Subgroups — C₃₁×#24273 #21481032

Order 7936.

1384 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#470	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#435	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2409	C₂	1	2	yes	maximal
5	C₃₁×#2056	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2001	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1971	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1807	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1787	C₂	1	2	yes	maximal, complemented
10	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₃₁	—	2	8	no	—
12	C₂²×Q₁₆×C₃₁	—	2	4	no	—
13	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
18	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#470	—	2	8	no	—
20	C₃₁×#435	—	2	4	no	—
21	C₂×C₃₁×#106	C₂²	1	2	yes	—
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#103	C₂²	1	4	yes	—
24	C₂×C₃₁×#103	C₂²	1	4	yes	—
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
28	C₃₁×#395	—	2	2	no	—
29	C₃₁×#393	—	2	2	no	—
30	C₃₁×#391	C₂²	1	4	yes	—
31	C₃₁×#386	—	2	2	no	—
32	C₃₁×#385	—	2	2	no	—
33	C₃₁×#380	C₂²	1	2	yes	—
34	C₃₁×#340	C₂²	1	2	yes	—
35	C₂²×Q₈×C₃₁	—	2	2	no	—
36	C₂²×Q₈×C₃₁	—	2	8	no	—
37	C₂²×Q₈×C₃₁	—	2	4	no	—
38	C₂²×Q₈×C₃₁	—	2	4	no	—
39	C₂²×Q₈×C₃₁	—	4	4	no	—
40	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
41	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₃₁	—	2	2	no	—
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×Q₁₆×C₃₁	—	2	32	no	—
46	C₂×Q₁₆×C₃₁	—	2	8	no	—
47	C₂×Q₁₆×C₃₁	—	2	4	no	—
48	C₂×Q₁₆×C₃₁	—	2	4	no	—
49	C₂×Q₁₆×C₃₁	—	2	4	no	—
50	C₂×Q₁₆×C₃₁	D₄	1	8	yes	complemented
51	C₂×M₁₆×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	—	2	4	no	—
53	C₂²×C₈×C₃₁	—	2	4	no	—
54	C₂²×C₈×C₃₁	D₄	1	4	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×C₂²⋊Q₈	—	2	8	no	—
60	C₃₁×C₂²⋊Q₈	—	4	4	no	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
64	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#106	—	2	4	no	—
72	C₃₁×C₄⋊C₈	—	2	2	no	—
73	C₃₁×#103	—	2	8	no	—
74	C₃₁×#103	—	4	4	no	—
75	C₃₁×C₂²⋊C₈	—	2	2	no	—
76	C₃₁×C₂²⋊C₈	—	2	2	no	—
77	C₃₁×#97	—	2	2	no	—
78	C₃₁×#95	—	2	2	no	—
79	C₃₁×#95	—	2	4	no	—
80	C₃₁×#95	—	2	4	no	—
81	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂×Q₈×C₃₁	—	2	4	no	—
84	C₂×Q₈×C₃₁	—	2	16	no	—
85	C₂×Q₈×C₃₁	—	2	4	no	—
86	C₂×Q₈×C₃₁	—	2	4	no	—
87	C₂×Q₈×C₃₁	—	2	4	no	—
88	C₂×Q₈×C₃₁	—	2	4	no	—
89	C₂×Q₈×C₃₁	—	4	8	no	—
90	C₂×Q₈×C₃₁	—	4	16	no	—
91	C₂×Q₈×C₃₁	—	4	8	no	—
92	C₂×Q₈×C₃₁	—	4	4	no	—
93	C₂×Q₈×C₃₁	C₂×D₄	1	8	yes	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	4	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	4	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	2	4	no	—
104	C₂²×C₄×C₃₁	—	4	8	no	—
105	C₂²×C₄×C₃₁	—	4	4	no	—
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₃₁	#41	1	2	yes	—
112	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
113	Q₁₆×C₃₁	—	2	16	no	—
114	Q₁₆×C₃₁	—	2	16	no	—
115	M₁₆×C₃₁	—	4	4	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	8	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
