Subgroups — C₃₁×#24745 #21481504

Order 7936.

1008 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#463	C₂	1	2	yes	maximal
2	C₂×C₃₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#442	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2547	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2349	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2066	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2042	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1960	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1877	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1804	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#124	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#463	—	2	8	no	—
20	C₃₁×#461	—	2	4	no	—
21	C₃₁×#442	—	2	4	no	—
22	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
25	C₂×C₃₁×#103	C₂²	1	2	yes	—
26	C₂×C₃₁×#103	C₂²	1	2	yes	—
27	C₂×C₃₁×#103	C₂²	1	2	yes	—
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
32	C₃₁×#397	—	2	1	no	—
33	C₃₁×#397	—	2	1	no	—
34	C₃₁×#391	C₂²	1	2	yes	—
35	C₃₁×#389	C₂²	1	2	yes	—
36	C₃₁×#385	—	2	2	no	—
37	C₃₁×#382	C₂²	1	1	yes	characteristic
38	C₃₁×#380	—	2	2	no	—
39	C₃₁×#380	—	2	1	no	—
40	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×C₃₁×#95	—	2	1	no	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	—	2	1	no	—
48	C₂³×C₄×C₃₁	—	2	1	no	—
49	C₂³×C₄×C₃₁	—	2	2	no	—
50	C₂×SD₁₆×C₃₁	—	2	4	no	—
51	C₂×SD₁₆×C₃₁	—	2	4	no	—
52	C₂×SD₁₆×C₃₁	—	2	4	no	—
53	C₂×D₈×C₃₁	—	2	2	no	—
54	C₂×D₈×C₃₁	—	2	2	no	—
55	C₂×D₈×C₃₁	D₄	1	4	yes	—
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	—	2	2	no	—
61	C₃₁×#124	—	2	4	no	—
62	C₃₁×#124	—	4	2	no	—
63	C₄×D₄×C₃₁	—	2	4	no	—
64	C₄×D₄×C₃₁	—	2	2	no	—
65	C₄×D₄×C₃₁	—	2	2	no	—
66	C₄×D₄×C₃₁	—	2	2	no	—
67	C₄×D₄×C₃₁	—	4	2	no	—
68	C₄×D₄×C₃₁	D₄	1	4	yes	—
69	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
81	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×#106	—	2	2	no	—
84	C₃₁×C₄⋊C₈	—	2	4	no	—
85	C₃₁×C₄⋊C₈	—	2	2	no	—
86	C₃₁×#103	—	2	4	no	—
87	C₃₁×#103	—	2	4	no	—
88	C₃₁×#103	—	4	2	no	—
89	C₃₁×#102	—	2	2	no	—
90	C₃₁×#102	—	2	4	no	—
91	C₃₁×#102	—	2	2	no	—
92	C₃₁×#97	—	2	2	no	—
93	C₃₁×#95	—	2	2	no	—
94	C₃₁×#95	—	2	2	no	—
95	C₃₁×#95	—	2	1	no	—
96	C₃₁×#95	—	2	2	no	—
97	C₃₁×#95	—	2	1	no	—
98	C₃₁×#95	D₄	1	1	yes	characteristic
99	C₃₁×#95	D₄	1	1	yes	characteristic
100	C₂⁴×C₃₁	—	2	1	no	—
101	C₂⁴×C₃₁	—	2	1	no	—
102	C₂×Q₈×C₃₁	—	2	2	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	—	2	1	no	—
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
117	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
118	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
119	C₂×D₄×C₃₁	#41	1	2	yes	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	4	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	4	4	no	—
126	C₂²×C₄×C₃₁	—	4	4	no	—
127	C₂²×C₄×C₃₁	—	4	4	no	—
128	C₂²×C₄×C₃₁	—	4	2	no	—
129	C₂²×C₄×C₃₁	—	8	2	no	—
130	C₂²×C₄×C₃₁	—	8	2	no	—
131	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
136	SD₁₆×C₃₁	—	4	8	no	—
137	D₈×C₃₁	—	2	8	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	2	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	C₂×D₄	1	2	yes	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₄⋊C₄×C₃₁	—	2	1	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	4	2	no	—
