—
291	C₄×C₃₁	—	2	1	no	—
292	C₄×C₃₁	—	2	1	no	—
293	C₄×C₃₁	—	2	1	no	—
294	C₄×C₃₁	—	4	1	no	—
295	C₄×C₃₁	—	4	1	no	—
296	C₄×C₃₁	—	8	1	no	—
297	C₄×C₃₁	#472	1	2	yes	complemented
298	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
299	C₄×C₃₁	#537	1	1	yes	characteristic
300	C₄×C₃₁	D₄²	1	1	yes	characteristic
301	C₄×C₃₁	#548	1	1	yes	characteristic
302	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
303	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
304	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
305	C₂×#131	C₂²×C₃₁	1	2	yes	—
306	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
307	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
308	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
309	#472	—	2	8	no	—
310	#460	—	2	1	no	—
311	#460	—	2	1	no	—
312	#459	—	2	1	no	—
313	#459	—	2	1	no	—
314	#459	—	2	1	no	—
315	#459	—	2	1	no	—
316	#459	—	2	2	no	—
317	#442	—	2	1	no	—
318	#440	—	2	1	no	—
319	C₄×SD₁₆	—	2	1	no	—
320	C₄×D₈	—	2	1	no	—
321	C₂×#108	C₂²×C₃₁	1	2	yes	—
322	#428	C₂²×C₃₁	1	1	yes	characteristic
323	#424	—	2	2	no	—
324	#424	C₂²×C₃₁	1	2	yes	—
325	#423	—	2	1	no	—
326	#423	C₂²×C₃₁	1	1	yes	characteristic
327	#423	C₂²×C₃₁	1	1	yes	characteristic
328	C₂×C₄⋊C₈	—	2	1	no	—
329	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
330	#418	C₂²×C₃₁	1	2	yes	—
331	#418	C₂²×C₃₁	1	1	yes	characteristic
332	#417	C₂²×C₃₁	1	1	yes	characteristic
333	C₂×#102	C₂²×C₃₁	1	2	yes	—
334	C₂×#100	C₂²×C₃₁	1	2	yes	—
335	C₂×#100	C₂²×C₃₁	1	2	yes	—
336	C₂×#100	C₂²×C₃₁	1	2	yes	—
337	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
338	#343	C₂²×C₃₁	1	2	yes	—
339	#341	C₂²×C₃₁	1	2	yes	—
340	C₂×C₃₁	—	2	1	no	—
341	C₂×C₃₁	—	2	1	no	—
342	C₂×C₃₁	—	4	2	no	—
343	C₂×C₃₁	—	8	2	no	—
344	C₂×C₃₁	—	16	2	no	—
345	C₂×C₃₁	#2416	1	1	yes	characteristic
346	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
347	C₂×C₃₁	#3068	1	1	yes	characteristic
348	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
349	C₂²×D₄	—	2	1	no	—
350	C₂²×D₄	—	2	1	no	—
351	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
352	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
353	C₂³×C₄	—	2	1	no	—
354	AGL(1,ℤ/8)	—	2	2	no	—
355	AGL(1,ℤ/8)	—	2	4	no	—
356	AGL(1,ℤ/8)	—	2	4	no	—
357	AGL(1,ℤ/8)	D₄×C₃₁	1	4	yes	complemented
358	C₂×SD₁₆	—	2	1	no	—
359	C₂×SD₁₆	—	2	2	no	—
360	C₂×D₈	—	2	1	no	—
361	C₂×D₈	—	2	4	no	—
362	C₂×D₈	—	2	2	no	—
363	C₂×D₈	—	2	2	no	—
364	C₂×D₈	—	2	2	no	—
365	C₂×D₈	—	4	2	no	—
366	#131	—	2	4	no	—
367	#131	—	4	2	no	—
368	C₂×M₁₆	—	2	2	no	—
369	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
370	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
371	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
372	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
373	C₂²×C₈	—	2	2	no	—
374	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
375	#127	—	2	1	no	—
376	#127	—	2	1	no	—
377	#127	—	2	1	no	—
378	#127	—	2	1	no	—
379	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
380	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
381	C₄×D₄	—	2	1	no	—
382	C₄×D₄	—	2	1	no	—
383	C₄×D₄	—	2	1	no	—
384	C₄×D₄	—	2	1	no	—
385	C₄×D₄	—	2	1	no	—
386	C₄×D₄	—	2	1	no	—
387	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
