Subgroups — C₃₁×#18612 #21475371

Order 7936.

876 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	C₃₁×#3502	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3216	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3182	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3103	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1648	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1645	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1626	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1623	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1554	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1491	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1485	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈²×C₃₁	—	2	1	no	—
13	Q₈²×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#557	—	2	1	no	—
15	C₃₁×#557	—	2	2	no	—
16	C₃₁×#557	—	2	2	no	—
17	C₃₁×#557	C₂²	1	2	yes	—
18	C₃₁×#544	—	2	2	no	—
19	C₃₁×#544	—	2	1	no	—
20	C₃₁×#533	—	2	2	no	—
21	C₃₁×#533	C₂²	1	1	yes	characteristic
22	C₃₁×#533	C₂²	1	1	yes	characteristic
23	C₃₁×#532	C₂²	1	1	yes	characteristic
24	C₃₁×#519	—	2	1	no	—
25	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#500	C₂²	1	1	yes	characteristic
27	C₃₁×#497	—	2	1	no	—
28	C₃₁×#492	C₂²	1	1	yes	characteristic
29	C₃₁×#489	—	2	1	no	—
30	C₃₁×#487	C₂²	1	1	yes	characteristic
31	C₃₁×#486	C₂²	1	1	yes	characteristic
32	C₃₁×#481	—	2	1	no	—
33	C₃₁×#475	C₂²	1	2	yes	—
34	C₃₁×#468	—	2	2	no	—
35	C₃₁×#462	C₂²	1	2	yes	—
36	C₃₁×#460	C₂²	1	2	yes	—
37	C₃₁×#460	C₂²	1	2	yes	—
38	C₃₁×#452	—	2	2	no	—
39	C₃₁×#448	—	2	1	no	—
40	C₃₁×#440	—	2	2	no	—
41	C₄×SD₁₆×C₃₁	—	2	2	no	—
42	C₃₁×#432	C₂²	1	1	yes	characteristic
43	C₃₁×#431	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#333	C₂²	1	2	yes	—
46	C₃₁×#332	C₂²	1	2	yes	—
47	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
51	C₂×#41×C₃₁	—	2	1	no	—
52	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
53	C₂×Q₁₆×C₃₁	—	2	2	no	—
54	C₂×SD₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×SD₁₆×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	—	2	1	no	—
58	C₃₁×C₄⋊Q₈	—	2	4	no	—
59	C₃₁×C₄⋊Q₈	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	—	2	2	no	—
61	C₃₁×C₄⋊Q₈	—	2	4	no	—
62	C₃₁×C₄⋊Q₈	—	2	2	no	—
63	C₃₁×C₄⋊Q₈	—	2	2	no	—
64	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
65	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
66	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
67	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
68	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
69	C₃₁×#127	C₂³	1	1	yes	characteristic
70	C₃₁×#125	—	2	4	no	—
71	C₃₁×#125	—	2	4	no	—
72	C₃₁×#125	—	2	2	no	—
73	C₃₁×#125	—	2	4	no	—
74	C₃₁×#125	—	2	2	no	—
75	C₃₁×#125	—	2	1	no	—
76	C₃₁×#125	—	2	1	no	—
77	C₃₁×#124	—	2	1	no	—
78	C₃₁×C₂²⋊Q₈	—	2	4	no	—
79	C₃₁×C₂²⋊Q₈	—	2	1	no	—
80	C₃₁×C₂²⋊Q₈	—	2	1	no	—
81	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
82	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
83	C₃₁×#121	—	2	1	no	—
84	C₃₁×#121	—	2	1	no	—
85	C₄×Q₈×C₃₁	—	2	2	no	—
86	C₄×Q₈×C₃₁	—	2	2	no	—
