Subgroups — C₃₁×#19725 #21476484

Order 7936.

764 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3230	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3228	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3227	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3207	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#3182	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3152	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1723	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1722	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1720	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1717	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1714	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1714	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1707	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1453	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#137	—	2	2	no	—
17	C₃₁×#533	C₂²	1	1	yes	characteristic
18	C₃₁×#532	C₂²	1	1	yes	characteristic
19	C₃₁×#532	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₃₁×#500	C₂²	1	1	yes	characteristic
22	C₃₁×#500	C₂²	1	1	yes	characteristic
23	C₃₁×#500	C₂²	1	1	yes	characteristic
24	C₃₁×#500	C₂²	1	1	yes	characteristic
25	C₃₁×#498	C₂²	1	1	yes	characteristic
26	C₃₁×#498	C₂²	1	1	yes	characteristic
27	C₃₁×#498	C₂²	1	1	yes	characteristic
28	C₃₁×#497	—	2	2	no	—
29	C₃₁×#494	C₂²	1	1	yes	characteristic
30	C₃₁×#494	C₂²	1	1	yes	characteristic
31	C₃₁×#493	C₂²	1	1	yes	characteristic
32	C₃₁×#492	C₂²	1	1	yes	characteristic
33	C₃₁×#492	C₂²	1	1	yes	characteristic
34	C₃₁×#492	C₂²	1	1	yes	characteristic
35	C₃₁×#492	C₂²	1	1	yes	characteristic
36	C₃₁×#487	C₂²	1	1	yes	characteristic
37	C₃₁×#487	C₂²	1	1	yes	characteristic
38	C₃₁×#487	C₂²	1	1	yes	characteristic
39	C₃₁×#487	C₂²	1	1	yes	characteristic
40	C₃₁×#486	C₂²	1	1	yes	characteristic
41	C₃₁×#486	C₂²	1	1	yes	characteristic
42	C₃₁×#486	C₂²	1	1	yes	characteristic
43	C₃₁×#486	C₂²	1	1	yes	characteristic
44	C₃₁×#486	C₂²	1	1	yes	characteristic
45	C₃₁×#486	C₂²	1	1	yes	characteristic
46	C₃₁×#484	—	2	2	no	—
47	C₃₁×#482	—	2	2	no	—
48	C₃₁×#481	—	2	2	no	—
49	C₃₁×#478	—	2	2	no	—
50	C₃₁×#477	—	2	2	no	—
51	C₃₁×#476	—	2	2	no	—
52	C₃₁×#468	—	2	2	no	—
53	C₃₁×#463	—	2	2	no	—
54	C₃₁×#449	—	2	2	no	—
55	C₃₁×#440	—	2	2	no	—
56	C₃₁×#431	C₂²	1	1	yes	characteristic
57	C₃₁×#431	C₂²	1	1	yes	characteristic
58	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
59	C₃₁×#322	C₂²	1	1	yes	characteristic
60	C₃₁×#322	C₂²	1	1	yes	characteristic
61	C₃₁×#322	C₂²	1	1	yes	characteristic
62	C₈²×C₃₁	C₂²	1	1	yes	characteristic
63	C₂×#41×C₃₁	—	2	1	no	—
64	C₂×#41×C₃₁	—	2	1	no	—
65	C₂²×Q₈×C₃₁	—	2	1	no	—
66	C₃₁×#137	—	4	8	no	—
67	C₂×Q₁₆×C₃₁	—	2	2	no	—
68	C₂×Q₁₆×C₃₁	—	2	2	no	—
69	C₂×SD₁₆×C₃₁	—	2	2	no	—
70	C₂×SD₁₆×C₃₁	—	2	2	no	—
71	C₂×SD₁₆×C₃₁	—	2	2	no	—
72	C₂×SD₁₆×C₃₁	—	2	2	no	—
73	C₂×D₈×C₃₁	—	2	2	no	—
