Subgroups — C₃₁×#24781 #21481540

Order 7936.

1010 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₃₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2417	C₂	1	2	yes	maximal
6	C₃₁×#2068	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2036	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1969	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1942	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1872	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#467	—	2	4	no	—
20	C₃₁×#467	—	2	4	no	—
21	C₈×D₄×C₃₁	—	2	4	no	—
22	C₂×C₃₁×#107	—	2	4	no	—
23	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#107	C₂²	1	2	yes	—
26	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#103	C₂²	1	2	yes	—
29	C₂×C₃₁×#103	C₂²	1	2	yes	—
30	C₂×C₃₁×#103	C₂²	1	2	yes	—
31	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#398	—	2	2	no	—
35	C₃₁×#396	—	2	2	no	—
36	C₃₁×#391	C₂²	1	2	yes	—
37	C₃₁×#389	C₂²	1	2	yes	—
38	C₃₁×#386	—	2	2	no	—
39	C₃₁×#380	—	2	2	no	—
40	C₃₁×#380	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
44	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	—	2	2	no	—
46	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×Q₁₆×C₃₁	—	2	2	no	—
49	C₂×Q₁₆×C₃₁	—	2	2	no	—
50	C₂×Q₁₆×C₃₁	—	2	2	no	—
51	C₂²×C₈×C₃₁	—	2	2	no	—
52	C₂²×C₈×C₃₁	—	2	2	no	—
53	C₂²×C₈×C₃₁	—	2	2	no	—
54	C₂²×C₈×C₃₁	—	2	2	no	—
55	C₂²×C₈×C₃₁	—	2	2	no	—
56	C₂²×C₈×C₃₁	D₄	1	4	yes	—
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×C₂²⋊Q₈	—	2	4	no	—
62	C₃₁×C₂²⋊Q₈	—	2	4	no	—
63	C₃₁×C₂²⋊Q₈	—	2	4	no	—
64	C₃₁×C₂²⋊Q₈	—	4	2	no	—
65	C₄×D₄×C₃₁	—	2	2	no	—
66	C₄×D₄×C₃₁	—	2	2	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
73	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
81	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×#107	—	2	2	no	—
84	C₃₁×#107	—	2	2	no	—
85	C₃₁×#107	—	2	4	no	—
86	C₃₁×#107	D₄	1	4	yes	complemented
87	C₃₁×C₄⋊C₈	—	2	2	no	—
88	C₃₁×#103	—	2	4	no	—
89	C₃₁×#103	—	2	4	no	—
90	C₃₁×#103	—	4	2	no	—
91	C₃₁×C₂²⋊C₈	—	2	2	no	—
92	C₃₁×C₂²⋊C₈	—	2	2	no	—
93	C₄×C₈×C₃₁	—	2	2	no	—
94	C₃₁×#95	—	2	2	no	—
95	C₃₁×#95	—	2	2	no	—
96	C₃₁×#95	—	2	2	no	—
97	C₃₁×#95	—	2	2	no	—
98	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×Q₈×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	—	2	2	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×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	Q₁₆	1	4	yes	complemented
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	1	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	4	4	no	—
124	C₂²×C₄×C₃₁	—	4	4	no	—
125	C₂²×C₄×C₃₁	—	8	2	no	—
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	2	yes	—
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	#41	1	2	yes	—
132	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
133	Q₁₆×C₃₁	—	4	4	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	8	no	—
137	C₂×C₈×C₃₁	—	2	1	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	1	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	4	4	no	—
144	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
145	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
146	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
147	C₂×C₈×C₃₁	#41	1	2	yes	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	2	2	no	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	4	4	no	—
155	C₄⋊C₄×C₃₁	—	4	4	no	—
156	C₄⋊C₄×C₃₁	—	4	4	no	—
157	C₄⋊C₄×C₃₁	—	8	2	no	—
158	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
159	C₂²⋊C₄×C₃₁	—	2	2	no	—
160	C₂²⋊C₄×C₃₁	—	2	2	no	—
161	C₂²⋊C₄×C₃₁	—	4	4	no	—
162	C₂²⋊C₄×C₃₁	—	4	4	no	—
163	C₂²⋊C₄×C₃₁	—	8	2	no	—
164	C₄²×C₃₁	—	2	1	no	—
165	C₄²×C₃₁	—	2	1	no	—