87	C₄×Q₈×C₃₁	—	2	4	no	—
88	C₄×Q₈×C₃₁	—	2	2	no	—
89	C₄×Q₈×C₃₁	—	2	2	no	—
90	C₄×Q₈×C₃₁	—	2	2	no	—
91	C₄×Q₈×C₃₁	—	2	2	no	—
92	C₄×Q₈×C₃₁	—	2	1	no	—
93	C₄×Q₈×C₃₁	—	2	1	no	—
94	C₄×Q₈×C₃₁	—	2	1	no	—
95	C₄×Q₈×C₃₁	—	2	1	no	—
96	C₄×Q₈×C₃₁	D₄	1	2	yes	—
97	C₄×Q₈×C₃₁	D₄	1	2	yes	—
98	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
99	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
100	C₄×D₄×C₃₁	—	2	1	no	—
101	C₄×D₄×C₃₁	—	2	1	no	—
102	C₃₁×#117	—	2	4	no	—
103	C₃₁×#117	—	2	1	no	—
104	C₃₁×#117	—	2	1	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
106	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
107	C₂×C₄²×C₃₁	—	2	1	no	—
108	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
109	C₃₁×#107	—	2	2	no	—
110	C₃₁×#106	—	2	2	no	—
111	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
112	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
113	C₃₁×#103	—	2	2	no	—
114	C₃₁×#103	—	2	2	no	—
115	C₃₁×#103	—	2	2	no	—
116	C₃₁×#102	—	2	2	no	—
117	C₃₁×#102	—	2	2	no	—
118	C₃₁×#102	—	2	2	no	—
119	C₃₁×C₂²⋊C₈	—	2	1	no	—
120	C₃₁×C₂²⋊C₈	—	2	1	no	—
121	C₃₁×#97	—	2	1	no	—
122	C₃₁×#97	—	2	1	no	—
123	C₄×C₈×C₃₁	—	2	1	no	—
124	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
126	#41×C₃₁	—	8	2	no	—
127	C₂×Q₈×C₃₁	—	2	4	no	—
128	C₂×Q₈×C₃₁	—	2	4	no	—
129	C₂×Q₈×C₃₁	—	2	1	no	—
130	C₂×Q₈×C₃₁	—	2	1	no	—
131	C₂×Q₈×C₃₁	—	2	1	no	—
132	C₂×Q₈×C₃₁	—	2	1	no	—
133	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
134	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
135	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
136	C₂×D₄×C₃₁	—	2	1	no	—
137	C₂×D₄×C₃₁	—	2	1	no	—
138	C₂×D₄×C₃₁	—	2	1	no	—
139	C₂×D₄×C₃₁	—	4	1	no	—
140	C₂²×C₄×C₃₁	—	2	1	no	—
141	C₂²×C₄×C₃₁	—	2	1	no	—
142	C₂²×C₄×C₃₁	—	4	1	no	—
143	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	Q₁₆×C₃₁	—	8	2	no	—
147	SD₁₆×C₃₁	—	4	4	no	—
148	SD₁₆×C₃₁	—	4	4	no	—
149	SD₁₆×C₃₁	—	8	2	no	—
150	M₁₆×C₃₁	—	4	2	no	—
151	C₂×C₈×C₃₁	—	2	1	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₂×C₈×C₃₁	—	2	1	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	2	4	no	—
156	C₄⋊C₄×C₃₁	—	2	4	no	—
157	C₄⋊C₄×C₃₁	—	2	4	no	—
158	C₄⋊C₄×C₃₁	—	2	4	no	—
159	C₄⋊C₄×C₃₁	—	2	4	no	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₄⋊C₄×C₃₁	—	2	1	no	—
164	C₄⋊C₄×C₃₁	—	2	2	no	—
165	C₄⋊C₄×C₃₁	—	2	2	no	—
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	—	2	2	no	—
168	C₄⋊C₄×C₃₁	—	2	2	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	—	2	2	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
176	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
177	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
178	C₂²⋊C₄×C₃₁	—	2	2	no	—
179	C₂²⋊C₄×C₃₁	—	2	2	no	—
180	C₂²⋊C₄×C₃₁	—	4	1	no	—
181	C₂²⋊C₄×C₃₁	—	4	1	no	—
182	C₄²×C₃₁	—	2	4	no	—
183	C₄²×C₃₁	—	2	4	no	—
184	C₄²×C₃₁	—	2	2	no	—
185	C₄²×C₃₁	—	2	2	no	—
186	C₄²×C₃₁	—	2	2	no	—
187	C₄²×C₃₁	—	2	2	no	—
188	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
189	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
191	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
192	#18612	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂³×C₃₁	—	4	1	no	—