74	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
75	C₃₁×C₄⋊Q₈	—	2	1	no	—
76	C₃₁×C₄⋊Q₈	—	2	1	no	—
77	C₃₁×C₄⋊Q₈	—	2	1	no	—
78	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₃₁×#127	C₂³	1	1	yes	characteristic
85	C₃₁×#127	C₂³	1	1	yes	characteristic
86	C₃₁×#125	—	2	1	no	—
87	C₃₁×#124	—	2	1	no	—
88	C₃₁×#124	—	2	1	no	—
89	C₃₁×C₂²⋊Q₈	—	2	1	no	—
90	C₃₁×C₂²⋊Q₈	—	2	1	no	—
91	C₃₁×#121	—	2	1	no	—
92	C₃₁×#121	—	2	1	no	—
93	C₃₁×#121	—	2	1	no	—
94	C₃₁×#121	—	2	1	no	—
95	C₄×Q₈×C₃₁	—	2	1	no	—
96	C₄×Q₈×C₃₁	—	2	1	no	—
97	C₄×D₄×C₃₁	—	2	1	no	—
98	C₄×D₄×C₃₁	—	2	1	no	—
99	C₄×D₄×C₃₁	—	2	1	no	—
100	C₄×D₄×C₃₁	—	2	1	no	—
101	C₃₁×#117	—	2	1	no	—
102	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
104	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
105	C₃₁×#107	—	2	2	no	—
106	C₃₁×#107	—	2	2	no	—
107	C₃₁×#107	—	2	2	no	—
108	C₃₁×#107	—	2	2	no	—
109	C₃₁×#106	—	2	2	no	—
110	C₃₁×#106	—	2	2	no	—
111	C₃₁×#106	—	2	2	no	—
112	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
113	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
114	C₃₁×#103	—	2	2	no	—
115	C₃₁×#103	—	2	2	no	—
116	C₃₁×#103	—	2	2	no	—
117	C₃₁×#103	—	2	2	no	—
118	C₃₁×#102	—	2	2	no	—
119	C₃₁×#102	—	2	2	no	—
120	C₃₁×#102	—	2	2	no	—
121	C₃₁×#102	—	2	2	no	—
122	C₃₁×#102	—	2	2	no	—
123	C₃₁×#102	—	2	2	no	—
124	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
125	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
126	C₃₁×#97	D₄	1	2	yes	—
127	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
128	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
129	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
130	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
131	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
132	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
133	#41×C₃₁	—	4	4	no	—
134	#41×C₃₁	—	8	2	no	—
135	C₂×Q₈×C₃₁	—	2	1	no	—
136	C₂×Q₈×C₃₁	—	2	1	no	—
137	C₂×Q₈×C₃₁	—	2	1	no	—
138	C₂×Q₈×C₃₁	—	2	1	no	—
139	C₂×Q₈×C₃₁	—	2	1	no	—
140	C₂×Q₈×C₃₁	—	2	1	no	—
141	C₂×Q₈×C₃₁	—	4	4	no	—
142	C₂×Q₈×C₃₁	—	4	1	no	—
143	C₂×D₄×C₃₁	—	2	1	no	—
144	C₂×D₄×C₃₁	—	2	1	no	—
145	C₂×D₄×C₃₁	—	2	1	no	—
146	C₂×D₄×C₃₁	—	2	1	no	—
147	C₂×D₄×C₃₁	—	2	1	no	—
148	C₂×D₄×C₃₁	—	2	1	no	—
149	C₂×D₄×C₃₁	—	4	1	no	—
150	C₂×D₄×C₃₁	—	4	1	no	—
151	C₂²×C₄×C₃₁	—	4	1	no	—
152	C₂²×C₄×C₃₁	—	4	1	no	—
153	C₂²×C₄×C₃₁	—	4	1	no	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	Q₁₆×C₃₁	—	4	4	no	—
158	Q₁₆×C₃₁	—	4	4	no	—
159	SD₁₆×C₃₁	—	4	4	no	—
160	SD₁₆×C₃₁	—	4	4	no	—
161	SD₁₆×C₃₁	—	4	4	no	—
162	SD₁₆×C₃₁	—	8	2	no	—
163	D₈×C₃₁	—	8	2	no	—
164	M₁₆×C₃₁	—	2	4	no	—
165	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
166	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
167	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
168	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
169	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
170	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	2	1	no	—