388	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
389	#117	—	2	1	no	—
390	#117	C₂³×C₃₁	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
392	C₂×C₂²⋊C₄	—	2	1	no	—
393	C₂×C₄²	—	2	1	no	—
394	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
395	#108	—	2	4	no	—
396	#107	—	2	1	no	—
397	#106	—	2	1	no	—
398	C₄⋊C₈	—	2	2	no	—
399	C₄⋊C₈	—	2	2	no	—
400	C₄⋊C₈	—	2	1	no	—
401	C₄⋊C₈	—	2	1	no	—
402	C₄⋊C₈	—	2	1	no	—
403	C₄⋊C₈	—	2	1	no	—
404	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
405	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
406	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
407	#103	—	2	1	no	—
408	#102	—	2	1	no	—
409	#102	—	2	2	no	—
410	#102	—	2	2	no	—
411	#102	—	2	1	no	—
412	#102	—	2	1	no	—
413	#102	—	2	2	no	—
414	#102	—	2	2	no	—
415	#100	—	2	4	no	—
416	#100	—	2	4	no	—
417	#100	—	4	2	no	—
418	#97	—	2	1	no	—
419	C₄×C₈	—	2	1	no	—
420	C₃₁	#24770	1	1	yes	characteristic, direct product factor, Sylow
421	C₂⁴	—	2	1	no	—
422	C₂⁴	—	2	2	no	—
423	#41	—	2	1	no	—
424	#41	—	2	1	no	—
425	#41	C₂×D₄×C₃₁	1	2	yes	—
426	#41	#41×C₃₁	1	2	yes	—
427	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	2	2	no	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	1	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	4	1	no	—
441	C₂×D₄	—	4	1	no	—
442	C₂×D₄	—	4	1	no	—
443	C₂×D₄	—	4	1	no	—
444	C₂×D₄	—	8	2	no	—
445	C₂×D₄	—	8	2	no	—
446	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
447	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	4	1	no	—
453	C₂²×C₄	—	4	1	no	—
454	C₂²×C₄	—	8	2	no	—
455	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
456	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
457	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
458	SD₁₆	—	2	4	no	—
459	SD₁₆	—	4	4	no	—
460	D₈	—	2	4	no	—
461	D₈	—	4	4	no	—
462	D₈	—	4	4	no	—
463	D₈	—	4	4	no	—
464	M₁₆	—	2	1	no	—
465	M₁₆	—	2	2	no	—
466	M₁₆	—	2	2	no	—
467	M₁₆	—	2	2	no	—
468	M₁₆	—	2	1	no	—
469	M₁₆	—	2	2	no	—
470	M₁₆	—	2	1	no	—
471	M₁₆	—	4	4	no	—
472	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	2	no	—
475	C₂×C₈	—	2	2	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	2	no	—
478	C₂×C₈	—	2	2	no	—
479	C₂×C₈	—	4	4	no	—
480	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
481	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
487	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
488	C₂²⋊C₄	—	2	1	no	—
489	C₂²⋊C₄	—	2	1	no	—
490	C₂²⋊C₄	—	2	1	no	—
491	C₂²⋊C₄	—	4	1	no	—
492	C₂²⋊C₄	—	4	1	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	2	1	no	—
495	C₄²	—	2	1	no	—
496	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
497	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
498	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
499	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
500	C₂³	—	2	1	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	4	1	no	—
503	C₂³	—	4	1	no	—
504	C₂³	—	4	2	no	—
505	C₂³	—	4	2	no	—
506	C₂³	—	4	2	no	—
507	C₂³	—	8	1	no	—
508	C₂³	—	8	1	no	—
509	C₂³	—	8	2	no	—
510	C₂³	—	8	2	no	—
511	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
512	Q₈	—	2	1	no	—
513	Q₈	C₃₁×#121	1	2	yes	—
514	D₄	—	2	2	no	—
515	D₄	—	2	2	no	—
516	D₄	—	2	2	no	—
517	D₄	—	2	1	no	—
518	D₄	—	2	2	no	—