194	C₂³×C₃₁	#120	1	1	yes	characteristic
195	Q₈×C₃₁	—	2	4	no	—
196	Q₈×C₃₁	—	2	2	no	—
197	Q₈×C₃₁	—	2	2	no	—
198	Q₈×C₃₁	—	2	2	no	—
199	Q₈×C₃₁	—	4	2	no	—
200	Q₈×C₃₁	—	4	2	no	—
201	Q₈×C₃₁	—	4	2	no	—
202	Q₈×C₃₁	—	8	1	no	—
203	Q₈×C₃₁	—	8	4	no	—
204	Q₈×C₃₁	#137	1	4	yes	—
205	D₄×C₃₁	—	4	2	no	—
206	D₄×C₃₁	—	4	2	no	—
207	D₄×C₃₁	—	4	2	no	—
208	D₄×C₃₁	—	8	2	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	4	1	no	—
219	C₂×C₄×C₃₁	—	4	1	no	—
220	C₂×C₄×C₃₁	—	8	2	no	—
221	C₂×C₄×C₃₁	—	8	1	no	—
222	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
223	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
224	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₈×C₃₁	—	2	2	no	—
229	C₈×C₃₁	—	4	1	no	—
230	C₈×C₃₁	—	4	2	no	—
231	C₈×C₃₁	—	4	2	no	—
232	#3502	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#3216	C₂×C₃₁	1	1	yes	characteristic
234	#3182	C₂×C₃₁	1	1	yes	characteristic
235	#3103	C₂×C₃₁	1	1	yes	characteristic
236	#1648	C₂×C₃₁	1	2	yes	complemented
237	#1645	C₂×C₃₁	1	2	yes	complemented
238	#1626	C₂×C₃₁	1	2	yes	complemented
239	#1623	C₂×C₃₁	1	2	yes	complemented
240	#1554	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#1491	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#1485	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂²×C₃₁	—	2	1	no	—
244	C₂²×C₃₁	—	2	1	no	—
245	C₂²×C₃₁	—	2	1	no	—
246	C₂²×C₃₁	—	8	1	no	—
247	C₂²×C₃₁	—	8	1	no	—
248	C₂²×C₃₁	—	8	1	no	—
249	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
250	C₄×C₃₁	—	2	4	no	—
251	C₄×C₃₁	—	2	1	no	—
252	C₄×C₃₁	—	2	1	no	—
253	C₄×C₃₁	—	2	1	no	—
254	C₄×C₃₁	—	2	1	no	—
255	C₄×C₃₁	—	4	2	no	—
256	C₄×C₃₁	—	4	2	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	—	8	1	no	—
261	C₄×C₃₁	C₂×#137	1	2	yes	—
262	C₄×C₃₁	#577	1	2	yes	—
263	Q₈²	—	2	1	no	—
264	Q₈²	C₂²×C₃₁	1	2	yes	—
265	#557	—	2	1	no	—
266	#557	—	2	2	no	—
267	#557	—	2	2	no	—
268	#557	C₂²×C₃₁	1	2	yes	—
269	#544	—	2	2	no	—
270	#544	—	2	1	no	—
271	#533	—	2	2	no	—
272	#533	C₂²×C₃₁	1	1	yes	characteristic
273	#533	C₂²×C₃₁	1	1	yes	characteristic
274	#532	C₂²×C₃₁	1	1	yes	characteristic
275	#519	—	2	1	no	—
276	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
277	#500	C₂²×C₃₁	1	1	yes	characteristic
278	#497	—	2	1	no	—
279	#492	C₂²×C₃₁	1	1	yes	characteristic
280	#489	—	2	1	no	—
281	#487	C₂²×C₃₁	1	1	yes	characteristic
282	#486	C₂²×C₃₁	1	1	yes	characteristic
283	#481	—	2	1	no	—
284	#475	C₂²×C₃₁	1	2	yes	—
285	#468	—	2	2	no	—
286	#462	C₂²×C₃₁	1	2	yes	—
287	#460	C₂²×C₃₁	1	2	yes	—
288	#460	C₂²×C₃₁	1	2	yes	—
289	#452	—	2	2	no	—
290	#448	—	2	1	no	—
291	#440	—	2	2	no	—
292	C₄×SD₁₆	—	2	2	no	—
293	#432	C₂²×C₃₁	1	1	yes	characteristic
294	#431	C₂²×C₃₁	1	1	yes	characteristic
295	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
296	#333	C₂²×C₃₁	1	2	yes	—
297	#332	C₂²×C₃₁	1	2	yes	—
298	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
299	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
300	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
301	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
302	C₂×C₃₁	—	4	1	no	—
303	C₂×C₃₁	—	16	1	no	—
304	C₂×C₃₁	#3004	1	1	yes	characteristic
