Subgroups — C₃₁×#24749 #21481508

Order 7936.

1044 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#463	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#459	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#440	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2548	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2349	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2067	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2042	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1961	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1879	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1805	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×D₈×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×D₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#463	—	2	8	no	—
20	C₃₁×#459	—	2	4	no	—
21	C₃₁×#440	—	2	4	no	—
22	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
25	C₂×C₃₁×#103	C₂²	1	2	yes	—
26	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#102	C₂²	1	2	yes	—
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#102	C₂²	1	2	yes	—
31	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
32	C₃₁×#394	—	2	1	no	—
33	C₃₁×#394	—	2	1	no	—
34	C₃₁×#390	—	2	2	no	—
35	C₃₁×#390	C₂²	1	2	yes	—
36	C₃₁×#389	C₂²	1	2	yes	—
37	C₃₁×#385	—	2	2	no	—
38	C₃₁×#382	C₂²	1	1	yes	characteristic
39	C₃₁×#380	—	2	1	no	—
40	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×C₃₁×#95	—	2	1	no	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
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	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	—	2	1	no	—
50	C₂³×C₄×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×SD₁₆×C₃₁	D₄	1	4	yes	complemented
54	C₂×D₈×C₃₁	—	2	4	no	—
55	C₂×D₈×C₃₁	—	2	4	no	—
56	C₂×D₈×C₃₁	—	2	4	no	—
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
59	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×#127	—	2	2	no	—
61	C₃₁×#127	—	2	2	no	—
62	C₃₁×#124	—	2	4	no	—
63	C₃₁×#124	—	4	2	no	—
64	C₄×Q₈×C₃₁	—	2	2	no	—
65	C₄×Q₈×C₃₁	—	2	2	no	—
66	C₄×D₄×C₃₁	—	2	4	no	—
67	C₄×D₄×C₃₁	—	2	2	no	—
68	C₄×D₄×C₃₁	—	4	2	no	—
69	C₄×D₄×C₃₁	D₄	1	4	yes	—
70	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
82	C₂×C₄²×C₃₁	—	2	1	no	—
83	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₃₁×#107	—	2	2	no	—
86	C₃₁×C₄⋊C₈	—	2	4	no	—
87	C₃₁×C₄⋊C₈	—	2	2	no	—
88	C₃₁×#103	—	2	4	no	—
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#102	—	2	2	no	—
91	C₃₁×#102	—	2	4	no	—
92	C₃₁×#102	—	2	4	no	—
93	C₃₁×#102	—	4	2	no	—
94	C₃₁×#97	—	2	2	no	—
95	C₃₁×#95	—	2	1	no	—
96	C₃₁×#95	—	2	2	no	—
97	C₃₁×#95	—	2	1	no	—
98	C₃₁×#95	D₄	1	1	yes	characteristic
99	C₃₁×#95	D₄	1	1	yes	characteristic
100	C₂⁴×C₃₁	—	2	1	no	—
101	C₂⁴×C₃₁	—	2	2	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	C₂×D₄	1	2	yes	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	4	4	no	—
118	C₂×D₄×C₃₁	—	4	2	no	—
119	C₂×D₄×C₃₁	—	4	2	no	—
120	C₂×D₄×C₃₁	—	8	2	no	—
121	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
122	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
123	C₂×D₄×C₃₁	#41	1	2	yes	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	4	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	2	1	no	—
129	C₂²×C₄×C₃₁	—	4	4	no	—
130	C₂²×C₄×C₃₁	—	4	4	no	—
131	C₂²×C₄×C₃₁	—	8	2	no	—
132	C₂²×C₄×C₃₁	—	8	2	no	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
138	SD₁₆×C₃₁	—	2	8	no	—
139	D₈×C₃₁	—	4	8	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	1	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₄⋊C₄×C₃₁	—	2	1	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	4	2	no	—