123	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	4	no	—
127	C₄⋊C₄×C₃₁	—	2	4	no	—
128	C₄⋊C₄×C₃₁	—	4	8	no	—
129	C₂²⋊C₄×C₃₁	—	2	2	no	—
130	C₂²⋊C₄×C₃₁	—	2	2	no	—
131	C₂²⋊C₄×C₃₁	—	4	8	no	—
132	C₂²⋊C₄×C₃₁	—	4	4	no	—
133	C₄²×C₃₁	—	2	1	no	—
134	C₄²×C₃₁	—	2	1	no	—
135	#24273	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂³×C₃₁	—	2	2	no	—
137	C₂³×C₃₁	—	2	2	no	—
138	C₂³×C₃₁	—	2	1	no	—
139	C₂³×C₃₁	—	2	2	no	—
140	C₂³×C₃₁	—	2	2	no	—
141	C₂³×C₃₁	—	2	2	no	—
142	C₂³×C₃₁	#121	1	2	yes	—
143	C₂³×C₃₁	#137	1	4	yes	—
144	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
145	Q₈×C₃₁	—	2	8	no	—
146	Q₈×C₃₁	—	2	8	no	—
147	Q₈×C₃₁	—	4	16	no	—
148	Q₈×C₃₁	—	4	4	no	—
149	Q₈×C₃₁	—	4	4	no	—
150	D₄×C₃₁	—	4	4	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	4	no	—
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	8	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	4	16	no	—
163	C₂×C₄×C₃₁	—	4	4	no	—
164	C₂×C₄×C₃₁	—	4	4	no	—
165	C₂×C₄×C₃₁	—	4	1	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	4	no	—
169	C₂×C₄×C₃₁	—	4	4	no	—
170	C₂×C₄×C₃₁	—	4	4	no	—
171	C₂×C₄×C₃₁	#121	1	2	yes	—
172	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
174	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
175	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
176	C₈×C₃₁	—	2	2	no	—
177	C₈×C₃₁	—	4	1	no	—
178	C₈×C₃₁	—	4	1	no	—
179	C₈×C₃₁	#120	1	4	yes	—
180	C₂³×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
181	C₂×#470	C₂×C₃₁	1	2	yes	complemented
182	C₂×#435	C₂×C₃₁	1	1	yes	characteristic
183	#2409	C₂×C₃₁	1	2	yes	—
184	#2056	C₂×C₃₁	1	2	yes	complemented
185	#2001	C₂×C₃₁	1	2	yes	complemented
186	#1971	C₂×C₃₁	1	2	yes	complemented
187	#1807	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#1787	C₂×C₃₁	1	2	yes	complemented
189	C₂²×C₃₁	—	2	2	no	—
190	C₂²×C₃₁	—	2	8	no	—
191	C₂²×C₃₁	—	2	2	no	—
192	C₂²×C₃₁	—	4	1	no	—
193	C₂²×C₃₁	—	4	4	no	—
194	C₂²×C₃₁	—	4	1	no	—
195	C₂²×C₃₁	—	4	1	no	—
196	C₂²×C₃₁	#470	1	4	yes	complemented
197	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
198	C₂²×C₃₁	D₄²	1	2	yes	—
199	C₂²×C₃₁	C₂×#137	1	2	yes	—
200	C₂²×C₃₁	#578	1	2	yes	—
201	C₄×C₃₁	—	2	2	no	—
202	C₄×C₃₁	—	4	1	no	—
203	C₄×C₃₁	—	4	8	no	—
204	C₄×C₃₁	—	4	1	no	—
205	C₄×C₃₁	—	4	1	no	—
206	C₄×C₃₁	—	8	4	no	—
207	C₄×C₃₁	C₂×#120	1	1	yes	characteristic
208	C₄×C₃₁	#535	1	1	yes	characteristic
209	C₄×C₃₁	#546	1	2	yes	—
210	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
211	C₂²×Q₁₆	—	2	8	no	—
212	C₂²×Q₁₆	—	2	4	no	—
213	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
214	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
215	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
216	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
217	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
218	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
219	#470	—	2	8	no	—
220	#435	—	2	4	no	—
221	C₂×#106	C₂²×C₃₁	1	2	yes	—
222	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×#103	C₂²×C₃₁	1	4	yes	—
