Subgroups — C₃₁×#22432 #21479191

Order 7936.

1582 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#550	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#417	C₂	1	2	yes	maximal, complemented
4	D₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1982	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1895	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1874	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1788	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#103	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#550	—	2	4	no	—
15	C₃₁×#550	C₄	1	8	yes	complemented
16	C₂×C₃₁×#127	—	2	1	no	—
17	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#121	—	2	2	no	—
19	C₂×C₃₁×#121	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₃₁	—	2	1	no	—
22	C₂×C₄×D₄×C₃₁	—	2	1	no	—
23	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#436	—	2	4	no	—
26	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₃₁×#417	—	2	8	no	—
28	C₂×C₃₁×#103	C₂²	1	2	yes	—
29	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#102	C₂²	1	2	yes	—
32	C₂×C₃₁×#102	C₂²	1	2	yes	—
33	C₂×C₃₁×#102	C₂²	1	2	yes	—
34	C₂×C₃₁×#102	C₂²	1	2	yes	—
35	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#389	—	2	1	no	—
38	C₃₁×#385	—	2	2	no	—
39	C₃₁×#384	C₂²	1	1	yes	characteristic
40	C₃₁×#380	C₂²	1	2	yes	—
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#340	C₂²	1	1	yes	characteristic
44	C₃₁×#340	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	—	2	4	no	—
46	C₂×#41×C₃₁	—	2	4	no	—
47	C₂×#41×C₃₁	—	2	4	no	—
48	C₂×#41×C₃₁	—	4	2	no	—
49	C₂×#41×C₃₁	—	4	2	no	—
50	C₂×#41×C₃₁	C₂×C₄	1	8	yes	complemented
51	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×D₄×C₃₁	—	2	2	no	—
53	C₂²×D₄×C₃₁	—	2	2	no	—
54	C₂²×D₄×C₃₁	—	2	1	no	—
55	C₂²×D₄×C₃₁	—	2	1	no	—
56	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
57	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
58	C₂³×C₄×C₃₁	—	2	2	no	—
59	C₂³×C₄×C₃₁	—	2	2	no	—
60	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×M₁₆×C₃₁	—	2	4	no	—
62	C₂×M₁₆×C₃₁	—	2	4	no	—
63	C₂×M₁₆×C₃₁	—	4	2	no	—
64	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
66	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#127	—	2	8	no	—
68	C₃₁×#127	—	2	2	no	—
69	C₃₁×#127	C₂×C₄	1	4	yes	complemented
70	C₃₁×#121	—	2	4	no	—
71	C₃₁×#121	—	2	16	no	—
72	C₃₁×#121	C₂×C₄	1	8	yes	complemented
73	C₄×Q₈×C₃₁	—	2	2	no	—
74	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	complemented
75	C₄×D₄×C₃₁	—	2	8	no	—
76	C₄×D₄×C₃₁	—	4	4	no	—
77	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
78	C₄×D₄×C₃₁	D₄	1	4	yes	—
79	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
81	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
82	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
85	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
93	C₂×C₄²×C₃₁	—	2	1	no	—
94	C₂×C₄²×C₃₁	—	2	1	no	—
95	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
96	C₃₁×C₄⋊C₈	—	2	2	no	—
97	C₃₁×#103	—	2	4	no	—
98	C₃₁×#103	—	2	2	no	—
99	C₃₁×#103	D₄	1	4	yes	complemented
100	C₃₁×#102	—	2	4	no	—
101	C₃₁×#102	—	2	4	no	—
102	C₃₁×#102	—	4	2	no	—
103	C₃₁×#102	—	4	2	no	—
104	C₃₁×C₂²⋊C₈	—	2	4	no	—
105	C₄×C₈×C₃₁	—	2	2	no	—
106	C₃₁×#95	—	2	1	no	—