153	C₄⋊C₄×C₃₁	—	4	2	no	—
154	C₄⋊C₄×C₃₁	—	4	2	no	—
155	C₄⋊C₄×C₃₁	—	8	2	no	—
156	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
157	C₂²⋊C₄×C₃₁	—	2	4	no	—
158	C₂²⋊C₄×C₃₁	—	4	4	no	—
159	C₂²⋊C₄×C₃₁	—	4	4	no	—
160	C₂²⋊C₄×C₃₁	—	4	4	no	—
161	C₂²⋊C₄×C₃₁	—	4	2	no	—
162	C₄²×C₃₁	—	2	2	no	—
163	C₄²×C₃₁	—	2	2	no	—
164	C₄²×C₃₁	C₂×D₄	1	2	yes	—
165	C₄²×C₃₁	#41	1	2	yes	—
166	#24745	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₃₁	—	2	2	no	—
168	C₂³×C₃₁	—	2	2	no	—
169	C₂³×C₃₁	—	2	2	no	—
170	C₂³×C₃₁	—	2	2	no	—
171	C₂³×C₃₁	—	4	2	no	—
172	C₂³×C₃₁	—	4	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	—	4	2	no	—
179	Q₈×C₃₁	—	4	2	no	—
180	Q₈×C₃₁	—	4	2	no	—
181	Q₈×C₃₁	—	4	2	no	—
182	D₄×C₃₁	—	2	2	no	—
183	D₄×C₃₁	—	2	2	no	—
184	D₄×C₃₁	—	2	2	no	—
185	D₄×C₃₁	—	2	2	no	—
186	D₄×C₃₁	—	2	2	no	—
187	D₄×C₃₁	—	4	1	no	—
188	D₄×C₃₁	—	4	1	no	—
189	D₄×C₃₁	#121	1	4	yes	—
190	C₂×C₄×C₃₁	—	2	2	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	2	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	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	1	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	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	2	no	—
208	C₂×C₄×C₃₁	—	8	4	no	—
209	C₂×C₄×C₃₁	—	8	4	no	—
210	C₂×C₄×C₃₁	—	8	8	no	—
211	C₂×C₄×C₃₁	#121	1	2	yes	—
212	C₂×C₄×C₃₁	#121	1	2	yes	—
213	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
218	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
219	C₈×C₃₁	—	2	2	no	—
220	C₈×C₃₁	—	4	2	no	—
221	C₈×C₃₁	—	4	1	no	—
222	C₂×#463	C₂×C₃₁	1	2	yes	—
223	C₂×#461	C₂×C₃₁	1	1	yes	characteristic, complemented
224	C₂×#442	C₂×C₃₁	1	1	yes	characteristic
225	#2547	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#2349	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#2066	C₂×C₃₁	1	2	yes	complemented
228	#2042	C₂×C₃₁	1	2	yes	complemented
229	#1960	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#1877	C₂×C₃₁	1	2	yes	complemented
232	#1804	C₂×C₃₁	1	1	yes	characteristic, complemented
233	C₂²×C₃₁	—	2	4	no	—
234	C₂²×C₃₁	—	4	4	no	—
235	C₂²×C₃₁	—	4	4	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	4	4	no	—
240	C₂²×C₃₁	—	8	2	no	—
241	C₂²×C₃₁	—	8	1	no	—
242	C₂²×C₃₁	—	8	1	no	—
243	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
244	C₂²×C₃₁	D₄²	1	1	yes	characteristic
245	C₂²×C₃₁	#548	1	1	yes	characteristic
246	C₂²×C₃₁	#575	1	2	yes	—
247	C₂²×C₃₁	#577	1	2	yes	—
248	C₄×C₃₁	—	2	2	no	—
249	C₄×C₃₁	—	2	2	no	—
250	C₄×C₃₁	—	4	2	no	—
251	C₄×C₃₁	—	4	2	no	—
252	C₄×C₃₁	—	4	2	no	—
253	C₄×C₃₁	—	8	2	no	—
254	C₄×C₃₁	—	8	2	no	—
255	C₄×C₃₁	—	8	1	no	—
256	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
257	C₄×C₃₁	#537	1	1	yes	characteristic
258	C₄×C₃₁	#547	1	1	yes	characteristic
259	C₄×C₃₁	#553	1	1	yes	characteristic
260	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
261	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×#124	C₂²×C₃₁	1	2	yes	—
264	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
266	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
267	#463	—	2	8	no	—
268	#461	—	2	4	no	—
269	#442	—	2	4	no	—
270	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
273	C₂×#103	C₂²×C₃₁	1	2	yes	—
274	C₂×#103	C₂²×C₃₁	1	2	yes	—
275	C₂×#103	C₂²×C₃₁	1	2	yes	—
276	C₂×#102	C₂²×C₃₁	1	2	yes	—
277	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
