Subgroups — C₂×C₃₁×#1931 #21477833

Order 7936.

1154 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#428	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#418	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₃₁×#100	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1931	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₃₁×#343	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#341	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#428	—	2	2	no	—
17	C₃₁×#428	C₂²	1	4	yes	complemented
18	C₂×C₃₁×#107	—	2	1	no	—
19	C₂×C₃₁×#106	—	2	1	no	—
20	C₃₁×#418	—	2	2	no	—
21	C₃₁×#418	C₂²	1	4	yes	—
22	C₂×C₃₁×#102	—	2	1	no	—
23	C₂×C₃₁×#102	—	2	1	no	—
24	C₂×C₃₁×#102	—	2	2	no	—
25	C₂×C₃₁×#102	—	2	1	no	—
26	C₂×C₃₁×#102	—	2	1	no	—
27	C₂×C₃₁×#102	C₂²	1	4	yes	—
28	C₂×C₃₁×#100	C₄	1	8	yes	complemented
29	C₂×C₃₁×#100	C₂²	1	4	yes	—
30	C₃₁×#405	—	2	2	no	—
31	C₃₁×#405	C₂²	1	4	yes	complemented
32	C₂×C₃₁×#97	—	2	1	no	—
33	C₂×C₄×C₈×C₃₁	—	2	1	no	—
34	C₃₁×#343	C₂²	1	4	yes	complemented
35	C₃₁×#341	C₂²	1	4	yes	complemented
36	C₂⁵×C₃₁	—	2	1	no	—
37	C₂²×D₄×C₃₁	—	2	1	no	—
38	C₂²×D₄×C₃₁	—	2	1	no	—
39	C₂²×D₄×C₃₁	—	2	1	no	—
40	C₂²×D₄×C₃₁	—	2	1	no	—
41	C₂²×D₄×C₃₁	—	2	2	no	—
42	C₂²×D₄×C₃₁	—	2	2	no	—
43	C₂²×D₄×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	—	2	2	no	—
45	C₂²×D₄×C₃₁	C₂×C₄	1	4	yes	—
46	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
47	C₂³×C₄×C₃₁	—	2	1	no	—
48	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₃₁	—	2	1	no	—
50	C₂×M₁₆×C₃₁	—	2	1	no	—
51	C₂×M₁₆×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂×M₁₆×C₃₁	—	2	1	no	—
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
57	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
58	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
59	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
60	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂²×C₈×C₃₁	—	2	1	no	—
62	C₂²×C₈×C₃₁	—	2	1	no	—
63	C₂²×C₈×C₃₁	—	2	2	no	—
64	C₂²×C₈×C₃₁	—	2	2	no	—
65	C₂²×C₈×C₃₁	—	2	1	no	—
66	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
68	C₂²×C₈×C₃₁	D₄	1	2	yes	—
69	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
70	C₃₁×#117	—	2	1	no	—
71	C₃₁×#117	—	2	1	no	—
72	C₃₁×#117	D₄	1	2	yes	—
73	C₃₁×#117	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
82	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₄²×C₃₁	—	2	1	no	—
84	C₃₁×#107	—	2	4	no	—
85	C₃₁×#106	—	2	4	no	—
86	C₃₁×#102	—	2	4	no	—
87	C₃₁×#102	—	2	4	no	—
88	C₃₁×#102	—	2	4	no	—
89	C₃₁×#102	—	2	4	no	—
90	C₃₁×#100	C₂×C₄	1	16	yes	complemented
91	C₃₁×#97	—	2	4	no	—
92	C₄×C₈×C₃₁	—	2	4	no	—
93	C₂⁴×C₃₁	—	2	4	no	—
94	C₂⁴×C₃₁	—	2	2	no	—
95	C₂⁴×C₃₁	—	4	1	no	—
96	C₂⁴×C₃₁	—	4	1	no	—
97	C₂⁴×C₃₁	—	4	2	no	—
98	C₂⁴×C₃₁	—	8	2	no	—
99	C₂⁴×C₃₁	—	8	2	no	—
100	C₂⁴×C₃₁	#41	1	1	yes	characteristic
101	C₂×D₄×C₃₁	—	2	4	no	—
102	C₂×D₄×C₃₁	—	2	4	no	—
103	C₂×D₄×C₃₁	—	2	4	no	—
104	C₂×D₄×C₃₁	—	2	4	no	—
105	C₂×D₄×C₃₁	—	2	4	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	4	1	no	—
112	C₂×D₄×C₃₁	—	4	1	no	—
113	C₂×D₄×C₃₁	—	4	2	no	—
114	C₂×D₄×C₃₁	—	4	2	no	—
115	C₂×D₄×C₃₁	—	4	2	no	—
116	C₂×D₄×C₃₁	—	4	2	no	—
117	C₂×D₄×C₃₁	—	4	2	no	—
118	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	4	1	no	—