155	C₄⋊C₄×C₃₁	—	4	2	no	—
156	C₄⋊C₄×C₃₁	—	8	2	no	—
157	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
158	C₂²⋊C₄×C₃₁	—	2	4	no	—
159	C₂²⋊C₄×C₃₁	—	4	4	no	—
160	C₂²⋊C₄×C₃₁	—	4	4	no	—
161	C₂²⋊C₄×C₃₁	—	4	4	no	—
162	C₂²⋊C₄×C₃₁	—	8	2	no	—
163	C₂²⋊C₄×C₃₁	—	8	2	no	—
164	C₄²×C₃₁	—	2	2	no	—
165	C₄²×C₃₁	—	2	2	no	—
166	C₄²×C₃₁	—	4	2	no	—
167	C₄²×C₃₁	C₂×D₄	1	2	yes	—
168	C₄²×C₃₁	#41	1	2	yes	—
169	#24749	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂³×C₃₁	—	2	2	no	—
171	C₂³×C₃₁	—	2	2	no	—
172	C₂³×C₃₁	—	2	2	no	—
173	C₂³×C₃₁	—	4	2	no	—
174	C₂³×C₃₁	—	4	2	no	—
175	C₂³×C₃₁	—	4	2	no	—
176	C₂³×C₃₁	—	4	2	no	—
177	C₂³×C₃₁	—	4	2	no	—
178	C₂³×C₃₁	—	4	2	no	—
179	C₂³×C₃₁	—	4	2	no	—
180	C₂³×C₃₁	—	4	2	no	—
181	C₂³×C₃₁	—	4	2	no	—
182	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
183	Q₈×C₃₁	—	2	2	no	—
184	Q₈×C₃₁	—	2	2	no	—
185	Q₈×C₃₁	—	4	1	no	—
186	Q₈×C₃₁	—	4	1	no	—
187	D₄×C₃₁	—	2	2	no	—
188	D₄×C₃₁	—	2	2	no	—
189	D₄×C₃₁	—	2	2	no	—
190	D₄×C₃₁	—	4	2	no	—
191	D₄×C₃₁	—	4	2	no	—
192	D₄×C₃₁	—	4	2	no	—
193	D₄×C₃₁	—	4	2	no	—
194	D₄×C₃₁	—	8	4	no	—
195	D₄×C₃₁	—	8	4	no	—
196	D₄×C₃₁	#121	1	4	yes	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	2	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	1	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	8	4	no	—
216	C₂×C₄×C₃₁	—	8	8	no	—
217	C₂×C₄×C₃₁	#121	1	2	yes	—
218	C₂×C₄×C₃₁	#121	1	2	yes	—
219	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
224	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
225	C₈×C₃₁	—	2	2	no	—
226	C₈×C₃₁	—	4	2	no	—
227	C₈×C₃₁	—	4	1	no	—
228	C₂×#463	C₂×C₃₁	1	2	yes	complemented
229	C₂×#459	C₂×C₃₁	1	1	yes	characteristic
230	C₂×#440	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#2548	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#2349	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#2067	C₂×C₃₁	1	2	yes	complemented
234	#2042	C₂×C₃₁	1	2	yes	complemented
235	#1961	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#1879	C₂×C₃₁	1	2	yes	complemented
238	#1805	C₂×C₃₁	1	1	yes	characteristic, complemented
239	C₂²×C₃₁	—	2	4	no	—
240	C₂²×C₃₁	—	4	4	no	—
241	C₂²×C₃₁	—	4	4	no	—
242	C₂²×C₃₁	—	4	2	no	—
243	C₂²×C₃₁	—	4	2	no	—
244	C₂²×C₃₁	—	8	2	no	—
245	C₂²×C₃₁	—	8	2	no	—
246	C₂²×C₃₁	—	8	2	no	—
247	C₂²×C₃₁	—	8	2	no	—
248	C₂²×C₃₁	—	8	2	no	—
249	C₂²×C₃₁	—	8	2	no	—
250	C₂²×C₃₁	—	8	2	no	—
251	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
252	C₂²×C₃₁	D₄²	1	1	yes	characteristic
253	C₂²×C₃₁	#548	1	1	yes	characteristic
254	C₂²×C₃₁	#575	1	2	yes	—
255	C₂²×C₃₁	#576	1	2	yes	—
256	C₄×C₃₁	—	2	2	no	—
257	C₄×C₃₁	—	2	2	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	—	4	2	no	—
261	C₄×C₃₁	—	4	2	no	—
262	C₄×C₃₁	—	8	2	no	—
263	C₄×C₃₁	—	8	1	no	—
264	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
265	C₄×C₃₁	#537	1	1	yes	characteristic
266	C₄×C₃₁	#547	1	1	yes	characteristic
267	C₄×C₃₁	#553	1	1	yes	characteristic
268	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
269	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
270	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×#124	C₂²×C₃₁	1	2	yes	—
272	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
274	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
275	#463	—	2	8	no	—
276	#459	—	2	4	no	—
277	#440	—	2	4	no	—
278	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
281	C₂×#103	C₂²×C₃₁	1	2	yes	—
282	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×#102	C₂²×C₃₁	1	2	yes	—
284	C₂×#102	C₂²×C₃₁	1	2	yes	—
285	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
