Subgroups — C₃₁×#24328 #21481087

Order 7936.

1070 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#474	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2538	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2059	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2025	C₂	1	2	yes	maximal
7	C₃₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1880	C₂	1	2	yes	maximal
10	C₃₁×#1863	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#557	—	2	4	no	—
14	C₃₁×#557	—	2	4	no	—
15	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#125	—	2	2	no	—
18	C₂×C₃₁×#125	C₂²	1	2	yes	—
19	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
20	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
21	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
22	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#474	—	2	4	no	—
26	C₄×SD₁₆×C₃₁	—	2	4	no	—
27	C₂×C₃₁×#107	C₂²	1	2	yes	—
28	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#103	C₂²	1	2	yes	—
31	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#102	C₂²	1	2	yes	—
33	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#396	—	2	2	no	—
36	C₃₁×#390	C₂²	1	2	yes	—
37	C₃₁×#389	C₂²	1	2	yes	—
38	C₃₁×#384	C₂²	1	1	yes	characteristic
39	C₃₁×#380	—	2	1	no	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₃₁×#336	C₂²	1	2	yes	—
45	C₂²×Q₈×C₃₁	—	2	1	no	—
46	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
47	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	—	2	1	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×SD₁₆×C₃₁	Q₈	1	4	yes	complemented
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×C₄⋊Q₈	—	2	8	no	—
57	C₃₁×C₄⋊Q₈	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
59	C₃₁×#125	—	2	16	no	—
60	C₃₁×#125	—	2	4	no	—
61	C₃₁×#125	—	2	4	no	—
62	C₃₁×C₂²⋊Q₈	—	4	4	no	—
63	C₄×Q₈×C₃₁	—	2	8	no	—
64	C₄×Q₈×C₃₁	—	2	4	no	—
65	C₄×Q₈×C₃₁	—	2	4	no	—
66	C₄×Q₈×C₃₁	—	2	2	no	—
67	C₄×Q₈×C₃₁	—	2	2	no	—
68	C₄×Q₈×C₃₁	—	2	2	no	—
69	C₄×Q₈×C₃₁	D₄	1	4	yes	—
70	C₄×D₄×C₃₁	—	2	2	no	—
71	C₄×D₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
81	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
89	C₂×C₄²×C₃₁	—	2	2	no	—
90	C₂×C₄²×C₃₁	—	2	1	no	—
91	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
92	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
93	C₃₁×#107	—	2	4	no	—
94	C₃₁×#106	—	2	2	no	—
95	C₃₁×C₄⋊C₈	—	2	2	no	—
96	C₃₁×#103	—	2	2	no	—
97	C₃₁×#103	—	2	4	no	—
98	C₃₁×#102	—	2	2	no	—
99	C₃₁×#102	—	2	4	no	—
100	C₄×C₈×C₃₁	—	2	2	no	—
101	C₃₁×#95	—	2	2	no	—
102	C₂⁴×C₃₁	—	2	1	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	4	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	—	2	1	no	—
110	C₂×Q₈×C₃₁	—	4	2	no	—
111	C₂×Q₈×C₃₁	—	4	2	no	—
112	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
113	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
114	C₂×Q₈×C₃₁	#41	1	2	yes	—
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	1	no	—
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	4	2	no	—
127	C₂²×C₄×C₃₁	—	4	2	no	—
128	C₂²×C₄×C₃₁	—	4	2	no	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
134	SD₁₆×C₃₁	—	2	8	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	2	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
144	C₄⋊C₄×C₃₁	—	2	8	no	—
145	C₄⋊C₄×C₃₁	—	2	8	no	—
146	C₄⋊C₄×C₃₁	—	2	8	no	—
147	C₄⋊C₄×C₃₁	—	2	4	no	—
148	C₄⋊C₄×C₃₁	—	2	4	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	2	4	no	—
151	C₄⋊C₄×C₃₁	—	2	1	no	—
152	C₄⋊C₄×C₃₁	—	2	2	no	—
153	C₄⋊C₄×C₃₁	—	2	1	no	—
154	C₄⋊C₄×C₃₁	—	2	1	no	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	—	4	4	no	—
158	C₄⋊C₄×C₃₁	—	4	2	no	—
