Subgroups — C₃₁×#22271 #21479030

Order 7936.

950 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#495	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#124	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1958	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1880	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1879	C₂	1	2	yes	maximal
6	C₃₁×#1878	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1877	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1835	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#124	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#495	C₄	1	8	yes	complemented
17	C₂×C₃₁×#103	C₂²	1	2	yes	—
18	C₂×C₃₁×#103	C₂²	1	2	yes	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#391	C₂²	1	2	yes	—
24	C₃₁×#390	C₂²	1	2	yes	—
25	C₃₁×#388	—	2	2	no	—
26	C₃₁×#388	—	2	2	no	—
27	C₃₁×#382	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
29	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
30	C₄³×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#340	C₂²	1	1	yes	characteristic
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#340	C₂²	1	1	yes	characteristic
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
38	C₂³×C₄×C₃₁	—	2	2	no	—
39	C₂×Q₁₆×C₃₁	—	4	2	no	—
40	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
41	C₂×SD₁₆×C₃₁	—	4	2	no	—
42	C₂×SD₁₆×C₃₁	—	4	2	no	—
43	C₂×SD₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
44	C₂×D₈×C₃₁	—	4	2	no	—
45	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×#124	—	2	4	no	—
50	C₃₁×#124	C₂×C₄	1	8	yes	complemented
51	C₄×Q₈×C₃₁	—	4	2	no	—
52	C₄×Q₈×C₃₁	—	4	2	no	—
53	C₄×D₄×C₃₁	—	4	2	no	—
54	C₄×D₄×C₃₁	—	4	2	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
58	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₄²×C₃₁	—	2	2	no	—
64	C₂×C₄²×C₃₁	D₄	1	2	yes	—
65	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
68	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
69	C₃₁×#103	—	4	2	no	—
70	C₃₁×#103	—	4	2	no	—
71	C₃₁×#102	—	4	2	no	—
72	C₃₁×#102	—	4	2	no	—
73	C₄×C₈×C₃₁	—	2	2	no	—
74	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
75	C₃₁×#95	—	2	2	no	—
76	C₃₁×#95	—	2	2	no	—
77	C₃₁×#95	—	2	2	no	—
78	C₃₁×#95	—	2	2	no	—
79	C₂⁴×C₃₁	—	2	2	no	—
80	C₂×Q₈×C₃₁	—	2	2	no	—
81	C₂×Q₈×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	—	2	2	no	—
85	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	2	2	no	—
90	C₂×D₄×C₃₁	—	2	2	no	—
91	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
92	C₂²×C₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	2	2	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	4	4	no	—
96	C₂²×C₄×C₃₁	—	4	4	no	—
97	C₂²×C₄×C₃₁	—	4	4	no	—
98	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
103	Q₁₆×C₃₁	—	4	4	no	—
104	SD₁₆×C₃₁	—	4	8	no	—
105	D₈×C₃₁	—	4	4	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
113	C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	4	4	no	—
118	C₂²⋊C₄×C₃₁	—	2	8	no	—
119	C₂²⋊C₄×C₃₁	—	2	8	no	—
120	C₂²⋊C₄×C₃₁	—	4	4	no	—
121	C₂²⋊C₄×C₃₁	—	4	4	no	—
122	C₂²⋊C₄×C₃₁	—	4	4	no	—
123	C₄²×C₃₁	—	2	4	no	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	—	2	2	no	—
126	C₄²×C₃₁	—	4	1	no	—
127	C₄²×C₃₁	—	4	1	no	—
128	C₄²×C₃₁	—	4	4	no	—
129	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
130	C₄²×C₃₁	#41	1	2	yes	—
131	#22271	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂³×C₃₁	—	2	4	no	—
133	C₂³×C₃₁	—	4	2	no	—
134	C₂³×C₃₁	—	4	2	no	—
135	C₂³×C₃₁	—	4	2	no	—
136	C₂³×C₃₁	—	4	2	no	—
137	C₂³×C₃₁	—	4	2	no	—
138	C₂³×C₃₁	—	4	2	no	—
139	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	Q₈×C₃₁	—	2	4	no	—
141	Q₈×C₃₁	—	2	4	no	—
142	Q₈×C₃₁	—	4	2	no	—
143	Q₈×C₃₁	—	4	2	no	—
144	D₄×C₃₁	—	2	4	no	—
145	D₄×C₃₁	—	2	4	no	—
146	D₄×C₃₁	—	4	2	no	—
147	D₄×C₃₁	—	4	2	no	—
148	C₂×C₄×C₃₁	—	2	2	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	4	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	4	2	no	—
159	C₂×C₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	4	2	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	—	8	8	no	—
