Subgroups — C₃₁×#60724 #21517483

Order 7936.

1874 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3512	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3510	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3183	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3179	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3178	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3168	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3166	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3153	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#488	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3063	C₂	1	2	yes	maximal, complemented
12	C₃₁×#3062	C₂	1	2	yes	maximal, complemented
13	C₃₁×#2947	C₂	1	2	yes	maximal, complemented
14	C₃₁×#2946	C₂	1	2	yes	maximal, complemented
15	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#137	C₂²	1	2	yes	—
17	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
18	C₂×C₃₁×#131	C₂²	1	2	yes	—
19	C₃₁×#559	—	2	2	no	—
20	C₃₁×#556	—	2	2	no	—
21	C₃₁×#553	—	2	4	no	—
22	C₃₁×#553	—	2	2	no	—
23	C₃₁×#552	—	2	2	no	—
24	C₃₁×#551	—	2	4	no	—
25	C₃₁×#542	C₂²	1	4	yes	—
26	C₃₁×#541	C₂²	1	2	yes	complemented
27	C₃₁×#540	C₂²	1	2	yes	complemented
28	C₃₁×#537	C₂²	1	2	yes	—
29	C₃₁×#536	C₂²	1	1	yes	characteristic
30	C₃₁×#535	C₂²	1	1	yes	characteristic
31	C₃₁×#533	C₂²	1	1	yes	characteristic
32	C₃₁×#532	C₂²	1	1	yes	characteristic
33	C₃₁×#529	C₂²	1	4	yes	—
34	C₃₁×#529	C₂²	1	4	yes	complemented
35	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
37	C₃₁×#520	C₂²	1	2	yes	—
38	C₃₁×#519	C₂²	1	1	yes	characteristic
39	C₃₁×#518	C₂²	1	1	yes	characteristic
40	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#501	C₂²	1	4	yes	complemented
42	C₃₁×#499	C₂²	1	4	yes	complemented
43	C₃₁×#497	C₂²	1	2	yes	—
44	C₃₁×#496	C₂²	1	2	yes	—
45	C₃₁×#495	C₂²	1	4	yes	—
46	C₃₁×#488	C₂²	1	2	yes	—
47	C₃₁×#488	C₂²	1	4	yes	—
48	C₃₁×#488	C₂²	1	2	yes	—
49	C₃₁×#488	C₂²	1	2	yes	complemented
50	C₃₁×#488	C₂²	1	2	yes	complemented
51	C₃₁×#487	C₂²	1	2	yes	complemented
52	C₃₁×#486	C₂²	1	2	yes	complemented
53	C₃₁×#470	C₂²	1	4	yes	complemented
54	C₃₁×#469	C₂²	1	4	yes	—
55	C₃₁×#468	C₂²	1	4	yes	—
56	C₃₁×#468	C₂²	1	4	yes	complemented
57	C₃₁×#467	C₂²	1	4	yes	complemented
58	C₃₁×#466	C₂²	1	4	yes	—
59	C₃₁×#465	C₂²	1	4	yes	—
60	C₃₁×#465	C₂²	1	4	yes	complemented
61	C₃₁×#442	C₂²	1	4	yes	—
62	C₃₁×#441	C₂²	1	4	yes	complemented
63	C₃₁×#440	C₂²	1	4	yes	—
64	C₃₁×#440	C₂²	1	4	yes	complemented
65	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	complemented
66	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
67	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
68	C₄×D₈×C₃₁	C₂²	1	4	yes	—
69	C₃₁×#428	C₂²	1	2	yes	—
70	C₃₁×#428	C₂²	1	2	yes	—
71	C₃₁×#416	C₂²	1	2	yes	—
72	C₃₁×#416	C₂²	1	2	yes	—
73	C₂×C₃₁×#103	C₂²	1	2	yes	—
74	C₂×C₃₁×#102	C₂²	1	2	yes	—
75	C₃₁×#405	C₂²	1	2	yes	—
76	C₃₁×#405	C₂²	1	4	yes	complemented
77	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