107	C₃₁×#95	—	2	2	no	—
108	C₂⁴×C₃₁	—	2	2	no	—
109	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
110	#41×C₃₁	—	2	16	no	—
111	#41×C₃₁	—	4	8	no	—
112	#41×C₃₁	—	4	8	no	—
113	#41×C₃₁	—	4	8	no	—
114	C₂×Q₈×C₃₁	—	2	1	no	—
115	C₂×Q₈×C₃₁	—	2	1	no	—
116	C₂×Q₈×C₃₁	—	2	1	no	—
117	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
118	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
119	C₂×Q₈×C₃₁	#41	1	2	yes	—
120	C₂×D₄×C₃₁	—	2	8	no	—
121	C₂×D₄×C₃₁	—	2	8	no	—
122	C₂×D₄×C₃₁	—	2	4	no	—
123	C₂×D₄×C₃₁	—	2	4	no	—
124	C₂×D₄×C₃₁	—	2	2	no	—
125	C₂×D₄×C₃₁	—	2	2	no	—
126	C₂×D₄×C₃₁	—	2	2	no	—
127	C₂×D₄×C₃₁	—	2	2	no	—
128	C₂×D₄×C₃₁	—	2	2	no	—
129	C₂×D₄×C₃₁	—	2	2	no	—
130	C₂×D₄×C₃₁	—	2	2	no	—
131	C₂×D₄×C₃₁	—	4	4	no	—
132	C₂×D₄×C₃₁	—	4	4	no	—
133	C₂×D₄×C₃₁	—	4	2	no	—
134	C₂×D₄×C₃₁	—	4	2	no	—
135	C₂×D₄×C₃₁	—	8	1	no	—
136	C₂×D₄×C₃₁	—	8	1	no	—
137	C₂×D₄×C₃₁	—	8	2	no	—
138	C₂×D₄×C₃₁	—	8	2	no	—
139	C₂×D₄×C₃₁	C₂²⋊C₄	1	4	yes	—
140	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
141	C₂²×C₄×C₃₁	—	2	1	no	—
142	C₂²×C₄×C₃₁	—	2	2	no	—
143	C₂²×C₄×C₃₁	—	2	2	no	—
144	C₂²×C₄×C₃₁	—	2	2	no	—
145	C₂²×C₄×C₃₁	—	2	2	no	—
146	C₂²×C₄×C₃₁	—	2	1	no	—
147	C₂²×C₄×C₃₁	—	2	1	no	—
148	C₂²×C₄×C₃₁	—	2	8	no	—
149	C₂²×C₄×C₃₁	—	2	1	no	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	4	4	no	—
152	C₂²×C₄×C₃₁	—	4	4	no	—
153	C₂²×C₄×C₃₁	—	4	4	no	—
154	C₂²×C₄×C₃₁	—	4	4	no	—
155	C₂²×C₄×C₃₁	—	8	2	no	—
156	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
162	M₁₆×C₃₁	—	4	8	no	—
163	C₂×C₈×C₃₁	—	2	1	no	—
164	C₂×C₈×C₃₁	—	2	1	no	—
165	C₂×C₈×C₃₁	—	2	1	no	—
166	C₂×C₈×C₃₁	—	2	2	no	—
167	C₂×C₈×C₃₁	—	2	2	no	—
168	C₂×C₈×C₃₁	—	2	1	no	—
169	C₂×C₈×C₃₁	—	2	1	no	—
170	C₂×C₈×C₃₁	—	2	2	no	—
171	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
172	C₄⋊C₄×C₃₁	—	2	4	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	—	2	2	no	—
176	C₄⋊C₄×C₃₁	—	2	1	no	—
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	2	2	no	—
179	C₄⋊C₄×C₃₁	—	4	4	no	—
180	C₄⋊C₄×C₃₁	—	4	2	no	—
181	C₄⋊C₄×C₃₁	—	8	1	no	—
182	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
183	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
184	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
185	C₂²⋊C₄×C₃₁	—	2	8	no	—
186	C₂²⋊C₄×C₃₁	—	4	4	no	—
187	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
188	C₄²×C₃₁	—	2	4	no	—
189	C₄²×C₃₁	—	4	2	no	—
190	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
191	C₄²×C₃₁	C₂×D₄	1	2	yes	—
192	#22432	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂³×C₃₁	—	2	2	no	—
194	C₂³×C₃₁	—	2	2	no	—
195	C₂³×C₃₁	—	2	2	no	—
196	C₂³×C₃₁	—	2	2	no	—
197	C₂³×C₃₁	—	2	2	no	—
198	C₂³×C₃₁	—	2	2	no	—
199	C₂³×C₃₁	—	2	4	no	—
200	C₂³×C₃₁	—	4	2	no	—
201	C₂³×C₃₁	—	4	2	no	—
202	C₂³×C₃₁	—	4	2	no	—
203	C₂³×C₃₁	—	4	2	no	—
204	C₂³×C₃₁	—	4	2	no	—
205	C₂³×C₃₁	—	4	2	no	—
206	C₂³×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
207	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
208	Q₈×C₃₁	—	2	2	no	—
209	Q₈×C₃₁	—	2	2	no	—
210	Q₈×C₃₁	—	2	2	no	—
211	Q₈×C₃₁	C₄×D₄	1	4	yes	complemented
212	D₄×C₃₁	—	2	4	no	—
213	D₄×C₃₁	—	2	4	no	—
214	D₄×C₃₁	—	2	8	no	—
215	D₄×C₃₁	—	4	16	no	—
216	D₄×C₃₁	—	4	2	no	—
217	D₄×C₃₁	—	4	2	no	—
218	D₄×C₃₁	—	4	8	no	—
219	D₄×C₃₁	—	8	4	no	—