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	2	1	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	—	2	1	no	—
181	C₄⋊C₄×C₃₁	—	2	1	no	—
182	C₄⋊C₄×C₃₁	—	2	1	no	—
183	C₄⋊C₄×C₃₁	—	4	1	no	—
184	C₄⋊C₄×C₃₁	—	4	1	no	—
185	C₄⋊C₄×C₃₁	—	4	1	no	—
186	C₄⋊C₄×C₃₁	—	4	1	no	—
187	C₂²⋊C₄×C₃₁	—	4	1	no	—
188	C₂²⋊C₄×C₃₁	—	4	1	no	—
189	C₂²⋊C₄×C₃₁	—	4	1	no	—
190	C₂²⋊C₄×C₃₁	—	4	1	no	—
191	C₂²⋊C₄×C₃₁	—	4	1	no	—
192	C₄²×C₃₁	—	4	1	no	—
193	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
196	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
197	#19725	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂³×C₃₁	—	4	1	no	—
199	C₂³×C₃₁	—	4	1	no	—
200	C₂³×C₃₁	#127	1	1	yes	characteristic
201	Q₈×C₃₁	—	4	2	no	—
202	Q₈×C₃₁	—	4	2	no	—
203	Q₈×C₃₁	—	4	2	no	—
204	Q₈×C₃₁	—	4	2	no	—
205	Q₈×C₃₁	—	4	2	no	—
206	Q₈×C₃₁	—	4	2	no	—
207	Q₈×C₃₁	—	8	2	no	—
208	D₄×C₃₁	—	4	2	no	—
209	D₄×C₃₁	—	4	2	no	—
210	D₄×C₃₁	—	4	2	no	—
211	D₄×C₃₁	—	4	2	no	—
212	D₄×C₃₁	—	4	2	no	—
213	D₄×C₃₁	—	4	2	no	—
214	D₄×C₃₁	—	8	2	no	—
215	D₄×C₃₁	—	8	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	4	1	no	—
220	C₂×C₄×C₃₁	—	4	1	no	—
221	C₂×C₄×C₃₁	—	4	1	no	—
222	C₂×C₄×C₃₁	—	4	1	no	—
223	C₂×C₄×C₃₁	—	4	1	no	—
224	C₂×C₄×C₃₁	—	4	1	no	—
225	C₂×C₄×C₃₁	—	8	2	no	—
226	C₂×C₄×C₃₁	—	8	2	no	—
227	C₂×C₄×C₃₁	—	8	2	no	—
228	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
229	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
230	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
234	C₈×C₃₁	—	2	2	no	—
235	C₈×C₃₁	—	2	2	no	—
236	C₈×C₃₁	—	2	2	no	—
237	C₈×C₃₁	—	2	2	no	—
238	C₈×C₃₁	—	2	2	no	—
239	C₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
240	#3230	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#3228	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#3227	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#3207	C₂×C₃₁	1	1	yes	characteristic
244	#3182	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#3152	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#1723	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#1722	C₂×C₃₁	1	1	yes	characteristic, complemented
248	#1720	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#1717	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#1714	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#1714	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1707	C₂×C₃₁	1	1	yes	characteristic, complemented
253	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#1453	C₂×C₃₁	1	1	yes	characteristic, complemented
255	C₂²×C₃₁	—	2	1	no	—
256	C₂²×C₃₁	—	2	1	no	—
257	C₂²×C₃₁	—	2	1	no	—
258	C₂²×C₃₁	—	8	1	no	—
259	C₂²×C₃₁	—	8	1	no	—
260	C₂²×C₃₁	—	8	1	no	—
261	C₂²×C₃₁	—	8	1	no	—
262	C₂²×C₃₁	—	8	1	no	—
263	C₂²×C₃₁	—	8	1	no	—
264	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
265	C₄×C₃₁	—	2	1	no	—
266	C₄×C₃₁	—	2	1	no	—
267	C₄×C₃₁	—	2	1	no	—
268	C₄×C₃₁	—	8	1	no	—
269	C₄×C₃₁	—	8	1	no	—
270	C₄×C₃₁	—	8	1	no	—
