Subgroups — C₃₁×#24201 #21480960

Order 7936.

1068 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#556	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#462	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#440	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2581	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2042	C₂	1	2	yes	maximal
6	C₃₁×#2015	C₂	1	2	yes	maximal
7	C₃₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1960	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1878	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1863	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1804	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#556	—	2	4	no	—
15	C₃₁×#556	—	2	4	no	—
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#126	—	2	2	no	—
18	C₂×C₃₁×#124	C₂²	1	2	yes	—
19	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
20	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
22	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#117	—	2	1	no	—
25	C₃₁×#462	—	2	4	no	—
26	C₃₁×#440	—	2	4	no	—
27	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#103	C₂²	1	2	yes	—
30	C₂×C₃₁×#103	C₂²	1	2	yes	—
31	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#103	C₂²	1	2	yes	—
33	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#102	C₂²	1	2	yes	—
35	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
36	C₃₁×#400	—	2	1	no	—
37	C₃₁×#395	—	2	1	no	—
38	C₃₁×#391	C₂²	1	2	yes	—
39	C₃₁×#389	C₂²	1	2	yes	—
40	C₃₁×#387	—	2	2	no	—
41	C₃₁×#382	C₂²	1	1	yes	characteristic
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₃₁×#336	C₂²	1	2	yes	—
45	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	—	2	1	no	—
50	C₂×Q₁₆×C₃₁	—	2	4	no	—
51	C₂×Q₁₆×C₃₁	—	2	4	no	—
52	C₂×Q₁₆×C₃₁	—	4	2	no	—
53	C₂×SD₁₆×C₃₁	—	2	2	no	—
54	C₂×SD₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×C₄⋊Q₈	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
61	C₃₁×#126	—	2	16	no	—
62	C₃₁×#124	—	2	4	no	—
63	C₃₁×#124	—	2	4	no	—
64	C₃₁×C₂²⋊Q₈	—	2	8	no	—
65	C₄×Q₈×C₃₁	—	2	4	no	—
66	C₄×Q₈×C₃₁	—	2	4	no	—
67	C₄×Q₈×C₃₁	—	2	2	no	—
68	C₄×Q₈×C₃₁	—	2	2	no	—
69	C₄×Q₈×C₃₁	—	2	2	no	—
70	C₄×Q₈×C₃₁	—	2	2	no	—
71	C₄×D₄×C₃₁	—	2	2	no	—
72	C₄×D₄×C₃₁	D₄	1	4	yes	—
73	C₃₁×#117	—	2	8	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
87	C₂×C₄²×C₃₁	—	2	2	no	—
88	C₂×C₄²×C₃₁	—	2	1	no	—
89	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
90	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
91	C₃₁×#107	—	2	2	no	—
92	C₃₁×C₄⋊C₈	—	2	2	no	—
93	C₃₁×#103	—	2	4	no	—
94	C₃₁×#103	—	2	4	no	—
95	C₃₁×#103	—	2	2	no	—
96	C₃₁×#103	—	4	2	no	—
97	C₃₁×#102	—	2	2	no	—
98	C₃₁×#102	—	4	2	no	—
99	C₃₁×#97	—	2	2	no	—
100	C₃₁×#95	—	2	1	no	—
101	C₃₁×#95	—	2	2	no	—
102	C₃₁×#95	—	2	2	no	—
103	C₃₁×#95	—	2	2	no	—
104	C₃₁×#95	—	2	1	no	—
105	C₂⁴×C₃₁	—	2	1	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	—	2	1	no	—
110	C₂×Q₈×C₃₁	—	2	1	no	—
111	C₂×Q₈×C₃₁	—	2	2	no	—
112	C₂×Q₈×C₃₁	—	2	1	no	—
113	C₂×Q₈×C₃₁	—	2	2	no	—
114	C₂×Q₈×C₃₁	—	2	2	no	—
115	C₂×Q₈×C₃₁	—	2	1	no	—
116	C₂×Q₈×C₃₁	—	2	2	no	—
117	C₂×Q₈×C₃₁	—	4	2	no	—
118	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
119	C₂×Q₈×C₃₁	#41	1	2	yes	—
120	C₂×D₄×C₃₁	—	2	1	no	—
121	C₂×D₄×C₃₁	—	2	1	no	—
122	C₂×D₄×C₃₁	—	2	1	no	—
123	C₂×D₄×C₃₁	—	2	1	no	—
124	C₂×D₄×C₃₁	—	2	1	no	—