220	D₄×C₃₁	—	8	8	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	2	no	—
226	C₂×C₄×C₃₁	—	2	1	no	—
227	C₂×C₄×C₃₁	—	2	1	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	4	no	—
230	C₂×C₄×C₃₁	—	2	4	no	—
231	C₂×C₄×C₃₁	—	2	4	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	2	2	no	—
234	C₂×C₄×C₃₁	—	4	16	no	—
235	C₂×C₄×C₃₁	—	4	1	no	—
236	C₂×C₄×C₃₁	—	4	1	no	—
237	C₂×C₄×C₃₁	—	4	1	no	—
238	C₂×C₄×C₃₁	—	4	1	no	—
239	C₂×C₄×C₃₁	—	4	2	no	—
240	C₂×C₄×C₃₁	—	4	1	no	—
241	C₂×C₄×C₃₁	—	4	1	no	—
242	C₂×C₄×C₃₁	—	4	2	no	—
243	C₂×C₄×C₃₁	—	4	2	no	—
244	C₂×C₄×C₃₁	—	4	1	no	—
245	C₂×C₄×C₃₁	—	8	8	no	—
246	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
247	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
248	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
249	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
250	C₂×C₄×C₃₁	#137	1	2	yes	—
251	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
252	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
255	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
256	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
257	C₈×C₃₁	—	2	2	no	—
258	C₈×C₃₁	—	4	2	no	—
259	C₈×C₃₁	—	4	1	no	—
260	C₂×#550	C₂×C₃₁	1	1	yes	characteristic
261	C₂×#436	C₂×C₃₁	1	1	yes	characteristic, complemented
262	C₂×#417	C₂×C₃₁	1	2	yes	complemented
263	D₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
264	#1982	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#1957	C₂×C₃₁	1	1	yes	characteristic, complemented
266	#1895	C₂×C₃₁	1	2	yes	complemented
267	#1874	C₂×C₃₁	1	2	yes	complemented
268	#1803	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#1788	C₂×C₃₁	1	2	yes	complemented
270	C₄×#103	C₂×C₃₁	1	1	yes	characteristic, complemented
271	C₂²×C₃₁	—	2	4	no	—
272	C₂²×C₃₁	—	2	4	no	—
273	C₂²×C₃₁	—	2	4	no	—
274	C₂²×C₃₁	—	4	2	no	—
275	C₂²×C₃₁	—	4	2	no	—
276	C₂²×C₃₁	—	4	2	no	—
277	C₂²×C₃₁	—	4	2	no	—
278	C₂²×C₃₁	—	4	4	no	—
279	C₂²×C₃₁	—	4	4	no	—
280	C₂²×C₃₁	—	4	4	no	—
281	C₂²×C₃₁	—	8	2	no	—
282	C₂²×C₃₁	—	8	2	no	—
283	C₂²×C₃₁	—	8	2	no	—
284	C₂²×C₃₁	—	8	2	no	—
285	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
286	C₂²×C₃₁	D₄²	1	1	yes	characteristic
287	C₂²×C₃₁	#546	1	1	yes	characteristic
288	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
289	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
290	C₂²×C₃₁	#576	1	1	yes	characteristic
291	C₂²×C₃₁	#577	1	1	yes	characteristic
292	C₄×C₃₁	—	2	4	no	—
293	C₄×C₃₁	—	2	4	no	—
294	C₄×C₃₁	—	2	2	no	—
295	C₄×C₃₁	—	4	2	no	—
296	C₄×C₃₁	—	4	2	no	—
297	C₄×C₃₁	—	8	1	no	—
298	C₄×C₃₁	—	8	2	no	—
299	C₄×C₃₁	#417	1	4	yes	complemented
300	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
301	C₄×C₃₁	#518	1	1	yes	characteristic
302	C₄×C₃₁	D₄²	1	1	yes	characteristic
303	C₄×C₃₁	#546	1	1	yes	characteristic
304	C₂²×#41	C₂²×C₃₁	1	2	yes	—
305	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
306	#550	—	2	4	no	—
307	#550	C₄×C₃₁	1	8	yes	complemented
308	C₂×#127	—	2	1	no	—
309	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
310	C₂×#121	—	2	2	no	—
311	C₂×#121	C₂²×C₃₁	1	2	yes	—
312	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
313	C₂×C₄×D₄	—	2	1	no	—
314	C₂×C₄×D₄	—	2	1	no	—
315	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