280	#397	—	2	1	no	—
281	#397	—	2	1	no	—
282	#391	C₂²×C₃₁	1	2	yes	—
283	#389	C₂²×C₃₁	1	2	yes	—
284	#385	—	2	2	no	—
285	#382	C₂²×C₃₁	1	1	yes	characteristic
286	#380	—	2	2	no	—
287	#380	—	2	1	no	—
288	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×#95	—	2	1	no	—
290	#340	C₂²×C₃₁	1	2	yes	—
291	#336	C₂²×C₃₁	1	2	yes	—
292	C₂×C₃₁	—	4	4	no	—
293	C₂×C₃₁	—	8	2	no	—
294	C₂×C₃₁	#2416	1	1	yes	characteristic
295	C₂×C₃₁	#3040	1	1	yes	characteristic
296	C₂×C₃₁	#3042	1	1	yes	characteristic
297	C₂×C₃₁	#3267	1	2	yes	—
298	C₂×C₃₁	#3376	1	2	yes	—
299	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
300	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
301	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
302	C₂³×C₄	—	2	1	no	—
303	C₂³×C₄	—	2	1	no	—
304	C₂³×C₄	—	2	2	no	—
305	C₂×SD₁₆	—	2	4	no	—
306	C₂×SD₁₆	—	2	4	no	—
307	C₂×SD₁₆	—	2	4	no	—
308	C₂×D₈	—	2	2	no	—
309	C₂×D₈	—	2	2	no	—
310	C₂×D₈	D₄×C₃₁	1	4	yes	—
311	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
312	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
313	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
314	C₄⋊Q₈	—	2	2	no	—
315	C₄⋊Q₈	—	2	2	no	—
316	#124	—	2	4	no	—
317	#124	—	4	2	no	—
318	C₄×D₄	—	2	4	no	—
319	C₄×D₄	—	2	2	no	—
320	C₄×D₄	—	2	2	no	—
321	C₄×D₄	—	2	2	no	—
322	C₄×D₄	—	4	2	no	—
323	C₄×D₄	D₄×C₃₁	1	4	yes	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	—	2	2	no	—
327	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
328	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
329	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
330	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
331	C₂×C₂²⋊C₄	—	2	1	no	—
332	C₂×C₂²⋊C₄	—	2	2	no	—
333	C₂×C₂²⋊C₄	—	2	1	no	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
337	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
338	#106	—	2	2	no	—
339	C₄⋊C₈	—	2	4	no	—
340	C₄⋊C₈	—	2	2	no	—
341	#103	—	2	4	no	—
342	#103	—	2	4	no	—
343	#103	—	4	2	no	—
344	#102	—	2	2	no	—
345	#102	—	2	4	no	—
346	#102	—	2	2	no	—
347	#97	—	2	2	no	—
348	#95	—	2	1	no	—
349	#95	—	2	2	no	—
350	#95	—	2	2	no	—
351	#95	—	2	2	no	—
352	#95	—	2	1	no	—
353	#95	D₄×C₃₁	1	1	yes	characteristic
354	#95	D₄×C₃₁	1	1	yes	characteristic
355	C₃₁	#24745	1	1	yes	characteristic, direct product factor, Sylow
356	C₂⁴	—	2	1	no	—
357	C₂⁴	—	2	1	no	—
358	C₂×Q₈	—	2	2	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	—	2	2	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	—	2	1	no	—
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
373	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
374	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
375	C₂×D₄	#41×C₃₁	1	2	yes	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	4	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	4	4	no	—
382	C₂²×C₄	—	4	4	no	—
383	C₂²×C₄	—	4	4	no	—
384	C₂²×C₄	—	4	2	no	—
385	C₂²×C₄	—	8	2	no	—
386	C₂²×C₄	—	8	2	no	—
387	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
388	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
390	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
391	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
392	SD₁₆	—	4	8	no	—
393	D₈	—	2	8	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	4	2	no	—
409	C₄⋊C₄	—	4	2	no	—
410	C₄⋊C₄	—	4	2	no	—
411	C₄⋊C₄	—	8	2	no	—