224	C₂×#103	C₂²×C₃₁	1	4	yes	—
225	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
226	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
227	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
228	#395	—	2	2	no	—
229	#393	—	2	2	no	—
230	#391	C₂²×C₃₁	1	4	yes	—
231	#386	—	2	2	no	—
232	#385	—	2	2	no	—
233	#380	C₂²×C₃₁	1	2	yes	—
234	#340	C₂²×C₃₁	1	2	yes	—
235	C₂×C₃₁	—	2	4	no	—
236	C₂×C₃₁	—	8	1	no	—
237	C₂×C₃₁	#2408	1	1	yes	characteristic
238	C₂×C₃₁	C₂×#470	1	1	yes	characteristic
239	C₂×C₃₁	#3059	1	1	yes	characteristic
240	C₂×C₃₁	#3270	1	4	yes	—
241	C₂²×Q₈	—	2	2	no	—
242	C₂²×Q₈	—	2	8	no	—
243	C₂²×Q₈	—	2	4	no	—
244	C₂²×Q₈	—	2	4	no	—
245	C₂²×Q₈	—	4	4	no	—
246	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
247	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
248	C₂³×C₄	—	2	2	no	—
249	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
250	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
251	C₂×Q₁₆	—	2	32	no	—
252	C₂×Q₁₆	—	2	8	no	—
253	C₂×Q₁₆	—	2	4	no	—
254	C₂×Q₁₆	—	2	4	no	—
255	C₂×Q₁₆	—	2	4	no	—
256	C₂×Q₁₆	D₄×C₃₁	1	8	yes	complemented
257	C₂×M₁₆	—	2	2	no	—
258	C₂×M₁₆	—	2	4	no	—
259	C₂²×C₈	—	2	4	no	—
260	C₂²×C₈	D₄×C₃₁	1	4	yes	—
261	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
263	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
264	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
265	C₂²⋊Q₈	—	2	8	no	—
266	C₂²⋊Q₈	—	4	4	no	—
267	C₄×D₄	—	2	2	no	—
268	C₄×D₄	—	2	2	no	—
269	C₂×C₄⋊C₄	—	2	4	no	—
270	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
271	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
272	C₂×C₂²⋊C₄	—	2	4	no	—
273	C₂×C₂²⋊C₄	—	2	2	no	—
274	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
275	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
276	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
277	#106	—	2	4	no	—
278	C₄⋊C₈	—	2	2	no	—
279	#103	—	2	8	no	—
280	#103	—	4	4	no	—
281	C₂²⋊C₈	—	2	2	no	—
282	C₂²⋊C₈	—	2	2	no	—
283	#97	—	2	2	no	—
284	#95	—	2	2	no	—
285	#95	—	2	4	no	—
286	#95	—	2	4	no	—
287	C₃₁	#24273	1	1	yes	characteristic, direct product factor, Sylow
288	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
289	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
290	C₂×Q₈	—	2	4	no	—
291	C₂×Q₈	—	2	16	no	—
292	C₂×Q₈	—	2	4	no	—
293	C₂×Q₈	—	2	4	no	—
294	C₂×Q₈	—	2	4	no	—
295	C₂×Q₈	—	2	4	no	—
296	C₂×Q₈	—	4	8	no	—
297	C₂×Q₈	—	4	16	no	—
298	C₂×Q₈	—	4	8	no	—
299	C₂×Q₈	—	4	4	no	—
300	C₂×Q₈	C₂×D₄×C₃₁	1	8	yes	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	—	2	4	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	4	no	—
311	C₂²×C₄	—	4	8	no	—
312	C₂²×C₄	—	4	4	no	—
313	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
314	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
315	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
316	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
318	C₂²×C₄	#41×C₃₁	1	2	yes	—
319	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
320	Q₁₆	—	2	16	no	—
321	Q₁₆	—	2	16	no	—
