Subgroups — C₃₁×#2480 #16842413

Order 3968.

664 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#126	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#397	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#397	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#396	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#388	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#388	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#388	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#385	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#385	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#380	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#379	C₂	1	1	yes	maximal, characteristic
14	C₃₁×#379	C₂	1	1	yes	maximal, characteristic
15	C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂⁵×C₃₁	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#126	—	2	4	no	—
20	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
27	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic, complemented
44	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic, complemented
45	C₃₁×#95	C₂²	1	1	yes	characteristic, complemented
46	C₃₁×#95	C₂²	1	1	yes	characteristic, complemented
47	C₃₁×#95	C₂²	1	1	yes	characteristic, complemented
48	C₃₁×#95	C₂²	1	1	yes	characteristic, complemented
49	C₃₁×#95	C₂²	1	1	yes	characteristic, complemented
50	C₃₁×#95	C₂²	1	1	yes	characteristic, complemented
51	C₃₁×#95	C₂²	1	1	yes	characteristic, complemented
52	C₃₁×#95	C₂²	1	1	yes	characteristic, complemented
53	C₂⁴×C₃₁	—	2	2	no	—
54	C₂⁴×C₃₁	—	2	2	no	—
55	C₂⁴×C₃₁	—	2	2	no	—
56	C₂⁴×C₃₁	—	2	2	no	—
57	C₂⁴×C₃₁	—	2	2	no	—
58	C₂⁴×C₃₁	—	4	1	no	—
59	C₂⁴×C₃₁	—	4	1	no	—
60	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
61	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
62	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	—	2	2	no	—
64	C₂²×C₄×C₃₁	—	2	2	no	—
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	—	4	1	no	—
68	C₂²×C₄×C₃₁	—	4	1	no	—
69	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₃₁	—	2	2	no	—
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₄⋊C₄×C₃₁	—	2	2	no	—
84	C₄⋊C₄×C₃₁	—	4	1	no	—
85	C₄⋊C₄×C₃₁	—	4	1	no	—
86	C₄⋊C₄×C₃₁	—	4	1	no	—
87	C₂²⋊C₄×C₃₁	—	2	2	no	—
88	C₂²⋊C₄×C₃₁	—	2	2	no	—
89	C₂²⋊C₄×C₃₁	—	2	2	no	—
90	C₂²⋊C₄×C₃₁	—	2	2	no	—
91	C₂²⋊C₄×C₃₁	—	2	2	no	—
92	C₂²⋊C₄×C₃₁	—	2	2	no	—
93	C₂²⋊C₄×C₃₁	—	2	2	no	—
94	C₂²⋊C₄×C₃₁	—	2	2	no	—
95	C₂²⋊C₄×C₃₁	—	2	2	no	—
96	C₂²⋊C₄×C₃₁	—	2	2	no	—
97	C₂²⋊C₄×C₃₁	—	2	2	no	—
98	C₂²⋊C₄×C₃₁	—	4	1	no	—
99	C₂²⋊C₄×C₃₁	—	4	1	no	—
100	C₂²⋊C₄×C₃₁	—	4	1	no	—
101	C₂²⋊C₄×C₃₁	—	4	1	no	—
102	C₂²⋊C₄×C₃₁	D₄	1	4	yes	—
103	C₄²×C₃₁	—	2	2	no	—
104	C₄²×C₃₁	—	4	1	no	—
105	C₂³×C₃₁	—	2	1	no	—
106	C₂³×C₃₁	—	2	1	no	—
107	C₂³×C₃₁	—	2	1	no	—
108	C₂³×C₃₁	—	2	1	no	—
109	C₂³×C₃₁	—	2	8	no	—
110	C₂³×C₃₁	—	2	1	no	—
111	C₂³×C₃₁	—	2	1	no	—
112	C₂³×C₃₁	—	2	1	no	—
113	C₂³×C₃₁	—	2	1	no	—
114	C₂³×C₃₁	—	2	1	no	—
115	C₂³×C₃₁	—	2	1	no	—
116	C₂³×C₃₁	—	2	1	no	—
117	C₂³×C₃₁	—	2	1	no	—
118	C₂³×C₃₁	—	2	1	no	—
119	C₂³×C₃₁	—	2	1	no	—
120	C₂³×C₃₁	—	2	1	no	—
121	C₂³×C₃₁	—	2	1	no	—
122	C₂³×C₃₁	—	4	4	no	—
123	C₂³×C₃₁	—	4	4	no	—
124	C₂³×C₃₁	—	4	4	no	—
125	C₂³×C₃₁	—	4	4	no	—
126	C₂³×C₃₁	—	4	4	no	—
127	C₂³×C₃₁	—	4	4	no	—
128	C₂³×C₃₁	C₂×D₄	1	2	yes	—
129	C₂³×C₃₁	#41	1	2	yes	—
130	C₂³×C₃₁	#41	1	2	yes	—
131	C₂³×C₃₁	#41	1	2	yes	—
132	C₂³×C₃₁	#41	1	2	yes	—
133	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