166	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
167	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
168	C₂×C₄×C₃₁	#135	1	4	yes	—
169	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
171	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
172	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
173	C₈×C₃₁	—	2	2	no	—
174	C₈×C₃₁	—	2	2	no	—
175	C₈×C₃₁	—	4	2	no	—
176	C₂×#495	C₂×C₃₁	1	1	yes	characteristic
177	C₄×#124	C₂×C₃₁	1	2	yes	complemented
178	#1958	C₂×C₃₁	1	2	yes	complemented
179	#1880	C₂×C₃₁	1	2	yes	complemented
180	#1879	C₂×C₃₁	1	2	yes	—
181	#1878	C₂×C₃₁	1	2	yes	complemented
182	#1877	C₂×C₃₁	1	2	yes	complemented
183	#1835	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
185	C₂²×C₃₁	—	4	4	no	—
186	C₂²×C₃₁	—	4	4	no	—
187	C₂²×C₃₁	—	4	4	no	—
188	C₂²×C₃₁	—	8	2	no	—
189	C₂²×C₃₁	—	8	2	no	—
190	C₂²×C₃₁	—	8	2	no	—
191	C₂²×C₃₁	—	8	2	no	—
192	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
193	C₂²×C₃₁	#535	1	2	yes	—
194	C₂²×C₃₁	C₂×#135	1	2	yes	—
195	C₂²×C₃₁	#575	1	1	yes	characteristic
196	C₂²×C₃₁	#575	1	1	yes	characteristic
197	C₄×C₃₁	—	2	2	no	—
198	C₄×C₃₁	—	2	2	no	—
199	C₄×C₃₁	—	2	4	no	—
200	C₄×C₃₁	—	4	4	no	—
201	C₄×C₃₁	—	4	2	no	—
202	C₄×C₃₁	—	8	2	no	—
203	C₄×C₃₁	—	8	2	no	—
204	C₄×C₃₁	#518	1	1	yes	characteristic
205	C₄×C₃₁	#518	1	1	yes	characteristic
206	C₄×C₃₁	#540	1	2	yes	—
207	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
208	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
209	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
210	C₂×#124	C₂²×C₃₁	1	2	yes	—
211	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
212	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
213	#495	C₄×C₃₁	1	8	yes	complemented
214	C₂×#103	C₂²×C₃₁	1	2	yes	—
215	C₂×#103	C₂²×C₃₁	1	2	yes	—
216	C₂×#102	C₂²×C₃₁	1	2	yes	—
217	C₂×#102	C₂²×C₃₁	1	2	yes	—
218	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
219	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
220	#391	C₂²×C₃₁	1	2	yes	—
221	#390	C₂²×C₃₁	1	2	yes	—
222	#388	—	2	2	no	—
223	#388	—	2	2	no	—
224	#382	C₂²×C₃₁	1	2	yes	—
225	C₄×C₂²⋊C₄	—	2	2	no	—
226	C₄×C₂²⋊C₄	—	2	2	no	—
227	C₄³	C₂²×C₃₁	1	1	yes	characteristic
228	#340	C₂²×C₃₁	1	1	yes	characteristic
229	#340	C₂²×C₃₁	1	2	yes	—
230	#340	C₂²×C₃₁	1	1	yes	characteristic
231	#336	C₂²×C₃₁	1	2	yes	—
232	#336	C₂²×C₃₁	1	2	yes	—
233	C₂×C₃₁	—	8	4	no	—
234	C₂×C₃₁	#2328	1	1	yes	characteristic
235	C₂×C₃₁	C₄×#135	1	1	yes	characteristic
236	C₂×C₃₁	#2943	1	1	yes	characteristic
237	C₂×C₃₁	#3211	1	2	yes	—
238	C₂×C₃₁	#3212	1	2	yes	—
239	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
240	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
241	C₂³×C₄	—	2	2	no	—
242	C₂×Q₁₆	—	4	2	no	—
243	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
244	C₂×SD₁₆	—	4	2	no	—
245	C₂×SD₁₆	—	4	2	no	—
246	C₂×SD₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
247	C₂×D₈	—	4	2	no	—
248	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
249	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
250	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
251	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
252	#124	—	2	4	no	—
253	#124	C₂×C₄×C₃₁	1	8	yes	complemented
254	C₄×Q₈	—	4	2	no	—
255	C₄×Q₈	—	4	2	no	—
256	C₄×D₄	—	4	2	no	—
257	C₄×D₄	—	4	2	no	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
260	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
261	C₂×C₂²⋊C₄	—	2	2	no	—
262	C₂×C₂²⋊C₄	—	2	2	no	—
263	C₂×C₂²⋊C₄	—	2	2	no	—
264	C₂×C₂²⋊C₄	—	2	2	no	—
265	C₂×C₂²⋊C₄	—	2	2	no	—
266	C₂×C₄²	—	2	2	no	—
267	C₂×C₄²	D₄×C₃₁	1	2	yes	—
268	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
269	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
270	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
271	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
272	#103	—	4	2	no	—
273	#103	—	4	2	no	—
274	#102	—	4	2	no	—
275	#102	—	4	2	no	—
276	C₄×C₈	—	2	2	no	—
277	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