305	C₂×C₃₁	#3010	1	1	yes	characteristic
306	C₂×C₃₁	#3018	1	1	yes	characteristic
307	C₂×#41	—	2	1	no	—
308	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
309	C₂×Q₁₆	—	2	2	no	—
310	C₂×SD₁₆	—	2	2	no	—
311	C₂×SD₁₆	—	2	2	no	—
312	C₂×SD₁₆	—	2	2	no	—
313	C₂×M₁₆	—	2	1	no	—
314	C₄⋊Q₈	—	2	4	no	—
315	C₄⋊Q₈	—	2	2	no	—
316	C₄⋊Q₈	—	2	2	no	—
317	C₄⋊Q₈	—	2	4	no	—
318	C₄⋊Q₈	—	2	2	no	—
319	C₄⋊Q₈	—	2	2	no	—
320	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
321	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
322	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
323	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
324	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
325	#127	C₂³×C₃₁	1	1	yes	characteristic
326	#125	—	2	4	no	—
327	#125	—	2	4	no	—
328	#125	—	2	2	no	—
329	#125	—	2	4	no	—
330	#125	—	2	2	no	—
331	#125	—	2	1	no	—
332	#125	—	2	1	no	—
333	#124	—	2	1	no	—
334	C₂²⋊Q₈	—	2	4	no	—
335	C₂²⋊Q₈	—	2	1	no	—
336	C₂²⋊Q₈	—	2	1	no	—
337	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
338	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
339	#121	—	2	1	no	—
340	#121	—	2	1	no	—
341	C₄×Q₈	—	2	2	no	—
342	C₄×Q₈	—	2	2	no	—
343	C₄×Q₈	—	2	4	no	—
344	C₄×Q₈	—	2	2	no	—
345	C₄×Q₈	—	2	2	no	—
346	C₄×Q₈	—	2	2	no	—
347	C₄×Q₈	—	2	2	no	—
348	C₄×Q₈	—	2	1	no	—
349	C₄×Q₈	—	2	1	no	—
350	C₄×Q₈	—	2	1	no	—
351	C₄×Q₈	—	2	1	no	—
352	C₄×Q₈	D₄×C₃₁	1	2	yes	—
353	C₄×Q₈	D₄×C₃₁	1	2	yes	—
354	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
355	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
356	C₄×D₄	—	2	1	no	—
357	C₄×D₄	—	2	1	no	—
358	#117	—	2	4	no	—
359	#117	—	2	1	no	—
360	#117	—	2	1	no	—
361	C₂×C₄⋊C₄	—	2	1	no	—
362	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
363	C₂×C₄²	—	2	1	no	—
364	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
365	#107	—	2	2	no	—
366	#106	—	2	2	no	—
367	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
368	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
369	#103	—	2	2	no	—
370	#103	—	2	2	no	—
371	#103	—	2	2	no	—
372	#102	—	2	2	no	—
373	#102	—	2	2	no	—
374	#102	—	2	2	no	—
375	C₂²⋊C₈	—	2	1	no	—
376	C₂²⋊C₈	—	2	1	no	—
377	#97	—	2	1	no	—
378	#97	—	2	1	no	—
379	C₄×C₈	—	2	1	no	—
380	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
381	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
382	C₃₁	#18612	1	1	yes	characteristic, direct product factor, Sylow
383	#41	—	8	2	no	—
384	C₂×Q₈	—	2	4	no	—
385	C₂×Q₈	—	2	4	no	—
386	C₂×Q₈	—	2	1	no	—
387	C₂×Q₈	—	2	1	no	—
388	C₂×Q₈	—	2	1	no	—
389	C₂×Q₈	—	2	1	no	—
390	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
391	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
392	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	—	2	1	no	—
396	C₂×D₄	—	4	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	4	1	no	—
400	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
401	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
403	Q₁₆	—	8	2	no	—
404	SD₁₆	—	4	4	no	—
405	SD₁₆	—	4	4	no	—
406	SD₁₆	—	8	2	no	—
407	M₁₆	—	4	2	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	2	no	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	2	4	no	—