125	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	2	1	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	4	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	4	4	no	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
138	Q₁₆×C₃₁	—	4	8	no	—
139	SD₁₆×C₃₁	—	2	8	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	1	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	#41	1	2	yes	—
149	C₄⋊C₄×C₃₁	—	2	8	no	—
150	C₄⋊C₄×C₃₁	—	2	4	no	—
151	C₄⋊C₄×C₃₁	—	2	4	no	—
152	C₄⋊C₄×C₃₁	—	2	2	no	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	—	2	1	no	—
155	C₄⋊C₄×C₃₁	—	2	1	no	—
156	C₄⋊C₄×C₃₁	—	2	1	no	—
157	C₄⋊C₄×C₃₁	—	2	2	no	—
158	C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₄⋊C₄×C₃₁	—	4	4	no	—
160	C₄⋊C₄×C₃₁	—	4	2	no	—
161	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
162	C₂²⋊C₄×C₃₁	—	2	8	no	—
163	C₂²⋊C₄×C₃₁	—	2	8	no	—
164	C₂²⋊C₄×C₃₁	—	2	4	no	—
165	C₄²×C₃₁	—	2	8	no	—
166	C₄²×C₃₁	—	2	2	no	—
167	C₄²×C₃₁	—	2	2	no	—
168	C₄²×C₃₁	—	4	2	no	—
169	C₄²×C₃₁	C₂×D₄	1	2	yes	—
170	C₄²×C₃₁	C₂×D₄	1	2	yes	—
171	#24201	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂³×C₃₁	—	2	2	no	—
173	C₂³×C₃₁	—	4	1	no	—
174	C₂³×C₃₁	—	4	1	no	—
175	C₂³×C₃₁	—	4	2	no	—
176	C₂³×C₃₁	—	4	2	no	—
177	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	Q₈×C₃₁	—	2	2	no	—
179	Q₈×C₃₁	—	2	2	no	—
180	Q₈×C₃₁	—	2	4	no	—
181	Q₈×C₃₁	—	2	4	no	—
182	Q₈×C₃₁	—	4	4	no	—
183	Q₈×C₃₁	—	4	2	no	—
184	Q₈×C₃₁	—	4	2	no	—
185	Q₈×C₃₁	—	4	1	no	—
186	Q₈×C₃₁	—	4	1	no	—
187	D₄×C₃₁	—	2	2	no	—
188	D₄×C₃₁	—	2	2	no	—
189	D₄×C₃₁	—	4	1	no	—
190	D₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	4	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	4	8	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	1	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	1	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	#120	1	2	yes	—
213	C₂×C₄×C₃₁	#120	1	2	yes	—
214	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
218	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
219	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
220	C₈×C₃₁	—	2	2	no	—
221	C₈×C₃₁	—	4	1	no	—
222	C₈×C₃₁	—	4	2	no	—
223	C₂×#556	C₂×C₃₁	1	1	yes	characteristic
224	C₂×#462	C₂×C₃₁	1	1	yes	characteristic, complemented
225	C₂×#440	C₂×C₃₁	1	1	yes	characteristic
226	#2581	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#2042	C₂×C₃₁	1	2	yes	—
228	#2015	C₂×C₃₁	1	2	yes	—
229	#1985	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1960	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#1878	C₂×C₃₁	1	2	yes	complemented
232	#1863	C₂×C₃₁	1	2	yes	complemented
233	#1804	C₂×C₃₁	1	1	yes	characteristic
234	C₂²×C₃₁	—	4	2	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	8	2	no	—
238	C₂²×C₃₁	—	8	2	no	—
239	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
240	C₂²×C₃₁	#548	1	1	yes	characteristic
241	C₂²×C₃₁	#550	1	1	yes	characteristic
242	C₂²×C₃₁	#575	1	2	yes	—
243	C₂²×C₃₁	#578	1	2	yes	—
244	C₄×C₃₁	—	2	2	no	—
245	C₄×C₃₁	—	2	2	no	—
246	C₄×C₃₁	—	4	4	no	—
247	C₄×C₃₁	—	4	2	no	—
248	C₄×C₃₁	—	4	2	no	—
249	C₄×C₃₁	—	4	2	no	—
250	C₄×C₃₁	—	4	2	no	—
251	C₄×C₃₁	—	8	2	no	—
252	C₄×C₃₁	—	8	1	no	—
253	C₄×C₃₁	#525	1	1	yes	characteristic
254	C₄×C₃₁	#537	1	1	yes	characteristic
255	C₄×C₃₁	#546	1	1	yes	characteristic
256	C₄×C₃₁	#547	1	1	yes	characteristic
257	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
258	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
259	#556	—	2	4	no	—
