Subgroups — C₃₁×#19401 #21476160

Order 7936.

632 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3207	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3112	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2982	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1684	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1678	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1668	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1666	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1572	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1553	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1543	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#532	C₂²	1	1	yes	characteristic
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#493	C₂²	1	1	yes	characteristic
16	C₃₁×#492	C₂²	1	1	yes	characteristic
17	C₃₁×#487	C₂²	1	1	yes	characteristic
18	C₃₁×#486	C₂²	1	1	yes	characteristic
19	C₃₁×#481	—	2	1	no	—
20	C₃₁×#480	—	2	1	no	—
21	C₃₁×#477	C₂²	1	2	yes	—
22	C₃₁×#475	C₂²	1	2	yes	—
23	C₃₁×#463	—	2	2	no	—
24	C₃₁×#460	C₂²	1	2	yes	—
25	C₃₁×#460	C₂²	1	2	yes	—
26	C₃₁×#449	—	2	2	no	—
27	C₃₁×#448	—	2	1	no	—
28	C₃₁×#448	—	2	1	no	—
29	C₈×Q₈×C₃₁	—	2	2	no	—
30	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
31	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
32	C₃₁×#440	—	2	2	no	—
33	C₄×SD₁₆×C₃₁	—	2	2	no	—
34	C₃₁×#432	C₂²	1	1	yes	characteristic
35	C₃₁×#431	—	2	2	no	—
36	C₃₁×#431	C₂²	1	1	yes	characteristic
37	C₃₁×#431	C₂²	1	1	yes	characteristic
38	C₃₁×#431	C₂²	1	1	yes	characteristic
39	C₃₁×#335	C₂²	1	2	yes	—
40	C₃₁×#334	C₂²	1	2	yes	—
41	C₃₁×#333	C₂²	1	2	yes	—
42	C₃₁×#332	C₂²	1	2	yes	—
43	C₃₁×#331	C₂²	1	2	yes	—
44	C₃₁×#331	C₂²	1	2	yes	—
45	C₂×#41×C₃₁	—	2	1	no	—
46	C₂×#41×C₃₁	—	2	1	no	—
47	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
48	C₂×SD₁₆×C₃₁	—	2	2	no	—
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×D₈×C₃₁	—	2	2	no	—
52	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
53	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
54	C₃₁×#127	C₂³	1	1	yes	characteristic
55	C₃₁×#127	C₂³	1	1	yes	characteristic
56	C₃₁×#124	—	2	1	no	—
57	C₃₁×#124	—	2	1	no	—
58	C₃₁×#121	—	2	1	no	—
59	C₃₁×#121	—	2	1	no	—
60	C₃₁×#121	—	2	1	no	—
61	C₃₁×#121	—	2	1	no	—
62	C₄×Q₈×C₃₁	—	2	1	no	—
63	C₄×Q₈×C₃₁	—	2	1	no	—
64	C₄×Q₈×C₃₁	D₄	1	2	yes	—
65	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
66	C₄×D₄×C₃₁	—	2	1	no	—
67	C₄×D₄×C₃₁	—	2	1	no	—
68	C₄×D₄×C₃₁	—	2	1	no	—
69	C₄×D₄×C₃₁	—	2	1	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	—	2	1	no	—
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#107	—	2	2	no	—
75	C₃₁×#106	—	2	2	no	—
76	C₃₁×C₄⋊C₈	—	2	2	no	—
77	C₃₁×C₄⋊C₈	—	2	2	no	—
78	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
79	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
80	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
81	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
82	C₃₁×#103	—	2	2	no	—
83	C₃₁×#103	—	2	2	no	—
84	C₃₁×#103	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	—	2	2	no	—
88	C₃₁×C₂²⋊C₈	—	2	1	no	—
89	C₃₁×C₂²⋊C₈	—	2	1	no	—
90	C₃₁×C₂²⋊C₈	—	2	1	no	—
91	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
92	C₃₁×#97	—	2	1	no	—
93	C₄×C₈×C₃₁	—	2	2	no	—
