Subgroups — C₃₁×#21131 #21477890

Order 7936.

862 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	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2319	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2276	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1926	C₂	1	2	yes	maximal
5	C₃₁×#1920	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1880	C₂	1	2	yes	maximal
7	C₃₁×#1865	C₂	1	2	yes	maximal, complemented
8	C₄×C₃₁×#106	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1763	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1762	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×SD₁₆×C₃₁	C₄	1	8	yes	complemented
17	C₂×C₃₁×#106	C₂²	1	2	yes	—
18	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#103	C₂²	1	2	yes	—
20	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#390	C₂²	1	2	yes	—
26	C₃₁×#389	C₂²	1	2	yes	—
27	C₃₁×#388	—	2	2	no	—
28	C₃₁×#381	—	2	1	no	—
29	C₃₁×#380	—	2	1	no	—
30	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
31	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#376	—	2	1	no	—
36	C₃₁×#376	C₂²	1	1	yes	characteristic
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	—	2	1	no	—
45	C₂×SD₁₆×C₃₁	—	2	4	no	—
46	C₂×SD₁₆×C₃₁	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₃₁	—	2	4	no	—
51	C₄×Q₈×C₃₁	—	2	4	no	—
52	C₄×Q₈×C₃₁	—	2	2	no	—
53	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
54	C₄×D₄×C₃₁	—	2	2	no	—
55	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₄²×C₃₁	—	2	2	no	—
67	C₂×C₄²×C₃₁	—	2	1	no	—
68	C₂×C₄²×C₃₁	—	2	2	no	—
69	C₂×C₄²×C₃₁	—	2	1	no	—
70	C₂×C₄²×C₃₁	—	2	1	no	—
71	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×#106	—	2	4	no	—
74	C₃₁×#106	C₂×C₄	1	4	yes	—
75	C₃₁×#103	—	2	4	no	—
76	C₃₁×#103	C₂×C₄	1	4	yes	—
77	C₃₁×#102	—	2	4	no	—
78	C₃₁×#102	C₂×C₄	1	4	yes	—
79	C₃₁×#97	—	2	2	no	—
80	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
81	C₃₁×#95	—	2	2	no	—
82	C₃₁×#95	—	2	1	no	—
83	C₃₁×#95	—	2	1	no	—
84	C₃₁×#95	—	2	2	no	—
85	C₃₁×#95	D₄	1	1	yes	characteristic
86	C₃₁×#95	D₄	1	1	yes	characteristic
87	C₃₁×#95	D₄	1	2	yes	—
88	C₂⁴×C₃₁	—	2	1	no	—
89	C₂×Q₈×C₃₁	—	2	1	no	—
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
92	C₂×Q₈×C₃₁	#41	1	4	yes	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	1	no	—
96	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	4	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	8	2	no	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
109	SD₁₆×C₃₁	—	2	8	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
116	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
117	C₄⋊C₄×C₃₁	—	2	8	no	—
118	C₄⋊C₄×C₃₁	—	2	4	no	—
119	C₄⋊C₄×C₃₁	—	2	1	no	—
120	C₄⋊C₄×C₃₁	—	2	1	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	8	2	no	—
125	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
126	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₃₁	#41	1	4	yes	—
128	C₂²⋊C₄×C₃₁	—	2	4	no	—
129	C₂²⋊C₄×C₃₁	—	8	2	no	—
130	C₂²⋊C₄×C₃₁	—	8	2	no	—
131	C₄²×C₃₁	—	2	8	no	—
132	C₄²×C₃₁	—	2	4	no	—
133	C₄²×C₃₁	—	2	2	no	—
134	C₄²×C₃₁	—	2	2	no	—
135	C₄²×C₃₁	—	8	1	no	—
136	C₄²×C₃₁	—	8	1	no	—
137	C₄²×C₃₁	—	8	2	no	—
138	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
139	C₄²×C₃₁	#41	1	2	yes	—
140	#21131	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	—	2	2	no	—
142	C₂³×C₃₁	—	4	2	no	—
143	C₂³×C₃₁	—	4	1	no	—
144	C₂³×C₃₁	—	4	2	no	—
145	C₂³×C₃₁	—	4	1	no	—
146	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
147	Q₈×C₃₁	—	2	2	no	—
148	Q₈×C₃₁	—	2	4	no	—
149	Q₈×C₃₁	#117	1	4	yes	—
150	D₄×C₃₁	—	2	2	no	—
151	D₄×C₃₁	—	2	2	no	—
152	D₄×C₃₁	—	4	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	2	no	—
158	C₂×C₄×C₃₁	—	2	2	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	—	2	4	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	8	4	no	—
176	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
177	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