260	#556	—	2	4	no	—
261	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×#126	—	2	2	no	—
263	C₂×#124	C₂²×C₃₁	1	2	yes	—
264	C₂×C₂²⋊Q₈	—	2	1	no	—
265	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
267	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×#117	—	2	1	no	—
270	#462	—	2	4	no	—
271	#440	—	2	4	no	—
272	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×#103	C₂²×C₃₁	1	2	yes	—
275	C₂×#103	C₂²×C₃₁	1	2	yes	—
276	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×#103	C₂²×C₃₁	1	2	yes	—
278	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×#102	C₂²×C₃₁	1	2	yes	—
280	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
281	#400	—	2	1	no	—
282	#395	—	2	1	no	—
283	#391	C₂²×C₃₁	1	2	yes	—
284	#389	C₂²×C₃₁	1	2	yes	—
285	#387	—	2	2	no	—
286	#382	C₂²×C₃₁	1	1	yes	characteristic
287	#340	C₂²×C₃₁	1	2	yes	—
288	#336	C₂²×C₃₁	1	2	yes	—
289	#336	C₂²×C₃₁	1	2	yes	—
290	C₂×C₃₁	—	8	2	no	—
291	C₂×C₃₁	#2407	1	1	yes	characteristic
292	C₂×C₃₁	#3005	1	1	yes	characteristic
293	C₂×C₃₁	#3011	1	1	yes	characteristic
294	C₂×C₃₁	#3377	1	2	yes	—
295	C₂×C₃₁	#3402	1	2	yes	—
296	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
297	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
298	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
299	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
300	C₂³×C₄	—	2	1	no	—
301	C₂×Q₁₆	—	2	4	no	—
302	C₂×Q₁₆	—	2	4	no	—
303	C₂×Q₁₆	—	4	2	no	—
304	C₂×SD₁₆	—	2	2	no	—
305	C₂×SD₁₆	—	2	2	no	—
306	C₂×SD₁₆	—	2	2	no	—
307	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
308	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
309	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
310	C₄⋊Q₈	—	2	2	no	—
311	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
312	#126	—	2	16	no	—
313	#124	—	2	4	no	—
314	#124	—	2	4	no	—
315	C₂²⋊Q₈	—	2	8	no	—
316	C₄×Q₈	—	2	4	no	—
317	C₄×Q₈	—	2	4	no	—
318	C₄×Q₈	—	2	2	no	—
319	C₄×Q₈	—	2	2	no	—
320	C₄×Q₈	—	2	2	no	—
321	C₄×Q₈	—	2	2	no	—
322	C₄×D₄	—	2	2	no	—
323	C₄×D₄	D₄×C₃₁	1	4	yes	—
324	#117	—	2	8	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	—	2	2	no	—
327	C₂×C₄⋊C₄	—	2	1	no	—
328	C₂×C₄⋊C₄	—	2	1	no	—
329	C₂×C₄⋊C₄	—	2	2	no	—
330	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
331	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
332	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
333	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
334	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₂²⋊C₄	—	2	1	no	—
337	C₂×C₂²⋊C₄	—	2	2	no	—
338	C₂×C₄²	—	2	2	no	—
339	C₂×C₄²	—	2	1	no	—
340	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
341	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
342	#107	—	2	2	no	—
343	C₄⋊C₈	—	2	2	no	—
344	#103	—	2	4	no	—
345	#103	—	2	4	no	—
346	#103	—	2	2	no	—
347	#103	—	4	2	no	—
348	#102	—	2	2	no	—
349	#102	—	4	2	no	—
350	#97	—	2	2	no	—
351	#95	—	2	1	no	—
352	#95	—	2	1	no	—
353	#95	—	2	2	no	—
354	#95	—	2	2	no	—
355	#95	—	2	2	no	—
356	C₃₁	#24201	1	1	yes	characteristic, direct product factor, Sylow
357	C₂⁴	—	2	1	no	—
358	C₂×Q₈	—	2	2	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	2	1	no	—
362	C₂×Q₈	—	2	1	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	2	1	no	—
365	C₂×Q₈	—	2	2	no	—
366	C₂×Q₈	—	2	2	no	—
367	C₂×Q₈	—	2	1	no	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	4	2	no	—
370	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