166	#24781	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₃₁	—	2	1	no	—
168	C₂³×C₃₁	—	2	1	no	—
169	C₂³×C₃₁	—	2	1	no	—
170	C₂³×C₃₁	—	2	1	no	—
171	C₂³×C₃₁	—	2	1	no	—
172	C₂³×C₃₁	—	2	1	no	—
173	C₂³×C₃₁	—	2	1	no	—
174	C₂³×C₃₁	—	2	1	no	—
175	C₂³×C₃₁	—	2	1	no	—
176	C₂³×C₃₁	—	2	1	no	—
177	C₂³×C₃₁	#121	1	2	yes	—
178	C₂³×C₃₁	C₂×Q₁₆	1	2	yes	—
179	C₂³×C₃₁	C₂×Q₁₆	1	2	yes	—
180	C₂³×C₃₁	#135	1	2	yes	—
181	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
182	Q₈×C₃₁	—	4	2	no	—
183	Q₈×C₃₁	—	4	2	no	—
184	Q₈×C₃₁	—	4	2	no	—
185	Q₈×C₃₁	—	4	2	no	—
186	D₄×C₃₁	—	2	8	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	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	1	no	—
196	C₂×C₄×C₃₁	—	2	1	no	—
197	C₂×C₄×C₃₁	—	2	4	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	1	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	1	no	—
205	C₂×C₄×C₃₁	—	4	1	no	—
206	C₂×C₄×C₃₁	—	4	1	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	—	4	2	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	—	8	8	no	—
218	C₂×C₄×C₃₁	#121	1	2	yes	—
219	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
220	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
225	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
226	C₈×C₃₁	—	2	2	no	—
227	C₈×C₃₁	—	4	1	no	—
228	C₈×C₃₁	—	4	1	no	—
229	C₈×C₃₁	#121	1	4	yes	—
230	C₂×#467	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂×#467	C₂×C₃₁	1	1	yes	characteristic, complemented
232	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
233	C₂²×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#2417	C₂×C₃₁	1	2	yes	—
235	#2068	C₂×C₃₁	1	2	yes	complemented
236	#2036	C₂×C₃₁	1	2	yes	complemented
237	#1969	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#1942	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#1872	C₂×C₃₁	1	2	yes	complemented
240	#1808	C₂×C₃₁	1	1	yes	characteristic, complemented
241	C₂²×C₃₁	—	2	2	no	—
242	C₂²×C₃₁	—	2	2	no	—
243	C₂²×C₃₁	—	2	2	no	—
244	C₂²×C₃₁	—	2	2	no	—
245	C₂²×C₃₁	—	2	2	no	—
246	C₂²×C₃₁	—	2	2	no	—
247	C₂²×C₃₁	—	2	2	no	—
248	C₂²×C₃₁	—	2	2	no	—
249	C₂²×C₃₁	—	2	2	no	—
250	C₂²×C₃₁	—	4	1	no	—
251	C₂²×C₃₁	—	4	1	no	—
252	C₂²×C₃₁	—	4	1	no	—
253	C₂²×C₃₁	—	4	1	no	—
254	C₂²×C₃₁	#467	1	4	yes	complemented
255	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
256	C₂²×C₃₁	D₄²	1	1	yes	characteristic
257	C₂²×C₃₁	#547	1	1	yes	characteristic
258	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
259	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
260	C₂²×C₃₁	#576	1	1	yes	characteristic
261	C₂²×C₃₁	#578	1	1	yes	characteristic
262	C₄×C₃₁	—	2	2	no	—
263	C₄×C₃₁	—	2	2	no	—
264	C₄×C₃₁	—	4	1	no	—
265	C₄×C₃₁	—	4	4	no	—
266	C₄×C₃₁	—	4	1	no	—
267	C₄×C₃₁	—	8	2	no	—
268	C₄×C₃₁	—	8	2	no	—
269	C₄×C₃₁	—	8	2	no	—
270	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
271	C₄×C₃₁	#537	1	1	yes	characteristic
272	C₄×C₃₁	#548	1	1	yes	characteristic
273	C₄×C₃₁	#553	1	1	yes	characteristic
274	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
275	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
276	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
278	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
279	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
280	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
281	#467	—	2	4	no	—
282	#467	—	2	4	no	—
283	C₈×D₄	—	2	4	no	—
284	C₂×#107	—	2	4	no	—
285	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
286	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×#107	C₂²×C₃₁	1	2	yes	—
288	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×#103	C₂²×C₃₁	1	2	yes	—
291	C₂×#103	C₂²×C₃₁	1	2	yes	—
