Subgroups — C₃₁×#19163 #21475922

Order 7936.

760 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3152	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3100	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#474	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1689	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1685	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1669	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1665	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1547	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1546	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	—	2	1	no	—
13	C₂×C₃₁×#137	—	2	1	no	—
14	C₂²×M₁₆×C₃₁	—	2	1	no	—
15	C₃₁×#532	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#497	—	2	2	no	—
19	C₃₁×#494	C₂²	1	2	yes	complemented
20	C₃₁×#492	C₂²	1	2	yes	complemented
21	C₃₁×#480	—	2	1	no	—
22	C₃₁×#474	—	2	1	no	—
23	C₃₁×#474	—	2	1	no	—
24	C₃₁×#474	C₂²	1	1	yes	characteristic
25	C₃₁×#474	C₂²	1	1	yes	characteristic
26	C₃₁×#474	C₂²	1	2	yes	—
27	C₃₁×#474	C₂²	1	2	yes	—
28	C₃₁×#459	C₂²	1	1	yes	characteristic
29	C₃₁×#459	C₂²	1	1	yes	characteristic
30	C₃₁×#447	—	2	1	no	—
31	C₄×D₈×C₃₁	—	2	1	no	—
32	C₄×D₈×C₃₁	—	2	1	no	—
33	C₃₁×#436	C₂²	1	2	yes	—
34	C₃₁×#435	—	2	2	no	—
35	C₈×D₄×C₃₁	C₂²	1	2	yes	—
36	C₃₁×#431	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×#107	—	2	1	no	—
38	C₃₁×#423	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#102	—	2	1	no	—
41	C₃₁×#407	—	2	1	no	—
42	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
43	C₃₁×#335	C₂²	1	2	yes	complemented
44	C₃₁×#334	C₂²	1	2	yes	complemented
45	C₃₁×#332	C₂²	1	2	yes	complemented
46	C₃₁×#332	C₂²	1	2	yes	complemented
47	C₃₁×#332	C₂²	1	2	yes	complemented
48	C₃₁×#331	C₂²	1	2	yes	complemented
49	C₂×#41×C₃₁	—	2	1	no	—
50	C₂²×Q₈×C₃₁	—	2	1	no	—
51	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₃₁	—	2	1	no	—
53	C₃₁×#137	—	4	4	no	—
54	C₃₁×#137	—	4	4	no	—
55	C₂×Q₁₆×C₃₁	—	2	2	no	—
56	C₂×Q₁₆×C₃₁	—	2	2	no	—
57	C₂×SD₁₆×C₃₁	—	2	2	no	—
58	C₂×SD₁₆×C₃₁	—	2	2	no	—
59	C₂×D₈×C₃₁	—	2	1	no	—
60	C₂×D₈×C₃₁	—	2	1	no	—
61	C₂×M₁₆×C₃₁	—	2	1	no	—
62	C₂×M₁₆×C₃₁	—	2	2	no	—
63	C₂×M₁₆×C₃₁	—	4	4	no	—
64	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
65	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₃₁	—	2	1	no	—
67	C₂²×C₈×C₃₁	—	2	2	no	—
68	C₃₁×C₄⋊Q₈	—	2	1	no	—
69	C₃₁×C₄⋊Q₈	—	2	1	no	—
70	C₃₁×C₄⋊Q₈	—	2	1	no	—
71	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
72	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
73	C₃₁×#127	C₂³	1	1	yes	characteristic
74	C₃₁×#124	—	2	1	no	—
75	C₃₁×#121	—	2	1	no	—
76	C₃₁×#121	—	2	1	no	—
77	C₄×D₄×C₃₁	—	2	1	no	—
78	C₄×D₄×C₃₁	—	2	1	no	—
79	C₄×D₄×C₃₁	—	2	1	no	—
80	C₄×D₄×C₃₁	—	2	1	no	—
81	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