94	C₄×C₈×C₃₁	—	2	2	no	—
95	C₄×C₈×C₃₁	—	2	1	no	—
96	C₄×C₈×C₃₁	—	2	1	no	—
97	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
98	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
99	#41×C₃₁	—	8	2	no	—
100	#41×C₃₁	—	8	2	no	—
101	C₂×Q₈×C₃₁	—	2	4	no	—
102	C₂×Q₈×C₃₁	—	2	1	no	—
103	C₂×Q₈×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
106	C₂×D₄×C₃₁	—	2	1	no	—
107	C₂×D₄×C₃₁	—	2	1	no	—
108	C₂×D₄×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	—	2	1	no	—
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	4	1	no	—
113	C₂×D₄×C₃₁	—	4	1	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	4	1	no	—
117	C₂²×C₄×C₃₁	—	4	1	no	—
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
121	SD₁₆×C₃₁	—	4	4	no	—
122	SD₁₆×C₃₁	—	4	4	no	—
123	SD₁₆×C₃₁	—	8	2	no	—
124	D₈×C₃₁	—	8	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
129	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	1	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	1	no	—
134	C₄⋊C₄×C₃₁	—	2	1	no	—
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
138	C₂²⋊C₄×C₃₁	—	4	1	no	—
139	C₂²⋊C₄×C₃₁	—	4	1	no	—
140	C₂²⋊C₄×C₃₁	—	4	1	no	—
141	C₂²⋊C₄×C₃₁	—	4	1	no	—
142	C₄²×C₃₁	—	2	2	no	—
143	C₄²×C₃₁	—	2	2	no	—
144	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₄²×C₃₁	#41	1	1	yes	characteristic
147	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
148	#19401	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂³×C₃₁	—	4	1	no	—
150	C₂³×C₃₁	—	4	1	no	—
151	C₂³×C₃₁	#121	1	1	yes	characteristic
152	Q₈×C₃₁	—	2	4	no	—
153	Q₈×C₃₁	—	2	2	no	—
154	Q₈×C₃₁	—	4	2	no	—
155	Q₈×C₃₁	—	8	1	no	—
156	Q₈×C₃₁	#135	1	4	yes	—
157	D₄×C₃₁	—	4	2	no	—
158	D₄×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	4	2	no	—
160	D₄×C₃₁	—	4	2	no	—
161	D₄×C₃₁	—	8	1	no	—
162	D₄×C₃₁	—	8	1	no	—
163	D₄×C₃₁	—	8	2	no	—
164	D₄×C₃₁	—	8	2	no	—
165	C₂×C₄×C₃₁	—	2	4	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	8	2	no	—
174	C₂×C₄×C₃₁	—	8	2	no	—
175	C₂×C₄×C₃₁	—	8	1	no	—
176	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
177	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
178	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
182	C₈×C₃₁	—	2	2	no	—
183	C₈×C₃₁	—	2	2	no	—
184	C₈×C₃₁	—	4	2	no	—
185	C₈×C₃₁	—	4	2	no	—
186	C₈×C₃₁	—	4	2	no	—
187	#3207	C₂×C₃₁	1	1	yes	characteristic
188	#3112	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#3103	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#2982	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#1684	C₂×C₃₁	1	2	yes	complemented
192	#1678	C₂×C₃₁	1	2	yes	complemented
193	#1668	C₂×C₃₁	1	2	yes	complemented
194	#1666	C₂×C₃₁	1	2	yes	complemented
195	#1572	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#1553	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1543	C₂×C₃₁	1	1	yes	characteristic, complemented
198	C₂²×C₃₁	—	2	1	no	—
199	C₂²×C₃₁	—	2	1	no	—
200	C₂²×C₃₁	—	2	1	no	—
201	C₂²×C₃₁	—	8	1	no	—
202	C₂²×C₃₁	—	8	1	no	—
203	C₂²×C₃₁	—	8	1	no	—
204	C₂²×C₃₁	—	8	1	no	—
205	C₂²×C₃₁	—	8	1	no	—
206	C₂²×C₃₁	—	8	1	no	—
207	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
208	C₄×C₃₁	—	2	1	no	—
209	C₄×C₃₁	—	2	4	no	—
210	C₄×C₃₁	—	2	1	no	—
211	C₄×C₃₁	—	2	1	no	—
212	C₄×C₃₁	—	2	1	no	—
213	C₄×C₃₁	—	4	2	no	—
214	C₄×C₃₁	—	8	1	no	—
