Subgroups — C₃₁×#25640 #21482399

Order 7936.

1102 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×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#497	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#464	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2592	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2016	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2013	C₂	1	2	yes	maximal
7	C₃₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1981	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1880	C₂	1	2	yes	maximal
10	C₃₁×#1878	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1763	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#557	—	2	4	no	—
17	C₃₁×#557	—	2	4	no	—
18	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#125	—	2	1	no	—
21	C₂×C₃₁×#125	—	2	2	no	—
22	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
24	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#497	—	2	4	no	—
28	C₃₁×#464	—	2	4	no	—
29	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#103	C₂²	1	2	yes	—
32	C₂×C₃₁×#103	C₂²	1	2	yes	—
33	C₂×C₃₁×#103	C₂²	1	2	yes	—
34	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
36	C₃₁×#397	—	2	1	no	—
37	C₃₁×#393	—	2	1	no	—
38	C₃₁×#391	C₂²	1	2	yes	—
39	C₃₁×#390	C₂²	1	2	yes	—
40	C₃₁×#388	—	2	2	no	—
41	C₄×C₄⋊C₄×C₃₁	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	1	yes	characteristic
47	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
49	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×Q₁₆×C₃₁	—	2	2	no	—
51	C₂×Q₁₆×C₃₁	—	2	4	no	—
52	C₂×Q₁₆×C₃₁	—	2	4	no	—
53	C₂×Q₁₆×C₃₁	—	4	2	no	—
54	C₂×Q₁₆×C₃₁	—	4	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×SD₁₆×C₃₁	—	2	4	no	—
57	C₂×SD₁₆×C₃₁	—	2	4	no	—
58	C₂×SD₁₆×C₃₁	—	4	2	no	—
59	C₂×SD₁₆×C₃₁	—	4	2	no	—
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
63	C₃₁×C₄⋊Q₈	—	2	4	no	—
64	C₃₁×C₄⋊Q₈	—	2	4	no	—
65	C₃₁×C₄⋊Q₈	—	2	2	no	—
66	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
67	C₃₁×#125	—	2	16	no	—
68	C₃₁×#125	—	2	8	no	—
69	C₃₁×#124	—	2	2	no	—
70	C₃₁×#124	—	2	2	no	—
71	C₄×Q₈×C₃₁	—	2	8	no	—
72	C₄×Q₈×C₃₁	—	2	4	no	—
73	C₄×Q₈×C₃₁	—	2	4	no	—
74	C₄×Q₈×C₃₁	—	2	2	no	—
75	C₄×Q₈×C₃₁	—	2	2	no	—
76	C₄×Q₈×C₃₁	—	2	2	no	—
77	C₄×Q₈×C₃₁	D₄	1	4	yes	—
78	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
88	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
92	C₂×C₄²×C₃₁	—	2	2	no	—
93	C₂×C₄²×C₃₁	—	2	1	no	—
94	C₂×C₄²×C₃₁	—	2	1	no	—
95	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
96	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
97	C₃₁×C₄⋊C₈	—	2	2	no	—
98	C₃₁×#103	—	2	4	no	—
99	C₃₁×#103	—	2	2	no	—
100	C₃₁×#103	—	4	2	no	—
101	C₃₁×#103	—	4	2	no	—
102	C₃₁×#102	—	2	2	no	—
103	C₃₁×#97	—	2	2	no	—
104	C₃₁×#95	—	2	1	no	—
105	C₃₁×#95	—	2	2	no	—
106	C₃₁×#95	—	2	2	no	—
107	C₂⁴×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	—	2	4	no	—
110	C₂×Q₈×C₃₁	—	2	2	no	—
111	C₂×Q₈×C₃₁	—	2	1	no	—
112	C₂×Q₈×C₃₁	—	2	1	no	—
113	C₂×Q₈×C₃₁	—	2	1	no	—
114	C₂×Q₈×C₃₁	—	2	2	no	—
115	C₂×Q₈×C₃₁	—	2	2	no	—
116	C₂×Q₈×C₃₁	—	2	2	no	—
117	C₂×Q₈×C₃₁	—	2	2	no	—
118	C₂×Q₈×C₃₁	—	2	2	no	—
119	C₂×Q₈×C₃₁	—	2	2	no	—
120	C₂×Q₈×C₃₁	—	2	2	no	—
121	C₂×Q₈×C₃₁	—	2	2	no	—
122	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
123	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
