Subgroups — C₃₁×#19625 #21476384

Order 7936.

836 conjugacy classes of subgroups.

The subgroup lattice has 454 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×#3205	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3167	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3141	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3138	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1719	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1717	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1708	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1706	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	—	2	2	no	—
13	C₃₁×#533	C₂²	1	1	yes	characteristic
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
17	C₃₁×#500	C₂²	1	2	yes	complemented
18	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#493	C₂²	1	2	yes	—
20	C₃₁×#492	—	2	2	no	—
21	C₃₁×#492	C₂²	1	2	yes	—
22	C₃₁×#492	C₂²	1	2	yes	—
23	C₃₁×#492	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×#490	—	2	2	no	—
25	C₃₁×#489	—	2	2	no	—
26	C₃₁×#487	C₂²	1	2	yes	complemented
27	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
28	C₃₁×#487	C₂²	1	2	yes	complemented
29	C₃₁×#486	C₂²	1	2	yes	—
30	C₃₁×#486	C₂²	1	2	yes	—
31	C₃₁×#486	C₂²	1	2	yes	—
32	C₃₁×#486	C₂²	1	2	yes	complemented
33	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×#484	—	2	2	no	—
35	C₃₁×#477	—	2	2	no	—
36	C₃₁×#465	—	2	2	no	—
37	C₃₁×#459	—	2	2	no	—
38	C₃₁×#447	—	2	2	no	—
39	C₄×SD₁₆×C₃₁	—	2	2	no	—
40	C₃₁×#431	C₂²	1	1	yes	characteristic
41	C₃₁×#418	—	2	2	no	—
42	C₃₁×#417	—	2	2	no	—
43	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#322	C₂²	1	2	yes	complemented
46	C₈²×C₃₁	C₂²	1	2	yes	complemented
47	C₂×#41×C₃₁	—	2	1	no	—
48	C₂²×Q₈×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×SD₁₆×C₃₁	—	2	4	no	—
54	C₂×SD₁₆×C₃₁	—	2	4	no	—
55	C₂×SD₁₆×C₃₁	—	4	8	no	—
56	C₂×D₈×C₃₁	—	2	4	no	—
57	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
58	C₂²×C₈×C₃₁	D₄	1	2	yes	—
59	C₃₁×C₄⋊Q₈	—	2	1	no	—
60	C₃₁×C₄⋊Q₈	—	2	1	no	—
61	C₃₁×C₄⋊Q₈	—	2	1	no	—
62	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
66	C₃₁×#127	—	2	1	no	—
67	C₃₁×#127	—	2	1	no	—
68	C₃₁×#127	—	2	1	no	—
69	C₃₁×#127	C₂³	1	2	yes	—
70	C₃₁×#127	C₂³	1	1	yes	characteristic
71	C₃₁×#125	—	2	1	no	—
72	C₃₁×#124	—	2	1	no	—
73	C₃₁×C₂²⋊Q₈	—	2	1	no	—
74	C₃₁×C₂²⋊Q₈	—	2	1	no	—
75	C₃₁×#121	—	2	1	no	—
76	C₃₁×#121	—	2	1	no	—
77	C₄×Q₈×C₃₁	—	2	1	no	—
78	C₄×Q₈×C₃₁	—	2	1	no	—
79	C₄×D₄×C₃₁	—	2	1	no	—
80	C₄×D₄×C₃₁	—	2	1	no	—
81	C₃₁×#117	—	2	1	no	—
82	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₃₁×#107	—	2	4	no	—
86	C₃₁×#106	—	2	2	no	—
87	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
88	C₃₁×#103	—	2	4	no	—
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#103	—	2	4	no	—