292	C₂×#103	C₂²×C₃₁	1	2	yes	—
293	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
294	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
296	#398	—	2	2	no	—
297	#396	—	2	2	no	—
298	#391	C₂²×C₃₁	1	2	yes	—
299	#389	C₂²×C₃₁	1	2	yes	—
300	#386	—	2	2	no	—
301	#380	—	2	2	no	—
302	#380	C₂²×C₃₁	1	2	yes	—
303	#340	C₂²×C₃₁	1	2	yes	—
304	#340	C₂²×C₃₁	1	2	yes	—
305	C₂×C₃₁	—	2	4	no	—
306	C₂×C₃₁	—	4	2	no	—
307	C₂×C₃₁	#2416	1	1	yes	characteristic
308	C₂×C₃₁	C₂×#467	1	1	yes	characteristic
309	C₂×C₃₁	#3061	1	1	yes	characteristic
310	C₂×C₃₁	#3280	1	1	yes	characteristic
311	C₂×C₃₁	D₄×Q₁₆	1	1	yes	characteristic
312	C₂×C₃₁	#3336	1	1	yes	characteristic
313	C₂×C₃₁	#3338	1	1	yes	characteristic
314	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
315	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
316	C₂³×C₄	—	2	2	no	—
317	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
318	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
319	C₂×Q₁₆	—	2	2	no	—
320	C₂×Q₁₆	—	2	2	no	—
321	C₂×Q₁₆	—	2	2	no	—
322	C₂²×C₈	—	2	2	no	—
323	C₂²×C₈	—	2	2	no	—
324	C₂²×C₈	—	2	2	no	—
325	C₂²×C₈	—	2	2	no	—
326	C₂²×C₈	—	2	2	no	—
327	C₂²×C₈	D₄×C₃₁	1	4	yes	—
328	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
329	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
330	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
331	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
332	C₂²⋊Q₈	—	2	4	no	—
333	C₂²⋊Q₈	—	2	4	no	—
334	C₂²⋊Q₈	—	2	4	no	—
335	C₂²⋊Q₈	—	4	2	no	—
336	C₄×D₄	—	2	2	no	—
337	C₄×D₄	—	2	2	no	—
338	C₂×C₄⋊C₄	—	2	4	no	—
339	C₂×C₄⋊C₄	—	2	2	no	—
340	C₂×C₄⋊C₄	—	2	2	no	—
341	C₂×C₄⋊C₄	—	2	2	no	—
342	C₂×C₄⋊C₄	—	2	2	no	—
343	C₂×C₄⋊C₄	—	4	2	no	—
344	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
345	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
346	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
347	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₂²⋊C₄	—	2	2	no	—
350	C₂×C₂²⋊C₄	—	2	2	no	—
351	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
352	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
353	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
354	#107	—	2	2	no	—
355	#107	—	2	2	no	—
356	#107	—	2	4	no	—
357	#107	D₄×C₃₁	1	4	yes	complemented
358	C₄⋊C₈	—	2	2	no	—
359	#103	—	2	4	no	—
360	#103	—	2	4	no	—
361	#103	—	4	2	no	—
362	C₂²⋊C₈	—	2	2	no	—
363	C₂²⋊C₈	—	2	2	no	—
364	C₄×C₈	—	2	2	no	—
365	#95	—	2	2	no	—
366	#95	—	2	2	no	—
367	#95	—	2	2	no	—
368	#95	—	2	2	no	—
369	C₃₁	#24781	1	1	yes	characteristic, direct product factor, Sylow
370	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
371	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	2	2	no	—
377	C₂×Q₈	—	2	2	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	Q₁₆×C₃₁	1	4	yes	complemented
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	4	4	no	—
396	C₂²×C₄	—	4	4	no	—
397	C₂²×C₄	—	8	2	no	—
398	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
401	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
402	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
403	C₂²×C₄	#41×C₃₁	1	2	yes	—
404	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
405	Q₁₆	—	4	4	no	—
406	C₂×C₈	—	2	1	no	—
407	C₂×C₈	—	2	1	no	—
408	C₂×C₈	—	2	8	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	4	4	no	—
416	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
417	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
418	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
419	C₂×C₈	#41×C₃₁	1	2	yes	—
420	C₄⋊C₄	—	2	2	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	4	4	no	—
427	C₄⋊C₄	—	4	4	no	—
428	C₄⋊C₄	—	4	4	no	—
429	C₄⋊C₄	—	8	2	no	—