271	C₄×C₃₁	—	8	1	no	—
272	C₄×C₃₁	—	8	1	no	—
273	C₄×C₃₁	—	8	1	no	—
274	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
275	C₄×C₃₁	#577	1	2	yes	—
276	C₄×C₃₁	#578	1	2	yes	—
277	C₂×#137	—	2	2	no	—
278	#533	C₂²×C₃₁	1	1	yes	characteristic
279	#532	C₂²×C₃₁	1	1	yes	characteristic
280	#532	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
282	#500	C₂²×C₃₁	1	1	yes	characteristic
283	#500	C₂²×C₃₁	1	1	yes	characteristic
284	#500	C₂²×C₃₁	1	1	yes	characteristic
285	#500	C₂²×C₃₁	1	1	yes	characteristic
286	#498	C₂²×C₃₁	1	1	yes	characteristic
287	#498	C₂²×C₃₁	1	1	yes	characteristic
288	#498	C₂²×C₃₁	1	1	yes	characteristic
289	#497	—	2	2	no	—
290	#494	C₂²×C₃₁	1	1	yes	characteristic
291	#494	C₂²×C₃₁	1	1	yes	characteristic
292	#493	C₂²×C₃₁	1	1	yes	characteristic
293	#492	C₂²×C₃₁	1	1	yes	characteristic
294	#492	C₂²×C₃₁	1	1	yes	characteristic
295	#492	C₂²×C₃₁	1	1	yes	characteristic
296	#492	C₂²×C₃₁	1	1	yes	characteristic
297	#487	C₂²×C₃₁	1	1	yes	characteristic
298	#487	C₂²×C₃₁	1	1	yes	characteristic
299	#487	C₂²×C₃₁	1	1	yes	characteristic
300	#487	C₂²×C₃₁	1	1	yes	characteristic
301	#486	C₂²×C₃₁	1	1	yes	characteristic
302	#486	C₂²×C₃₁	1	1	yes	characteristic
303	#486	C₂²×C₃₁	1	1	yes	characteristic
304	#486	C₂²×C₃₁	1	1	yes	characteristic
305	#486	C₂²×C₃₁	1	1	yes	characteristic
306	#486	C₂²×C₃₁	1	1	yes	characteristic
307	#484	—	2	2	no	—
308	#482	—	2	2	no	—
309	#481	—	2	2	no	—
310	#478	—	2	2	no	—
311	#477	—	2	2	no	—
312	#476	—	2	2	no	—
313	#468	—	2	2	no	—
314	#463	—	2	2	no	—
315	#449	—	2	2	no	—
316	#440	—	2	2	no	—
317	#431	C₂²×C₃₁	1	1	yes	characteristic
318	#431	C₂²×C₃₁	1	1	yes	characteristic
319	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
320	#322	C₂²×C₃₁	1	1	yes	characteristic
321	#322	C₂²×C₃₁	1	1	yes	characteristic
322	#322	C₂²×C₃₁	1	1	yes	characteristic
323	C₈²	C₂²×C₃₁	1	1	yes	characteristic
324	C₂×C₃₁	—	4	1	no	—
325	C₂×C₃₁	—	16	1	no	—
326	C₂×C₃₁	—	16	1	no	—
327	C₂×C₃₁	#3151	1	1	yes	characteristic
328	C₂×C₃₁	#3154	1	1	yes	characteristic
329	C₂×C₃₁	#3156	1	1	yes	characteristic
330	C₂×#41	—	2	1	no	—
331	C₂×#41	—	2	1	no	—
332	C₂²×Q₈	—	2	1	no	—
333	#137	—	4	8	no	—
334	C₂×Q₁₆	—	2	2	no	—
335	C₂×Q₁₆	—	2	2	no	—
336	C₂×SD₁₆	—	2	2	no	—
337	C₂×SD₁₆	—	2	2	no	—
338	C₂×SD₁₆	—	2	2	no	—
339	C₂×SD₁₆	—	2	2	no	—
340	C₂×D₈	—	2	2	no	—
341	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
342	C₄⋊Q₈	—	2	1	no	—
343	C₄⋊Q₈	—	2	1	no	—
344	C₄⋊Q₈	—	2	1	no	—
345	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
346	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
347	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
348	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
349	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
350	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
351	#127	C₂³×C₃₁	1	1	yes	characteristic
352	#127	C₂³×C₃₁	1	1	yes	characteristic
353	#125	—	2	1	no	—
354	#124	—	2	1	no	—