134	C₂×C₄×C₃₁	—	2	1	no	—
135	C₂×C₄×C₃₁	—	2	1	no	—
136	C₂×C₄×C₃₁	—	2	1	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	2	1	no	—
139	C₂×C₄×C₃₁	—	2	1	no	—
140	C₂×C₄×C₃₁	—	2	1	no	—
141	C₂×C₄×C₃₁	—	2	1	no	—
142	C₂×C₄×C₃₁	—	2	1	no	—
143	C₂×C₄×C₃₁	—	2	1	no	—
144	C₂×C₄×C₃₁	—	2	1	no	—
145	C₂×C₄×C₃₁	—	2	1	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	4	4	no	—
169	C₂×C₄×C₃₁	—	4	4	no	—
170	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
171	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
172	#2480	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂²×C₃₁	—	2	2	no	—
174	C₂²×C₃₁	—	2	2	no	—
175	C₂²×C₃₁	—	2	2	no	—
176	C₂²×C₃₁	—	2	2	no	—
177	C₂²×C₃₁	—	2	2	no	—
178	C₂²×C₃₁	—	2	2	no	—
179	C₂²×C₃₁	—	2	2	no	—
180	C₂²×C₃₁	—	2	2	no	—
181	C₂²×C₃₁	—	2	2	no	—
182	C₂²×C₃₁	—	2	2	no	—
183	C₂²×C₃₁	—	2	2	no	—
184	C₂²×C₃₁	—	2	2	no	—
185	C₂²×C₃₁	—	4	16	no	—
186	C₂²×C₃₁	—	4	1	no	—
187	C₂²×C₃₁	—	4	1	no	—
188	C₂²×C₃₁	—	4	1	no	—
189	C₂²×C₃₁	—	4	1	no	—
190	C₂²×C₃₁	—	4	1	no	—
191	C₂²×C₃₁	—	4	1	no	—
192	C₂²×C₃₁	—	4	1	no	—
193	C₂²×C₃₁	—	4	1	no	—
194	C₂²×C₃₁	#126	1	4	yes	complemented
195	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
196	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
197	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
198	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
199	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
200	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
201	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
202	C₄×C₃₁	—	2	2	no	—
203	C₄×C₃₁	—	2	2	no	—
204	C₄×C₃₁	—	4	1	no	—
205	C₄×C₃₁	—	4	1	no	—
206	C₄×C₃₁	—	4	1	no	—
207	C₄×C₃₁	—	4	1	no	—
208	C₄×C₃₁	—	4	1	no	—
209	C₄×C₃₁	—	4	1	no	—
210	C₄×C₃₁	—	4	1	no	—
211	C₄×C₃₁	—	4	1	no	—
212	C₄×C₃₁	—	4	1	no	—
213	C₄×C₃₁	—	4	1	no	—
214	C₂×#126	C₂×C₃₁	1	1	yes	characteristic, complemented
215	C₂²×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic
216	#397	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#397	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#397	C₂×C₃₁	1	1	yes	characteristic, complemented
219	#396	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#388	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#388	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#388	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#385	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#385	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#380	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#379	C₂×C₃₁	1	1	yes	characteristic
227	#379	C₂×C₃₁	1	1	yes	characteristic
228	C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
229	C₂×C₃₁	—	2	4	no	—
230	C₂×C₃₁	—	4	2	no	—
231	C₂×C₃₁	—	4	2	no	—
232	C₂×C₃₁	#525	1	1	yes	characteristic
233	C₂×C₃₁	C₂×#126	1	1	yes	characteristic
234	C₂×C₃₁	#538	1	1	yes	characteristic
235	C₂×C₃₁	#546	1	1	yes	characteristic
236	C₂×C₃₁	#546	1	1	yes	characteristic
237	C₂×C₃₁	#551	1	1	yes	characteristic
238	C₂×C₃₁	#551	1	1	yes	characteristic
239	C₂⁵	C₂²×C₃₁	1	1	yes	characteristic
240	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
241	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
242	#126	—	2	4	no	—
243	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
244	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
245	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
246	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
247	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