78	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
79	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
80	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
81	C₂×#41×C₃₁	C₂³	1	2	yes	—
82	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
83	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₃₁×#137	—	2	8	no	—
86	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
87	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
88	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
89	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
90	C₂×D₈×C₃₁	C₂³	1	4	yes	—
91	C₃₁×#131	—	2	8	no	—
92	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
93	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
94	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
95	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
96	C₃₁×C₄⋊Q₈	—	2	1	no	—
97	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
98	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
99	C₃₁×#127	—	2	1	no	—
100	C₃₁×#127	C₂³	1	1	yes	characteristic
101	C₃₁×#126	—	2	4	no	—
102	C₃₁×#126	—	2	4	no	—
103	C₃₁×#126	—	2	4	no	—
104	C₃₁×#126	—	2	4	no	—
105	C₃₁×#125	—	2	4	no	—
106	C₃₁×#125	—	2	2	no	—
107	C₃₁×#125	C₂³	1	2	yes	—
108	C₃₁×#125	C₂³	1	2	yes	—
109	C₃₁×#125	C₂³	1	2	yes	—
110	C₃₁×#124	—	2	1	no	—
111	C₃₁×#124	—	2	1	no	—
112	C₃₁×#124	—	2	4	no	—
113	C₃₁×#124	—	2	2	no	—
114	C₃₁×#124	C₂³	1	2	yes	—
115	C₃₁×#124	C₂³	1	2	yes	—
116	C₃₁×#124	C₂³	1	2	yes	—
117	C₃₁×#123	—	2	2	no	—
118	C₃₁×#123	—	2	4	no	—
119	C₃₁×#123	—	2	4	no	—
120	C₃₁×#123	—	2	4	no	—
121	C₃₁×#123	—	2	4	no	—
122	C₃₁×C₂²⋊Q₈	—	2	2	no	—
123	C₃₁×C₂²⋊Q₈	—	2	2	no	—
124	C₃₁×C₂²⋊Q₈	—	2	4	no	—
125	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
126	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
127	C₃₁×#121	—	2	2	no	—
128	C₃₁×#121	—	2	4	no	—
129	C₃₁×#121	—	2	2	no	—
130	C₃₁×#121	—	2	4	no	—
131	C₃₁×#121	C₂³	1	4	yes	—
132	C₃₁×#121	C₂³	1	4	yes	—
133	C₃₁×#120	—	2	2	no	—
134	C₄×Q₈×C₃₁	—	2	2	no	—
135	C₄×Q₈×C₃₁	—	2	2	no	—
136	C₄×Q₈×C₃₁	D₄	1	2	yes	—
137	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
138	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
139	C₄×D₄×C₃₁	—	2	4	no	—
140	C₄×D₄×C₃₁	—	2	4	no	—
141	C₄×D₄×C₃₁	—	2	2	no	—
142	C₄×D₄×C₃₁	—	2	4	no	—
143	C₄×D₄×C₃₁	—	2	2	no	—
144	C₄×D₄×C₃₁	D₄	1	4	yes	—
145	C₄×D₄×C₃₁	D₄	1	2	yes	—
146	C₄×D₄×C₃₁	C₂³	1	4	yes	—
147	C₄×D₄×C₃₁	C₂³	1	4	yes	—
148	C₃₁×#117	—	2	2	no	—
149	C₃₁×#117	—	2	2	no	—
150	C₃₁×#117	—	2	4	no	—
151	C₃₁×#117	—	2	4	no	—
152	C₃₁×#117	C₂³	1	1	yes	characteristic
153	C₃₁×#117	C₂³	1	1	yes	characteristic
154	C₃₁×#117	C₂³	1	1	yes	characteristic
155	C₃₁×#117	C₂³	1	2	yes	—
156	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
158	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
159	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
160	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
161	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