124	C₂×D₄×C₃₁	—	2	1	no	—
125	C₂×D₄×C₃₁	—	2	1	no	—
126	C₂×D₄×C₃₁	—	2	2	no	—
127	C₂×D₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	2	1	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	2	2	no	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
138	Q₁₆×C₃₁	—	4	8	no	—
139	Q₁₆×C₃₁	—	4	4	no	—
140	SD₁₆×C₃₁	—	4	8	no	—
141	SD₁₆×C₃₁	—	4	4	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	2	2	no	—
148	C₂×C₈×C₃₁	#41	1	4	yes	—
149	C₄⋊C₄×C₃₁	—	2	8	no	—
150	C₄⋊C₄×C₃₁	—	2	8	no	—
151	C₄⋊C₄×C₃₁	—	2	8	no	—
152	C₄⋊C₄×C₃₁	—	2	4	no	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	—	2	4	no	—
155	C₄⋊C₄×C₃₁	—	2	4	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
161	C₂²⋊C₄×C₃₁	—	4	2	no	—
162	C₂²⋊C₄×C₃₁	—	4	2	no	—
163	C₂²⋊C₄×C₃₁	—	4	4	no	—
164	C₂²⋊C₄×C₃₁	—	8	2	no	—
165	C₄²×C₃₁	—	2	8	no	—
166	C₄²×C₃₁	—	2	4	no	—
167	C₄²×C₃₁	—	2	2	no	—
168	C₄²×C₃₁	—	2	2	no	—
169	C₄²×C₃₁	—	8	1	no	—
170	C₄²×C₃₁	C₂×D₄	1	2	yes	—
171	C₄²×C₃₁	C₂×D₄	1	2	yes	—
172	#25640	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂³×C₃₁	—	4	1	no	—
174	C₂³×C₃₁	—	4	2	no	—
175	C₂³×C₃₁	—	4	2	no	—
176	C₂³×C₃₁	—	4	1	no	—
177	C₂³×C₃₁	—	4	1	no	—
178	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	Q₈×C₃₁	—	2	8	no	—
180	Q₈×C₃₁	—	2	2	no	—
181	Q₈×C₃₁	—	2	2	no	—
182	Q₈×C₃₁	—	2	4	no	—
183	Q₈×C₃₁	—	2	4	no	—
184	Q₈×C₃₁	—	4	2	no	—
185	Q₈×C₃₁	—	4	2	no	—
186	Q₈×C₃₁	—	4	2	no	—
187	Q₈×C₃₁	—	4	1	no	—
188	Q₈×C₃₁	—	4	1	no	—
189	Q₈×C₃₁	—	4	2	no	—
190	D₄×C₃₁	—	4	1	no	—
191	D₄×C₃₁	—	4	1	no	—
192	D₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	1	no	—
197	C₂×C₄×C₃₁	—	2	1	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	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	4	2	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	1	no	—
209	C₂×C₄×C₃₁	—	4	1	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	#120	1	2	yes	—
215	C₂×C₄×C₃₁	#120	1	2	yes	—
216	C₂×C₄×C₃₁	#137	1	4	yes	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
220	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
221	C₈×C₃₁	—	2	4	no	—
222	C₈×C₃₁	—	4	1	no	—
223	C₈×C₃₁	—	4	1	no	—
224	C₂×#557	C₂×C₃₁	1	1	yes	characteristic, complemented
225	C₂×#497	C₂×C₃₁	1	1	yes	characteristic
226	C₂×#464	C₂×C₃₁	1	1	yes	characteristic
227	#2592	C₂×C₃₁	1	1	yes	characteristic
228	#2016	C₂×C₃₁	1	2	yes	complemented
229	#2013	C₂×C₃₁	1	2	yes	—
230	#1985	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#1981	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#1880	C₂×C₃₁	1	2	yes	—
233	#1878	C₂×C₃₁	1	2	yes	complemented
234	#1763	C₂×C₃₁	1	1	yes	characteristic, complemented
235	C₂²×C₃₁	—	8	2	no	—
236	C₂²×C₃₁	—	8	2	no	—
237	C₂²×C₃₁	—	8	1	no	—
238	C₂²×C₃₁	—	8	1	no	—
239	C₂²×C₃₁	—	8	1	no	—
240	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
241	C₂²×C₃₁	#550	1	2	yes	—
242	C₂²×C₃₁	C₂×#137	1	2	yes	—
243	C₂²×C₃₁	#577	1	1	yes	characteristic
244	C₂²×C₃₁	#578	1	1	yes	characteristic
245	C₄×C₃₁	—	2	2	no	—
246	C₄×C₃₁	—	2	4	no	—
247	C₄×C₃₁	—	2	2	no	—
248	C₄×C₃₁	—	4	2	no	—
249	C₄×C₃₁	—	4	4	no	—
250	C₄×C₃₁	—	4	2	no	—
251	C₄×C₃₁	—	4	2	no	—
252	C₄×C₃₁	—	8	2	no	—
253	C₄×C₃₁	—	8	1	no	—
254	C₄×C₃₁	#532	1	1	yes	characteristic
255	C₄×C₃₁	#535	1	1	yes	characteristic
256	C₄×C₃₁	#546	1	2	yes	—
257	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
258	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