91	C₃₁×#102	—	2	4	no	—
92	C₃₁×#102	—	2	4	no	—
93	C₃₁×#102	—	2	4	no	—
94	C₃₁×#102	—	2	2	no	—
95	C₃₁×#102	—	2	4	no	—
96	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
97	C₃₁×#97	D₄	1	2	yes	—
98	C₄×C₈×C₃₁	D₄	1	2	yes	—
99	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₃₁	C₂³	1	2	yes	—
102	C₄×C₈×C₃₁	C₂³	1	2	yes	—
103	C₂⁴×C₃₁	—	4	1	no	—
104	#41×C₃₁	—	4	4	no	—
105	C₂×Q₈×C₃₁	—	2	1	no	—
106	C₂×Q₈×C₃₁	—	2	1	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	—	4	4	no	—
110	C₂×Q₈×C₃₁	—	4	1	no	—
111	C₂×D₄×C₃₁	—	2	1	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	1	no	—
119	C₂×D₄×C₃₁	—	4	1	no	—
120	C₂×D₄×C₃₁	—	4	1	no	—
121	C₂×D₄×C₃₁	—	4	1	no	—
122	C₂×D₄×C₃₁	—	4	4	no	—
123	C₂×D₄×C₃₁	—	8	2	no	—
124	C₂²×C₄×C₃₁	—	4	1	no	—
125	C₂²×C₄×C₃₁	—	4	1	no	—
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
129	SD₁₆×C₃₁	—	4	8	no	—
130	SD₁₆×C₃₁	—	4	4	no	—
131	SD₁₆×C₃₁	—	4	8	no	—
132	D₈×C₃₁	—	8	4	no	—
133	M₁₆×C₃₁	—	2	4	no	—
134	C₂×C₈×C₃₁	—	2	4	no	—
135	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
136	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
137	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
138	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	2	1	no	—
142	C₄⋊C₄×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	2	1	no	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	4	1	no	—
148	C₄⋊C₄×C₃₁	—	4	1	no	—
149	C₄⋊C₄×C₃₁	—	4	1	no	—
150	C₄⋊C₄×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	C₄²×C₃₁	—	4	1	no	—
155	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
159	#19625	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₃₁	—	4	1	no	—
161	C₂³×C₃₁	—	4	2	no	—
162	C₂³×C₃₁	—	8	2	no	—
163	C₂³×C₃₁	—	8	2	no	—
164	C₂³×C₃₁	—	8	2	no	—
165	C₂³×C₃₁	#127	1	1	yes	characteristic
166	Q₈×C₃₁	—	4	2	no	—
167	Q₈×C₃₁	—	4	4	no	—
168	Q₈×C₃₁	—	4	2	no	—
169	Q₈×C₃₁	—	4	2	no	—
170	Q₈×C₃₁	—	8	2	no	—
171	D₄×C₃₁	—	4	4	no	—
172	D₄×C₃₁	—	4	2	no	—
173	D₄×C₃₁	—	4	4	no	—
174	D₄×C₃₁	—	4	2	no	—
175	D₄×C₃₁	—	4	2	no	—
176	D₄×C₃₁	—	4	4	no	—
177	D₄×C₃₁	—	8	2	no	—
178	D₄×C₃₁	—	8	2	no	—
179	D₄×C₃₁	—	8	2	no	—
180	D₄×C₃₁	—	8	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	4	1	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	8	2	no	—
188	C₂×C₄×C₃₁	—	8	2	no	—
189	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
190	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
191	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
192	C₂×C₄×C₃₁	#135	1	4	yes	—
193	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
196	C₈×C₃₁	—	2	4	no	—
197	C₈×C₃₁	—	2	2	no	—
198	C₈×C₃₁	—	2	4	no	—
199	C₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
200	#3230	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#3205	C₂×C₃₁	1	1	yes	characteristic