416	C₄⋊C₄	—	2	4	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	2	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
433	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
434	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
435	C₂²⋊C₄	—	2	2	no	—
436	C₂²⋊C₄	—	2	2	no	—
437	C₂²⋊C₄	—	4	1	no	—
438	C₂²⋊C₄	—	4	1	no	—
439	C₄²	—	2	4	no	—
440	C₄²	—	2	4	no	—
441	C₄²	—	2	2	no	—
442	C₄²	—	2	2	no	—
443	C₄²	—	2	2	no	—
444	C₄²	—	2	2	no	—
445	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
446	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
447	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
448	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
449	C₂³	—	4	1	no	—
450	C₂³	C₃₁×#120	1	1	yes	characteristic
451	Q₈	—	2	4	no	—
452	Q₈	—	2	2	no	—
453	Q₈	—	2	2	no	—
454	Q₈	—	2	2	no	—
455	Q₈	—	4	2	no	—
456	Q₈	—	4	2	no	—
457	Q₈	—	4	2	no	—
458	Q₈	—	8	1	no	—
459	Q₈	—	8	4	no	—
460	Q₈	C₃₁×#137	1	4	yes	—
461	D₄	—	4	2	no	—
462	D₄	—	4	2	no	—
463	D₄	—	4	2	no	—
464	D₄	—	8	2	no	—
465	C₂×C₄	—	2	4	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	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	8	2	no	—
477	C₂×C₄	—	8	1	no	—
478	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
479	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
480	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
481	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
482	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
483	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
484	C₈	—	2	2	no	—
485	C₈	—	4	1	no	—
486	C₈	—	4	2	no	—
487	C₈	—	4	2	no	—
488	C₂²	—	2	1	no	—
489	C₂²	—	2	1	no	—
490	C₂²	—	2	1	no	—
491	C₂²	—	8	1	no	—
492	C₂²	—	8	1	no	—
493	C₂²	—	8	1	no	—
494	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
495	C₄	—	2	4	no	—
496	C₄	—	2	1	no	—
497	C₄	—	2	1	no	—
498	C₄	—	2	1	no	—
499	C₄	—	2	1	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	2	no	—
505	C₄	—	8	1	no	—
506	C₄	C₂×C₃₁×#137	1	2	yes	—
507	C₄	C₃₁×#577	1	2	yes	—
508	C₂	—	4	1	no	—
509	C₂	—	16	1	no	—
510	C₂	C₃₁×#3004	1	1	yes	characteristic
511	C₂	C₃₁×#3010	1	1	yes	characteristic
512	C₂	C₃₁×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#532	C₂²
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×#3502	C₂	semidirect
5. C₃₁×#1648	C₂	semidirect
6. C₃₁×#1645	C₂	semidirect
7. C₃₁×#1626	C₂	semidirect
8. C₃₁×#1623	C₂	semidirect
9. C₃₁×#1554	C₂	semidirect
10. C₃₁×#1491	C₂	semidirect
11. C₃₁×#1485	C₂	semidirect
192. #18612	C₃₁	direct (N × H)
232. #3502	C₂×C₃₁	semidirect
236. #1648	C₂×C₃₁	semidirect
237. #1645	C₂×C₃₁	semidirect
238. #1626	C₂×C₃₁	semidirect
239. #1623	C₂×C₃₁	semidirect
240. #1554	C₂×C₃₁	semidirect
241. #1491	C₂×C₃₁	semidirect
242. #1485	C₂×C₃₁	semidirect
382. C₃₁	#18612	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×#3502	2
C₃₁×#3216	2
C₃₁×#3182	2
C₃₁×#3103	2
C₃₁×#1648	2
C₃₁×#1645	2
C₃₁×#1626	2
C₃₁×#1623	2
C₃₁×#1554	2
C₃₁×#1491	2
C₃₁×#1485	2
#18612	31

Subgroups — C31×#18612 #21475371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18612 #21475371