Subgroups — C₃₁×#24813 #21481572

Order 7936.

894 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2542	C₂	1	1	yes	maximal, characteristic
4	Q₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2084	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2065	C₂	1	2	yes	maximal
7	C₃₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1880	C₂	1	2	yes	maximal
10	C₃₁×#1803	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic
12	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
15	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
16	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#475	—	2	4	no	—
20	C₄×SD₁₆×C₃₁	—	2	4	no	—
21	C₂×C₃₁×#106	C₂²	1	2	yes	—
22	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
25	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#103	C₂²	1	2	yes	—
27	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#399	—	2	2	no	—
31	C₃₁×#390	C₂²	1	2	yes	—
32	C₃₁×#389	—	2	2	no	—
33	C₃₁×#389	C₂²	1	2	yes	—
34	C₃₁×#384	—	2	1	no	—
35	C₃₁×#384	C₂²	1	1	yes	characteristic
36	C₃₁×#384	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂×SD₁₆×C₃₁	—	2	2	no	—
48	C₂×SD₁₆×C₃₁	—	2	2	no	—
49	C₂×SD₁₆×C₃₁	Q₈	1	4	yes	complemented
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×C₄⋊Q₈	—	2	2	no	—
54	C₃₁×C₄⋊Q₈	—	2	2	no	—
55	C₃₁×C₂²⋊Q₈	—	4	4	no	—
56	C₃₁×C₂²⋊Q₈	—	4	4	no	—
57	C₄×Q₈×C₃₁	—	2	2	no	—
58	C₄×Q₈×C₃₁	—	2	4	no	—
59	C₄×Q₈×C₃₁	—	4	2	no	—
60	C₄×Q₈×C₃₁	D₄	1	4	yes	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₂×C₄²×C₃₁	—	2	1	no	—
81	C₂×C₄²×C₃₁	—	2	2	no	—
82	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#106	—	2	2	no	—
85	C₃₁×#106	—	2	4	no	—
86	C₃₁×C₄⋊C₈	—	2	2	no	—
87	C₃₁×C₄⋊C₈	—	4	2	no	—
88	C₃₁×#103	—	2	2	no	—
89	C₃₁×#103	—	2	4	no	—
90	C₃₁×#102	—	2	2	no	—
91	C₃₁×#102	—	4	2	no	—
92	C₄×C₈×C₃₁	—	2	2	no	—
93	C₃₁×#95	—	2	2	no	—
94	C₃₁×#95	D₄	1	2	yes	—
95	C₂⁴×C₃₁	—	2	1	no	—
96	C₂×Q₈×C₃₁	—	2	1	no	—
97	C₂×Q₈×C₃₁	—	2	1	no	—
98	C₂×Q₈×C₃₁	—	2	1	no	—
99	C₂×Q₈×C₃₁	—	4	2	no	—
100	C₂×Q₈×C₃₁	—	4	2	no	—
101	C₂×Q₈×C₃₁	—	4	2	no	—
102	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
103	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
104	C₂×Q₈×C₃₁	#41	1	2	yes	—
105	C₂×D₄×C₃₁	—	2	1	no	—
106	C₂×D₄×C₃₁	—	2	1	no	—
107	C₂×D₄×C₃₁	—	2	1	no	—
108	C₂×D₄×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	—	2	1	no	—
110	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
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	2	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	4	2	no	—
117	C₂²×C₄×C₃₁	—	4	2	no	—
118	C₂²×C₄×C₃₁	—	4	2	no	—
119	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
124	SD₁₆×C₃₁	—	2	8	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	1	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
134	C₄⋊C₄×C₃₁	—	2	4	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	—	2	1	no	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	—	2	2	no	—
141	C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₄⋊C₄×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	4	4	no	—
144	C₄⋊C₄×C₃₁	—	4	4	no	—
145	C₄⋊C₄×C₃₁	—	4	2	no	—
146	C₄⋊C₄×C₃₁	—	4	4	no	—
147	C₄⋊C₄×C₃₁	—	4	2	no	—
148	C₄⋊C₄×C₃₁	—	8	1	no	—
149	C₄⋊C₄×C₃₁	—	8	2	no	—
150	C₄⋊C₄×C₃₁	SD₁₆	1	4	yes	complemented
151	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
152	C₂²⋊C₄×C₃₁	—	4	4	no	—
153	C₂²⋊C₄×C₃₁	—	4	2	no	—