286	C₂×#102	C₂²×C₃₁	1	2	yes	—
287	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
288	#394	—	2	1	no	—
289	#394	—	2	1	no	—
290	#390	—	2	2	no	—
291	#390	C₂²×C₃₁	1	2	yes	—
292	#389	C₂²×C₃₁	1	2	yes	—
293	#385	—	2	2	no	—
294	#382	C₂²×C₃₁	1	1	yes	characteristic
295	#380	—	2	1	no	—
296	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
297	C₂×#95	—	2	1	no	—
298	#340	C₂²×C₃₁	1	2	yes	—
299	#336	C₂²×C₃₁	1	2	yes	—
300	C₂×C₃₁	—	4	4	no	—
301	C₂×C₃₁	—	16	2	no	—
302	C₂×C₃₁	#2416	1	1	yes	characteristic
303	C₂×C₃₁	#3040	1	1	yes	characteristic
304	C₂×C₃₁	#3041	1	1	yes	characteristic
305	C₂×C₃₁	#3268	1	2	yes	—
306	C₂×C₃₁	#3370	1	2	yes	—
307	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
308	C₂²×D₄	—	2	1	no	—
309	C₂²×D₄	—	2	1	no	—
310	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
311	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂³×C₄	—	2	1	no	—
313	C₂³×C₄	—	2	2	no	—
314	C₂×SD₁₆	—	2	2	no	—
315	C₂×SD₁₆	—	2	2	no	—
316	C₂×SD₁₆	D₄×C₃₁	1	4	yes	complemented
317	C₂×D₈	—	2	4	no	—
318	C₂×D₈	—	2	4	no	—
319	C₂×D₈	—	2	4	no	—
320	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
321	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
322	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
323	#127	—	2	2	no	—
324	#127	—	2	2	no	—
325	#124	—	2	4	no	—
326	#124	—	4	2	no	—
327	C₄×Q₈	—	2	2	no	—
328	C₄×Q₈	—	2	2	no	—
329	C₄×D₄	—	2	4	no	—
330	C₄×D₄	—	2	2	no	—
331	C₄×D₄	—	4	2	no	—
332	C₄×D₄	D₄×C₃₁	1	4	yes	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	2	no	—
335	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
336	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
337	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
338	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
339	C₂×C₂²⋊C₄	—	2	1	no	—
340	C₂×C₂²⋊C₄	—	2	2	no	—
341	C₂×C₂²⋊C₄	—	2	2	no	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₂²⋊C₄	—	2	2	no	—
344	C₂×C₂²⋊C₄	—	2	2	no	—
345	C₂×C₄²	—	2	1	no	—
346	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
347	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
348	#107	—	2	2	no	—
349	C₄⋊C₈	—	2	4	no	—
350	C₄⋊C₈	—	2	2	no	—
351	#103	—	2	4	no	—
352	#103	—	2	2	no	—
353	#102	—	2	2	no	—
354	#102	—	2	4	no	—
355	#102	—	2	4	no	—
356	#102	—	4	2	no	—
357	#97	—	2	2	no	—
358	#95	—	2	1	no	—
359	#95	—	2	2	no	—
360	#95	—	2	1	no	—
361	#95	D₄×C₃₁	1	1	yes	characteristic
362	#95	D₄×C₃₁	1	1	yes	characteristic
363	C₃₁	#24749	1	1	yes	characteristic, direct product factor, Sylow
364	C₂⁴	—	2	1	no	—
365	C₂⁴	—	2	2	no	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	—	2	1	no	—
368	C₂×Q₈	—	2	1	no	—
369	C₂×Q₈	—	2	1	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	1	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	1	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	4	4	no	—
382	C₂×D₄	—	4	2	no	—
383	C₂×D₄	—	4	2	no	—
384	C₂×D₄	—	8	2	no	—
385	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
386	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
387	C₂×D₄	#41×C₃₁	1	2	yes	—
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	4	4	no	—
394	C₂²×C₄	—	4	4	no	—
395	C₂²×C₄	—	8	2	no	—
396	C₂²×C₄	—	8	2	no	—
397	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
398	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
399	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
401	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
402	SD₁₆	—	2	8	no	—
403	D₈	—	4	8	no	—
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	2	1	no	—
406	C₂×C₈	—	2	1	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	4	2	no	—