519	D₄	—	4	2	no	—
520	D₄	—	4	1	no	—
521	D₄	—	4	2	no	—
522	D₄	—	4	2	no	—
523	D₄	—	4	1	no	—
524	D₄	—	4	2	no	—
525	D₄	—	8	2	no	—
526	D₄	—	8	2	no	—
527	D₄	—	8	2	no	—
528	D₄	—	8	2	no	—
529	D₄	C₃₁×#121	1	2	yes	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	8	1	no	—
540	C₂×C₄	—	8	2	no	—
541	C₂×C₄	—	8	2	no	—
542	C₂×C₄	—	8	1	no	—
543	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
544	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
545	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
546	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
547	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
548	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
549	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
550	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
551	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
552	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
553	C₈	—	2	2	no	—
554	C₈	—	2	2	no	—
555	C₈	—	2	2	no	—
556	C₈	—	4	2	no	—
557	C₈	—	4	1	no	—
558	C₈	—	4	2	no	—
559	C₂²	—	2	1	no	—
560	C₂²	—	2	1	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	4	1	no	—
563	C₂²	—	4	1	no	—
564	C₂²	—	4	2	no	—
565	C₂²	—	8	2	no	—
566	C₂²	—	8	2	no	—
567	C₂²	—	8	1	no	—
568	C₂²	—	8	1	no	—
569	C₂²	—	8	2	no	—
570	C₂²	—	8	2	no	—
571	C₂²	—	8	2	no	—
572	C₂²	—	16	2	no	—
573	C₂²	—	16	2	no	—
574	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
575	C₂²	C₃₁×#547	1	1	yes	characteristic
576	C₂²	C₃₁×#553	1	1	yes	characteristic
577	C₄	—	2	2	no	—
578	C₄	—	2	1	no	—
579	C₄	—	2	1	no	—
580	C₄	—	2	1	no	—
581	C₄	—	4	1	no	—
582	C₄	—	4	1	no	—
583	C₄	—	8	1	no	—
584	C₄	C₃₁×#472	1	2	yes	complemented
585	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
586	C₄	C₃₁×#537	1	1	yes	characteristic
587	C₄	D₄²×C₃₁	1	1	yes	characteristic
588	C₄	C₃₁×#548	1	1	yes	characteristic
589	C₂	—	2	1	no	—
590	C₂	—	2	1	no	—
591	C₂	—	4	2	no	—
592	C₂	—	8	2	no	—
593	C₂	—	16	2	no	—
594	C₂	C₃₁×#2416	1	1	yes	characteristic
595	C₂	C₂×C₃₁×#472	1	1	yes	characteristic
596	C₂	C₃₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#472	C₂	semidirect
2. C₃₁×#3038	C₂	semidirect
3. C₂×C₃₁×#459	C₂	semidirect
4. C₄×C₃₁×AGL(1,ℤ/8)	C₂	semidirect
5. C₃₁×#2906	C₂	semidirect
6. C₃₁×#2063	C₂	semidirect
7. C₃₁×#2045	C₂	semidirect
8. C₃₁×#1965	C₂	semidirect
9. C₃₁×#1950	C₂	semidirect
10. C₃₁×#1907	C₂	semidirect
11. C₃₁×#1800	C₂	semidirect
59. C₃₁×AGL(1,ℤ/8)	D₄	semidirect
201. #24770	C₃₁	direct (N × H)
261. C₂×#472	C₂×C₃₁	semidirect
262. #3038	C₂×C₃₁	semidirect
263. C₂×#459	C₂×C₃₁	semidirect
264. C₄×AGL(1,ℤ/8)	C₂×C₃₁	semidirect
265. #2906	C₂×C₃₁	semidirect
266. #2063	C₂×C₃₁	semidirect
267. #2045	C₂×C₃₁	semidirect
268. #1965	C₂×C₃₁	semidirect
269. #1950	C₂×C₃₁	semidirect
270. #1907	C₂×C₃₁	semidirect
271. #1800	C₂×C₃₁	semidirect
297. C₄×C₃₁	#472	semidirect
357. AGL(1,ℤ/8)	D₄×C₃₁	semidirect
420. C₃₁	#24770	direct (N × H)
584. C₄	C₃₁×#472	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#472	2
C₃₁×#3038	2
C₂×C₃₁×#459	2
C₄×C₃₁×AGL(1,ℤ/8)	2
C₃₁×#2906	2
C₃₁×#2063	2
C₃₁×#2045	2
C₃₁×#1965	2
C₃₁×#1950	2
C₃₁×#1907	2
C₃₁×#1800	2
#24770	31

Subgroups — C31×#24770 #21481529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24770 #21481529