Subgroups — C₃₁×#24800 #21481559

Order 7936.

842 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#473	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3059	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3046	C₂	1	2	yes	maximal
4	C₂×C₃₁×#444	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2912	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2064	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2038	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1971	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1946	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1882	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1815	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₂²	1	2	yes	—
13	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
14	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
17	C₃₁×#536	C₂²	1	2	yes	—
18	C₃₁×#520	C₂²	1	2	yes	—
19	C₃₁×#473	—	2	4	no	—
20	C₃₁×#470	—	2	1	no	—
21	C₃₁×#470	—	2	1	no	—
22	C₃₁×#467	—	2	2	no	—
23	C₃₁×#464	—	2	2	no	—
24	C₃₁×#463	—	2	2	no	—
25	C₃₁×#462	—	2	2	no	—
26	C₃₁×#461	—	2	2	no	—
27	C₃₁×#444	—	2	4	no	—
28	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
30	C₃₁×#428	—	2	2	no	—
31	C₃₁×#428	C₂²	1	2	yes	—
32	C₃₁×#427	—	2	1	no	—
33	C₃₁×#427	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#106	—	2	1	no	—
35	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
36	C₃₁×#424	C₂²	1	2	yes	—
37	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
39	C₃₁×#419	C₂²	1	2	yes	—
40	C₃₁×#417	C₂²	1	2	yes	—
41	C₂×C₃₁×#103	C₂²	1	2	yes	—
42	C₂×C₃₁×#101	C₂²	1	2	yes	—
43	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
44	C₃₁×#391	C₂²	1	2	yes	—
45	C₃₁×#343	C₂²	1	2	yes	—
46	C₃₁×#339	C₂²	1	2	yes	—
47	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
50	C₃₁×#137	—	2	4	no	—
51	C₃₁×#137	—	2	4	no	—
52	C₃₁×#135	—	2	2	no	—
53	C₃₁×#135	—	2	2	no	—
54	C₂×Q₁₆×C₃₁	—	2	2	no	—
55	C₂×Q₁₆×C₃₁	—	2	1	no	—
56	C₂×Q₁₆×C₃₁	—	2	1	no	—
57	C₂×Q₁₆×C₃₁	—	2	2	no	—
58	C₂×Q₁₆×C₃₁	—	2	2	no	—
59	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
60	C₂×SD₁₆×C₃₁	—	2	2	no	—
61	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
62	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
63	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
64	C₃₁×#131	—	2	2	no	—
65	C₃₁×#131	—	2	2	no	—
66	C₃₁×#131	—	2	2	no	—
67	C₃₁×#131	D₄	1	4	yes	—
68	C₂×M₁₆×C₃₁	—	2	1	no	—
69	C₂×M₁₆×C₃₁	—	2	1	no	—
70	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
71	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₃₁	—	2	1	no	—
74	C₂²×C₈×C₃₁	—	2	1	no	—
75	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×C₄⋊Q₈	—	2	2	no	—
78	C₃₁×#124	—	2	2	no	—
79	C₃₁×#123	—	4	2	no	—
80	C₃₁×C₂²⋊Q₈	—	2	2	no	—
81	C₃₁×C₂²⋊Q₈	—	2	2	no	—
82	C₄×Q₈×C₃₁	—	2	2	no	—
83	C₄×D₄×C₃₁	—	2	2	no	—
84	C₄×D₄×C₃₁	—	4	2	no	—
85	C₃₁×#117	—	2	2	no	—
86	C₃₁×#117	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
89	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