248	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
249	C₂×C₂²⋊C₄	—	2	4	no	—
250	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
254	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic, complemented
267	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic, complemented
268	#95	C₂²×C₃₁	1	1	yes	characteristic, complemented
269	#95	C₂²×C₃₁	1	1	yes	characteristic, complemented
270	#95	C₂²×C₃₁	1	1	yes	characteristic, complemented
271	#95	C₂²×C₃₁	1	1	yes	characteristic, complemented
272	#95	C₂²×C₃₁	1	1	yes	characteristic, complemented
273	#95	C₂²×C₃₁	1	1	yes	characteristic, complemented
274	#95	C₂²×C₃₁	1	1	yes	characteristic, complemented
275	#95	C₂²×C₃₁	1	1	yes	characteristic, complemented
276	C₃₁	#2480	1	1	yes	characteristic, direct product factor, Sylow
277	C₂⁴	—	2	2	no	—
278	C₂⁴	—	2	2	no	—
279	C₂⁴	—	2	2	no	—
280	C₂⁴	—	2	2	no	—
281	C₂⁴	—	2	2	no	—
282	C₂⁴	—	4	1	no	—
283	C₂⁴	—	4	1	no	—
284	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
285	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
286	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
294	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
295	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
296	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
297	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
298	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
299	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
300	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
301	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
302	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
303	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
304	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₂²⋊C₄	—	2	2	no	—
312	C₂²⋊C₄	—	2	2	no	—
313	C₂²⋊C₄	—	2	2	no	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₂²⋊C₄	—	2	2	no	—
317	C₂²⋊C₄	—	2	2	no	—
318	C₂²⋊C₄	—	2	2	no	—
319	C₂²⋊C₄	—	2	2	no	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	4	1	no	—
323	C₂²⋊C₄	—	4	1	no	—
324	C₂²⋊C₄	—	4	1	no	—
325	C₂²⋊C₄	—	4	1	no	—
326	C₂²⋊C₄	D₄×C₃₁	1	4	yes	—
327	C₄²	—	2	2	no	—
328	C₄²	—	4	1	no	—
329	C₂³	—	2	1	no	—
330	C₂³	—	2	1	no	—
331	C₂³	—	2	1	no	—
332	C₂³	—	2	1	no	—
333	C₂³	—	2	8	no	—
334	C₂³	—	2	1	no	—
335	C₂³	—	2	1	no	—
336	C₂³	—	2	1	no	—
337	C₂³	—	2	1	no	—
338	C₂³	—	2	1	no	—
339	C₂³	—	2	1	no	—
340	C₂³	—	2	1	no	—
341	C₂³	—	2	1	no	—
342	C₂³	—	2	1	no	—
343	C₂³	—	2	1	no	—
344	C₂³	—	2	1	no	—
345	C₂³	—	2	1	no	—
346	C₂³	—	4	4	no	—
347	C₂³	—	4	4	no	—
348	C₂³	—	4	4	no	—
349	C₂³	—	4	4	no	—
350	C₂³	—	4	4	no	—
351	C₂³	—	4	4	no	—
352	C₂³	C₂×D₄×C₃₁	1	2	yes	—
353	C₂³	#41×C₃₁	1	2	yes	—
354	C₂³	#41×C₃₁	1	2	yes	—
355	C₂³	#41×C₃₁	1	2	yes	—
356	C₂³	#41×C₃₁	1	2	yes	—
357	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	4	4	no	—
393	C₂×C₄	—	4	4	no	—
394	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
395	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
396	C₂²	—	2	2	no	—
397	C₂²	—	2	2	no	—
398	C₂²	—	2	2	no	—
399	C₂²	—	2	2	no	—
400	C₂²	—	2	2	no	—
401	C₂²	—	2	2	no	—
402	C₂²	—	2	2	no	—
403	C₂²	—	2	2	no	—
404	C₂²	—	2	2	no	—
405	C₂²	—	2	2	no	—
406	C₂²	—	2	2	no	—
407	C₂²	—	2	2	no	—
408	C₂²	—	4	16	no	—
409	C₂²	—	4	1	no	—
410	C₂²	—	4	1	no	—
411	C₂²	—	4	1	no	—
412	C₂²	—	4	1	no	—
413	C₂²	—	4	1	no	—
414	C₂²	—	4	1	no	—
415	C₂²	—	4	1	no	—
416	C₂²	—	4	1	no	—
417	C₂²	C₃₁×#126	1	4	yes	complemented