82	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
83	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
87	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
88	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
89	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
90	C₃₁×#107	—	2	2	no	—
91	C₃₁×#107	—	2	2	no	—
92	C₃₁×#107	—	2	1	no	—
93	C₃₁×#107	—	2	1	no	—
94	C₃₁×C₄⋊C₈	—	2	1	no	—
95	C₃₁×C₄⋊C₈	—	2	1	no	—
96	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
97	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
98	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
99	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
100	C₃₁×#102	—	2	2	no	—
101	C₃₁×#102	—	2	2	no	—
102	C₃₁×#102	—	2	1	no	—
103	C₃₁×#102	—	2	1	no	—
104	C₃₁×#102	—	2	1	no	—
105	C₃₁×#102	—	2	1	no	—
106	C₃₁×C₂²⋊C₈	—	2	1	no	—
107	C₃₁×C₂²⋊C₈	—	2	2	no	—
108	C₃₁×C₂²⋊C₈	—	2	2	no	—
109	C₃₁×#97	D₄	1	2	yes	—
110	C₄×C₈×C₃₁	—	2	1	no	—
111	C₄×C₈×C₃₁	C₂³	1	2	yes	—
112	C₂⁴×C₃₁	—	2	1	no	—
113	#41×C₃₁	—	4	2	no	—
114	#41×C₃₁	—	4	2	no	—
115	C₂×Q₈×C₃₁	—	2	1	no	—
116	C₂×Q₈×C₃₁	—	2	2	no	—
117	C₂×Q₈×C₃₁	—	4	1	no	—
118	C₂×Q₈×C₃₁	—	4	2	no	—
119	C₂×Q₈×C₃₁	—	4	2	no	—
120	C₂×D₄×C₃₁	—	2	2	no	—
121	C₂×D₄×C₃₁	—	2	1	no	—
122	C₂×D₄×C₃₁	—	2	1	no	—
123	C₂×D₄×C₃₁	—	2	1	no	—
124	C₂×D₄×C₃₁	—	2	1	no	—
125	C₂×D₄×C₃₁	—	4	1	no	—
126	C₂×D₄×C₃₁	—	4	1	no	—
127	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	—	2	1	no	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	2	1	no	—
134	C₂²×C₄×C₃₁	—	4	2	no	—
135	C₂²×C₄×C₃₁	—	4	2	no	—
136	C₂²×C₄×C₃₁	—	4	1	no	—
137	C₂²×C₄×C₃₁	—	4	1	no	—
138	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
141	Q₁₆×C₃₁	—	4	4	no	—
142	Q₁₆×C₃₁	—	4	4	no	—
143	SD₁₆×C₃₁	—	4	4	no	—
144	SD₁₆×C₃₁	—	4	4	no	—
145	D₈×C₃₁	—	4	2	no	—
146	D₈×C₃₁	—	4	2	no	—
147	M₁₆×C₃₁	—	2	2	no	—
148	M₁₆×C₃₁	—	4	2	no	—
149	M₁₆×C₃₁	—	4	4	no	—
150	M₁₆×C₃₁	D₈	1	4	yes	complemented
151	C₂×C₈×C₃₁	—	2	1	no	—
152	C₂×C₈×C₃₁	—	2	1	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	—	4	2	no	—
156	C₂×C₈×C₃₁	—	4	4	no	—
157	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
158	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
159	C₄⋊C₄×C₃₁	—	2	1	no	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	—	2	2	no	—
163	C₄⋊C₄×C₃₁	—	2	1	no	—
164	C₄⋊C₄×C₃₁	—	4	1	no	—
165	C₄⋊C₄×C₃₁	—	4	1	no	—
166	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
168	C₂²⋊C₄×C₃₁	—	2	1	no	—
169	C₂²⋊C₄×C₃₁	—	2	1	no	—
170	C₂²⋊C₄×C₃₁	—	2	1	no	—
171	C₂²⋊C₄×C₃₁	—	2	1	no	—
172	C₂²⋊C₄×C₃₁	—	4	1	no	—
173	C₂²⋊C₄×C₃₁	—	4	1	no	—
174	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₃₁	#41	1	1	yes	characteristic
177	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
178	#19163	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
179	C₂³×C₃₁	—	2	1	no	—
180	C₂³×C₃₁	—	2	1	no	—
181	C₂³×C₃₁	—	4	2	no	—