278	#95	—	2	2	no	—
279	#95	—	2	2	no	—
280	#95	—	2	2	no	—
281	#95	—	2	2	no	—
282	C₃₁	#22271	1	1	yes	characteristic, direct product factor, Sylow
283	C₂⁴	—	2	2	no	—
284	C₂×Q₈	—	2	2	no	—
285	C₂×Q₈	—	2	2	no	—
286	C₂×Q₈	—	2	2	no	—
287	C₂×Q₈	—	2	2	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	2	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	4	4	no	—
300	C₂²×C₄	—	4	4	no	—
301	C₂²×C₄	—	4	4	no	—
302	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
303	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
304	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
305	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
306	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
307	Q₁₆	—	4	4	no	—
308	SD₁₆	—	4	8	no	—
309	D₈	—	4	4	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
316	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	4	4	no	—
322	C₂²⋊C₄	—	2	8	no	—
323	C₂²⋊C₄	—	2	8	no	—
324	C₂²⋊C₄	—	4	4	no	—
325	C₂²⋊C₄	—	4	4	no	—
326	C₂²⋊C₄	—	4	4	no	—
327	C₄²	—	2	4	no	—
328	C₄²	—	2	2	no	—
329	C₄²	—	2	2	no	—
330	C₄²	—	4	1	no	—
331	C₄²	—	4	1	no	—
332	C₄²	—	4	4	no	—
333	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
334	C₄²	#41×C₃₁	1	2	yes	—
335	C₂³	—	2	4	no	—
336	C₂³	—	4	2	no	—
337	C₂³	—	4	2	no	—
338	C₂³	—	4	2	no	—
339	C₂³	—	4	2	no	—
340	C₂³	—	4	2	no	—
341	C₂³	—	4	2	no	—
342	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
343	Q₈	—	2	4	no	—
344	Q₈	—	2	4	no	—
345	Q₈	—	4	2	no	—
346	Q₈	—	4	2	no	—
347	D₄	—	2	4	no	—
348	D₄	—	2	4	no	—
349	D₄	—	4	2	no	—
350	D₄	—	4	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	8	8	no	—
369	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
370	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
371	C₂×C₄	C₃₁×#135	1	4	yes	—
372	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
373	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
374	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
375	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
376	C₈	—	2	2	no	—
377	C₈	—	2	2	no	—
378	C₈	—	4	2	no	—
379	C₂²	—	4	4	no	—
380	C₂²	—	4	4	no	—
381	C₂²	—	4	4	no	—
382	C₂²	—	8	2	no	—
383	C₂²	—	8	2	no	—
384	C₂²	—	8	2	no	—
385	C₂²	—	8	2	no	—
386	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
387	C₂²	C₃₁×#535	1	2	yes	—
388	C₂²	C₂×C₃₁×#135	1	2	yes	—
389	C₂²	C₃₁×#575	1	1	yes	characteristic
390	C₂²	C₃₁×#575	1	1	yes	characteristic
391	C₄	—	2	2	no	—
392	C₄	—	2	2	no	—
393	C₄	—	2	4	no	—
394	C₄	—	4	4	no	—
395	C₄	—	4	2	no	—
396	C₄	—	8	2	no	—
397	C₄	—	8	2	no	—
398	C₄	C₃₁×#518	1	1	yes	characteristic
399	C₄	C₃₁×#518	1	1	yes	characteristic
400	C₄	C₃₁×#540	1	2	yes	—
401	C₂	—	8	4	no	—
402	C₂	C₃₁×#2328	1	1	yes	characteristic
403	C₂	C₄×C₃₁×#135	1	1	yes	characteristic
404	C₂	C₃₁×#2943	1	1	yes	characteristic
405	C₂	C₃₁×#3211	1	2	yes	—
406	C₂	C₃₁×#3212	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₃₁×#124	C₂	semidirect
3. C₃₁×#1958	C₂	semidirect
4. C₃₁×#1880	C₂	semidirect
6. C₃₁×#1878	C₂	semidirect
7. C₃₁×#1877	C₂	semidirect
8. C₃₁×#1835	C₂	semidirect
9. C₃₁×#1744	C₂	semidirect
16. C₃₁×#495	C₄	semidirect
40. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
43. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
50. C₃₁×#124	C₂×C₄	semidirect
74. C₄×C₈×C₃₁	C₂×C₄	semidirect
131. #22271	C₃₁	direct (N × H)
177. C₄×#124	C₂×C₃₁	semidirect
178. #1958	C₂×C₃₁	semidirect
179. #1880	C₂×C₃₁	semidirect
181. #1878	C₂×C₃₁	semidirect
182. #1877	C₂×C₃₁	semidirect
183. #1835	C₂×C₃₁	semidirect
184. #1744	C₂×C₃₁	semidirect
213. #495	C₄×C₃₁	semidirect
243. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
246. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
253. #124	C₂×C₄×C₃₁	semidirect
277. C₄×C₈	C₂×C₄×C₃₁	semidirect
282. C₃₁	#22271	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#495	2
C₄×C₃₁×#124	2
C₃₁×#1958	2
C₃₁×#1880	2
C₃₁×#1879	2
C₃₁×#1878	2
C₃₁×#1877	2
C₃₁×#1835	2
C₃₁×#1744	2
#22271	31

Subgroups — C31×#22271 #21479030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22271 #21479030