90	C₃₁×#108	—	2	2	no	—
91	C₃₁×#108	—	2	2	no	—
92	C₃₁×#107	—	2	2	no	—
93	C₃₁×#107	D₄	1	2	yes	—
94	C₃₁×#106	—	2	1	no	—
95	C₃₁×#106	—	2	1	no	—
96	C₃₁×#106	—	2	2	no	—
97	C₃₁×#106	D₄	1	1	yes	characteristic
98	C₃₁×#106	D₄	1	1	yes	characteristic
99	C₃₁×C₄⋊C₈	—	2	2	no	—
100	C₃₁×C₄⋊C₈	—	2	2	no	—
101	C₃₁×#104	—	2	2	no	—
102	C₃₁×#104	—	2	2	no	—
103	C₃₁×#103	—	2	2	no	—
104	C₃₁×#103	—	2	2	no	—
105	C₃₁×#103	—	2	2	no	—
106	C₃₁×#103	—	2	2	no	—
107	C₃₁×#103	—	2	2	no	—
108	C₃₁×#102	—	2	2	no	—
109	C₃₁×#101	—	2	4	no	—
110	C₃₁×#97	—	2	2	no	—
111	C₃₁×#95	—	4	2	no	—
112	#41×C₃₁	—	2	1	no	—
113	#41×C₃₁	—	2	1	no	—
114	#41×C₃₁	—	2	1	no	—
115	#41×C₃₁	—	2	1	no	—
116	#41×C₃₁	—	2	1	no	—
117	#41×C₃₁	—	2	1	no	—
118	#41×C₃₁	C₂×D₄	1	2	yes	—
119	#41×C₃₁	#41	1	2	yes	—
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	—	2	2	no	—
124	C₂×Q₈×C₃₁	—	2	2	no	—
125	C₂×Q₈×C₃₁	—	2	2	no	—
126	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂×D₄×C₃₁	—	2	1	no	—
129	C₂×D₄×C₃₁	—	2	1	no	—
130	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	2	1	no	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	2	2	no	—
136	C₂²×C₄×C₃₁	—	2	2	no	—
137	C₂²×C₄×C₃₁	—	4	2	no	—
138	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
139	Q₁₆×C₃₁	—	2	2	no	—
140	Q₁₆×C₃₁	—	4	4	no	—
141	Q₁₆×C₃₁	—	4	2	no	—
142	Q₁₆×C₃₁	—	4	2	no	—
143	SD₁₆×C₃₁	—	2	2	no	—
144	SD₁₆×C₃₁	—	2	2	no	—
145	SD₁₆×C₃₁	—	4	4	no	—
146	D₈×C₃₁	—	2	2	no	—
147	M₁₆×C₃₁	—	2	1	no	—
148	M₁₆×C₃₁	—	2	4	no	—
149	M₁₆×C₃₁	—	2	2	no	—
150	M₁₆×C₃₁	—	2	1	no	—
151	M₁₆×C₃₁	—	2	1	no	—
152	M₁₆×C₃₁	—	2	2	no	—
153	M₁₆×C₃₁	—	4	2	no	—
154	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
155	C₂×C₈×C₃₁	—	2	1	no	—
156	C₂×C₈×C₃₁	—	2	1	no	—
157	C₂×C₈×C₃₁	—	2	1	no	—
158	C₂×C₈×C₃₁	—	2	4	no	—
159	C₂×C₈×C₃₁	—	2	2	no	—
160	C₂×C₈×C₃₁	—	2	1	no	—
161	C₂×C₈×C₃₁	—	4	2	no	—
162	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
165	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₂×C₈×C₃₁	#41	1	2	yes	—
168	C₄⋊C₄×C₃₁	—	2	2	no	—
169	C₄⋊C₄×C₃₁	—	2	2	no	—
170	C₄⋊C₄×C₃₁	—	4	2	no	—
171	C₄⋊C₄×C₃₁	—	4	2	no	—
172	C₄⋊C₄×C₃₁	—	4	2	no	—
173	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₂²⋊C₄×C₃₁	—	2	2	no	—
176	C₂²⋊C₄×C₃₁	—	4	2	no	—
177	C₂²⋊C₄×C₃₁	—	4	2	no	—
178	C₂²⋊C₄×C₃₁	—	4	2	no	—
179	C₄²×C₃₁	—	2	2	no	—
180	C₄²×C₃₁	—	2	1	no	—
181	C₄²×C₃₁	—	2	1	no	—
182	C₄²×C₃₁	—	4	2	no	—
183	#24800	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂³×C₃₁	—	2	1	no	—
185	C₂³×C₃₁	—	2	1	no	—
186	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
187	Q₈×C₃₁	—	2	1	no	—
188	Q₈×C₃₁	—	2	1	no	—
189	Q₈×C₃₁	—	2	1	no	—
190	Q₈×C₃₁	—	4	2	no	—
191	Q₈×C₃₁	—	4	2	no	—
192	Q₈×C₃₁	—	4	2	no	—
193	Q₈×C₃₁	—	4	2	no	—
194	Q₈×C₃₁	#121	1	2	yes	—
195	D₄×C₃₁	—	2	2	no	—
196	D₄×C₃₁	—	2	2	no	—
197	D₄×C₃₁	—	2	1	no	—
198	D₄×C₃₁	—	2	1	no	—
199	D₄×C₃₁	—	2	1	no	—
200	D₄×C₃₁	—	4	1	no	—
201	D₄×C₃₁	—	4	1	no	—
202	D₄×C₃₁	#121	1	2	yes	—
203	C₂×C₄×C₃₁	—	2	1	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	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	4	1	no	—