162	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
163	C₃₁×#107	C₂³	1	4	yes	—
164	C₃₁×#107	C₂³	1	4	yes	—
165	C₃₁×#106	C₂³	1	4	yes	—
166	C₃₁×#106	C₂³	1	4	yes	—
167	C₃₁×#103	C₂³	1	4	yes	—
168	C₃₁×#103	C₂³	1	4	yes	—
169	C₃₁×#103	C₂³	1	2	yes	—
170	C₃₁×#103	C₂³	1	2	yes	—
171	C₃₁×#103	C₂³	1	2	yes	—
172	C₃₁×#103	C₂³	1	2	yes	—
173	C₃₁×#102	C₂³	1	4	yes	—
174	C₃₁×#102	C₂³	1	4	yes	—
175	C₃₁×#102	C₂³	1	2	yes	—
176	C₃₁×#102	C₂³	1	2	yes	—
177	C₃₁×#102	C₂³	1	2	yes	—
178	C₃₁×#102	C₂³	1	2	yes	—
179	C₃₁×#97	C₂³	1	2	yes	—
180	C₃₁×#97	C₂³	1	4	yes	—
181	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
182	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
183	C₄×C₈×C₃₁	C₂³	1	2	yes	—
184	C₄×C₈×C₃₁	C₂³	1	2	yes	—
185	C₄×C₈×C₃₁	C₂³	1	4	yes	—
186	C₂⁴×C₃₁	—	2	1	no	—
187	#41×C₃₁	—	2	4	no	—
188	#41×C₃₁	—	2	4	no	—
189	#41×C₃₁	—	2	4	no	—
190	C₂×Q₈×C₃₁	—	2	1	no	—
191	C₂×Q₈×C₃₁	—	2	2	no	—
192	C₂×Q₈×C₃₁	—	2	4	no	—
193	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
195	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
196	C₂×D₄×C₃₁	—	2	2	no	—
197	C₂×D₄×C₃₁	—	2	4	no	—
198	C₂×D₄×C₃₁	—	2	4	no	—
199	C₂×D₄×C₃₁	—	2	1	no	—
200	C₂×D₄×C₃₁	—	2	2	no	—
201	C₂×D₄×C₃₁	—	2	4	no	—
202	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
203	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
204	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
205	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
206	C₂²×C₄×C₃₁	—	2	1	no	—
207	C₂²×C₄×C₃₁	—	2	2	no	—
208	C₂²×C₄×C₃₁	—	2	2	no	—
209	C₂²×C₄×C₃₁	—	2	4	no	—
210	C₂²×C₄×C₃₁	—	2	4	no	—
211	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
212	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
213	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
214	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
215	Q₁₆×C₃₁	—	2	8	no	—
216	SD₁₆×C₃₁	—	2	8	no	—
217	SD₁₆×C₃₁	—	2	8	no	—
218	D₈×C₃₁	—	2	8	no	—
219	M₁₆×C₃₁	—	2	8	no	—
220	M₁₆×C₃₁	#41	1	8	yes	—
221	C₂×C₈×C₃₁	—	2	4	no	—
222	C₂×C₈×C₃₁	—	2	8	no	—
223	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
224	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
225	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
226	C₄⋊C₄×C₃₁	—	2	2	no	—
227	C₄⋊C₄×C₃₁	—	2	4	no	—
228	C₄⋊C₄×C₃₁	—	2	4	no	—
229	C₄⋊C₄×C₃₁	—	2	4	no	—
230	C₄⋊C₄×C₃₁	—	2	2	no	—
231	C₄⋊C₄×C₃₁	—	2	4	no	—
232	C₄⋊C₄×C₃₁	—	2	4	no	—
233	C₄⋊C₄×C₃₁	—	2	4	no	—
234	C₄⋊C₄×C₃₁	—	2	2	no	—
235	C₄⋊C₄×C₃₁	—	2	2	no	—
236	C₄⋊C₄×C₃₁	—	2	2	no	—
237	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
238	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
239	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
240	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
241	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
242	C₂²⋊C₄×C₃₁	—	2	1	no	—
243	C₂²⋊C₄×C₃₁	—	2	1	no	—
244	C₂²⋊C₄×C₃₁	—	2	4	no	—
245	C₂²⋊C₄×C₃₁	—	2	4	no	—
246	C₂²⋊C₄×C₃₁	—	2	4	no	—
247	C₂²⋊C₄×C₃₁	—	2	4	no	—
248	C₂²⋊C₄×C₃₁	—	2	4	no	—