159	C₄⋊C₄×C₃₁	—	8	1	no	—
160	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
161	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
162	C₂²⋊C₄×C₃₁	—	4	4	no	—
163	C₂²⋊C₄×C₃₁	—	4	2	no	—
164	C₂²⋊C₄×C₃₁	—	8	2	no	—
165	C₄²×C₃₁	—	2	8	no	—
166	C₄²×C₃₁	—	2	4	no	—
167	C₄²×C₃₁	—	2	2	no	—
168	C₄²×C₃₁	—	2	2	no	—
169	C₄²×C₃₁	C₂×D₄	1	2	yes	—
170	C₄²×C₃₁	C₂×D₄	1	2	yes	—
171	#24328	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂³×C₃₁	—	2	2	no	—
173	C₂³×C₃₁	—	4	1	no	—
174	C₂³×C₃₁	—	4	1	no	—
175	C₂³×C₃₁	—	4	1	no	—
176	C₂³×C₃₁	—	4	1	no	—
177	C₂³×C₃₁	—	4	1	no	—
178	C₂³×C₃₁	—	4	1	no	—
179	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
180	Q₈×C₃₁	—	2	2	no	—
181	Q₈×C₃₁	—	2	2	no	—
182	Q₈×C₃₁	—	2	2	no	—
183	Q₈×C₃₁	—	4	4	no	—
184	Q₈×C₃₁	—	4	8	no	—
185	Q₈×C₃₁	C₂²⋊Q₈	1	4	yes	complemented
186	D₄×C₃₁	—	2	2	no	—
187	D₄×C₃₁	—	2	2	no	—
188	D₄×C₃₁	—	4	1	no	—
189	D₄×C₃₁	—	4	1	no	—
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	1	no	—
194	C₂×C₄×C₃₁	—	2	1	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	4	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	2	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	1	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	8	4	no	—
214	C₂×C₄×C₃₁	#120	1	2	yes	—
215	C₂×C₄×C₃₁	#120	1	2	yes	—
216	C₂×C₄×C₃₁	#135	1	2	yes	—
217	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
222	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
223	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
224	C₈×C₃₁	—	2	2	no	—
225	C₈×C₃₁	—	4	1	no	—
226	C₈×C₃₁	—	4	2	no	—
227	C₂×#557	C₂×C₃₁	1	1	yes	characteristic, complemented
228	C₂×#474	C₂×C₃₁	1	1	yes	characteristic
229	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
230	#2538	C₂×C₃₁	1	1	yes	characteristic
231	#2059	C₂×C₃₁	1	2	yes	complemented
232	#2025	C₂×C₃₁	1	2	yes	—
233	#1990	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#1880	C₂×C₃₁	1	2	yes	—
236	#1863	C₂×C₃₁	1	2	yes	complemented
237	#1803	C₂×C₃₁	1	1	yes	characteristic, complemented
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	4	2	no	—
240	C₂²×C₃₁	—	4	2	no	—
241	C₂²×C₃₁	—	8	1	no	—
242	C₂²×C₃₁	—	8	1	no	—
243	C₂²×C₃₁	—	8	1	no	—
244	C₂²×C₃₁	—	8	1	no	—
245	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
246	C₂²×C₃₁	#548	1	1	yes	characteristic
247	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
248	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
249	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
250	C₂²×C₃₁	#577	1	1	yes	characteristic
251	C₂²×C₃₁	#578	1	1	yes	characteristic
252	C₄×C₃₁	—	2	4	no	—
253	C₄×C₃₁	—	2	2	no	—
254	C₄×C₃₁	—	4	2	no	—
255	C₄×C₃₁	—	4	4	no	—
256	C₄×C₃₁	—	4	2	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	8	2	no	—
259	C₄×C₃₁	—	8	1	no	—
260	C₄×C₃₁	#449	1	4	yes	—
261	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
262	C₄×C₃₁	#536	1	1	yes	characteristic
263	C₄×C₃₁	#548	1	1	yes	characteristic
264	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
265	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
266	#557	—	2	4	no	—
267	#557	—	2	4	no	—
268	C₂×C₄⋊Q₈	—	2	1	no	—
269	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×#125	—	2	2	no	—
271	C₂×#125	C₂²×C₃₁	1	2	yes	—
272	C₂×C₂²⋊Q₈	—	2	1	no	—
273	C₂×C₄×Q₈	—	2	1	no	—
274	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
275	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
278	#474	—	2	4	no	—
279	C₄×SD₁₆	—	2	4	no	—
280	C₂×#107	C₂²×C₃₁	1	2	yes	—
281	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×#103	C₂²×C₃₁	1	2	yes	—
284	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
285	C₂×#102	C₂²×C₃₁	1	2	yes	—