154	C₂²⋊C₄×C₃₁	—	4	4	no	—
155	C₄²×C₃₁	—	2	4	no	—
156	C₄²×C₃₁	—	2	2	no	—
157	C₄²×C₃₁	—	2	2	no	—
158	C₄²×C₃₁	—	4	4	no	—
159	C₄²×C₃₁	C₂×D₄	1	2	yes	—
160	C₄²×C₃₁	#41	1	2	yes	—
161	#24813	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂³×C₃₁	—	2	2	no	—
163	C₂³×C₃₁	—	4	1	no	—
164	C₂³×C₃₁	—	4	1	no	—
165	C₂³×C₃₁	—	4	1	no	—
166	C₂³×C₃₁	—	4	1	no	—
167	C₂³×C₃₁	—	4	1	no	—
168	C₂³×C₃₁	—	4	1	no	—
169	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
170	Q₈×C₃₁	—	2	2	no	—
171	Q₈×C₃₁	—	2	2	no	—
172	Q₈×C₃₁	—	2	2	no	—
173	Q₈×C₃₁	—	8	4	no	—
174	Q₈×C₃₁	C₂²⋊Q₈	1	4	yes	complemented
175	D₄×C₃₁	—	2	2	no	—
176	D₄×C₃₁	—	2	2	no	—
177	D₄×C₃₁	—	4	1	no	—
178	D₄×C₃₁	—	4	1	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	2	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	4	2	no	—
198	C₂×C₄×C₃₁	—	4	1	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	8	4	no	—
202	C₂×C₄×C₃₁	#121	1	2	yes	—
203	C₂×C₄×C₃₁	#121	1	2	yes	—
204	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
205	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
206	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
207	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
209	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
213	C₈×C₃₁	—	2	2	no	—
214	C₈×C₃₁	—	4	1	no	—
215	C₈×C₃₁	—	4	2	no	—
216	C₂×#475	C₂×C₃₁	1	1	yes	characteristic, complemented
217	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
218	#2542	C₂×C₃₁	1	1	yes	characteristic
219	Q₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#2084	C₂×C₃₁	1	2	yes	complemented
221	#2065	C₂×C₃₁	1	2	yes	—
222	#1990	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#1880	C₂×C₃₁	1	2	yes	—
225	#1803	C₂×C₃₁	1	2	yes	complemented
226	#1802	C₂×C₃₁	1	1	yes	characteristic
227	C₂²×C₃₁	—	4	2	no	—
228	C₂²×C₃₁	—	4	2	no	—
229	C₂²×C₃₁	—	4	2	no	—
230	C₂²×C₃₁	—	8	1	no	—
231	C₂²×C₃₁	—	8	1	no	—
232	C₂²×C₃₁	—	8	1	no	—
233	C₂²×C₃₁	—	8	1	no	—
234	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
235	C₂²×C₃₁	#546	1	1	yes	characteristic
236	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
237	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
238	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
239	C₂²×C₃₁	#577	1	1	yes	characteristic
240	C₂²×C₃₁	#578	1	1	yes	characteristic
241	C₄×C₃₁	—	2	4	no	—
242	C₄×C₃₁	—	2	4	no	—
243	C₄×C₃₁	—	2	2	no	—
244	C₄×C₃₁	—	4	2	no	—
245	C₄×C₃₁	—	4	2	no	—
246	C₄×C₃₁	—	8	2	no	—
247	C₄×C₃₁	—	8	1	no	—
248	C₄×C₃₁	—	8	2	no	—
249	C₄×C₃₁	#460	1	4	yes	—
250	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
251	C₄×C₃₁	#537	1	1	yes	characteristic
252	C₄×C₃₁	#552	1	1	yes	characteristic
253	C₄×C₃₁	#554	1	1	yes	characteristic
254	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₂²⋊Q₈	—	2	1	no	—
257	C₂×C₂²⋊Q₈	—	2	1	no	—
258	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
259	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
261	#475	—	2	4	no	—
262	C₄×SD₁₆	—	2	4	no	—
263	C₂×#106	C₂²×C₃₁	1	2	yes	—
264	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
267	C₂×#103	C₂²×C₃₁	1	2	yes	—
268	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×#102	C₂²×C₃₁	1	2	yes	—
271	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
272	#399	—	2	2	no	—
273	#390	C₂²×C₃₁	1	2	yes	—
274	#389	—	2	2	no	—
275	#389	C₂²×C₃₁	1	2	yes	—
276	#384	—	2	1	no	—