210	C₂×C₄×C₃₁	—	4	1	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	—	8	2	no	—
218	C₂×C₄×C₃₁	—	16	2	no	—
219	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
225	C₈×C₃₁	—	2	2	no	—
226	C₈×C₃₁	—	2	1	no	—
227	C₈×C₃₁	—	2	1	no	—
228	C₈×C₃₁	—	4	1	no	—
229	C₈×C₃₁	—	4	2	no	—
230	C₈×C₃₁	#121	1	2	yes	—
231	C₈×C₃₁	#121	1	2	yes	—
232	C₂×#473	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#3059	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#3046	C₂×C₃₁	1	2	yes	—
235	C₂×#444	C₂×C₃₁	1	1	yes	characteristic
236	#2912	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#2064	C₂×C₃₁	1	2	yes	complemented
238	#2038	C₂×C₃₁	1	2	yes	complemented
239	#1971	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#1946	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#1882	C₂×C₃₁	1	2	yes	complemented
242	#1815	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂²×C₃₁	—	2	2	no	—
244	C₂²×C₃₁	—	2	2	no	—
245	C₂²×C₃₁	—	4	2	no	—
246	C₂²×C₃₁	—	4	1	no	—
247	C₂²×C₃₁	—	4	2	no	—
248	C₂²×C₃₁	#537	1	1	yes	characteristic
249	C₂²×C₃₁	D₄²	1	1	yes	characteristic
250	C₂²×C₃₁	#553	1	1	yes	characteristic
251	C₄×C₃₁	—	2	2	no	—
252	C₄×C₃₁	—	4	1	no	—
253	C₄×C₃₁	—	4	2	no	—
254	C₄×C₃₁	—	4	2	no	—
255	C₄×C₃₁	—	8	1	no	—
256	C₄×C₃₁	—	8	2	no	—
257	C₄×C₃₁	—	8	2	no	—
258	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
259	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
260	C₄×C₃₁	#547	1	1	yes	characteristic
261	C₄×C₃₁	#548	1	1	yes	characteristic
262	C₂×#137	C₂²×C₃₁	1	2	yes	—
263	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
264	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
267	#536	C₂²×C₃₁	1	2	yes	—
268	#520	C₂²×C₃₁	1	2	yes	—
269	#473	—	2	4	no	—
270	#470	—	2	1	no	—
271	#470	—	2	1	no	—
272	#467	—	2	2	no	—
273	#464	—	2	2	no	—
274	#463	—	2	2	no	—
275	#462	—	2	2	no	—
276	#461	—	2	2	no	—
277	#444	—	2	4	no	—
278	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
280	#428	—	2	2	no	—
281	#428	C₂²×C₃₁	1	2	yes	—
282	#427	—	2	1	no	—
283	#427	C₂²×C₃₁	1	1	yes	characteristic
284	C₂×#106	—	2	1	no	—
285	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
286	#424	C₂²×C₃₁	1	2	yes	—
287	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
289	#419	C₂²×C₃₁	1	2	yes	—
290	#417	C₂²×C₃₁	1	2	yes	—
291	C₂×#103	C₂²×C₃₁	1	2	yes	—
292	C₂×#101	C₂²×C₃₁	1	2	yes	—
293	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
294	#391	C₂²×C₃₁	1	2	yes	—
295	#343	C₂²×C₃₁	1	2	yes	—
296	#339	C₂²×C₃₁	1	2	yes	—
297	C₂×C₃₁	—	2	1	no	—
298	C₂×C₃₁	—	2	1	no	—
299	C₂×C₃₁	—	4	2	no	—
300	C₂×C₃₁	—	8	1	no	—
301	C₂×C₃₁	#2416	1	1	yes	characteristic
302	C₂×C₃₁	#3289	1	2	yes	—
303	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
304	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
305	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
306	#137	—	2	4	no	—
307	#137	—	2	4	no	—
308	#135	—	2	2	no	—
309	#135	—	2	2	no	—
310	C₂×Q₁₆	—	2	2	no	—
311	C₂×Q₁₆	—	2	1	no	—