182	C₂³×C₃₁	—	4	2	no	—
183	C₂³×C₃₁	—	4	1	no	—
184	C₂³×C₃₁	—	8	1	no	—
185	C₂³×C₃₁	#121	1	1	yes	characteristic
186	Q₈×C₃₁	—	4	2	no	—
187	Q₈×C₃₁	—	4	1	no	—
188	Q₈×C₃₁	—	4	2	no	—
189	Q₈×C₃₁	—	4	1	no	—
190	Q₈×C₃₁	—	8	2	no	—
191	D₄×C₃₁	—	2	1	no	—
192	D₄×C₃₁	—	2	1	no	—
193	D₄×C₃₁	—	2	1	no	—
194	D₄×C₃₁	—	4	2	no	—
195	D₄×C₃₁	—	4	1	no	—
196	D₄×C₃₁	—	4	1	no	—
197	D₄×C₃₁	—	4	1	no	—
198	D₄×C₃₁	—	4	1	no	—
199	D₄×C₃₁	—	8	1	no	—
200	D₄×C₃₁	—	8	2	no	—
201	D₄×C₃₁	#137	1	2	yes	complemented
202	C₂×C₄×C₃₁	—	2	1	no	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	4	1	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	4	1	no	—
210	C₂×C₄×C₃₁	—	4	1	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	4	1	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	—	8	2	no	—
217	C₂×C₄×C₃₁	—	8	2	no	—
218	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
219	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
220	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
222	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
223	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
226	C₈×C₃₁	—	2	2	no	—
227	C₈×C₃₁	—	4	2	no	—
228	C₈×C₃₁	—	4	1	no	—
229	C₈×C₃₁	—	4	2	no	—
230	C₈×C₃₁	—	4	1	no	—
231	C₈×C₃₁	C₂×D₈	1	4	yes	—
232	#3152	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#3100	C₂×C₃₁	1	1	yes	characteristic
234	C₂×#474	C₂×C₃₁	1	1	yes	characteristic, complemented
235	D₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#1689	C₂×C₃₁	1	2	yes	complemented
237	#1685	C₂×C₃₁	1	2	yes	complemented
238	#1669	C₂×C₃₁	1	2	yes	complemented
239	#1665	C₂×C₃₁	1	2	yes	complemented
240	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#1547	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#1546	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂²×C₃₁	—	2	1	no	—
244	C₂²×C₃₁	—	2	1	no	—
245	C₂²×C₃₁	—	2	2	no	—
246	C₂²×C₃₁	—	4	1	no	—
247	C₂²×C₃₁	—	4	1	no	—
248	C₂²×C₃₁	—	4	1	no	—
249	C₂²×C₃₁	—	4	1	no	—
250	C₂²×C₃₁	—	4	1	no	—
251	C₂²×C₃₁	—	8	4	no	—
252	C₂²×C₃₁	—	8	1	no	—
253	C₂²×C₃₁	—	8	1	no	—
254	C₂²×C₃₁	—	8	1	no	—
255	C₂²×C₃₁	#459	1	2	yes	—
256	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
257	C₄×C₃₁	—	2	1	no	—
258	C₄×C₃₁	—	2	1	no	—
259	C₄×C₃₁	—	2	1	no	—
260	C₄×C₃₁	—	4	1	no	—
261	C₄×C₃₁	—	4	1	no	—
262	C₄×C₃₁	—	8	1	no	—
263	C₄×C₃₁	—	8	2	no	—
264	C₄×C₃₁	#459	1	2	yes	—
265	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
266	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
267	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
268	C₄×C₃₁	#575	1	1	yes	characteristic
269	C₂×#137	—	2	1	no	—
270	C₂×#137	—	2	1	no	—
271	C₂²×M₁₆	—	2	1	no	—
272	#532	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
275	#497	—	2	2	no	—
276	#494	C₂²×C₃₁	1	2	yes	complemented
277	#492	C₂²×C₃₁	1	2	yes	complemented
278	#480	—	2	1	no	—