418	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
419	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
420	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
421	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
422	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
423	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
424	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
425	C₄	—	2	2	no	—
426	C₄	—	2	2	no	—
427	C₄	—	4	1	no	—
428	C₄	—	4	1	no	—
429	C₄	—	4	1	no	—
430	C₄	—	4	1	no	—
431	C₄	—	4	1	no	—
432	C₄	—	4	1	no	—
433	C₄	—	4	1	no	—
434	C₄	—	4	1	no	—
435	C₄	—	4	1	no	—
436	C₄	—	4	1	no	—
437	C₂	—	2	4	no	—
438	C₂	—	4	2	no	—
439	C₂	—	4	2	no	—
440	C₂	C₃₁×#525	1	1	yes	characteristic
441	C₂	C₂×C₃₁×#126	1	1	yes	characteristic
442	C₂	C₃₁×#538	1	1	yes	characteristic
443	C₂	C₃₁×#546	1	1	yes	characteristic
444	C₂	C₃₁×#546	1	1	yes	characteristic
445	C₂	C₃₁×#551	1	1	yes	characteristic
446	C₂	C₃₁×#551	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂⁵×C₃₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#126	C₂	semidirect
3. C₃₁×#397	C₂	semidirect
4. C₃₁×#397	C₂	semidirect
5. C₃₁×#397	C₂	semidirect
6. C₃₁×#396	C₂	semidirect
7. C₃₁×#388	C₂	semidirect
8. C₃₁×#388	C₂	semidirect
9. C₃₁×#388	C₂	semidirect
10. C₃₁×#385	C₂	semidirect
11. C₃₁×#385	C₂	semidirect
12. C₃₁×#380	C₂	semidirect
15. C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
20. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
21. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
22. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
23. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
24. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
25. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
43. C₂×C₄²×C₃₁	C₂²	semidirect
44. C₂×C₄²×C₃₁	C₂²	semidirect
45. C₃₁×#95	C₂²	semidirect
46. C₃₁×#95	C₂²	semidirect
47. C₃₁×#95	C₂²	semidirect
48. C₃₁×#95	C₂²	semidirect
49. C₃₁×#95	C₂²	semidirect
50. C₃₁×#95	C₂²	semidirect
51. C₃₁×#95	C₂²	semidirect
52. C₃₁×#95	C₂²	semidirect
172. #2480	C₃₁	direct (N × H)
194. C₂²×C₃₁	#126	semidirect
214. C₂×#126	C₂×C₃₁	semidirect
216. #397	C₂×C₃₁	semidirect
217. #397	C₂×C₃₁	semidirect
218. #397	C₂×C₃₁	semidirect
219. #396	C₂×C₃₁	semidirect
220. #388	C₂×C₃₁	semidirect
221. #388	C₂×C₃₁	semidirect
222. #388	C₂×C₃₁	semidirect
223. #385	C₂×C₃₁	semidirect
224. #385	C₂×C₃₁	semidirect
225. #380	C₂×C₃₁	semidirect
228. C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
243. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
244. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
245. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
246. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
247. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
248. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
266. C₂×C₄²	C₂²×C₃₁	semidirect
267. C₂×C₄²	C₂²×C₃₁	semidirect
268. #95	C₂²×C₃₁	semidirect
269. #95	C₂²×C₃₁	semidirect
270. #95	C₂²×C₃₁	semidirect
271. #95	C₂²×C₃₁	semidirect
272. #95	C₂²×C₃₁	semidirect
273. #95	C₂²×C₃₁	semidirect
274. #95	C₂²×C₃₁	semidirect
275. #95	C₂²×C₃₁	semidirect
276. C₃₁	#2480	direct (N × H)
417. C₂²	C₃₁×#126	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#126	2
C₂²×C₂²⋊C₄×C₃₁	2
C₃₁×#397	2
C₃₁×#396	2
C₃₁×#388	2
C₃₁×#385	2
C₃₁×#380	2
C₃₁×#379	2
C₄×C₂²⋊C₄×C₃₁	2
#2480	31

Subgroups — C31×#2480 #16842413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2480 #16842413