412	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
413	C₂²⋊C₄	—	2	4	no	—
414	C₂²⋊C₄	—	4	4	no	—
415	C₂²⋊C₄	—	4	4	no	—
416	C₂²⋊C₄	—	4	4	no	—
417	C₂²⋊C₄	—	4	2	no	—
418	C₄²	—	2	2	no	—
419	C₄²	—	2	2	no	—
420	C₄²	C₂×D₄×C₃₁	1	2	yes	—
421	C₄²	#41×C₃₁	1	2	yes	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	2	no	—
426	C₂³	—	4	2	no	—
427	C₂³	—	4	2	no	—
428	C₂³	—	4	1	no	—
429	C₂³	—	4	1	no	—
430	C₂³	—	4	2	no	—
431	C₂³	—	4	2	no	—
432	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
433	Q₈	—	4	2	no	—
434	Q₈	—	4	2	no	—
435	Q₈	—	4	2	no	—
436	Q₈	—	4	2	no	—
437	D₄	—	2	2	no	—
438	D₄	—	2	2	no	—
439	D₄	—	2	2	no	—
440	D₄	—	2	2	no	—
441	D₄	—	2	2	no	—
442	D₄	—	4	1	no	—
443	D₄	—	4	1	no	—
444	D₄	C₃₁×#121	1	4	yes	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	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	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	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	2	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	—	8	4	no	—
464	C₂×C₄	—	8	4	no	—
465	C₂×C₄	—	8	8	no	—
466	C₂×C₄	C₃₁×#121	1	2	yes	—
467	C₂×C₄	C₃₁×#121	1	2	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	1	yes	characteristic
471	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
472	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
473	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
474	C₈	—	2	2	no	—
475	C₈	—	4	2	no	—
476	C₈	—	4	1	no	—
477	C₂²	—	2	4	no	—
478	C₂²	—	4	4	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	4	2	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	4	no	—
484	C₂²	—	8	2	no	—
485	C₂²	—	8	1	no	—
486	C₂²	—	8	1	no	—
487	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
488	C₂²	D₄²×C₃₁	1	1	yes	characteristic
489	C₂²	C₃₁×#548	1	1	yes	characteristic
490	C₂²	C₃₁×#575	1	2	yes	—
491	C₂²	C₃₁×#577	1	2	yes	—
492	C₄	—	2	2	no	—
493	C₄	—	2	2	no	—
494	C₄	—	4	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	2	no	—
497	C₄	—	8	2	no	—
498	C₄	—	8	2	no	—
499	C₄	—	8	1	no	—
500	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
501	C₄	C₃₁×#537	1	1	yes	characteristic
502	C₄	C₃₁×#547	1	1	yes	characteristic
503	C₄	C₃₁×#553	1	1	yes	characteristic
504	C₂	—	4	4	no	—
505	C₂	—	8	2	no	—
506	C₂	C₃₁×#2416	1	1	yes	characteristic
507	C₂	C₃₁×#3040	1	1	yes	characteristic
508	C₂	C₃₁×#3042	1	1	yes	characteristic
509	C₂	C₃₁×#3267	1	2	yes	—
510	C₂	C₃₁×#3376	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#461	C₂	semidirect
4. C₃₁×#2547	C₂	semidirect
5. C₃₁×#2349	C₂	semidirect
6. C₃₁×#2066	C₂	semidirect
7. C₃₁×#2042	C₂	semidirect
8. C₃₁×#1960	C₂	semidirect
9. C₃₁×#1940	C₂	semidirect
10. C₃₁×#1877	C₂	semidirect
11. C₃₁×#1804	C₂	semidirect
166. #24745	C₃₁	direct (N × H)
223. C₂×#461	C₂×C₃₁	semidirect
225. #2547	C₂×C₃₁	semidirect
226. #2349	C₂×C₃₁	semidirect
227. #2066	C₂×C₃₁	semidirect
228. #2042	C₂×C₃₁	semidirect
229. #1960	C₂×C₃₁	semidirect
230. #1940	C₂×C₃₁	semidirect
231. #1877	C₂×C₃₁	semidirect
232. #1804	C₂×C₃₁	semidirect
355. C₃₁	#24745	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×#463	2
C₂×C₃₁×#461	2
C₂×C₃₁×#442	2
C₃₁×#2547	2
C₃₁×#2349	2
C₃₁×#2066	2
C₃₁×#2042	2
C₃₁×#1960	2
C₃₁×#1940	2
C₃₁×#1877	2
C₃₁×#1804	2
#24745	31

Subgroups — C31×#24745 #21481504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24745 #21481504