371	C₂×Q₈	#41×C₃₁	1	2	yes	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	—	2	1	no	—
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
386	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
387	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
389	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
390	Q₁₆	—	4	8	no	—
391	SD₁₆	—	2	8	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	#41×C₃₁	1	2	yes	—
401	C₄⋊C₄	—	2	8	no	—
402	C₄⋊C₄	—	2	4	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	4	4	no	—
412	C₄⋊C₄	—	4	2	no	—
413	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
414	C₂²⋊C₄	—	2	8	no	—
415	C₂²⋊C₄	—	2	8	no	—
416	C₂²⋊C₄	—	2	4	no	—
417	C₄²	—	2	8	no	—
418	C₄²	—	2	2	no	—
419	C₄²	—	2	2	no	—
420	C₄²	—	4	2	no	—
421	C₄²	C₂×D₄×C₃₁	1	2	yes	—
422	C₄²	C₂×D₄×C₃₁	1	2	yes	—
423	C₂³	—	2	2	no	—
424	C₂³	—	4	1	no	—
425	C₂³	—	4	1	no	—
426	C₂³	—	4	2	no	—
427	C₂³	—	4	2	no	—
428	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
429	Q₈	—	2	2	no	—
430	Q₈	—	2	2	no	—
431	Q₈	—	2	4	no	—
432	Q₈	—	2	4	no	—
433	Q₈	—	4	4	no	—
434	Q₈	—	4	2	no	—
435	Q₈	—	4	2	no	—
436	Q₈	—	4	1	no	—
437	Q₈	—	4	1	no	—
438	D₄	—	2	2	no	—
439	D₄	—	2	2	no	—
440	D₄	—	4	1	no	—
441	D₄	—	4	1	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	4	8	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	C₃₁×#120	1	2	yes	—
464	C₂×C₄	C₃₁×#120	1	2	yes	—
465	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
466	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
467	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
468	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
469	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
470	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
471	C₈	—	2	2	no	—
472	C₈	—	4	1	no	—
473	C₈	—	4	2	no	—
474	C₂²	—	4	2	no	—
475	C₂²	—	4	2	no	—
476	C₂²	—	4	2	no	—
477	C₂²	—	8	2	no	—
478	C₂²	—	8	2	no	—
479	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
480	C₂²	C₃₁×#548	1	1	yes	characteristic
481	C₂²	C₃₁×#550	1	1	yes	characteristic
482	C₂²	C₃₁×#575	1	2	yes	—
483	C₂²	C₃₁×#578	1	2	yes	—
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	—	4	4	no	—
487	C₄	—	4	2	no	—
488	C₄	—	4	2	no	—
489	C₄	—	4	2	no	—
490	C₄	—	4	2	no	—
491	C₄	—	8	2	no	—
492	C₄	—	8	1	no	—
493	C₄	C₃₁×#525	1	1	yes	characteristic
494	C₄	C₃₁×#537	1	1	yes	characteristic
495	C₄	C₃₁×#546	1	1	yes	characteristic
496	C₄	C₃₁×#547	1	1	yes	characteristic
497	C₂	—	8	2	no	—
498	C₂	C₃₁×#2407	1	1	yes	characteristic
499	C₂	C₃₁×#3005	1	1	yes	characteristic
500	C₂	C₃₁×#3011	1	1	yes	characteristic
501	C₂	C₃₁×#3377	1	2	yes	—
502	C₂	C₃₁×#3402	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#462	C₂	semidirect
4. C₃₁×#2581	C₂	semidirect
7. C₃₁×#1985	C₂	semidirect
8. C₃₁×#1960	C₂	semidirect
9. C₃₁×#1878	C₂	semidirect
10. C₃₁×#1863	C₂	semidirect
171. #24201	C₃₁	direct (N × H)
224. C₂×#462	C₂×C₃₁	semidirect
226. #2581	C₂×C₃₁	semidirect
229. #1985	C₂×C₃₁	semidirect
230. #1960	C₂×C₃₁	semidirect
231. #1878	C₂×C₃₁	semidirect
232. #1863	C₂×C₃₁	semidirect
356. C₃₁	#24201	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×#556	2
C₂×C₃₁×#462	2
C₂×C₃₁×#440	2
C₃₁×#2581	2
C₃₁×#2042	2
C₃₁×#2015	2
C₃₁×#1985	2
C₃₁×#1960	2
C₃₁×#1878	2
C₃₁×#1863	2
C₃₁×#1804	2
#24201	31

Subgroups — C31×#24201 #21480960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24201 #21480960