277	#384	C₂²×C₃₁	1	1	yes	characteristic
278	#384	C₂²×C₃₁	1	2	yes	—
279	C₄×C₄⋊C₄	—	2	1	no	—
280	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
281	#340	C₂²×C₃₁	1	2	yes	—
282	#340	C₂²×C₃₁	1	2	yes	—
283	#340	C₂²×C₃₁	1	2	yes	—
284	#336	C₂²×C₃₁	1	2	yes	—
285	C₂×C₃₁	—	8	2	no	—
286	C₂×C₃₁	#2417	1	1	yes	characteristic
287	C₂×C₃₁	C₂×#460	1	1	yes	characteristic
288	C₂×C₃₁	#3046	1	1	yes	characteristic
289	C₂×C₃₁	#3298	1	1	yes	characteristic
290	C₂×C₃₁	#3332	1	1	yes	characteristic
291	C₂×C₃₁	Q₈×SD₁₆	1	1	yes	characteristic
292	C₂×C₃₁	#3386	1	1	yes	characteristic
293	C₂²×Q₈	—	2	1	no	—
294	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
295	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
296	C₂³×C₄	—	2	1	no	—
297	C₂×SD₁₆	—	2	2	no	—
298	C₂×SD₁₆	—	2	2	no	—
299	C₂×SD₁₆	Q₈×C₃₁	1	4	yes	complemented
300	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
301	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
302	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
303	C₄⋊Q₈	—	2	2	no	—
304	C₄⋊Q₈	—	2	2	no	—
305	C₂²⋊Q₈	—	4	4	no	—
306	C₂²⋊Q₈	—	4	4	no	—
307	C₄×Q₈	—	2	2	no	—
308	C₄×Q₈	—	2	4	no	—
309	C₄×Q₈	—	4	2	no	—
310	C₄×Q₈	D₄×C₃₁	1	4	yes	—
311	C₄×D₄	—	2	2	no	—
312	C₄×D₄	—	2	2	no	—
313	C₂×C₄⋊C₄	—	2	1	no	—
314	C₂×C₄⋊C₄	—	2	2	no	—
315	C₂×C₄⋊C₄	—	2	2	no	—
316	C₂×C₄⋊C₄	—	2	2	no	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	2	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
322	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
325	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
326	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
327	C₂×C₂²⋊C₄	—	2	1	no	—
328	C₂×C₂²⋊C₄	—	2	2	no	—
329	C₂×C₂²⋊C₄	—	2	2	no	—
330	C₂×C₄²	—	2	1	no	—
331	C₂×C₄²	—	2	2	no	—
332	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
333	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
334	#106	—	2	2	no	—
335	#106	—	2	4	no	—
336	C₄⋊C₈	—	2	2	no	—
337	C₄⋊C₈	—	4	2	no	—
338	#103	—	2	2	no	—
339	#103	—	2	4	no	—
340	#102	—	2	2	no	—
341	#102	—	4	2	no	—
342	C₄×C₈	—	2	2	no	—
343	#95	—	2	2	no	—
344	#95	D₄×C₃₁	1	2	yes	—
345	C₃₁	#24813	1	1	yes	characteristic, direct product factor, Sylow
346	C₂⁴	—	2	1	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	1	no	—
350	C₂×Q₈	—	4	2	no	—
351	C₂×Q₈	—	4	2	no	—
352	C₂×Q₈	—	4	2	no	—
353	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
354	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
355	C₂×Q₈	#41×C₃₁	1	2	yes	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	2	1	no	—
358	C₂×D₄	—	2	1	no	—
359	C₂×D₄	—	2	1	no	—
360	C₂×D₄	—	2	1	no	—
361	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	—	4	2	no	—
369	C₂²×C₄	—	4	2	no	—
370	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
371	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
373	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
375	SD₁₆	—	2	8	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
385	C₄⋊C₄	—	2	4	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	4	4	no	—
395	C₄⋊C₄	—	4	4	no	—
396	C₄⋊C₄	—	4	2	no	—
397	C₄⋊C₄	—	4	4	no	—
398	C₄⋊C₄	—	4	2	no	—
399	C₄⋊C₄	—	8	1	no	—
400	C₄⋊C₄	—	8	2	no	—
401	C₄⋊C₄	SD₁₆×C₃₁	1	4	yes	complemented
402	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
403	C₂²⋊C₄	—	4	4	no	—
404	C₂²⋊C₄	—	4	2	no	—
405	C₂²⋊C₄	—	4	4	no	—
406	C₄²	—	2	4	no	—
407	C₄²	—	2	2	no	—