249	C₂²⋊C₄×C₃₁	—	2	2	no	—
250	C₂²⋊C₄×C₃₁	—	2	4	no	—
251	C₂²⋊C₄×C₃₁	—	2	4	no	—
252	C₂²⋊C₄×C₃₁	—	2	4	no	—
253	C₂²⋊C₄×C₃₁	—	2	2	no	—
254	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
255	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
256	C₄²×C₃₁	—	2	4	no	—
257	C₄²×C₃₁	—	2	4	no	—
258	C₄²×C₃₁	—	2	1	no	—
259	C₄²×C₃₁	—	2	1	no	—
260	C₄²×C₃₁	C₂×D₄	1	4	yes	—
261	C₄²×C₃₁	C₂×D₄	1	2	yes	—
262	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
263	C₄²×C₃₁	C₂⁴	1	2	yes	—
264	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
265	#60724	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₂³×C₃₁	—	2	2	no	—
267	C₂³×C₃₁	—	2	2	no	—
268	C₂³×C₃₁	—	4	2	no	—
269	C₂³×C₃₁	—	8	2	no	—
270	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
271	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
272	Q₈×C₃₁	—	2	2	no	—
273	Q₈×C₃₁	—	2	2	no	—
274	Q₈×C₃₁	—	2	4	no	—
275	Q₈×C₃₁	—	2	2	no	—
276	Q₈×C₃₁	—	4	2	no	—
277	Q₈×C₃₁	—	8	2	no	—
278	D₄×C₃₁	—	2	4	no	—
279	D₄×C₃₁	—	2	2	no	—
280	D₄×C₃₁	—	2	2	no	—
281	D₄×C₃₁	—	2	4	no	—
282	D₄×C₃₁	—	2	2	no	—
283	D₄×C₃₁	—	4	4	no	—
284	D₄×C₃₁	—	4	2	no	—
285	D₄×C₃₁	—	8	2	no	—
286	D₄×C₃₁	—	8	4	no	—
287	D₄×C₃₁	—	8	4	no	—
288	C₂×C₄×C₃₁	—	2	4	no	—
289	C₂×C₄×C₃₁	—	2	2	no	—
290	C₂×C₄×C₃₁	—	2	2	no	—
291	C₂×C₄×C₃₁	—	2	2	no	—
292	C₂×C₄×C₃₁	—	2	2	no	—
293	C₂×C₄×C₃₁	—	2	4	no	—
294	C₂×C₄×C₃₁	—	4	4	no	—
295	C₂×C₄×C₃₁	—	4	2	no	—
296	C₂×C₄×C₃₁	—	4	2	no	—
297	C₂×C₄×C₃₁	—	4	4	no	—
298	C₂×C₄×C₃₁	—	8	4	no	—
299	C₂×C₄×C₃₁	—	8	4	no	—
300	C₂×C₄×C₃₁	—	8	2	no	—
301	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
302	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
303	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
304	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
305	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
306	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
307	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
308	C₈×C₃₁	—	2	2	no	—
309	C₈×C₃₁	—	2	2	no	—
310	C₈×C₃₁	C₂×#41	1	8	yes	—
311	#3512	C₂×C₃₁	1	1	yes	characteristic, complemented
312	#3510	C₂×C₃₁	1	1	yes	characteristic, complemented
313	#3183	C₂×C₃₁	1	4	yes	complemented
314	#3179	C₂×C₃₁	1	4	yes	complemented
315	#3178	C₂×C₃₁	1	4	yes	complemented
316	#3168	C₂×C₃₁	1	4	yes	complemented
317	#3166	C₂×C₃₁	1	1	yes	characteristic, complemented
318	#3153	C₂×C₃₁	1	1	yes	characteristic, complemented
319	#3131	C₂×C₃₁	1	1	yes	characteristic, complemented
320	C₂×#488	C₂×C₃₁	1	1	yes	characteristic, complemented
321	#3063	C₂×C₃₁	1	2	yes	complemented
322	#3062	C₂×C₃₁	1	2	yes	complemented
323	#2947	C₂×C₃₁	1	2	yes	complemented
324	#2946	C₂×C₃₁	1	2	yes	complemented
325	C₄×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
326	C₂²×C₃₁	—	2	2	no	—
327	C₂²×C₃₁	—	2	4	no	—
328	C₂²×C₃₁	—	2	2	no	—
329	C₂²×C₃₁	—	4	2	no	—
330	C₂²×C₃₁	—	4	4	no	—
331	C₂²×C₃₁	—	4	2	no	—
332	C₂²×C₃₁	—	4	2	no	—
333	C₂²×C₃₁	—	4	2	no	—
334	C₂²×C₃₁	—	16	2	no	—
335	C₂²×C₃₁	#532	1	2	yes	—