316	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
317	#436	—	2	4	no	—
318	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
319	#417	—	2	8	no	—
320	C₂×#103	C₂²×C₃₁	1	2	yes	—
321	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
322	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
323	C₂×#102	C₂²×C₃₁	1	2	yes	—
324	C₂×#102	C₂²×C₃₁	1	2	yes	—
325	C₂×#102	C₂²×C₃₁	1	2	yes	—
326	C₂×#102	C₂²×C₃₁	1	2	yes	—
327	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
328	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
329	#389	—	2	1	no	—
330	#385	—	2	2	no	—
331	#384	C₂²×C₃₁	1	1	yes	characteristic
332	#380	C₂²×C₃₁	1	2	yes	—
333	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
334	#340	C₂²×C₃₁	1	2	yes	—
335	#340	C₂²×C₃₁	1	1	yes	characteristic
336	#340	C₂²×C₃₁	1	1	yes	characteristic
337	C₂×C₃₁	—	4	4	no	—
338	C₂×C₃₁	—	8	4	no	—
339	C₂×C₃₁	D₄×C₂²⋊C₄	1	1	yes	characteristic
340	C₂×C₃₁	C₂×#417	1	1	yes	characteristic
341	C₂×C₃₁	#2897	1	1	yes	characteristic
342	C₂×C₃₁	#3274	1	1	yes	characteristic
343	C₂×C₃₁	#3278	1	1	yes	characteristic
344	C₂×C₃₁	#3332	1	1	yes	characteristic
345	C₂×C₃₁	#3333	1	1	yes	characteristic
346	C₂×#41	—	2	4	no	—
347	C₂×#41	—	2	4	no	—
348	C₂×#41	—	2	4	no	—
349	C₂×#41	—	4	2	no	—
350	C₂×#41	—	4	2	no	—
351	C₂×#41	C₂×C₄×C₃₁	1	8	yes	complemented
352	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
353	C₂²×D₄	—	2	2	no	—
354	C₂²×D₄	—	2	2	no	—
355	C₂²×D₄	—	2	1	no	—
356	C₂²×D₄	—	2	1	no	—
357	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
358	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
359	C₂³×C₄	—	2	2	no	—
360	C₂³×C₄	—	2	2	no	—
361	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
362	C₂×M₁₆	—	2	4	no	—
363	C₂×M₁₆	—	2	4	no	—
364	C₂×M₁₆	—	4	2	no	—
365	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
366	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
367	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
368	#127	—	2	8	no	—
369	#127	—	2	2	no	—
370	#127	C₂×C₄×C₃₁	1	4	yes	complemented
371	#121	—	2	4	no	—
372	#121	—	2	16	no	—
373	#121	C₂×C₄×C₃₁	1	8	yes	complemented
374	C₄×Q₈	—	2	2	no	—
375	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
376	C₄×D₄	—	2	8	no	—
377	C₄×D₄	—	4	4	no	—
378	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
379	C₄×D₄	D₄×C₃₁	1	4	yes	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	4	no	—
382	C₂×C₄⋊C₄	—	2	2	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	1	no	—
385	C₂×C₄⋊C₄	—	4	2	no	—
386	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
388	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
390	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
391	C₂×C₂²⋊C₄	—	2	2	no	—
392	C₂×C₂²⋊C₄	—	2	2	no	—
393	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
394	C₂×C₄²	—	2	1	no	—
395	C₂×C₄²	—	2	1	no	—
396	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
397	C₄⋊C₈	—	2	2	no	—
398	#103	—	2	4	no	—
399	#103	—	2	2	no	—
400	#103	D₄×C₃₁	1	4	yes	complemented
401	#102	—	2	4	no	—
402	#102	—	2	4	no	—
403	#102	—	4	2	no	—
404	#102	—	4	2	no	—
405	C₂²⋊C₈	—	2	4	no	—
406	C₄×C₈	—	2	2	no	—
407	#95	—	2	1	no	—
408	#95	—	2	2	no	—
409	C₃₁	#22432	1	1	yes	characteristic, direct product factor, Sylow
410	C₂⁴	—	2	2	no	—