279	#474	—	2	1	no	—
280	#474	—	2	1	no	—
281	#474	C₂²×C₃₁	1	1	yes	characteristic
282	#474	C₂²×C₃₁	1	1	yes	characteristic
283	#474	C₂²×C₃₁	1	2	yes	—
284	#474	C₂²×C₃₁	1	2	yes	—
285	#459	C₂²×C₃₁	1	1	yes	characteristic
286	#459	C₂²×C₃₁	1	1	yes	characteristic
287	#447	—	2	1	no	—
288	C₄×D₈	—	2	1	no	—
289	C₄×D₈	—	2	1	no	—
290	#436	C₂²×C₃₁	1	2	yes	—
291	#435	—	2	2	no	—
292	C₈×D₄	C₂²×C₃₁	1	2	yes	—
293	#431	C₂²×C₃₁	1	1	yes	characteristic
294	C₂×#107	—	2	1	no	—
295	#423	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
297	C₂×#102	—	2	1	no	—
298	#407	—	2	1	no	—
299	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
300	#335	C₂²×C₃₁	1	2	yes	complemented
301	#334	C₂²×C₃₁	1	2	yes	complemented
302	#332	C₂²×C₃₁	1	2	yes	complemented
303	#332	C₂²×C₃₁	1	2	yes	complemented
304	#332	C₂²×C₃₁	1	2	yes	complemented
305	#331	C₂²×C₃₁	1	2	yes	complemented
306	C₂×C₃₁	—	2	2	no	—
307	C₂×C₃₁	—	4	2	no	—
308	C₂×C₃₁	—	8	1	no	—
309	C₂×C₃₁	—	16	1	no	—
310	C₂×C₃₁	C₂×#459	1	1	yes	characteristic
311	C₂×C₃₁	#3056	1	1	yes	characteristic
312	C₂×C₃₁	#3069	1	1	yes	characteristic
313	C₂×#41	—	2	1	no	—
314	C₂²×Q₈	—	2	1	no	—
315	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
316	C₂³×C₄	—	2	1	no	—
317	#137	—	4	4	no	—
318	#137	—	4	4	no	—
319	C₂×Q₁₆	—	2	2	no	—
320	C₂×Q₁₆	—	2	2	no	—
321	C₂×SD₁₆	—	2	2	no	—
322	C₂×SD₁₆	—	2	2	no	—
323	C₂×D₈	—	2	1	no	—
324	C₂×D₈	—	2	1	no	—
325	C₂×M₁₆	—	2	1	no	—
326	C₂×M₁₆	—	2	2	no	—
327	C₂×M₁₆	—	4	4	no	—
328	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
329	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
330	C₂²×C₈	—	2	1	no	—
331	C₂²×C₈	—	2	2	no	—
332	C₄⋊Q₈	—	2	1	no	—
333	C₄⋊Q₈	—	2	1	no	—
334	C₄⋊Q₈	—	2	1	no	—
335	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
336	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
337	#127	C₂³×C₃₁	1	1	yes	characteristic
338	#124	—	2	1	no	—
339	#121	—	2	1	no	—
340	#121	—	2	1	no	—
341	C₄×D₄	—	2	1	no	—
342	C₄×D₄	—	2	1	no	—
343	C₄×D₄	—	2	1	no	—
344	C₄×D₄	—	2	1	no	—
345	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
346	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
347	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
348	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
349	C₂×C₄⋊C₄	—	2	1	no	—
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
352	C₂×C₂²⋊C₄	—	2	1	no	—
353	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
354	#107	—	2	2	no	—
355	#107	—	2	2	no	—
356	#107	—	2	1	no	—
357	#107	—	2	1	no	—
358	C₄⋊C₈	—	2	1	no	—
359	C₄⋊C₈	—	2	1	no	—
360	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
361	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
362	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
363	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
364	#102	—	2	2	no	—
365	#102	—	2	2	no	—
366	#102	—	2	1	no	—