336	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
337	C₄×C₃₁	—	2	2	no	—
338	C₄×C₃₁	—	2	4	no	—
339	C₄×C₃₁	—	2	2	no	—
340	C₄×C₃₁	—	2	2	no	—
341	C₄×C₃₁	—	4	4	no	—
342	C₄×C₃₁	—	4	2	no	—
343	C₄×C₃₁	—	4	2	no	—
344	C₄×C₃₁	—	4	2	no	—
345	C₄×C₃₁	—	4	2	no	—
346	C₄×C₃₁	#532	1	2	yes	—
347	C₄×C₃₁	C₂²×#41	1	2	yes	—
348	C₂×#137	C₂²×C₃₁	1	2	yes	—
349	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
350	C₂×#131	C₂²×C₃₁	1	2	yes	—
351	#559	—	2	2	no	—
352	#556	—	2	2	no	—
353	#553	—	2	4	no	—
354	#553	—	2	2	no	—
355	#552	—	2	2	no	—
356	#551	—	2	4	no	—
357	#542	C₂²×C₃₁	1	4	yes	—
358	#541	C₂²×C₃₁	1	2	yes	complemented
359	#540	C₂²×C₃₁	1	2	yes	complemented
360	#537	C₂²×C₃₁	1	2	yes	—
361	#536	C₂²×C₃₁	1	1	yes	characteristic
362	#535	C₂²×C₃₁	1	1	yes	characteristic
363	#533	C₂²×C₃₁	1	1	yes	characteristic
364	#532	C₂²×C₃₁	1	1	yes	characteristic
365	#529	C₂²×C₃₁	1	4	yes	—
366	#529	C₂²×C₃₁	1	4	yes	complemented
367	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
368	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
369	#520	C₂²×C₃₁	1	2	yes	—
370	#519	C₂²×C₃₁	1	1	yes	characteristic
371	#518	C₂²×C₃₁	1	1	yes	characteristic
372	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
373	#501	C₂²×C₃₁	1	4	yes	complemented
374	#499	C₂²×C₃₁	1	4	yes	complemented
375	#497	C₂²×C₃₁	1	2	yes	—
376	#496	C₂²×C₃₁	1	2	yes	—
377	#495	C₂²×C₃₁	1	4	yes	—
378	#488	C₂²×C₃₁	1	2	yes	—
379	#488	C₂²×C₃₁	1	4	yes	—
380	#488	C₂²×C₃₁	1	2	yes	—
381	#488	C₂²×C₃₁	1	2	yes	complemented
382	#488	C₂²×C₃₁	1	2	yes	complemented
383	#487	C₂²×C₃₁	1	2	yes	complemented
384	#486	C₂²×C₃₁	1	2	yes	complemented
385	#470	C₂²×C₃₁	1	4	yes	complemented
386	#469	C₂²×C₃₁	1	4	yes	—
387	#468	C₂²×C₃₁	1	4	yes	—
388	#468	C₂²×C₃₁	1	4	yes	complemented
389	#467	C₂²×C₃₁	1	4	yes	complemented
390	#466	C₂²×C₃₁	1	4	yes	—
391	#465	C₂²×C₃₁	1	4	yes	—
392	#465	C₂²×C₃₁	1	4	yes	complemented
393	#442	C₂²×C₃₁	1	4	yes	—
394	#441	C₂²×C₃₁	1	4	yes	complemented
395	#440	C₂²×C₃₁	1	4	yes	—
396	#440	C₂²×C₃₁	1	4	yes	complemented
397	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	complemented
398	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
399	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
400	C₄×D₈	C₂²×C₃₁	1	4	yes	—
401	#428	C₂²×C₃₁	1	2	yes	—
402	#428	C₂²×C₃₁	1	2	yes	—
403	#416	C₂²×C₃₁	1	2	yes	—
404	#416	C₂²×C₃₁	1	2	yes	—
405	C₂×#103	C₂²×C₃₁	1	2	yes	—
406	C₂×#102	C₂²×C₃₁	1	2	yes	—
407	#405	C₂²×C₃₁	1	2	yes	—
408	#405	C₂²×C₃₁	1	4	yes	complemented
409	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
410	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
411	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
412	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
413	C₂×C₃₁	—	2	2	no	—
414	C₂×C₃₁	—	4	2	no	—
415	C₂×C₃₁	—	8	2	no	—
416	C₂×C₃₁	—	8	2	no	—
417	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
418	C₂×C₃₁	#3584	1	2	yes	—
419	C₂×#41	C₂³×C₃₁	1	2	yes	—
420	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