178	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
180	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
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	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
186	#2319	C₂×C₃₁	1	1	yes	characteristic
187	#2276	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#1926	C₂×C₃₁	1	2	yes	—
189	#1920	C₂×C₃₁	1	2	yes	complemented
190	#1880	C₂×C₃₁	1	2	yes	—
191	#1865	C₂×C₃₁	1	2	yes	complemented
192	C₄×#106	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1763	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1762	C₂×C₃₁	1	1	yes	characteristic
195	#1746	C₂×C₃₁	1	1	yes	characteristic, complemented
196	C₂²×C₃₁	—	4	2	no	—
197	C₂²×C₃₁	—	4	2	no	—
198	C₂²×C₃₁	—	4	2	no	—
199	C₂²×C₃₁	—	8	2	no	—
200	C₂²×C₃₁	—	8	2	no	—
201	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
202	C₂²×C₃₁	#525	1	2	yes	—
203	C₂²×C₃₁	#577	1	1	yes	characteristic
204	C₂²×C₃₁	#577	1	2	yes	—
205	C₂²×C₃₁	#577	1	1	yes	characteristic
206	C₄×C₃₁	—	2	2	no	—
207	C₄×C₃₁	—	2	2	no	—
208	C₄×C₃₁	—	2	4	no	—
209	C₄×C₃₁	—	4	2	no	—
210	C₄×C₃₁	—	4	4	no	—
211	C₄×C₃₁	—	4	2	no	—
212	C₄×C₃₁	—	4	2	no	—
213	C₄×C₃₁	—	4	2	no	—
214	C₄×C₃₁	—	8	2	no	—
215	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
216	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
217	C₄×C₃₁	#529	1	2	yes	—
218	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
219	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
220	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
221	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
222	C₄×SD₁₆	C₄×C₃₁	1	8	yes	complemented
223	C₂×#106	C₂²×C₃₁	1	2	yes	—
224	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
225	C₂×#103	C₂²×C₃₁	1	2	yes	—
226	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
227	C₂×#102	C₂²×C₃₁	1	2	yes	—
228	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
229	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
231	#390	C₂²×C₃₁	1	2	yes	—
232	#389	C₂²×C₃₁	1	2	yes	—
233	#388	—	2	2	no	—
234	#381	—	2	1	no	—
235	#380	—	2	1	no	—
236	C₄×C₄⋊C₄	—	2	1	no	—
237	C₄×C₄⋊C₄	—	2	2	no	—
238	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
239	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
240	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
241	#376	—	2	1	no	—
242	#376	C₂²×C₃₁	1	1	yes	characteristic
243	#340	C₂²×C₃₁	1	2	yes	—
244	#340	C₂²×C₃₁	1	2	yes	—
245	#340	C₂²×C₃₁	1	2	yes	—
246	#336	C₂²×C₃₁	1	2	yes	—
247	#336	C₂²×C₃₁	1	2	yes	—
248	C₂×C₃₁	—	8	2	no	—
249	C₂×C₃₁	#2290	1	1	yes	characteristic
250	C₂×C₃₁	#2949	1	1	yes	characteristic
251	C₂×C₃₁	#2952	1	1	yes	characteristic
252	C₂×C₃₁	#3125	1	2	yes	—
253	C₂×C₃₁	#3126	1	2	yes	—
254	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
255	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
256	C₂³×C₄	—	2	1	no	—
257	C₂×SD₁₆	—	2	4	no	—
258	C₂×SD₁₆	C₂×C₄×C₃₁	1	4	yes	—
259	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
260	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
261	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
262	C₄×Q₈	—	2	4	no	—
263	C₄×Q₈	—	2	4	no	—
264	C₄×Q₈	—	2	2	no	—
265	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
266	C₄×D₄	—	2	2	no	—
267	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
272	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
273	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
274	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
275	C₂×C₂²⋊C₄	—	2	1	no	—
276	C₂×C₂²⋊C₄	—	2	2	no	—
277	C₂×C₂²⋊C₄	—	2	2	no	—
278	C₂×C₄²	—	2	2	no	—
279	C₂×C₄²	—	2	1	no	—
280	C₂×C₄²	—	2	2	no	—
281	C₂×C₄²	—	2	1	no	—
282	C₂×C₄²	—	2	1	no	—
283	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
284	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
285	#106	—	2	4	no	—
286	#106	C₂×C₄×C₃₁	1	4	yes	—
287	#103	—	2	4	no	—
288	#103	C₂×C₄×C₃₁	1	4	yes	—
289	#102	—	2	4	no	—
290	#102	C₂×C₄×C₃₁	1	4	yes	—