312	C₂×Q₁₆	—	2	1	no	—
313	C₂×Q₁₆	—	2	2	no	—
314	C₂×Q₁₆	—	2	2	no	—
315	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
316	C₂×SD₁₆	—	2	2	no	—
317	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
318	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
319	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
320	#131	—	2	2	no	—
321	#131	—	2	2	no	—
322	#131	—	2	2	no	—
323	#131	D₄×C₃₁	1	4	yes	—
324	C₂×M₁₆	—	2	1	no	—
325	C₂×M₁₆	—	2	1	no	—
326	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
327	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
328	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
329	C₂²×C₈	—	2	1	no	—
330	C₂²×C₈	—	2	1	no	—
331	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
332	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
333	C₄⋊Q₈	—	2	2	no	—
334	#124	—	2	2	no	—
335	#123	—	4	2	no	—
336	C₂²⋊Q₈	—	2	2	no	—
337	C₂²⋊Q₈	—	2	2	no	—
338	C₄×Q₈	—	2	2	no	—
339	C₄×D₄	—	2	2	no	—
340	C₄×D₄	—	4	2	no	—
341	#117	—	2	2	no	—
342	#117	C₂³×C₃₁	1	2	yes	—
343	C₂×C₄⋊C₄	—	2	2	no	—
344	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
345	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
346	#108	—	2	2	no	—
347	#108	—	2	2	no	—
348	#107	—	2	2	no	—
349	#107	D₄×C₃₁	1	2	yes	—
350	#106	—	2	1	no	—
351	#106	—	2	1	no	—
352	#106	—	2	2	no	—
353	#106	D₄×C₃₁	1	1	yes	characteristic
354	#106	D₄×C₃₁	1	1	yes	characteristic
355	C₄⋊C₈	—	2	2	no	—
356	C₄⋊C₈	—	2	2	no	—
357	#104	—	2	2	no	—
358	#104	—	2	2	no	—
359	#103	—	2	2	no	—
360	#103	—	2	2	no	—
361	#103	—	2	2	no	—
362	#103	—	2	2	no	—
363	#103	—	2	2	no	—
364	#102	—	2	2	no	—
365	#101	—	2	4	no	—
366	#97	—	2	2	no	—
367	#95	—	4	2	no	—
368	C₃₁	#24800	1	1	yes	characteristic, direct product factor, Sylow
369	#41	—	2	1	no	—
370	#41	—	2	1	no	—
371	#41	—	2	1	no	—
372	#41	—	2	1	no	—
373	#41	—	2	1	no	—
374	#41	—	2	1	no	—
375	#41	C₂×D₄×C₃₁	1	2	yes	—
376	#41	#41×C₃₁	1	2	yes	—
377	C₂×Q₈	—	2	2	no	—
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	—	2	2	no	—
380	C₂×Q₈	—	2	2	no	—
381	C₂×Q₈	—	2	2	no	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	4	2	no	—
395	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
396	Q₁₆	—	2	2	no	—
397	Q₁₆	—	4	4	no	—
398	Q₁₆	—	4	2	no	—
399	Q₁₆	—	4	2	no	—
400	SD₁₆	—	2	2	no	—
401	SD₁₆	—	2	2	no	—
402	SD₁₆	—	4	4	no	—
403	D₈	—	2	2	no	—
404	M₁₆	—	2	1	no	—
405	M₁₆	—	2	4	no	—
406	M₁₆	—	2	2	no	—
407	M₁₆	—	2	1	no	—
408	M₁₆	—	2	1	no	—
409	M₁₆	—	2	2	no	—
410	M₁₆	—	4	2	no	—
411	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	4	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	4	2	no	—
419	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
420	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
421	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
422	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
423	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
424	C₂×C₈	#41×C₃₁	1	2	yes	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	2	no	—
427	C₄⋊C₄	—	4	2	no	—