411	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
412	#41	—	2	16	no	—
413	#41	—	4	8	no	—
414	#41	—	4	8	no	—
415	#41	—	4	8	no	—
416	C₂×Q₈	—	2	1	no	—
417	C₂×Q₈	—	2	1	no	—
418	C₂×Q₈	—	2	1	no	—
419	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
420	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
421	C₂×Q₈	#41×C₃₁	1	2	yes	—
422	C₂×D₄	—	2	8	no	—
423	C₂×D₄	—	2	8	no	—
424	C₂×D₄	—	2	4	no	—
425	C₂×D₄	—	2	4	no	—
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	2	2	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	2	2	no	—
433	C₂×D₄	—	4	4	no	—
434	C₂×D₄	—	4	4	no	—
435	C₂×D₄	—	4	2	no	—
436	C₂×D₄	—	4	2	no	—
437	C₂×D₄	—	8	1	no	—
438	C₂×D₄	—	8	1	no	—
439	C₂×D₄	—	8	2	no	—
440	C₂×D₄	—	8	2	no	—
441	C₂×D₄	C₂²⋊C₄×C₃₁	1	4	yes	—
442	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	2	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	8	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	2	no	—
453	C₂²×C₄	—	4	4	no	—
454	C₂²×C₄	—	4	4	no	—
455	C₂²×C₄	—	4	4	no	—
456	C₂²×C₄	—	4	4	no	—
457	C₂²×C₄	—	8	2	no	—
458	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
459	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
461	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
462	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
464	M₁₆	—	4	8	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	2	no	—
469	C₂×C₈	—	2	2	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	2	no	—
473	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	2	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	4	4	no	—
482	C₄⋊C₄	—	4	2	no	—
483	C₄⋊C₄	—	8	1	no	—
484	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
485	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
486	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
487	C₂²⋊C₄	—	2	8	no	—
488	C₂²⋊C₄	—	4	4	no	—
489	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
490	C₄²	—	2	4	no	—
491	C₄²	—	4	2	no	—
492	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
493	C₄²	C₂×D₄×C₃₁	1	2	yes	—
494	C₂³	—	2	2	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	2	2	no	—
497	C₂³	—	2	2	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	2	2	no	—
500	C₂³	—	2	4	no	—
501	C₂³	—	4	2	no	—
502	C₂³	—	4	2	no	—
503	C₂³	—	4	2	no	—
504	C₂³	—	4	2	no	—
505	C₂³	—	4	2	no	—
506	C₂³	—	4	2	no	—
507	C₂³	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
508	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
509	Q₈	—	2	2	no	—
510	Q₈	—	2	2	no	—
511	Q₈	—	2	2	no	—
512	Q₈	C₄×D₄×C₃₁	1	4	yes	complemented
513	D₄	—	2	8	no	—
514	D₄	—	2	4	no	—
515	D₄	—	2	4	no	—
516	D₄	—	4	16	no	—
517	D₄	—	4	8	no	—
518	D₄	—	4	2	no	—
519	D₄	—	4	2	no	—
520	D₄	—	8	4	no	—
521	D₄	—	8	8	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	4	16	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	2	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	8	8	no	—
547	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
548	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
549	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
550	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
551	C₂×C₄	C₃₁×#137	1	2	yes	—
552	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