123	C₂²×C₄×C₃₁	—	4	1	no	—
124	C₂²×C₄×C₃₁	—	8	2	no	—
125	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	#41	1	2	yes	—
132	C₂²×C₄×C₃₁	#41	1	2	yes	—
133	C₂²×C₄×C₃₁	#41	1	2	yes	—
134	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
135	M₁₆×C₃₁	—	2	4	no	—
136	M₁₆×C₃₁	—	2	4	no	—
137	M₁₆×C₃₁	—	2	2	no	—
138	M₁₆×C₃₁	—	2	2	no	—
139	M₁₆×C₃₁	—	2	2	no	—
140	M₁₆×C₃₁	—	2	2	no	—
141	M₁₆×C₃₁	—	4	2	no	—
142	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
143	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
144	C₂×C₈×C₃₁	—	2	4	no	—
145	C₂×C₈×C₃₁	—	2	4	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	2	2	no	—
148	C₂×C₈×C₃₁	—	2	4	no	—
149	C₂×C₈×C₃₁	—	2	2	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	4	2	no	—
155	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
156	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
157	C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	—	4	1	no	—
162	C₄⋊C₄×C₃₁	—	4	1	no	—
163	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
164	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
165	C₂²⋊C₄×C₃₁	—	2	2	no	—
166	C₂²⋊C₄×C₃₁	—	2	2	no	—
167	C₄²×C₃₁	—	2	2	no	—
168	C₄²×C₃₁	—	2	2	no	—
169	C₂×#1931	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂³×C₃₁	—	2	2	no	—
171	C₂³×C₃₁	—	2	2	no	—
172	C₂³×C₃₁	—	2	4	no	—
173	C₂³×C₃₁	—	4	2	no	—
174	C₂³×C₃₁	—	4	2	no	—
175	C₂³×C₃₁	—	4	4	no	—
176	C₂³×C₃₁	—	4	4	no	—
177	C₂³×C₃₁	—	8	8	no	—
178	C₂³×C₃₁	—	8	2	no	—
179	C₂³×C₃₁	—	8	2	no	—
180	C₂³×C₃₁	—	8	8	no	—
181	C₂³×C₃₁	—	8	2	no	—
182	C₂³×C₃₁	—	8	2	no	—
183	C₂³×C₃₁	—	8	2	no	—
184	C₂³×C₃₁	—	8	2	no	—
185	C₂³×C₃₁	#117	1	2	yes	—
186	C₂³×C₃₁	#121	1	1	yes	characteristic
187	C₂³×C₃₁	#121	1	1	yes	characteristic
188	C₂³×C₃₁	#126	1	2	yes	—
189	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
190	D₄×C₃₁	—	2	4	no	—
191	D₄×C₃₁	—	2	4	no	—
192	D₄×C₃₁	—	4	2	no	—
193	D₄×C₃₁	—	4	2	no	—
194	D₄×C₃₁	—	4	4	no	—
195	D₄×C₃₁	—	4	4	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	8	2	no	—
202	C₂×C₄×C₃₁	—	8	2	no	—
203	C₂×C₄×C₃₁	—	8	2	no	—
204	C₂×C₄×C₃₁	#117	1	2	yes	—
205	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
206	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
207	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
208	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
209	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
210	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
211	C₂×C₄×C₃₁	#123	1	2	yes	—
212	C₂×C₄×C₃₁	#123	1	2	yes	—
213	C₂×C₄×C₃₁	#124	1	2	yes	—
214	C₂×C₄×C₃₁	#124	1	2	yes	—
215	C₂×C₄×C₃₁	#126	1	2	yes	—
216	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
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	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
222	C₈×C₃₁	—	2	4	no	—
223	C₈×C₃₁	—	2	4	no	—
224	C₈×C₃₁	—	2	2	no	—
225	C₈×C₃₁	—	2	2	no	—
226	C₈×C₃₁	—	4	2	no	—
227	C₂×#428	C₂×C₃₁	1	1	yes	characteristic, complemented
228	C₂×#418	C₂×C₃₁	1	1	yes	characteristic
229	C₂²×#102	C₂×C₃₁	1	1	yes	characteristic
230	C₂²×#100	C₂×C₃₁	1	1	yes	characteristic
231	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#1931	C₂×C₃₁	1	8	yes	direct product factor
233	C₂×#343	C₂×C₃₁	1	1	yes	characteristic, complemented