421	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
422	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
423	#137	—	2	8	no	—
424	AGL(1,ℤ/8)	—	2	8	no	—
425	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
426	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
427	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
428	C₂×D₈	C₂³×C₃₁	1	4	yes	—
429	#131	—	2	8	no	—
430	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
431	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
432	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
433	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
434	C₄⋊Q₈	—	2	1	no	—
435	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
436	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
437	#127	—	2	1	no	—
438	#127	C₂³×C₃₁	1	1	yes	characteristic
439	#126	—	2	4	no	—
440	#126	—	2	4	no	—
441	#126	—	2	4	no	—
442	#126	—	2	4	no	—
443	#125	—	2	4	no	—
444	#125	—	2	2	no	—
445	#125	C₂³×C₃₁	1	2	yes	—
446	#125	C₂³×C₃₁	1	2	yes	—
447	#125	C₂³×C₃₁	1	2	yes	—
448	#124	—	2	1	no	—
449	#124	—	2	1	no	—
450	#124	—	2	4	no	—
451	#124	—	2	2	no	—
452	#124	C₂³×C₃₁	1	2	yes	—
453	#124	C₂³×C₃₁	1	2	yes	—
454	#124	C₂³×C₃₁	1	2	yes	—
455	#123	—	2	2	no	—
456	#123	—	2	4	no	—
457	#123	—	2	4	no	—
458	#123	—	2	4	no	—
459	#123	—	2	4	no	—
460	C₂²⋊Q₈	—	2	2	no	—
461	C₂²⋊Q₈	—	2	2	no	—
462	C₂²⋊Q₈	—	2	4	no	—
463	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
464	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
465	#121	—	2	2	no	—
466	#121	—	2	4	no	—
467	#121	—	2	2	no	—
468	#121	—	2	4	no	—
469	#121	C₂³×C₃₁	1	4	yes	—
470	#121	C₂³×C₃₁	1	4	yes	—
471	#120	—	2	2	no	—
472	C₄×Q₈	—	2	2	no	—
473	C₄×Q₈	—	2	2	no	—
474	C₄×Q₈	D₄×C₃₁	1	2	yes	—
475	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
476	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
477	C₄×D₄	—	2	4	no	—
478	C₄×D₄	—	2	4	no	—
479	C₄×D₄	—	2	2	no	—
480	C₄×D₄	—	2	4	no	—
481	C₄×D₄	—	2	2	no	—
482	C₄×D₄	D₄×C₃₁	1	4	yes	—
483	C₄×D₄	D₄×C₃₁	1	2	yes	—
484	C₄×D₄	C₂³×C₃₁	1	4	yes	—
485	C₄×D₄	C₂³×C₃₁	1	4	yes	—
486	#117	—	2	2	no	—
487	#117	—	2	2	no	—
488	#117	—	2	4	no	—
489	#117	—	2	4	no	—
490	#117	C₂³×C₃₁	1	1	yes	characteristic
491	#117	C₂³×C₃₁	1	1	yes	characteristic
492	#117	C₂³×C₃₁	1	1	yes	characteristic
493	#117	C₂³×C₃₁	1	2	yes	—
494	C₂×C₄⋊C₄	—	2	2	no	—
495	C₂×C₄⋊C₄	—	2	2	no	—
496	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
497	C₂×C₂²⋊C₄	—	2	2	no	—
498	C₂×C₂²⋊C₄	—	2	2	no	—
499	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
500	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
501	#107	C₂³×C₃₁	1	4	yes	—
502	#107	C₂³×C₃₁	1	4	yes	—
503	#106	C₂³×C₃₁	1	4	yes	—
504	#106	C₂³×C₃₁	1	4	yes	—
505	#103	C₂³×C₃₁	1	4	yes	—
506	#103	C₂³×C₃₁	1	4	yes	—
507	#103	C₂³×C₃₁	1	2	yes	—
508	#103	C₂³×C₃₁	1	2	yes	—
509	#103	C₂³×C₃₁	1	2	yes	—
510	#103	C₂³×C₃₁	1	2	yes	—
511	#102	C₂³×C₃₁	1	4	yes	—
512	#102	C₂³×C₃₁	1	4	yes	—
513	#102	C₂³×C₃₁	1	2	yes	—
514	#102	C₂³×C₃₁	1	2	yes	—
515	#102	C₂³×C₃₁	1	2	yes	—