553	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
554	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
555	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
556	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
557	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
558	C₈	—	2	2	no	—
559	C₈	—	4	2	no	—
560	C₈	—	4	1	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	2	4	no	—
563	C₂²	—	2	4	no	—
564	C₂²	—	4	2	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	4	4	no	—
569	C₂²	—	4	4	no	—
570	C₂²	—	4	4	no	—
571	C₂²	—	8	2	no	—
572	C₂²	—	8	2	no	—
573	C₂²	—	8	2	no	—
574	C₂²	—	8	2	no	—
575	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
576	C₂²	D₄²×C₃₁	1	1	yes	characteristic
577	C₂²	C₃₁×#546	1	1	yes	characteristic
578	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
579	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
580	C₂²	C₃₁×#576	1	1	yes	characteristic
581	C₂²	C₃₁×#577	1	1	yes	characteristic
582	C₄	—	2	4	no	—
583	C₄	—	2	4	no	—
584	C₄	—	2	2	no	—
585	C₄	—	4	2	no	—
586	C₄	—	4	2	no	—
587	C₄	—	8	1	no	—
588	C₄	—	8	2	no	—
589	C₄	C₃₁×#417	1	4	yes	complemented
590	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
591	C₄	C₃₁×#518	1	1	yes	characteristic
592	C₄	D₄²×C₃₁	1	1	yes	characteristic
593	C₄	C₃₁×#546	1	1	yes	characteristic
594	C₂	—	4	4	no	—
595	C₂	—	8	4	no	—
596	C₂	D₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
597	C₂	C₂×C₃₁×#417	1	1	yes	characteristic
598	C₂	C₃₁×#2897	1	1	yes	characteristic
599	C₂	C₃₁×#3274	1	1	yes	characteristic
600	C₂	C₃₁×#3278	1	1	yes	characteristic
601	C₂	C₃₁×#3332	1	1	yes	characteristic
602	C₂	C₃₁×#3333	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#550	C₂
LCS	G ▷ C₂×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
2. C₂×C₃₁×#436	C₂	semidirect
3. C₂×C₃₁×#417	C₂	semidirect
4. D₄×C₄⋊C₄×C₃₁	C₂	semidirect
5. C₃₁×#1982	C₂	semidirect
6. C₃₁×#1957	C₂	semidirect
7. C₃₁×#1895	C₂	semidirect
8. C₃₁×#1874	C₂	semidirect
9. C₃₁×#1803	C₂	semidirect
10. C₃₁×#1788	C₂	semidirect
11. C₄×C₃₁×#103	C₂	semidirect
15. C₃₁×#550	C₄	semidirect
50. C₂×#41×C₃₁	C₂×C₄	semidirect
69. C₃₁×#127	C₂×C₄	semidirect
72. C₃₁×#121	C₂×C₄	semidirect
74. C₄×Q₈×C₃₁	C₂×C₄	semidirect
99. C₃₁×#103	D₄	semidirect
192. #22432	C₃₁	direct (N × H)
211. Q₈×C₃₁	C₄×D₄	semidirect
261. C₂×#436	C₂×C₃₁	semidirect
262. C₂×#417	C₂×C₃₁	semidirect
263. D₄×C₄⋊C₄	C₂×C₃₁	semidirect
264. #1982	C₂×C₃₁	semidirect
265. #1957	C₂×C₃₁	semidirect
266. #1895	C₂×C₃₁	semidirect
267. #1874	C₂×C₃₁	semidirect
268. #1803	C₂×C₃₁	semidirect
269. #1788	C₂×C₃₁	semidirect
270. C₄×#103	C₂×C₃₁	semidirect
299. C₄×C₃₁	#417	semidirect
307. #550	C₄×C₃₁	semidirect
351. C₂×#41	C₂×C₄×C₃₁	semidirect
370. #127	C₂×C₄×C₃₁	semidirect
373. #121	C₂×C₄×C₃₁	semidirect
375. C₄×Q₈	C₂×C₄×C₃₁	semidirect
400. #103	D₄×C₃₁	semidirect
409. C₃₁	#22432	direct (N × H)
512. Q₈	C₄×D₄×C₃₁	semidirect
589. C₄	C₃₁×#417	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#550	2
C₂×C₃₁×#436	2
C₂×C₃₁×#417	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#1982	2
C₃₁×#1957	2
C₃₁×#1895	2
C₃₁×#1874	2
C₃₁×#1803	2
C₃₁×#1788	2
C₄×C₃₁×#103	2
#22432	31

Subgroups — C31×#22432 #21479191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22432 #21479191