202	#3167	C₂×C₃₁	1	1	yes	characteristic, complemented
203	C₂×#492	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#3141	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#3138	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1719	C₂×C₃₁	1	2	yes	complemented
207	#1717	C₂×C₃₁	1	2	yes	complemented
208	#1708	C₂×C₃₁	1	2	yes	complemented
209	#1706	C₂×C₃₁	1	2	yes	complemented
210	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
211	C₂²×C₃₁	—	2	1	no	—
212	C₂²×C₃₁	—	2	1	no	—
213	C₂²×C₃₁	—	8	1	no	—
214	C₂²×C₃₁	—	8	2	no	—
215	C₂²×C₃₁	—	8	2	no	—
216	C₂²×C₃₁	—	8	1	no	—
217	C₂²×C₃₁	—	8	2	no	—
218	C₂²×C₃₁	—	8	1	no	—
219	C₂²×C₃₁	—	16	4	no	—
220	C₂²×C₃₁	#495	1	2	yes	—
221	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
222	C₄×C₃₁	—	2	1	no	—
223	C₄×C₃₁	—	2	1	no	—
224	C₄×C₃₁	—	8	1	no	—
225	C₄×C₃₁	—	8	2	no	—
226	C₄×C₃₁	—	8	1	no	—
227	C₄×C₃₁	—	8	1	no	—
228	C₄×C₃₁	#495	1	2	yes	—
229	C₄×C₃₁	C₂×#135	1	2	yes	—
230	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
231	C₄×C₃₁	#577	1	2	yes	—
232	C₂²×SD₁₆	—	2	2	no	—
233	#533	C₂²×C₃₁	1	1	yes	characteristic
234	#532	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
237	#500	C₂²×C₃₁	1	2	yes	complemented
238	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
239	#493	C₂²×C₃₁	1	2	yes	—
240	#492	—	2	2	no	—
241	#492	C₂²×C₃₁	1	2	yes	—
242	#492	C₂²×C₃₁	1	2	yes	—
243	#492	C₂²×C₃₁	1	1	yes	characteristic, complemented
244	#490	—	2	2	no	—
245	#489	—	2	2	no	—
246	#487	C₂²×C₃₁	1	2	yes	complemented
247	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
248	#487	C₂²×C₃₁	1	2	yes	complemented
249	#486	C₂²×C₃₁	1	2	yes	—
250	#486	C₂²×C₃₁	1	2	yes	—
251	#486	C₂²×C₃₁	1	2	yes	—
252	#486	C₂²×C₃₁	1	2	yes	complemented
253	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
254	#484	—	2	2	no	—
255	#477	—	2	2	no	—
256	#465	—	2	2	no	—
257	#459	—	2	2	no	—
258	#447	—	2	2	no	—
259	C₄×SD₁₆	—	2	2	no	—
260	#431	C₂²×C₃₁	1	1	yes	characteristic
261	#418	—	2	2	no	—
262	#417	—	2	2	no	—
263	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
265	#322	C₂²×C₃₁	1	2	yes	complemented
266	C₈²	C₂²×C₃₁	1	2	yes	complemented
267	C₂×C₃₁	—	2	2	no	—
268	C₂×C₃₁	—	16	1	no	—
269	C₂×C₃₁	—	16	2	no	—
270	C₂×C₃₁	C₂×#495	1	1	yes	characteristic
271	C₂×C₃₁	#3151	1	1	yes	characteristic
272	C₂×C₃₁	#3154	1	1	yes	characteristic
273	C₂×#41	—	2	1	no	—
274	C₂²×Q₈	—	2	1	no	—
275	C₂²×D₄	—	2	1	no	—
276	C₂²×D₄	—	2	1	no	—
277	C₂²×D₄	—	2	1	no	—
278	C₂×SD₁₆	—	2	4	no	—
279	C₂×SD₁₆	—	2	4	no	—
280	C₂×SD₁₆	—	2	2	no	—
281	C₂×SD₁₆	—	4	8	no	—
282	C₂×D₈	—	2	4	no	—
283	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
284	C₂²×C₈	D₄×C₃₁	1	2	yes	—
285	C₄⋊Q₈	—	2	1	no	—
286	C₄⋊Q₈	—	2	1	no	—
287	C₄⋊Q₈	—	2	1	no	—
288	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
289	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