322	M₁₆	—	4	4	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	8	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
330	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
331	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	4	no	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	4	8	no	—
336	C₂²⋊C₄	—	2	2	no	—
337	C₂²⋊C₄	—	2	2	no	—
338	C₂²⋊C₄	—	4	8	no	—
339	C₂²⋊C₄	—	4	4	no	—
340	C₄²	—	2	1	no	—
341	C₄²	—	2	1	no	—
342	C₂³	—	2	2	no	—
343	C₂³	—	2	2	no	—
344	C₂³	—	2	1	no	—
345	C₂³	—	2	2	no	—
346	C₂³	—	2	2	no	—
347	C₂³	—	2	2	no	—
348	C₂³	C₃₁×#121	1	2	yes	—
349	C₂³	C₃₁×#137	1	4	yes	—
350	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
351	Q₈	—	2	8	no	—
352	Q₈	—	2	8	no	—
353	Q₈	—	4	16	no	—
354	Q₈	—	4	4	no	—
355	Q₈	—	4	4	no	—
356	D₄	—	4	4	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	8	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	4	16	no	—
369	C₂×C₄	—	4	4	no	—
370	C₂×C₄	—	4	4	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	4	no	—
375	C₂×C₄	—	4	4	no	—
376	C₂×C₄	—	4	4	no	—
377	C₂×C₄	C₃₁×#121	1	2	yes	—
378	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
379	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
380	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
381	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
382	C₈	—	2	2	no	—
383	C₈	—	4	1	no	—
384	C₈	—	4	1	no	—
385	C₈	C₃₁×#120	1	4	yes	—
386	C₂²	—	2	2	no	—
387	C₂²	—	2	8	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	4	no	—
391	C₂²	—	4	1	no	—
392	C₂²	—	4	1	no	—
393	C₂²	C₃₁×#470	1	4	yes	complemented
394	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
395	C₂²	D₄²×C₃₁	1	2	yes	—
396	C₂²	C₂×C₃₁×#137	1	2	yes	—
397	C₂²	C₃₁×#578	1	2	yes	—
398	C₄	—	2	2	no	—
399	C₄	—	4	1	no	—
400	C₄	—	4	8	no	—
401	C₄	—	4	1	no	—
402	C₄	—	4	1	no	—
403	C₄	—	8	4	no	—
404	C₄	C₂×C₃₁×#120	1	1	yes	characteristic
405	C₄	C₃₁×#535	1	1	yes	characteristic
406	C₄	C₃₁×#546	1	2	yes	—
407	C₂	—	2	4	no	—
408	C₂	—	8	1	no	—
409	C₂	C₃₁×#2408	1	1	yes	characteristic
410	C₂	C₂×C₃₁×#470	1	1	yes	characteristic
411	C₂	C₃₁×#3059	1	1	yes	characteristic
412	C₂	C₃₁×#3270	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂³×Q₁₆×C₃₁	C₂	semidirect
2. C₂×C₃₁×#470	C₂	semidirect
5. C₃₁×#2056	C₂	semidirect
6. C₃₁×#2001	C₂	semidirect
7. C₃₁×#1971	C₂	semidirect
8. C₃₁×#1807	C₂	semidirect
9. C₃₁×#1787	C₂	semidirect
50. C₂×Q₁₆×C₃₁	D₄	semidirect
135. #24273	C₃₁	direct (N × H)
180. C₂³×Q₁₆	C₂×C₃₁	semidirect
181. C₂×#470	C₂×C₃₁	semidirect
184. #2056	C₂×C₃₁	semidirect
185. #2001	C₂×C₃₁	semidirect
186. #1971	C₂×C₃₁	semidirect
187. #1807	C₂×C₃₁	semidirect
188. #1787	C₂×C₃₁	semidirect
196. C₂²×C₃₁	#470	semidirect
256. C₂×Q₁₆	D₄×C₃₁	semidirect
287. C₃₁	#24273	direct (N × H)
393. C₂²	C₃₁×#470	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₃₁	2
C₂×C₃₁×#470	2
C₂×C₃₁×#435	2
C₃₁×#2409	2
C₃₁×#2056	2
C₃₁×#2001	2
C₃₁×#1971	2
C₃₁×#1807	2
C₃₁×#1787	2
#24273	31

Subgroups — C31×#24273 #21481032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24273 #21481032