259	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
260	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
261	#557	—	2	4	no	—
262	#557	—	2	4	no	—
263	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
264	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×#125	—	2	2	no	—
266	C₂×#125	—	2	1	no	—
267	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×C₄×Q₈	—	2	1	no	—
269	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
270	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
272	#497	—	2	4	no	—
273	#464	—	2	4	no	—
274	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×#103	C₂²×C₃₁	1	2	yes	—
277	C₂×#103	C₂²×C₃₁	1	2	yes	—
278	C₂×#103	C₂²×C₃₁	1	2	yes	—
279	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
281	#397	—	2	1	no	—
282	#393	—	2	1	no	—
283	#391	C₂²×C₃₁	1	2	yes	—
284	#390	C₂²×C₃₁	1	2	yes	—
285	#388	—	2	2	no	—
286	C₄×C₄⋊C₄	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	—	16	1	no	—
291	C₂×C₃₁	#2431	1	1	yes	characteristic
292	C₂×C₃₁	#3004	1	1	yes	characteristic
293	C₂×C₃₁	#3013	1	1	yes	characteristic
294	C₂×C₃₁	#3398	1	2	yes	—
295	C₂×C₃₁	#3400	1	2	yes	—
296	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
297	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
299	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
300	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
301	C₂×Q₁₆	—	2	2	no	—
302	C₂×Q₁₆	—	2	4	no	—
303	C₂×Q₁₆	—	2	4	no	—
304	C₂×Q₁₆	—	4	2	no	—
305	C₂×Q₁₆	—	4	2	no	—
306	C₂×SD₁₆	—	2	2	no	—
307	C₂×SD₁₆	—	2	4	no	—
308	C₂×SD₁₆	—	2	4	no	—
309	C₂×SD₁₆	—	4	2	no	—
310	C₂×SD₁₆	—	4	2	no	—
311	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
312	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
313	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
314	C₄⋊Q₈	—	2	4	no	—
315	C₄⋊Q₈	—	2	4	no	—
316	C₄⋊Q₈	—	2	2	no	—
317	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
318	#125	—	2	16	no	—
319	#125	—	2	8	no	—
320	#124	—	2	2	no	—
321	#124	—	2	2	no	—
322	C₄×Q₈	—	2	8	no	—
323	C₄×Q₈	—	2	4	no	—
324	C₄×Q₈	—	2	4	no	—
325	C₄×Q₈	—	2	2	no	—
326	C₄×Q₈	—	2	2	no	—
327	C₄×Q₈	—	2	2	no	—
328	C₄×Q₈	D₄×C₃₁	1	4	yes	—
329	C₂×C₄⋊C₄	—	2	2	no	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	—	2	2	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	2	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
336	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
337	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
338	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
339	C₂×C₂²⋊C₄	—	2	1	no	—
340	C₂×C₂²⋊C₄	—	2	1	no	—
341	C₂×C₂²⋊C₄	—	2	2	no	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₄²	—	2	2	no	—
344	C₂×C₄²	—	2	1	no	—
345	C₂×C₄²	—	2	1	no	—
346	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
347	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
348	C₄⋊C₈	—	2	2	no	—
349	#103	—	2	4	no	—
350	#103	—	2	2	no	—
351	#103	—	4	2	no	—
352	#103	—	4	2	no	—
353	#102	—	2	2	no	—
354	#97	—	2	2	no	—
355	#95	—	2	1	no	—
356	#95	—	2	2	no	—
357	#95	—	2	2	no	—
358	C₃₁	#25640	1	1	yes	characteristic, direct product factor, Sylow
359	C₂⁴	—	2	1	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	—	2	4	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	1	no	—
364	C₂×Q₈	—	2	1	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	—	2	2	no	—