355	#124	—	2	1	no	—
356	C₂²⋊Q₈	—	2	1	no	—
357	C₂²⋊Q₈	—	2	1	no	—
358	#121	—	2	1	no	—
359	#121	—	2	1	no	—
360	#121	—	2	1	no	—
361	#121	—	2	1	no	—
362	C₄×Q₈	—	2	1	no	—
363	C₄×Q₈	—	2	1	no	—
364	C₄×D₄	—	2	1	no	—
365	C₄×D₄	—	2	1	no	—
366	C₄×D₄	—	2	1	no	—
367	C₄×D₄	—	2	1	no	—
368	#117	—	2	1	no	—
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
372	#107	—	2	2	no	—
373	#107	—	2	2	no	—
374	#107	—	2	2	no	—
375	#107	—	2	2	no	—
376	#106	—	2	2	no	—
377	#106	—	2	2	no	—
378	#106	—	2	2	no	—
379	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
380	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
381	#103	—	2	2	no	—
382	#103	—	2	2	no	—
383	#103	—	2	2	no	—
384	#103	—	2	2	no	—
385	#102	—	2	2	no	—
386	#102	—	2	2	no	—
387	#102	—	2	2	no	—
388	#102	—	2	2	no	—
389	#102	—	2	2	no	—
390	#102	—	2	2	no	—
391	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
392	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
393	#97	D₄×C₃₁	1	2	yes	—
394	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
395	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
396	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
397	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
398	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
399	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
400	C₃₁	#19725	1	1	yes	characteristic, direct product factor, Sylow
401	#41	—	4	4	no	—
402	#41	—	8	2	no	—
403	C₂×Q₈	—	2	1	no	—
404	C₂×Q₈	—	2	1	no	—
405	C₂×Q₈	—	2	1	no	—
406	C₂×Q₈	—	2	1	no	—
407	C₂×Q₈	—	2	1	no	—
408	C₂×Q₈	—	2	1	no	—
409	C₂×Q₈	—	4	4	no	—
410	C₂×Q₈	—	4	1	no	—
411	C₂×D₄	—	2	1	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	1	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	—	4	1	no	—
418	C₂×D₄	—	4	1	no	—
419	C₂²×C₄	—	4	1	no	—
420	C₂²×C₄	—	4	1	no	—
421	C₂²×C₄	—	4	1	no	—
422	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
423	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
425	Q₁₆	—	4	4	no	—
426	Q₁₆	—	4	4	no	—
427	SD₁₆	—	4	4	no	—
428	SD₁₆	—	4	4	no	—
429	SD₁₆	—	4	4	no	—
430	SD₁₆	—	8	2	no	—
431	D₈	—	8	2	no	—
432	M₁₆	—	2	4	no	—
433	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
434	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
435	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
436	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
437	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
438	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
439	C₄⋊C₄	—	2	1	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	4	1	no	—
452	C₄⋊C₄	—	4	1	no	—
453	C₄⋊C₄	—	4	1	no	—
454	C₄⋊C₄	—	4	1	no	—
455	C₂²⋊C₄	—	4	1	no	—
456	C₂²⋊C₄	—	4	1	no	—
457	C₂²⋊C₄	—	4	1	no	—
458	C₂²⋊C₄	—	4	1	no	—
459	C₂²⋊C₄	—	4	1	no	—
460	C₄²	—	4	1	no	—
461	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
462	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
464	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