419	C₄⋊C₄	—	4	2	no	—
420	C₄⋊C₄	—	8	2	no	—
421	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
422	C₂²⋊C₄	—	2	4	no	—
423	C₂²⋊C₄	—	4	4	no	—
424	C₂²⋊C₄	—	4	4	no	—
425	C₂²⋊C₄	—	4	4	no	—
426	C₂²⋊C₄	—	8	2	no	—
427	C₂²⋊C₄	—	8	2	no	—
428	C₄²	—	2	2	no	—
429	C₄²	—	2	2	no	—
430	C₄²	—	4	2	no	—
431	C₄²	C₂×D₄×C₃₁	1	2	yes	—
432	C₄²	#41×C₃₁	1	2	yes	—
433	C₂³	—	2	2	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	2	2	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	4	2	no	—
439	C₂³	—	4	2	no	—
440	C₂³	—	4	2	no	—
441	C₂³	—	4	2	no	—
442	C₂³	—	4	2	no	—
443	C₂³	—	4	2	no	—
444	C₂³	—	4	2	no	—
445	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
446	Q₈	—	2	2	no	—
447	Q₈	—	2	2	no	—
448	Q₈	—	4	1	no	—
449	Q₈	—	4	1	no	—
450	D₄	—	2	2	no	—
451	D₄	—	2	2	no	—
452	D₄	—	2	2	no	—
453	D₄	—	4	2	no	—
454	D₄	—	4	2	no	—
455	D₄	—	4	2	no	—
456	D₄	—	4	2	no	—
457	D₄	—	8	4	no	—
458	D₄	—	8	4	no	—
459	D₄	C₃₁×#121	1	4	yes	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	8	4	no	—
479	C₂×C₄	—	8	8	no	—
480	C₂×C₄	C₃₁×#121	1	2	yes	—
481	C₂×C₄	C₃₁×#121	1	2	yes	—
482	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
483	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
484	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
485	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
486	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
487	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
488	C₈	—	2	2	no	—
489	C₈	—	4	2	no	—
490	C₈	—	4	1	no	—
491	C₂²	—	2	4	no	—
492	C₂²	—	4	4	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	8	2	no	—
497	C₂²	—	8	2	no	—
498	C₂²	—	8	2	no	—
499	C₂²	—	8	2	no	—
500	C₂²	—	8	2	no	—
501	C₂²	—	8	2	no	—
502	C₂²	—	8	2	no	—
503	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
504	C₂²	D₄²×C₃₁	1	1	yes	characteristic
505	C₂²	C₃₁×#548	1	1	yes	characteristic
506	C₂²	C₃₁×#575	1	2	yes	—
507	C₂²	C₃₁×#576	1	2	yes	—
508	C₄	—	2	2	no	—
509	C₄	—	2	2	no	—
510	C₄	—	4	2	no	—
511	C₄	—	4	2	no	—
512	C₄	—	4	2	no	—
513	C₄	—	4	2	no	—
514	C₄	—	8	2	no	—
515	C₄	—	8	1	no	—
516	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
517	C₄	C₃₁×#537	1	1	yes	characteristic
518	C₄	C₃₁×#547	1	1	yes	characteristic
519	C₄	C₃₁×#553	1	1	yes	characteristic
520	C₂	—	4	4	no	—
521	C₂	—	16	2	no	—
522	C₂	C₃₁×#2416	1	1	yes	characteristic
523	C₂	C₃₁×#3040	1	1	yes	characteristic
524	C₂	C₃₁×#3041	1	1	yes	characteristic
525	C₂	C₃₁×#3268	1	2	yes	—
526	C₂	C₃₁×#3370	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#459	C₂
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
1. C₂×C₃₁×#463	C₂	semidirect
3. C₂×C₃₁×#440	C₂	semidirect
4. C₃₁×#2548	C₂	semidirect
5. C₃₁×#2349	C₂	semidirect
6. C₃₁×#2067	C₂	semidirect
7. C₃₁×#2042	C₂	semidirect
8. C₃₁×#1961	C₂	semidirect
9. C₃₁×#1940	C₂	semidirect
10. C₃₁×#1879	C₂	semidirect
11. C₃₁×#1805	C₂	semidirect
53. C₂×SD₁₆×C₃₁	D₄	semidirect
169. #24749	C₃₁	direct (N × H)
228. C₂×#463	C₂×C₃₁	semidirect
230. C₂×#440	C₂×C₃₁	semidirect
231. #2548	C₂×C₃₁	semidirect
232. #2349	C₂×C₃₁	semidirect
233. #2067	C₂×C₃₁	semidirect
234. #2042	C₂×C₃₁	semidirect
235. #1961	C₂×C₃₁	semidirect
236. #1940	C₂×C₃₁	semidirect
237. #1879	C₂×C₃₁	semidirect
238. #1805	C₂×C₃₁	semidirect
316. C₂×SD₁₆	D₄×C₃₁	semidirect
363. C₃₁	#24749	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×#463	2
C₂×C₃₁×#459	2
C₂×C₃₁×#440	2
C₃₁×#2548	2
C₃₁×#2349	2
C₃₁×#2067	2
C₃₁×#2042	2
C₃₁×#1961	2
C₃₁×#1940	2
C₃₁×#1879	2
C₃₁×#1805	2
#24749	31

Subgroups — C31×#24749 #21481508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24749 #21481508