291	#97	—	2	2	no	—
292	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
293	#95	—	2	2	no	—
294	#95	—	2	1	no	—
295	#95	—	2	1	no	—
296	#95	—	2	2	no	—
297	#95	D₄×C₃₁	1	1	yes	characteristic
298	#95	D₄×C₃₁	1	1	yes	characteristic
299	#95	D₄×C₃₁	1	2	yes	—
300	C₃₁	#21131	1	1	yes	characteristic, direct product factor, Sylow
301	C₂⁴	—	2	1	no	—
302	C₂×Q₈	—	2	1	no	—
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
305	C₂×Q₈	#41×C₃₁	1	4	yes	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	1	no	—
309	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	4	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	8	2	no	—
317	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
318	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
319	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
322	SD₁₆	—	2	8	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
329	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
330	C₄⋊C₄	—	2	8	no	—
331	C₄⋊C₄	—	2	4	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	8	2	no	—
338	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
339	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
340	C₄⋊C₄	#41×C₃₁	1	4	yes	—
341	C₂²⋊C₄	—	2	4	no	—
342	C₂²⋊C₄	—	8	2	no	—
343	C₂²⋊C₄	—	8	2	no	—
344	C₄²	—	2	8	no	—
345	C₄²	—	2	4	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	8	1	no	—
349	C₄²	—	8	1	no	—
350	C₄²	—	8	2	no	—
351	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
352	C₄²	#41×C₃₁	1	2	yes	—
353	C₂³	—	2	2	no	—
354	C₂³	—	4	2	no	—
355	C₂³	—	4	1	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	4	1	no	—
358	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
359	Q₈	—	2	2	no	—
360	Q₈	—	2	4	no	—
361	Q₈	C₃₁×#117	1	4	yes	—
362	D₄	—	2	2	no	—
363	D₄	—	2	2	no	—
364	D₄	—	4	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	4	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	8	4	no	—
388	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
389	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
390	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
391	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
392	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
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	C₂²	—	4	2	no	—
398	C₂²	—	4	2	no	—
399	C₂²	—	4	2	no	—
400	C₂²	—	8	2	no	—
401	C₂²	—	8	2	no	—
402	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
403	C₂²	C₃₁×#525	1	2	yes	—
404	C₂²	C₃₁×#577	1	1	yes	characteristic
405	C₂²	C₃₁×#577	1	2	yes	—
406	C₂²	C₃₁×#577	1	1	yes	characteristic
407	C₄	—	2	2	no	—
408	C₄	—	2	2	no	—
409	C₄	—	2	4	no	—
410	C₄	—	4	2	no	—
411	C₄	—	4	4	no	—
412	C₄	—	4	2	no	—
413	C₄	—	4	2	no	—
414	C₄	—	4	2	no	—
415	C₄	—	8	2	no	—
416	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
417	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
418	C₄	C₃₁×#529	1	2	yes	—
419	C₂	—	8	2	no	—
420	C₂	C₃₁×#2290	1	1	yes	characteristic
421	C₂	C₃₁×#2949	1	1	yes	characteristic
422	C₂	C₃₁×#2952	1	1	yes	characteristic
423	C₂	C₃₁×#3125	1	2	yes	—
424	C₂	C₃₁×#3126	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×D₄×C₃₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2276	C₂	semidirect
5. C₃₁×#1920	C₂	semidirect
7. C₃₁×#1865	C₂	semidirect
8. C₄×C₃₁×#106	C₂	semidirect
9. C₃₁×#1763	C₂	semidirect
11. C₃₁×#1746	C₂	semidirect
16. C₄×SD₁₆×C₃₁	C₄	semidirect
140. #21131	C₃₁	direct (N × H)
187. #2276	C₂×C₃₁	semidirect
189. #1920	C₂×C₃₁	semidirect
191. #1865	C₂×C₃₁	semidirect
192. C₄×#106	C₂×C₃₁	semidirect
193. #1763	C₂×C₃₁	semidirect
195. #1746	C₂×C₃₁	semidirect
222. C₄×SD₁₆	C₄×C₃₁	semidirect
300. C₃₁	#21131	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×C₄×SD₁₆×C₃₁	2
C₃₁×#2319	2
C₃₁×#2276	2
C₃₁×#1926	2
C₃₁×#1920	2
C₃₁×#1880	2
C₃₁×#1865	2
C₄×C₃₁×#106	2
C₃₁×#1763	2
C₃₁×#1762	2
C₃₁×#1746	2
#21131	31

Subgroups — C31×#21131 #21477890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21131 #21477890