286	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
288	#396	—	2	2	no	—
289	#390	C₂²×C₃₁	1	2	yes	—
290	#389	C₂²×C₃₁	1	2	yes	—
291	#384	C₂²×C₃₁	1	1	yes	characteristic
292	#380	—	2	1	no	—
293	#340	C₂²×C₃₁	1	2	yes	—
294	#340	C₂²×C₃₁	1	2	yes	—
295	#340	C₂²×C₃₁	1	2	yes	—
296	#336	C₂²×C₃₁	1	2	yes	—
297	#336	C₂²×C₃₁	1	2	yes	—
298	C₂×C₃₁	—	8	2	no	—
299	C₂×C₃₁	#2409	1	1	yes	characteristic
300	C₂×C₃₁	C₂×#449	1	1	yes	characteristic
301	C₂×C₃₁	#3013	1	1	yes	characteristic
302	C₂×C₃₁	#3364	1	1	yes	characteristic
303	C₂×C₃₁	#3368	1	1	yes	characteristic
304	C₂×C₃₁	#3381	1	1	yes	characteristic
305	C₂×C₃₁	#3386	1	1	yes	characteristic
306	C₂²×Q₈	—	2	1	no	—
307	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
308	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
309	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
310	C₂³×C₄	—	2	1	no	—
311	C₂×SD₁₆	—	2	2	no	—
312	C₂×SD₁₆	—	2	2	no	—
313	C₂×SD₁₆	Q₈×C₃₁	1	4	yes	complemented
314	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
315	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
316	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
317	C₄⋊Q₈	—	2	8	no	—
318	C₄⋊Q₈	—	2	2	no	—
319	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
320	#125	—	2	16	no	—
321	#125	—	2	4	no	—
322	#125	—	2	4	no	—
323	C₂²⋊Q₈	—	4	4	no	—
324	C₄×Q₈	—	2	8	no	—
325	C₄×Q₈	—	2	4	no	—
326	C₄×Q₈	—	2	4	no	—
327	C₄×Q₈	—	2	2	no	—
328	C₄×Q₈	—	2	2	no	—
329	C₄×Q₈	—	2	2	no	—
330	C₄×Q₈	D₄×C₃₁	1	4	yes	—
331	C₄×D₄	—	2	2	no	—
332	C₄×D₄	—	2	2	no	—
333	C₂×C₄⋊C₄	—	2	2	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	2	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	2	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
342	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
343	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
344	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
345	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
346	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
347	C₂×C₂²⋊C₄	—	2	1	no	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₂²⋊C₄	—	2	2	no	—
350	C₂×C₄²	—	2	2	no	—
351	C₂×C₄²	—	2	1	no	—
352	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
353	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
354	#107	—	2	4	no	—
355	#106	—	2	2	no	—
356	C₄⋊C₈	—	2	2	no	—
357	#103	—	2	2	no	—
358	#103	—	2	4	no	—
359	#102	—	2	2	no	—
360	#102	—	2	4	no	—
361	C₄×C₈	—	2	2	no	—
362	#95	—	2	2	no	—
363	C₃₁	#24328	1	1	yes	characteristic, direct product factor, Sylow
364	C₂⁴	—	2	1	no	—
365	C₂×Q₈	—	2	2	no	—
366	C₂×Q₈	—	2	4	no	—
367	C₂×Q₈	—	2	2	no	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	2	1	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	2	1	no	—
372	C₂×Q₈	—	4	2	no	—
373	C₂×Q₈	—	4	2	no	—
374	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
375	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
376	C₂×Q₈	#41×C₃₁	1	2	yes	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	1	no	—
379	C₂×D₄	—	2	1	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
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	2	no	—
388	C₂²×C₄	—	4	2	no	—
389	C₂²×C₄	—	4	2	no	—
390	C₂²×C₄	—	4	2	no	—
391	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
392	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
393	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
395	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
396	SD₁₆	—	2	8	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	1	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
406	C₄⋊C₄	—	2	8	no	—
407	C₄⋊C₄	—	2	8	no	—