516	#102	C₂³×C₃₁	1	2	yes	—
517	#97	C₂³×C₃₁	1	2	yes	—
518	#97	C₂³×C₃₁	1	4	yes	—
519	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
520	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
521	C₄×C₈	C₂³×C₃₁	1	2	yes	—
522	C₄×C₈	C₂³×C₃₁	1	2	yes	—
523	C₄×C₈	C₂³×C₃₁	1	4	yes	—
524	C₃₁	#60724	1	1	yes	characteristic, direct product factor, Sylow
525	C₂⁴	—	2	1	no	—
526	#41	—	2	4	no	—
527	#41	—	2	4	no	—
528	#41	—	2	4	no	—
529	C₂×Q₈	—	2	1	no	—
530	C₂×Q₈	—	2	2	no	—
531	C₂×Q₈	—	2	4	no	—
532	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
533	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
534	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
535	C₂×D₄	—	2	2	no	—
536	C₂×D₄	—	2	4	no	—
537	C₂×D₄	—	2	4	no	—
538	C₂×D₄	—	2	1	no	—
539	C₂×D₄	—	2	2	no	—
540	C₂×D₄	—	2	4	no	—
541	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
542	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
543	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
544	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
545	C₂²×C₄	—	2	1	no	—
546	C₂²×C₄	—	2	2	no	—
547	C₂²×C₄	—	2	2	no	—
548	C₂²×C₄	—	2	4	no	—
549	C₂²×C₄	—	2	4	no	—
550	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
551	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
552	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
553	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
554	Q₁₆	—	2	8	no	—
555	SD₁₆	—	2	8	no	—
556	SD₁₆	—	2	8	no	—
557	D₈	—	2	8	no	—
558	M₁₆	—	2	8	no	—
559	M₁₆	#41×C₃₁	1	8	yes	—
560	C₂×C₈	—	2	4	no	—
561	C₂×C₈	—	2	8	no	—
562	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
563	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
564	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
565	C₄⋊C₄	—	2	2	no	—
566	C₄⋊C₄	—	2	4	no	—
567	C₄⋊C₄	—	2	4	no	—
568	C₄⋊C₄	—	2	4	no	—
569	C₄⋊C₄	—	2	2	no	—
570	C₄⋊C₄	—	2	4	no	—
571	C₄⋊C₄	—	2	4	no	—
572	C₄⋊C₄	—	2	4	no	—
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	—	2	2	no	—
575	C₄⋊C₄	—	2	2	no	—
576	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
577	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
578	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
579	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
580	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
581	C₂²⋊C₄	—	2	1	no	—
582	C₂²⋊C₄	—	2	1	no	—
583	C₂²⋊C₄	—	2	4	no	—
584	C₂²⋊C₄	—	2	4	no	—
585	C₂²⋊C₄	—	2	4	no	—
586	C₂²⋊C₄	—	2	4	no	—
587	C₂²⋊C₄	—	2	4	no	—
588	C₂²⋊C₄	—	2	2	no	—
589	C₂²⋊C₄	—	2	4	no	—
590	C₂²⋊C₄	—	2	4	no	—
591	C₂²⋊C₄	—	2	4	no	—
592	C₂²⋊C₄	—	2	2	no	—
593	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
594	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
595	C₄²	—	2	4	no	—
596	C₄²	—	2	4	no	—
597	C₄²	—	2	1	no	—
598	C₄²	—	2	1	no	—
599	C₄²	C₂×D₄×C₃₁	1	4	yes	—
600	C₄²	C₂×D₄×C₃₁	1	2	yes	—
601	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
602	C₄²	C₂⁴×C₃₁	1	2	yes	—
603	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