215	C₄×C₃₁	—	8	1	no	—
216	C₄×C₃₁	C₂×#135	1	2	yes	—
217	C₄×C₃₁	#577	1	2	yes	—
218	#532	C₂²×C₃₁	1	1	yes	characteristic
219	#532	C₂²×C₃₁	1	1	yes	characteristic
220	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
221	#493	C₂²×C₃₁	1	1	yes	characteristic
222	#492	C₂²×C₃₁	1	1	yes	characteristic
223	#487	C₂²×C₃₁	1	1	yes	characteristic
224	#486	C₂²×C₃₁	1	1	yes	characteristic
225	#481	—	2	1	no	—
226	#480	—	2	1	no	—
227	#477	C₂²×C₃₁	1	2	yes	—
228	#475	C₂²×C₃₁	1	2	yes	—
229	#463	—	2	2	no	—
230	#460	C₂²×C₃₁	1	2	yes	—
231	#460	C₂²×C₃₁	1	2	yes	—
232	#449	—	2	2	no	—
233	#448	—	2	1	no	—
234	#448	—	2	1	no	—
235	C₈×Q₈	—	2	2	no	—
236	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
237	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
238	#440	—	2	2	no	—
239	C₄×SD₁₆	—	2	2	no	—
240	#432	C₂²×C₃₁	1	1	yes	characteristic
241	#431	—	2	2	no	—
242	#431	C₂²×C₃₁	1	1	yes	characteristic
243	#431	C₂²×C₃₁	1	1	yes	characteristic
244	#431	C₂²×C₃₁	1	1	yes	characteristic
245	#335	C₂²×C₃₁	1	2	yes	—
246	#334	C₂²×C₃₁	1	2	yes	—
247	#333	C₂²×C₃₁	1	2	yes	—
248	#332	C₂²×C₃₁	1	2	yes	—
249	#331	C₂²×C₃₁	1	2	yes	—
250	#331	C₂²×C₃₁	1	2	yes	—
251	C₂×C₃₁	—	4	1	no	—
252	C₂×C₃₁	—	16	1	no	—
253	C₂×C₃₁	—	16	1	no	—
254	C₂×C₃₁	#3043	1	1	yes	characteristic
255	C₂×C₃₁	#3050	1	1	yes	characteristic
256	C₂×C₃₁	#3068	1	1	yes	characteristic
257	C₂×#41	—	2	1	no	—
258	C₂×#41	—	2	1	no	—
259	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
260	C₂×SD₁₆	—	2	2	no	—
261	C₂×SD₁₆	—	2	2	no	—
262	C₂×SD₁₆	—	2	2	no	—
263	C₂×D₈	—	2	2	no	—
264	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
265	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
266	#127	C₂³×C₃₁	1	1	yes	characteristic
267	#127	C₂³×C₃₁	1	1	yes	characteristic
268	#124	—	2	1	no	—
269	#124	—	2	1	no	—
270	#121	—	2	1	no	—
271	#121	—	2	1	no	—
272	#121	—	2	1	no	—
273	#121	—	2	1	no	—
274	C₄×Q₈	—	2	1	no	—
275	C₄×Q₈	—	2	1	no	—
276	C₄×Q₈	D₄×C₃₁	1	2	yes	—
277	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
278	C₄×D₄	—	2	1	no	—
279	C₄×D₄	—	2	1	no	—
280	C₄×D₄	—	2	1	no	—
281	C₄×D₄	—	2	1	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
284	C₂×C₄²	—	2	1	no	—
285	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
286	#107	—	2	2	no	—
287	#106	—	2	2	no	—
288	C₄⋊C₈	—	2	2	no	—
289	C₄⋊C₈	—	2	2	no	—
290	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
291	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
292	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
293	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
294	#103	—	2	2	no	—
295	#103	—	2	2	no	—
296	#103	—	2	2	no	—
297	#102	—	2	2	no	—
298	#102	—	2	2	no	—
299	#102	—	2	2	no	—
300	C₂²⋊C₈	—	2	1	no	—
301	C₂²⋊C₈	—	2	1	no	—
302	C₂²⋊C₈	—	2	1	no	—
303	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
304	#97	—	2	1	no	—
305	C₄×C₈	—	2	2	no	—
306	C₄×C₈	—	2	2	no	—
307	C₄×C₈	—	2	1	no	—
308	C₄×C₈	—	2	1	no	—
309	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
310	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
311	C₃₁	#19401	1	1	yes	characteristic, direct product factor, Sylow
312	#41	—	8	2	no	—
313	#41	—	8	2	no	—
314	C₂×Q₈	—	2	4	no	—
315	C₂×Q₈	—	2	1	no	—
316	C₂×Q₈	—	2	1	no	—
317	C₂×Q₈	—	2	1	no	—
318	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	—	4	1	no	—