234	C₂×#341	C₂×C₃₁	1	1	yes	characteristic, complemented
235	C₂²×C₃₁	—	2	4	no	—
236	C₂²×C₃₁	—	2	4	no	—
237	C₂²×C₃₁	—	2	1	no	—
238	C₂²×C₃₁	—	2	1	no	—
239	C₂²×C₃₁	—	2	1	no	—
240	C₂²×C₃₁	—	2	1	no	—
241	C₂²×C₃₁	—	4	4	no	—
242	C₂²×C₃₁	—	8	4	no	—
243	C₂²×C₃₁	—	8	4	no	—
244	C₂²×C₃₁	—	8	8	no	—
245	C₂²×C₃₁	—	8	8	no	—
246	C₂²×C₃₁	—	8	2	no	—
247	C₂²×C₃₁	—	8	2	no	—
248	C₂²×C₃₁	#388	1	2	yes	—
249	C₂²×C₃₁	#388	1	2	yes	—
250	C₂²×C₃₁	#471	1	2	yes	—
251	C₂²×C₃₁	#472	1	2	yes	—
252	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
253	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
254	C₂²×C₃₁	C₂×#126	1	1	yes	characteristic
255	C₄×C₃₁	—	4	2	no	—
256	C₄×C₃₁	—	4	2	no	—
257	C₄×C₃₁	—	8	2	no	—
258	C₄×C₃₁	#388	1	2	yes	—
259	C₄×C₃₁	#388	1	2	yes	—
260	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
261	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
262	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
263	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
264	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
265	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
266	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
267	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
268	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
270	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
271	#428	—	2	2	no	—
272	#428	C₂²×C₃₁	1	4	yes	complemented
273	C₂×#107	—	2	1	no	—
274	C₂×#106	—	2	1	no	—
275	#418	—	2	2	no	—
276	#418	C₂²×C₃₁	1	4	yes	—
277	C₂×#102	—	2	1	no	—
278	C₂×#102	—	2	1	no	—
279	C₂×#102	—	2	2	no	—
280	C₂×#102	—	2	1	no	—
281	C₂×#102	—	2	1	no	—
282	C₂×#102	C₂²×C₃₁	1	4	yes	—
283	C₂×#100	C₄×C₃₁	1	8	yes	complemented
284	C₂×#100	C₂²×C₃₁	1	4	yes	—
285	#405	—	2	2	no	—
286	#405	C₂²×C₃₁	1	4	yes	complemented
287	C₂×#97	—	2	1	no	—
288	C₂×C₄×C₈	—	2	1	no	—
289	#343	C₂²×C₃₁	1	4	yes	complemented
290	#341	C₂²×C₃₁	1	4	yes	complemented
291	C₂×C₃₁	—	2	1	no	—
292	C₂×C₃₁	—	2	1	no	—
293	C₂×C₃₁	—	2	2	no	—
294	C₂×C₃₁	—	8	4	no	—
295	C₂×C₃₁	#1931	1	4	yes	direct product factor
296	C₂×C₃₁	C₂×#388	1	1	yes	characteristic
297	C₂×C₃₁	C₂×#471	1	1	yes	characteristic
298	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
299	C₂⁵	—	2	1	no	—
300	C₂²×D₄	—	2	1	no	—
301	C₂²×D₄	—	2	1	no	—
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	—	2	1	no	—
304	C₂²×D₄	—	2	2	no	—
305	C₂²×D₄	—	2	2	no	—
306	C₂²×D₄	—	2	1	no	—
307	C₂²×D₄	—	2	2	no	—
308	C₂²×D₄	C₂×C₄×C₃₁	1	4	yes	—
309	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
310	C₂³×C₄	—	2	1	no	—
311	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂×M₁₆	—	2	1	no	—
313	C₂×M₁₆	—	2	1	no	—
314	C₂×M₁₆	—	2	2	no	—
315	C₂×M₁₆	—	2	2	no	—
316	C₂×M₁₆	—	2	1	no	—
317	C₂×M₁₆	—	2	2	no	—
318	C₂×M₁₆	—	2	2	no	—
319	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
320	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
321	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
322	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
323	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
324	C₂²×C₈	—	2	1	no	—
325	C₂²×C₈	—	2	1	no	—
326	C₂²×C₈	—	2	2	no	—
327	C₂²×C₈	—	2	2	no	—
328	C₂²×C₈	—	2	1	no	—
329	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