430	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
431	C₂²⋊C₄	—	2	2	no	—
432	C₂²⋊C₄	—	2	2	no	—
433	C₂²⋊C₄	—	4	4	no	—
434	C₂²⋊C₄	—	4	4	no	—
435	C₂²⋊C₄	—	8	2	no	—
436	C₄²	—	2	1	no	—
437	C₄²	—	2	1	no	—
438	C₂³	—	2	1	no	—
439	C₂³	—	2	1	no	—
440	C₂³	—	2	1	no	—
441	C₂³	—	2	1	no	—
442	C₂³	—	2	1	no	—
443	C₂³	—	2	1	no	—
444	C₂³	—	2	1	no	—
445	C₂³	—	2	1	no	—
446	C₂³	—	2	1	no	—
447	C₂³	—	2	1	no	—
448	C₂³	C₃₁×#121	1	2	yes	—
449	C₂³	C₂×Q₁₆×C₃₁	1	2	yes	—
450	C₂³	C₂×Q₁₆×C₃₁	1	2	yes	—
451	C₂³	C₃₁×#135	1	2	yes	—
452	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
453	Q₈	—	4	2	no	—
454	Q₈	—	4	2	no	—
455	Q₈	—	4	2	no	—
456	Q₈	—	4	2	no	—
457	D₄	—	2	8	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	8	8	no	—
489	C₂×C₄	C₃₁×#121	1	2	yes	—
490	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
491	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
492	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
493	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
494	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
495	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
496	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
497	C₈	—	2	2	no	—
498	C₈	—	4	1	no	—
499	C₈	—	4	1	no	—
500	C₈	C₃₁×#121	1	4	yes	—
501	C₂²	—	2	2	no	—
502	C₂²	—	2	2	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	2	2	no	—
505	C₂²	—	2	2	no	—
506	C₂²	—	2	2	no	—
507	C₂²	—	2	2	no	—
508	C₂²	—	2	2	no	—
509	C₂²	—	2	2	no	—
510	C₂²	—	4	1	no	—
511	C₂²	—	4	1	no	—
512	C₂²	—	4	1	no	—
513	C₂²	—	4	1	no	—
514	C₂²	C₃₁×#467	1	4	yes	complemented
515	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
516	C₂²	D₄²×C₃₁	1	1	yes	characteristic
517	C₂²	C₃₁×#547	1	1	yes	characteristic
518	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
519	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
520	C₂²	C₃₁×#576	1	1	yes	characteristic
521	C₂²	C₃₁×#578	1	1	yes	characteristic
522	C₄	—	2	2	no	—
523	C₄	—	2	2	no	—
524	C₄	—	4	1	no	—
525	C₄	—	4	4	no	—
526	C₄	—	4	1	no	—
527	C₄	—	8	2	no	—
528	C₄	—	8	2	no	—
529	C₄	—	8	2	no	—
530	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
531	C₄	C₃₁×#537	1	1	yes	characteristic
532	C₄	C₃₁×#548	1	1	yes	characteristic
533	C₄	C₃₁×#553	1	1	yes	characteristic
534	C₂	—	2	4	no	—
535	C₂	—	4	2	no	—
536	C₂	C₃₁×#2416	1	1	yes	characteristic
537	C₂	C₂×C₃₁×#467	1	1	yes	characteristic
538	C₂	C₃₁×#3061	1	1	yes	characteristic
539	C₂	C₃₁×#3280	1	1	yes	characteristic
540	C₂	D₄×Q₁₆×C₃₁	1	1	yes	characteristic
541	C₂	C₃₁×#3336	1	1	yes	characteristic
542	C₂	C₃₁×#3338	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#467	C₂	semidirect
2. C₂×C₃₁×#467	C₂	semidirect
4. C₂²×C₃₁×#107	C₂	semidirect
6. C₃₁×#2068	C₂	semidirect
7. C₃₁×#2036	C₂	semidirect
8. C₃₁×#1969	C₂	semidirect
9. C₃₁×#1942	C₂	semidirect
10. C₃₁×#1872	C₂	semidirect
11. C₃₁×#1808	C₂	semidirect
86. C₃₁×#107	D₄	semidirect
108. C₂×D₄×C₃₁	Q₁₆	semidirect
166. #24781	C₃₁	direct (N × H)
230. C₂×#467	C₂×C₃₁	semidirect
231. C₂×#467	C₂×C₃₁	semidirect
233. C₂²×#107	C₂×C₃₁	semidirect
235. #2068	C₂×C₃₁	semidirect
236. #2036	C₂×C₃₁	semidirect
237. #1969	C₂×C₃₁	semidirect
238. #1942	C₂×C₃₁	semidirect
239. #1872	C₂×C₃₁	semidirect
240. #1808	C₂×C₃₁	semidirect
254. C₂²×C₃₁	#467	semidirect
357. #107	D₄×C₃₁	semidirect
369. C₃₁	#24781	direct (N × H)
380. C₂×D₄	Q₁₆×C₃₁	semidirect
514. C₂²	C₃₁×#467	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#467	2
C₂×C₈×D₄×C₃₁	2
C₂²×C₃₁×#107	2
C₃₁×#2417	2
C₃₁×#2068	2
C₃₁×#2036	2
C₃₁×#1969	2
C₃₁×#1942	2
C₃₁×#1872	2
C₃₁×#1808	2
#24781	31

Subgroups — C31×#24781 #21481540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24781 #21481540