465	C₂³	—	4	1	no	—
466	C₂³	—	4	1	no	—
467	C₂³	C₃₁×#127	1	1	yes	characteristic
468	Q₈	—	4	2	no	—
469	Q₈	—	4	2	no	—
470	Q₈	—	4	2	no	—
471	Q₈	—	4	2	no	—
472	Q₈	—	4	2	no	—
473	Q₈	—	4	2	no	—
474	Q₈	—	8	2	no	—
475	D₄	—	4	2	no	—
476	D₄	—	4	2	no	—
477	D₄	—	4	2	no	—
478	D₄	—	4	2	no	—
479	D₄	—	4	2	no	—
480	D₄	—	4	2	no	—
481	D₄	—	8	2	no	—
482	D₄	—	8	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	8	2	no	—
493	C₂×C₄	—	8	2	no	—
494	C₂×C₄	—	8	2	no	—
495	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
496	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
497	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
498	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
499	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
500	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
501	C₈	—	2	2	no	—
502	C₈	—	2	2	no	—
503	C₈	—	2	2	no	—
504	C₈	—	2	2	no	—
505	C₈	—	2	2	no	—
506	C₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
507	C₂²	—	2	1	no	—
508	C₂²	—	2	1	no	—
509	C₂²	—	2	1	no	—
510	C₂²	—	8	1	no	—
511	C₂²	—	8	1	no	—
512	C₂²	—	8	1	no	—
513	C₂²	—	8	1	no	—
514	C₂²	—	8	1	no	—
515	C₂²	—	8	1	no	—
516	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
517	C₄	—	2	1	no	—
518	C₄	—	2	1	no	—
519	C₄	—	2	1	no	—
520	C₄	—	8	1	no	—
521	C₄	—	8	1	no	—
522	C₄	—	8	1	no	—
523	C₄	—	8	1	no	—
524	C₄	—	8	1	no	—
525	C₄	—	8	1	no	—
526	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
527	C₄	C₃₁×#577	1	2	yes	—
528	C₄	C₃₁×#578	1	2	yes	—
529	C₂	—	4	1	no	—
530	C₂	—	16	1	no	—
531	C₂	—	16	1	no	—
532	C₂	C₃₁×#3151	1	1	yes	characteristic
533	C₂	C₃₁×#3154	1	1	yes	characteristic
534	C₂	C₃₁×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3207	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×#3230	C₂	semidirect
2. C₃₁×#3228	C₂	semidirect
3. C₃₁×#3227	C₂	semidirect
5. C₃₁×#3182	C₂	semidirect
6. C₃₁×#3152	C₂	semidirect
7. C₃₁×#1723	C₂	semidirect
8. C₃₁×#1722	C₂	semidirect
9. C₃₁×#1720	C₂	semidirect
10. C₃₁×#1717	C₂	semidirect
11. C₃₁×#1714	C₂	semidirect
12. C₃₁×#1714	C₂	semidirect
13. C₃₁×#1707	C₂	semidirect
14. C₃₁×#1706	C₂	semidirect
15. C₃₁×#1453	C₂	semidirect
197. #19725	C₃₁	direct (N × H)
240. #3230	C₂×C₃₁	semidirect
241. #3228	C₂×C₃₁	semidirect
242. #3227	C₂×C₃₁	semidirect
244. #3182	C₂×C₃₁	semidirect
245. #3152	C₂×C₃₁	semidirect
246. #1723	C₂×C₃₁	semidirect
247. #1722	C₂×C₃₁	semidirect
248. #1720	C₂×C₃₁	semidirect
249. #1717	C₂×C₃₁	semidirect
250. #1714	C₂×C₃₁	semidirect
251. #1714	C₂×C₃₁	semidirect
252. #1707	C₂×C₃₁	semidirect
253. #1706	C₂×C₃₁	semidirect
254. #1453	C₂×C₃₁	semidirect
400. C₃₁	#19725	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×#3230	2
C₃₁×#3228	2
C₃₁×#3227	2
C₃₁×#3207	2
C₃₁×#3182	2
C₃₁×#3152	2
C₃₁×#1723	2
C₃₁×#1722	2
C₃₁×#1720	2
C₃₁×#1717	2
C₃₁×#1714	2
C₃₁×#1707	2
C₃₁×#1706	2
C₃₁×#1453	2
#19725	31

Subgroups — C31×#19725 #21476484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19725 #21476484