330	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
331	C₂²×C₈	D₄×C₃₁	1	2	yes	—
332	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
333	#117	—	2	1	no	—
334	#117	—	2	1	no	—
335	#117	D₄×C₃₁	1	2	yes	—
336	#117	C₂³×C₃₁	1	2	yes	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	—	2	1	no	—
342	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
343	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
344	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
345	C₂×C₂²⋊C₄	—	2	1	no	—
346	C₂×C₄²	—	2	1	no	—
347	#107	—	2	4	no	—
348	#106	—	2	4	no	—
349	#102	—	2	4	no	—
350	#102	—	2	4	no	—
351	#102	—	2	4	no	—
352	#102	—	2	4	no	—
353	#100	C₂×C₄×C₃₁	1	16	yes	complemented
354	#97	—	2	4	no	—
355	C₄×C₈	—	2	4	no	—
356	C₃₁	C₂×#1931	1	1	yes	characteristic, direct product factor, Sylow
357	C₂⁴	—	2	4	no	—
358	C₂⁴	—	2	2	no	—
359	C₂⁴	—	4	1	no	—
360	C₂⁴	—	4	1	no	—
361	C₂⁴	—	4	2	no	—
362	C₂⁴	—	8	2	no	—
363	C₂⁴	—	8	2	no	—
364	C₂⁴	#41×C₃₁	1	1	yes	characteristic
365	C₂×D₄	—	2	4	no	—
366	C₂×D₄	—	2	4	no	—
367	C₂×D₄	—	2	4	no	—
368	C₂×D₄	—	2	4	no	—
369	C₂×D₄	—	2	4	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	4	1	no	—
376	C₂×D₄	—	4	1	no	—
377	C₂×D₄	—	4	2	no	—
378	C₂×D₄	—	4	2	no	—
379	C₂×D₄	—	4	2	no	—
380	C₂×D₄	—	4	2	no	—
381	C₂×D₄	—	4	2	no	—
382	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	4	1	no	—
387	C₂²×C₄	—	4	1	no	—
388	C₂²×C₄	—	8	2	no	—
389	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
390	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
394	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
395	C₂²×C₄	#41×C₃₁	1	2	yes	—
396	C₂²×C₄	#41×C₃₁	1	2	yes	—
397	C₂²×C₄	#41×C₃₁	1	2	yes	—
398	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
399	M₁₆	—	2	4	no	—
400	M₁₆	—	2	4	no	—
401	M₁₆	—	2	2	no	—
402	M₁₆	—	2	2	no	—
403	M₁₆	—	2	2	no	—
404	M₁₆	—	2	2	no	—
405	M₁₆	—	4	2	no	—
406	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
407	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
408	C₂×C₈	—	2	4	no	—
409	C₂×C₈	—	2	4	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	4	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	4	2	no	—
419	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
420	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	4	1	no	—
426	C₄⋊C₄	—	4	1	no	—
427	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
428	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
429	C₂²⋊C₄	—	2	2	no	—
430	C₂²⋊C₄	—	2	2	no	—
431	C₄²	—	2	2	no	—
432	C₄²	—	2	2	no	—
433	C₂³	—	2	2	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	2	4	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	4	4	no	—
439	C₂³	—	4	4	no	—
440	C₂³	—	8	8	no	—
441	C₂³	—	8	2	no	—
442	C₂³	—	8	2	no	—
443	C₂³	—	8	8	no	—
444	C₂³	—	8	2	no	—
445	C₂³	—	8	2	no	—
446	C₂³	—	8	2	no	—
447	C₂³	—	8	2	no	—
448	C₂³	C₃₁×#117	1	2	yes	—
449	C₂³	C₃₁×#121	1	1	yes	characteristic
450	C₂³	C₃₁×#121	1	1	yes	characteristic
451	C₂³	C₃₁×#126	1	2	yes	—
452	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
453	D₄	—	2	4	no	—
454	D₄	—	2	4	no	—
455	D₄	—	4	2	no	—
456	D₄	—	4	2	no	—
457	D₄	—	4	4	no	—
458	D₄	—	4	4	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	8	2	no	—
465	C₂×C₄	—	8	2	no	—