290	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
292	#127	—	2	1	no	—
293	#127	—	2	1	no	—
294	#127	—	2	1	no	—
295	#127	C₂³×C₃₁	1	2	yes	—
296	#127	C₂³×C₃₁	1	1	yes	characteristic
297	#125	—	2	1	no	—
298	#124	—	2	1	no	—
299	C₂²⋊Q₈	—	2	1	no	—
300	C₂²⋊Q₈	—	2	1	no	—
301	#121	—	2	1	no	—
302	#121	—	2	1	no	—
303	C₄×Q₈	—	2	1	no	—
304	C₄×Q₈	—	2	1	no	—
305	C₄×D₄	—	2	1	no	—
306	C₄×D₄	—	2	1	no	—
307	#117	—	2	1	no	—
308	C₂×C₄⋊C₄	—	2	1	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
311	#107	—	2	4	no	—
312	#106	—	2	2	no	—
313	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
314	#103	—	2	4	no	—
315	#103	—	2	2	no	—
316	#103	—	2	4	no	—
317	#102	—	2	4	no	—
318	#102	—	2	4	no	—
319	#102	—	2	4	no	—
320	#102	—	2	2	no	—
321	#102	—	2	4	no	—
322	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
323	#97	D₄×C₃₁	1	2	yes	—
324	C₄×C₈	D₄×C₃₁	1	2	yes	—
325	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
326	C₄×C₈	C₂³×C₃₁	1	2	yes	—
327	C₄×C₈	C₂³×C₃₁	1	2	yes	—
328	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
329	C₃₁	#19625	1	1	yes	characteristic, direct product factor, Sylow
330	C₂⁴	—	4	1	no	—
331	#41	—	4	4	no	—
332	C₂×Q₈	—	2	1	no	—
333	C₂×Q₈	—	2	1	no	—
334	C₂×Q₈	—	2	1	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	4	4	no	—
337	C₂×Q₈	—	4	1	no	—
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	4	4	no	—
345	C₂×D₄	—	4	1	no	—
346	C₂×D₄	—	4	1	no	—
347	C₂×D₄	—	4	1	no	—
348	C₂×D₄	—	4	1	no	—
349	C₂×D₄	—	4	4	no	—
350	C₂×D₄	—	8	2	no	—
351	C₂²×C₄	—	4	1	no	—
352	C₂²×C₄	—	4	1	no	—
353	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
356	SD₁₆	—	4	8	no	—
357	SD₁₆	—	4	4	no	—
358	SD₁₆	—	4	8	no	—
359	D₈	—	8	4	no	—
360	M₁₆	—	2	4	no	—
361	C₂×C₈	—	2	4	no	—
362	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
363	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
364	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
365	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	—	4	1	no	—
375	C₄⋊C₄	—	4	1	no	—
376	C₄⋊C₄	—	4	1	no	—
377	C₄⋊C₄	—	4	1	no	—
378	C₂²⋊C₄	—	4	1	no	—
379	C₂²⋊C₄	—	4	1	no	—
380	C₂²⋊C₄	—	4	1	no	—
381	C₄²	—	4	1	no	—
382	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
383	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
386	C₂³	—	4	1	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	8	2	no	—
389	C₂³	—	8	2	no	—
390	C₂³	—	8	2	no	—
391	C₂³	C₃₁×#127	1	1	yes	characteristic
392	Q₈	—	4	2	no	—
393	Q₈	—	4	4	no	—
394	Q₈	—	4	2	no	—
395	Q₈	—	4	2	no	—
396	Q₈	—	8	2	no	—
397	D₄	—	4	4	no	—
398	D₄	—	4	2	no	—
399	D₄	—	4	4	no	—
400	D₄	—	4	2	no	—
401	D₄	—	4	2	no	—
402	D₄	—	4	4	no	—
403	D₄	—	8	2	no	—
404	D₄	—	8	2	no	—
405	D₄	—	8	2	no	—
406	D₄	—	8	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	8	2	no	—
414	C₂×C₄	—	8	2	no	—