326	C₂×D₄	—	4	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	4	1	no	—
330	C₂²×C₄	—	4	1	no	—
331	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
334	SD₁₆	—	4	4	no	—
335	SD₁₆	—	4	4	no	—
336	SD₁₆	—	8	2	no	—
337	D₈	—	8	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
342	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
351	C₂²⋊C₄	—	4	1	no	—
352	C₂²⋊C₄	—	4	1	no	—
353	C₂²⋊C₄	—	4	1	no	—
354	C₂²⋊C₄	—	4	1	no	—
355	C₄²	—	2	2	no	—
356	C₄²	—	2	2	no	—
357	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
358	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₄²	#41×C₃₁	1	1	yes	characteristic
360	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
361	C₂³	—	4	1	no	—
362	C₂³	—	4	1	no	—
363	C₂³	C₃₁×#121	1	1	yes	characteristic
364	Q₈	—	2	4	no	—
365	Q₈	—	2	2	no	—
366	Q₈	—	4	2	no	—
367	Q₈	—	8	1	no	—
368	Q₈	C₃₁×#135	1	4	yes	—
369	D₄	—	4	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	4	2	no	—
372	D₄	—	4	2	no	—
373	D₄	—	8	1	no	—
374	D₄	—	8	1	no	—
375	D₄	—	8	2	no	—
376	D₄	—	8	2	no	—
377	C₂×C₄	—	2	4	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	8	2	no	—
386	C₂×C₄	—	8	2	no	—
387	C₂×C₄	—	8	1	no	—
388	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
389	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
390	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
391	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
392	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
393	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
394	C₈	—	2	2	no	—
395	C₈	—	2	2	no	—
396	C₈	—	4	2	no	—
397	C₈	—	4	2	no	—
398	C₈	—	4	2	no	—
399	C₂²	—	2	1	no	—
400	C₂²	—	2	1	no	—
401	C₂²	—	2	1	no	—
402	C₂²	—	8	1	no	—
403	C₂²	—	8	1	no	—
404	C₂²	—	8	1	no	—
405	C₂²	—	8	1	no	—
406	C₂²	—	8	1	no	—
407	C₂²	—	8	1	no	—
408	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
409	C₄	—	2	1	no	—
410	C₄	—	2	4	no	—
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	4	2	no	—
415	C₄	—	8	1	no	—
416	C₄	—	8	1	no	—
417	C₄	C₂×C₃₁×#135	1	2	yes	—
418	C₄	C₃₁×#577	1	2	yes	—
419	C₂	—	4	1	no	—
420	C₂	—	16	1	no	—
421	C₂	—	16	1	no	—
422	C₂	C₃₁×#3043	1	1	yes	characteristic
423	C₂	C₃₁×#3050	1	1	yes	characteristic
424	C₂	C₃₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
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
2. C₃₁×#3112	C₂	semidirect
3. C₃₁×#3103	C₂	semidirect
4. C₃₁×#2982	C₂	semidirect
5. C₃₁×#1684	C₂	semidirect
6. C₃₁×#1678	C₂	semidirect
7. C₃₁×#1668	C₂	semidirect
8. C₃₁×#1666	C₂	semidirect
9. C₃₁×#1572	C₂	semidirect
10. C₃₁×#1553	C₂	semidirect
11. C₃₁×#1543	C₂	semidirect
148. #19401	C₃₁	direct (N × H)
188. #3112	C₂×C₃₁	semidirect
189. #3103	C₂×C₃₁	semidirect
190. #2982	C₂×C₃₁	semidirect
191. #1684	C₂×C₃₁	semidirect
192. #1678	C₂×C₃₁	semidirect
193. #1668	C₂×C₃₁	semidirect
194. #1666	C₂×C₃₁	semidirect
195. #1572	C₂×C₃₁	semidirect
196. #1553	C₂×C₃₁	semidirect
197. #1543	C₂×C₃₁	semidirect
311. C₃₁	#19401	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×#3207	2
C₃₁×#3112	2
C₃₁×#3103	2
C₃₁×#2982	2
C₃₁×#1684	2
C₃₁×#1678	2
C₃₁×#1668	2
C₃₁×#1666	2
C₃₁×#1572	2
C₃₁×#1553	2
C₃₁×#1543	2
#19401	31

Subgroups — C31×#19401 #21476160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19401 #21476160