367	C₂×Q₈	—	2	2	no	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	2	2	no	—
370	C₂×Q₈	—	2	2	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
375	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
386	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
387	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
389	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
390	Q₁₆	—	4	8	no	—
391	Q₁₆	—	4	4	no	—
392	SD₁₆	—	4	8	no	—
393	SD₁₆	—	4	4	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	#41×C₃₁	1	4	yes	—
401	C₄⋊C₄	—	2	8	no	—
402	C₄⋊C₄	—	2	8	no	—
403	C₄⋊C₄	—	2	8	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
413	C₂²⋊C₄	—	4	2	no	—
414	C₂²⋊C₄	—	4	2	no	—
415	C₂²⋊C₄	—	4	4	no	—
416	C₂²⋊C₄	—	8	2	no	—
417	C₄²	—	2	8	no	—
418	C₄²	—	2	4	no	—
419	C₄²	—	2	2	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	8	1	no	—
422	C₄²	C₂×D₄×C₃₁	1	2	yes	—
423	C₄²	C₂×D₄×C₃₁	1	2	yes	—
424	C₂³	—	4	1	no	—
425	C₂³	—	4	2	no	—
426	C₂³	—	4	2	no	—
427	C₂³	—	4	1	no	—
428	C₂³	—	4	1	no	—
429	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
430	Q₈	—	2	8	no	—
431	Q₈	—	2	2	no	—
432	Q₈	—	2	2	no	—
433	Q₈	—	2	4	no	—
434	Q₈	—	2	4	no	—
435	Q₈	—	4	2	no	—
436	Q₈	—	4	2	no	—
437	Q₈	—	4	2	no	—
438	Q₈	—	4	1	no	—
439	Q₈	—	4	1	no	—
440	Q₈	—	4	2	no	—
441	D₄	—	4	1	no	—
442	D₄	—	4	1	no	—
443	D₄	—	4	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	C₃₁×#120	1	2	yes	—
466	C₂×C₄	C₃₁×#120	1	2	yes	—
467	C₂×C₄	C₃₁×#137	1	4	yes	—
468	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
469	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
470	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
471	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
472	C₈	—	2	4	no	—
473	C₈	—	4	1	no	—
474	C₈	—	4	1	no	—
475	C₂²	—	8	2	no	—
476	C₂²	—	8	2	no	—
477	C₂²	—	8	1	no	—
478	C₂²	—	8	1	no	—
479	C₂²	—	8	1	no	—
480	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
481	C₂²	C₃₁×#550	1	2	yes	—
482	C₂²	C₂×C₃₁×#137	1	2	yes	—
483	C₂²	C₃₁×#577	1	1	yes	characteristic
484	C₂²	C₃₁×#578	1	1	yes	characteristic
485	C₄	—	2	2	no	—
486	C₄	—	2	4	no	—
487	C₄	—	2	2	no	—
488	C₄	—	4	2	no	—
489	C₄	—	4	4	no	—
490	C₄	—	4	2	no	—
491	C₄	—	4	2	no	—
492	C₄	—	8	2	no	—
493	C₄	—	8	1	no	—
494	C₄	C₃₁×#532	1	1	yes	characteristic
495	C₄	C₃₁×#535	1	1	yes	characteristic
496	C₄	C₃₁×#546	1	2	yes	—
497	C₂	—	16	1	no	—
498	C₂	C₃₁×#2431	1	1	yes	characteristic
499	C₂	C₃₁×#3004	1	1	yes	characteristic
500	C₂	C₃₁×#3013	1	1	yes	characteristic
501	C₂	C₃₁×#3398	1	2	yes	—
502	C₂	C₃₁×#3400	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
1. C₂×C₃₁×#557	C₂	semidirect
5. C₃₁×#2016	C₂	semidirect
7. C₃₁×#1985	C₂	semidirect
8. C₃₁×#1981	C₂	semidirect
10. C₃₁×#1878	C₂	semidirect
11. C₃₁×#1763	C₂	semidirect
172. #25640	C₃₁	direct (N × H)
224. C₂×#557	C₂×C₃₁	semidirect
228. #2016	C₂×C₃₁	semidirect
230. #1985	C₂×C₃₁	semidirect
231. #1981	C₂×C₃₁	semidirect
233. #1878	C₂×C₃₁	semidirect
234. #1763	C₂×C₃₁	semidirect
358. C₃₁	#25640	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×#557	2
C₂×C₃₁×#497	2
C₂×C₃₁×#464	2
C₃₁×#2592	2
C₃₁×#2016	2
C₃₁×#2013	2
C₃₁×#1985	2
C₃₁×#1981	2
C₃₁×#1880	2
C₃₁×#1878	2
C₃₁×#1763	2
#25640	31

Subgroups — C31×#25640 #21482399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25640 #21482399