367	#102	—	2	1	no	—
368	#102	—	2	1	no	—
369	#102	—	2	1	no	—
370	C₂²⋊C₈	—	2	1	no	—
371	C₂²⋊C₈	—	2	2	no	—
372	C₂²⋊C₈	—	2	2	no	—
373	#97	D₄×C₃₁	1	2	yes	—
374	C₄×C₈	—	2	1	no	—
375	C₄×C₈	C₂³×C₃₁	1	2	yes	—
376	C₃₁	#19163	1	1	yes	characteristic, direct product factor, Sylow
377	C₂⁴	—	2	1	no	—
378	#41	—	4	2	no	—
379	#41	—	4	2	no	—
380	C₂×Q₈	—	2	1	no	—
381	C₂×Q₈	—	2	2	no	—
382	C₂×Q₈	—	4	1	no	—
383	C₂×Q₈	—	4	2	no	—
384	C₂×Q₈	—	4	2	no	—
385	C₂×D₄	—	2	2	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	4	1	no	—
391	C₂×D₄	—	4	1	no	—
392	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
393	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	4	2	no	—
400	C₂²×C₄	—	4	2	no	—
401	C₂²×C₄	—	4	1	no	—
402	C₂²×C₄	—	4	1	no	—
403	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
404	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
405	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
406	Q₁₆	—	4	4	no	—
407	Q₁₆	—	4	4	no	—
408	SD₁₆	—	4	4	no	—
409	SD₁₆	—	4	4	no	—
410	D₈	—	4	2	no	—
411	D₈	—	4	2	no	—
412	M₁₆	—	2	2	no	—
413	M₁₆	—	4	2	no	—
414	M₁₆	—	4	4	no	—
415	M₁₆	D₈×C₃₁	1	4	yes	complemented
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	4	2	no	—
421	C₂×C₈	—	4	4	no	—
422	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
423	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	2	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	4	1	no	—
430	C₄⋊C₄	—	4	1	no	—
431	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
432	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
433	C₂²⋊C₄	—	2	1	no	—
434	C₂²⋊C₄	—	2	1	no	—
435	C₂²⋊C₄	—	2	1	no	—
436	C₂²⋊C₄	—	2	1	no	—
437	C₂²⋊C₄	—	4	1	no	—
438	C₂²⋊C₄	—	4	1	no	—
439	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
440	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
441	C₄²	#41×C₃₁	1	1	yes	characteristic
442	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
443	C₂³	—	2	1	no	—
444	C₂³	—	2	1	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	4	1	no	—
448	C₂³	—	8	1	no	—
449	C₂³	C₃₁×#121	1	1	yes	characteristic
450	Q₈	—	4	2	no	—
451	Q₈	—	4	1	no	—
452	Q₈	—	4	2	no	—
453	Q₈	—	4	1	no	—
454	Q₈	—	8	2	no	—
455	D₄	—	2	1	no	—
456	D₄	—	2	1	no	—
457	D₄	—	2	1	no	—
458	D₄	—	4	2	no	—
459	D₄	—	4	1	no	—
460	D₄	—	4	1	no	—
461	D₄	—	4	1	no	—
462	D₄	—	4	1	no	—
463	D₄	—	8	1	no	—
464	D₄	—	8	2	no	—
465	D₄	C₃₁×#137	1	2	yes	complemented
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	8	2	no	—
481	C₂×C₄	—	8	2	no	—
482	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
483	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
484	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
485	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
486	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
487	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