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	4	4	no	—
420	C₄⋊C₄	—	4	2	no	—
421	C₄⋊C₄	—	8	1	no	—
422	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
423	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
424	C₂²⋊C₄	—	4	4	no	—
425	C₂²⋊C₄	—	4	2	no	—
426	C₂²⋊C₄	—	8	2	no	—
427	C₄²	—	2	8	no	—
428	C₄²	—	2	4	no	—
429	C₄²	—	2	2	no	—
430	C₄²	—	2	2	no	—
431	C₄²	C₂×D₄×C₃₁	1	2	yes	—
432	C₄²	C₂×D₄×C₃₁	1	2	yes	—
433	C₂³	—	2	2	no	—
434	C₂³	—	4	1	no	—
435	C₂³	—	4	1	no	—
436	C₂³	—	4	1	no	—
437	C₂³	—	4	1	no	—
438	C₂³	—	4	1	no	—
439	C₂³	—	4	1	no	—
440	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
441	Q₈	—	2	2	no	—
442	Q₈	—	2	2	no	—
443	Q₈	—	2	2	no	—
444	Q₈	—	4	4	no	—
445	Q₈	—	4	8	no	—
446	Q₈	C₃₁×C₂²⋊Q₈	1	4	yes	complemented
447	D₄	—	2	2	no	—
448	D₄	—	2	2	no	—
449	D₄	—	4	1	no	—
450	D₄	—	4	1	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	1	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	8	4	no	—
475	C₂×C₄	C₃₁×#120	1	2	yes	—
476	C₂×C₄	C₃₁×#120	1	2	yes	—
477	C₂×C₄	C₃₁×#135	1	2	yes	—
478	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
479	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
480	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
481	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
482	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
483	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
484	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
485	C₈	—	2	2	no	—
486	C₈	—	4	1	no	—
487	C₈	—	4	2	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	8	1	no	—
492	C₂²	—	8	1	no	—
493	C₂²	—	8	1	no	—
494	C₂²	—	8	1	no	—
495	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
496	C₂²	C₃₁×#548	1	1	yes	characteristic
497	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
498	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
499	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
500	C₂²	C₃₁×#577	1	1	yes	characteristic
501	C₂²	C₃₁×#578	1	1	yes	characteristic
502	C₄	—	2	4	no	—
503	C₄	—	2	2	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	4	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	2	no	—
508	C₄	—	8	2	no	—
509	C₄	—	8	1	no	—
510	C₄	C₃₁×#449	1	4	yes	—
511	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
512	C₄	C₃₁×#536	1	1	yes	characteristic
513	C₄	C₃₁×#548	1	1	yes	characteristic
514	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
515	C₂	—	8	2	no	—
516	C₂	C₃₁×#2409	1	1	yes	characteristic
517	C₂	C₂×C₃₁×#449	1	1	yes	characteristic
518	C₂	C₃₁×#3013	1	1	yes	characteristic
519	C₂	C₃₁×#3364	1	1	yes	characteristic
520	C₂	C₃₁×#3368	1	1	yes	characteristic
521	C₂	C₃₁×#3381	1	1	yes	characteristic
522	C₂	C₃₁×#3386	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	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#557	C₂	semidirect
5. C₃₁×#2059	C₂	semidirect
7. C₃₁×#1990	C₂	semidirect
8. C₃₁×#1984	C₂	semidirect
10. C₃₁×#1863	C₂	semidirect
11. C₃₁×#1803	C₂	semidirect
52. C₂×SD₁₆×C₃₁	Q₈	semidirect
171. #24328	C₃₁	direct (N × H)
185. Q₈×C₃₁	C₂²⋊Q₈	semidirect
227. C₂×#557	C₂×C₃₁	semidirect
231. #2059	C₂×C₃₁	semidirect
233. #1990	C₂×C₃₁	semidirect
234. #1984	C₂×C₃₁	semidirect
236. #1863	C₂×C₃₁	semidirect
237. #1803	C₂×C₃₁	semidirect
313. C₂×SD₁₆	Q₈×C₃₁	semidirect
363. C₃₁	#24328	direct (N × H)
446. Q₈	C₃₁×C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#557	2
C₂×C₃₁×#474	2
C₂×C₄×SD₁₆×C₃₁	2
C₃₁×#2538	2
C₃₁×#2059	2
C₃₁×#2025	2
C₃₁×#1990	2
C₃₁×#1984	2
C₃₁×#1880	2
C₃₁×#1863	2
C₃₁×#1803	2
#24328	31

Subgroups — C31×#24328 #21481087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24328 #21481087