428	C₄⋊C₄	—	4	2	no	—
429	C₄⋊C₄	—	4	2	no	—
430	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
431	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
432	C₂²⋊C₄	—	2	2	no	—
433	C₂²⋊C₄	—	4	2	no	—
434	C₂²⋊C₄	—	4	2	no	—
435	C₂²⋊C₄	—	4	2	no	—
436	C₄²	—	2	2	no	—
437	C₄²	—	2	1	no	—
438	C₄²	—	2	1	no	—
439	C₄²	—	4	2	no	—
440	C₂³	—	2	1	no	—
441	C₂³	—	2	1	no	—
442	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
443	Q₈	—	2	1	no	—
444	Q₈	—	2	1	no	—
445	Q₈	—	2	1	no	—
446	Q₈	—	4	2	no	—
447	Q₈	—	4	2	no	—
448	Q₈	—	4	2	no	—
449	Q₈	—	4	2	no	—
450	Q₈	C₃₁×#121	1	2	yes	—
451	D₄	—	2	2	no	—
452	D₄	—	2	2	no	—
453	D₄	—	2	1	no	—
454	D₄	—	2	1	no	—
455	D₄	—	2	1	no	—
456	D₄	—	4	1	no	—
457	D₄	—	4	1	no	—
458	D₄	C₃₁×#121	1	2	yes	—
459	C₂×C₄	—	2	1	no	—
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	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	8	2	no	—
474	C₂×C₄	—	16	2	no	—
475	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
476	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
477	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
478	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
479	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
480	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
481	C₈	—	2	2	no	—
482	C₈	—	2	1	no	—
483	C₈	—	2	1	no	—
484	C₈	—	4	1	no	—
485	C₈	—	4	2	no	—
486	C₈	C₃₁×#121	1	2	yes	—
487	C₈	C₃₁×#121	1	2	yes	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	1	no	—
492	C₂²	—	4	2	no	—
493	C₂²	C₃₁×#537	1	1	yes	characteristic
494	C₂²	D₄²×C₃₁	1	1	yes	characteristic
495	C₂²	C₃₁×#553	1	1	yes	characteristic
496	C₄	—	2	2	no	—
497	C₄	—	4	1	no	—
498	C₄	—	4	2	no	—
499	C₄	—	4	2	no	—
500	C₄	—	8	1	no	—
501	C₄	—	8	2	no	—
502	C₄	—	8	2	no	—
503	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
504	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
505	C₄	C₃₁×#547	1	1	yes	characteristic
506	C₄	C₃₁×#548	1	1	yes	characteristic
507	C₂	—	2	1	no	—
508	C₂	—	2	1	no	—
509	C₂	—	4	2	no	—
510	C₂	—	8	1	no	—
511	C₂	C₃₁×#2416	1	1	yes	characteristic
512	C₂	C₃₁×#3289	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#473	C₂	semidirect
2. C₃₁×#3059	C₂	semidirect
5. C₃₁×#2912	C₂	semidirect
6. C₃₁×#2064	C₂	semidirect
7. C₃₁×#2038	C₂	semidirect
8. C₃₁×#1971	C₂	semidirect
9. C₃₁×#1946	C₂	semidirect
10. C₃₁×#1882	C₂	semidirect
11. C₃₁×#1815	C₂	semidirect
183. #24800	C₃₁	direct (N × H)
232. C₂×#473	C₂×C₃₁	semidirect
233. #3059	C₂×C₃₁	semidirect
236. #2912	C₂×C₃₁	semidirect
237. #2064	C₂×C₃₁	semidirect
238. #2038	C₂×C₃₁	semidirect
239. #1971	C₂×C₃₁	semidirect
240. #1946	C₂×C₃₁	semidirect
241. #1882	C₂×C₃₁	semidirect
242. #1815	C₂×C₃₁	semidirect
368. C₃₁	#24800	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×#473	2
C₃₁×#3059	2
C₃₁×#3046	2
C₂×C₃₁×#444	2
C₃₁×#2912	2
C₃₁×#2064	2
C₃₁×#2038	2
C₃₁×#1971	2
C₃₁×#1946	2
C₃₁×#1882	2
C₃₁×#1815	2
#24800	31

Subgroups — C31×#24800 #21481559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24800 #21481559