408	C₄²	—	2	2	no	—
409	C₄²	—	4	4	no	—
410	C₄²	C₂×D₄×C₃₁	1	2	yes	—
411	C₄²	#41×C₃₁	1	2	yes	—
412	C₂³	—	2	2	no	—
413	C₂³	—	4	1	no	—
414	C₂³	—	4	1	no	—
415	C₂³	—	4	1	no	—
416	C₂³	—	4	1	no	—
417	C₂³	—	4	1	no	—
418	C₂³	—	4	1	no	—
419	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
420	Q₈	—	2	2	no	—
421	Q₈	—	2	2	no	—
422	Q₈	—	2	2	no	—
423	Q₈	—	8	4	no	—
424	Q₈	C₃₁×C₂²⋊Q₈	1	4	yes	complemented
425	D₄	—	2	2	no	—
426	D₄	—	2	2	no	—
427	D₄	—	4	1	no	—
428	D₄	—	4	1	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	8	4	no	—
452	C₂×C₄	C₃₁×#121	1	2	yes	—
453	C₂×C₄	C₃₁×#121	1	2	yes	—
454	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
455	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
456	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
457	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
458	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
459	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
460	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
461	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
462	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
463	C₈	—	2	2	no	—
464	C₈	—	4	1	no	—
465	C₈	—	4	2	no	—
466	C₂²	—	4	2	no	—
467	C₂²	—	4	2	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	8	1	no	—
470	C₂²	—	8	1	no	—
471	C₂²	—	8	1	no	—
472	C₂²	—	8	1	no	—
473	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
474	C₂²	C₃₁×#546	1	1	yes	characteristic
475	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
476	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
477	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
478	C₂²	C₃₁×#577	1	1	yes	characteristic
479	C₂²	C₃₁×#578	1	1	yes	characteristic
480	C₄	—	2	4	no	—
481	C₄	—	2	4	no	—
482	C₄	—	2	2	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	2	no	—
485	C₄	—	8	2	no	—
486	C₄	—	8	1	no	—
487	C₄	—	8	2	no	—
488	C₄	C₃₁×#460	1	4	yes	—
489	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
490	C₄	C₃₁×#537	1	1	yes	characteristic
491	C₄	C₃₁×#552	1	1	yes	characteristic
492	C₄	C₃₁×#554	1	1	yes	characteristic
493	C₂	—	8	2	no	—
494	C₂	C₃₁×#2417	1	1	yes	characteristic
495	C₂	C₂×C₃₁×#460	1	1	yes	characteristic
496	C₂	C₃₁×#3046	1	1	yes	characteristic
497	C₂	C₃₁×#3298	1	1	yes	characteristic
498	C₂	C₃₁×#3332	1	1	yes	characteristic
499	C₂	Q₈×SD₁₆×C₃₁	1	1	yes	characteristic
500	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 ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#475	C₂	semidirect
4. Q₈×C₄⋊C₄×C₃₁	C₂	semidirect
5. C₃₁×#2084	C₂	semidirect
7. C₃₁×#1990	C₂	semidirect
8. C₃₁×#1939	C₂	semidirect
10. C₃₁×#1803	C₂	semidirect
49. C₂×SD₁₆×C₃₁	Q₈	semidirect
150. C₄⋊C₄×C₃₁	SD₁₆	semidirect
161. #24813	C₃₁	direct (N × H)
174. Q₈×C₃₁	C₂²⋊Q₈	semidirect
216. C₂×#475	C₂×C₃₁	semidirect
219. Q₈×C₄⋊C₄	C₂×C₃₁	semidirect
220. #2084	C₂×C₃₁	semidirect
222. #1990	C₂×C₃₁	semidirect
223. #1939	C₂×C₃₁	semidirect
225. #1803	C₂×C₃₁	semidirect
299. C₂×SD₁₆	Q₈×C₃₁	semidirect
345. C₃₁	#24813	direct (N × H)
401. C₄⋊C₄	SD₁₆×C₃₁	semidirect
424. 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×#475	2
C₂×C₄×SD₁₆×C₃₁	2
C₃₁×#2542	2
Q₈×C₄⋊C₄×C₃₁	2
C₃₁×#2084	2
C₃₁×#2065	2
C₃₁×#1990	2
C₃₁×#1939	2
C₃₁×#1880	2
C₃₁×#1803	2
C₃₁×#1802	2
#24813	31

Subgroups — C31×#24813 #21481572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24813 #21481572