415	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
416	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
417	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
418	C₂×C₄	C₃₁×#135	1	4	yes	—
419	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
420	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
421	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
422	C₈	—	2	4	no	—
423	C₈	—	2	2	no	—
424	C₈	—	2	4	no	—
425	C₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
426	C₂²	—	2	1	no	—
427	C₂²	—	2	1	no	—
428	C₂²	—	8	1	no	—
429	C₂²	—	8	2	no	—
430	C₂²	—	8	2	no	—
431	C₂²	—	8	1	no	—
432	C₂²	—	8	2	no	—
433	C₂²	—	8	1	no	—
434	C₂²	—	16	4	no	—
435	C₂²	C₃₁×#495	1	2	yes	—
436	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
437	C₄	—	2	1	no	—
438	C₄	—	2	1	no	—
439	C₄	—	8	1	no	—
440	C₄	—	8	2	no	—
441	C₄	—	8	1	no	—
442	C₄	—	8	1	no	—
443	C₄	C₃₁×#495	1	2	yes	—
444	C₄	C₂×C₃₁×#135	1	2	yes	—
445	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
446	C₄	C₃₁×#577	1	2	yes	—
447	C₂	—	2	2	no	—
448	C₂	—	16	2	no	—
449	C₂	—	16	1	no	—
450	C₂	C₂×C₃₁×#495	1	1	yes	characteristic
451	C₂	C₃₁×#3151	1	1	yes	characteristic
452	C₂	C₃₁×#3154	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3205	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
3. C₃₁×#3167	C₂	semidirect
4. C₂×C₃₁×#492	C₂	semidirect
5. C₃₁×#3141	C₂	semidirect
6. C₃₁×#3138	C₂	semidirect
7. C₃₁×#1719	C₂	semidirect
8. C₃₁×#1717	C₂	semidirect
9. C₃₁×#1708	C₂	semidirect
10. C₃₁×#1706	C₂	semidirect
11. C₈×M₁₆×C₃₁	C₂	semidirect
17. C₃₁×#500	C₂²	semidirect
18. C₃₁×#498	C₂²	semidirect
23. C₃₁×#492	C₂²	semidirect
26. C₃₁×#487	C₂²	semidirect
27. C₃₁×#487	C₂²	semidirect
28. C₃₁×#487	C₂²	semidirect
32. C₃₁×#486	C₂²	semidirect
33. C₃₁×#486	C₂²	semidirect
45. C₃₁×#322	C₂²	semidirect
46. C₈²×C₃₁	C₂²	semidirect
159. #19625	C₃₁	direct (N × H)
200. #3230	C₂×C₃₁	semidirect
202. #3167	C₂×C₃₁	semidirect
203. C₂×#492	C₂×C₃₁	semidirect
204. #3141	C₂×C₃₁	semidirect
205. #3138	C₂×C₃₁	semidirect
206. #1719	C₂×C₃₁	semidirect
207. #1717	C₂×C₃₁	semidirect
208. #1708	C₂×C₃₁	semidirect
209. #1706	C₂×C₃₁	semidirect
210. C₈×M₁₆	C₂×C₃₁	semidirect
237. #500	C₂²×C₃₁	semidirect
238. #498	C₂²×C₃₁	semidirect
243. #492	C₂²×C₃₁	semidirect
246. #487	C₂²×C₃₁	semidirect
247. #487	C₂²×C₃₁	semidirect
248. #487	C₂²×C₃₁	semidirect
252. #486	C₂²×C₃₁	semidirect
253. #486	C₂²×C₃₁	semidirect
265. #322	C₂²×C₃₁	semidirect
266. C₈²	C₂²×C₃₁	semidirect
329. C₃₁	#19625	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×#3205	2
C₃₁×#3167	2
C₂×C₃₁×#492	2
C₃₁×#3141	2
C₃₁×#3138	2
C₃₁×#1719	2
C₃₁×#1717	2
C₃₁×#1708	2
C₃₁×#1706	2
C₈×M₁₆×C₃₁	2
#19625	31

Subgroups — C31×#19625 #21476384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19625 #21476384