488	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
489	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
490	C₈	—	2	2	no	—
491	C₈	—	4	2	no	—
492	C₈	—	4	1	no	—
493	C₈	—	4	2	no	—
494	C₈	—	4	1	no	—
495	C₈	C₂×D₈×C₃₁	1	4	yes	—
496	C₂²	—	2	1	no	—
497	C₂²	—	2	1	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	4	1	no	—
500	C₂²	—	4	1	no	—
501	C₂²	—	4	1	no	—
502	C₂²	—	4	1	no	—
503	C₂²	—	4	1	no	—
504	C₂²	—	8	4	no	—
505	C₂²	—	8	1	no	—
506	C₂²	—	8	1	no	—
507	C₂²	—	8	1	no	—
508	C₂²	C₃₁×#459	1	2	yes	—
509	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
510	C₄	—	2	1	no	—
511	C₄	—	2	1	no	—
512	C₄	—	2	1	no	—
513	C₄	—	4	1	no	—
514	C₄	—	4	1	no	—
515	C₄	—	8	1	no	—
516	C₄	—	8	2	no	—
517	C₄	C₃₁×#459	1	2	yes	—
518	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
519	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
520	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
521	C₄	C₃₁×#575	1	1	yes	characteristic
522	C₂	—	2	2	no	—
523	C₂	—	4	2	no	—
524	C₂	—	8	1	no	—
525	C₂	—	16	1	no	—
526	C₂	C₂×C₃₁×#459	1	1	yes	characteristic
527	C₂	C₃₁×#3056	1	1	yes	characteristic
528	C₂	C₃₁×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3100	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ 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. C₃₁×#3152	C₂	semidirect
3. C₂×C₃₁×#474	C₂	semidirect
4. D₄×M₁₆×C₃₁	C₂	semidirect
5. C₃₁×#1689	C₂	semidirect
6. C₃₁×#1685	C₂	semidirect
7. C₃₁×#1669	C₂	semidirect
8. C₃₁×#1665	C₂	semidirect
9. C₃₁×#1565	C₂	semidirect
10. C₃₁×#1547	C₂	semidirect
11. C₃₁×#1546	C₂	semidirect
19. C₃₁×#494	C₂²	semidirect
20. C₃₁×#492	C₂²	semidirect
43. C₃₁×#335	C₂²	semidirect
44. C₃₁×#334	C₂²	semidirect
45. C₃₁×#332	C₂²	semidirect
46. C₃₁×#332	C₂²	semidirect
47. C₃₁×#332	C₂²	semidirect
48. C₃₁×#331	C₂²	semidirect
150. M₁₆×C₃₁	D₈	semidirect
178. #19163	C₃₁	direct (N × H)
201. D₄×C₃₁	#137	semidirect
232. #3152	C₂×C₃₁	semidirect
234. C₂×#474	C₂×C₃₁	semidirect
235. D₄×M₁₆	C₂×C₃₁	semidirect
236. #1689	C₂×C₃₁	semidirect
237. #1685	C₂×C₃₁	semidirect
238. #1669	C₂×C₃₁	semidirect
239. #1665	C₂×C₃₁	semidirect
240. #1565	C₂×C₃₁	semidirect
241. #1547	C₂×C₃₁	semidirect
242. #1546	C₂×C₃₁	semidirect
276. #494	C₂²×C₃₁	semidirect
277. #492	C₂²×C₃₁	semidirect
300. #335	C₂²×C₃₁	semidirect
301. #334	C₂²×C₃₁	semidirect
302. #332	C₂²×C₃₁	semidirect
303. #332	C₂²×C₃₁	semidirect
304. #332	C₂²×C₃₁	semidirect
305. #331	C₂²×C₃₁	semidirect
376. C₃₁	#19163	direct (N × H)
415. M₁₆	D₈×C₃₁	semidirect
465. D₄	C₃₁×#137	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3152	2
C₃₁×#3100	2
C₂×C₃₁×#474	2
D₄×M₁₆×C₃₁	2
C₃₁×#1689	2
C₃₁×#1685	2
C₃₁×#1669	2
C₃₁×#1665	2
C₃₁×#1565	2
C₃₁×#1547	2
C₃₁×#1546	2
#19163	31

Subgroups — C31×#19163 #21475922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19163 #21475922