466	C₂×C₄	—	8	2	no	—
467	C₂×C₄	C₃₁×#117	1	2	yes	—
468	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
469	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
470	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
471	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
472	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
473	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
474	C₂×C₄	C₃₁×#123	1	2	yes	—
475	C₂×C₄	C₃₁×#123	1	2	yes	—
476	C₂×C₄	C₃₁×#124	1	2	yes	—
477	C₂×C₄	C₃₁×#124	1	2	yes	—
478	C₂×C₄	C₃₁×#126	1	2	yes	—
479	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
480	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
481	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
482	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
483	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
484	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
485	C₈	—	2	4	no	—
486	C₈	—	2	4	no	—
487	C₈	—	2	2	no	—
488	C₈	—	2	2	no	—
489	C₈	—	4	2	no	—
490	C₂²	—	2	4	no	—
491	C₂²	—	2	4	no	—
492	C₂²	—	2	1	no	—
493	C₂²	—	2	1	no	—
494	C₂²	—	2	1	no	—
495	C₂²	—	2	1	no	—
496	C₂²	—	4	4	no	—
497	C₂²	—	8	4	no	—
498	C₂²	—	8	4	no	—
499	C₂²	—	8	8	no	—
500	C₂²	—	8	8	no	—
501	C₂²	—	8	2	no	—
502	C₂²	—	8	2	no	—
503	C₂²	C₃₁×#388	1	2	yes	—
504	C₂²	C₃₁×#388	1	2	yes	—
505	C₂²	C₃₁×#471	1	2	yes	—
506	C₂²	C₃₁×#472	1	2	yes	—
507	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
508	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
509	C₂²	C₂×C₃₁×#126	1	1	yes	characteristic
510	C₄	—	4	2	no	—
511	C₄	—	4	2	no	—
512	C₄	—	8	2	no	—
513	C₄	C₃₁×#388	1	2	yes	—
514	C₄	C₃₁×#388	1	2	yes	—
515	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
516	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
517	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
518	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
519	C₂	—	2	1	no	—
520	C₂	—	2	1	no	—
521	C₂	—	2	2	no	—
522	C₂	—	8	4	no	—
523	C₂	C₃₁×#1931	1	4	yes	direct product factor
524	C₂	C₂×C₃₁×#388	1	1	yes	characteristic
525	C₂	C₂×C₃₁×#471	1	1	yes	characteristic
526	C₂	C₂×C₃₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#428	C₂	semidirect
5. C₂×C₃₁×#405	C₂	semidirect
6. C₃₁×#1931	C₂	direct (N × H)
7. C₂×C₃₁×#343	C₂	semidirect
8. C₂×C₃₁×#341	C₂	semidirect
17. C₃₁×#428	C₂²	semidirect
28. C₂×C₃₁×#100	C₄	semidirect
31. C₃₁×#405	C₂²	semidirect
34. C₃₁×#343	C₂²	semidirect
35. C₃₁×#341	C₂²	semidirect
90. C₃₁×#100	C₂×C₄	semidirect
169. C₂×#1931	C₃₁	direct (N × H)
227. C₂×#428	C₂×C₃₁	semidirect
231. C₂×#405	C₂×C₃₁	semidirect
232. #1931	C₂×C₃₁	direct (N × H)
233. C₂×#343	C₂×C₃₁	semidirect
234. C₂×#341	C₂×C₃₁	semidirect
272. #428	C₂²×C₃₁	semidirect
283. C₂×#100	C₄×C₃₁	semidirect
286. #405	C₂²×C₃₁	semidirect
289. #343	C₂²×C₃₁	semidirect
290. #341	C₂²×C₃₁	semidirect
295. C₂×C₃₁	#1931	direct (N × H)
353. #100	C₂×C₄×C₃₁	semidirect
356. C₃₁	C₂×#1931	direct (N × H)
523. C₂	C₃₁×#1931	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×#428	2
C₂×C₃₁×#418	2
C₂²×C₃₁×#102	2
C₂²×C₃₁×#100	2
C₂×C₃₁×#405	2
C₃₁×#1931	2
C₂×C₃₁×#343	2
C₂×C₃₁×#341	2
C₂×#1931	31

Subgroups — C2×C31×#1931 #21477833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1931 #21477833