604	C₂³	—	2	2	no	—
605	C₂³	—	2	2	no	—
606	C₂³	—	4	2	no	—
607	C₂³	—	8	2	no	—
608	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
609	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
610	Q₈	—	2	2	no	—
611	Q₈	—	2	2	no	—
612	Q₈	—	2	4	no	—
613	Q₈	—	2	2	no	—
614	Q₈	—	4	2	no	—
615	Q₈	—	8	2	no	—
616	D₄	—	2	4	no	—
617	D₄	—	2	2	no	—
618	D₄	—	2	2	no	—
619	D₄	—	2	4	no	—
620	D₄	—	2	2	no	—
621	D₄	—	4	4	no	—
622	D₄	—	4	2	no	—
623	D₄	—	8	2	no	—
624	D₄	—	8	4	no	—
625	D₄	—	8	4	no	—
626	C₂×C₄	—	2	4	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	2	no	—
631	C₂×C₄	—	2	4	no	—
632	C₂×C₄	—	4	4	no	—
633	C₂×C₄	—	4	2	no	—
634	C₂×C₄	—	4	2	no	—
635	C₂×C₄	—	4	4	no	—
636	C₂×C₄	—	8	4	no	—
637	C₂×C₄	—	8	4	no	—
638	C₂×C₄	—	8	2	no	—
639	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
640	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
641	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
642	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
643	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
644	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
645	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
646	C₈	—	2	2	no	—
647	C₈	—	2	2	no	—
648	C₈	C₂×#41×C₃₁	1	8	yes	—
649	C₂²	—	2	2	no	—
650	C₂²	—	2	2	no	—
651	C₂²	—	2	4	no	—
652	C₂²	—	4	2	no	—
653	C₂²	—	4	2	no	—
654	C₂²	—	4	2	no	—
655	C₂²	—	4	4	no	—
656	C₂²	—	4	2	no	—
657	C₂²	—	16	2	no	—
658	C₂²	C₃₁×#532	1	2	yes	—
659	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
660	C₄	—	2	2	no	—
661	C₄	—	2	2	no	—
662	C₄	—	2	4	no	—
663	C₄	—	2	2	no	—
664	C₄	—	4	2	no	—
665	C₄	—	4	2	no	—
666	C₄	—	4	4	no	—
667	C₄	—	4	2	no	—
668	C₄	—	4	2	no	—
669	C₄	C₃₁×#532	1	2	yes	—
670	C₄	C₂²×#41×C₃₁	1	2	yes	—
671	C₂	—	2	2	no	—
672	C₂	—	4	2	no	—
673	C₂	—	8	2	no	—
674	C₂	—	8	2	no	—
675	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
676	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3512	C₂	semidirect
2. C₃₁×#3510	C₂	semidirect
3. C₃₁×#3183	C₂	semidirect
4. C₃₁×#3179	C₂	semidirect
5. C₃₁×#3178	C₂	semidirect
6. C₃₁×#3168	C₂	semidirect
7. C₃₁×#3166	C₂	semidirect
8. C₃₁×#3153	C₂	semidirect
9. C₃₁×#3131	C₂	semidirect
10. C₂×C₃₁×#488	C₂	semidirect
11. C₃₁×#3063	C₂	semidirect
12. C₃₁×#3062	C₂	semidirect
13. C₃₁×#2947	C₂	semidirect
14. C₃₁×#2946	C₂	semidirect
15. C₄×C₃₁×#131	C₂	semidirect
26. C₃₁×#541	C₂²	semidirect
27. C₃₁×#540	C₂²	semidirect
34. C₃₁×#529	C₂²	semidirect
41. C₃₁×#501	C₂²	semidirect
42. C₃₁×#499	C₂²	semidirect
49. C₃₁×#488	C₂²	semidirect
50. C₃₁×#488	C₂²	semidirect
51. C₃₁×#487	C₂²	semidirect
52. C₃₁×#486	C₂²	semidirect
53. C₃₁×#470	C₂²	semidirect
56. C₃₁×#468	C₂²	semidirect
57. C₃₁×#467	C₂²	semidirect
60. C₃₁×#465	C₂²	semidirect
62. C₃₁×#441	C₂²	semidirect
64. C₃₁×#440	C₂²	semidirect
65. C₄×Q₁₆×C₃₁	C₂²	semidirect
67. C₄×SD₁₆×C₃₁	C₂²	semidirect
76. C₃₁×#405	C₂²	semidirect
78. C₄×M₁₆×C₃₁	C₂²	semidirect
80. C₂×C₄×C₈×C₃₁	C₂²	semidirect
265. #60724	C₃₁	direct (N × H)
311. #3512	C₂×C₃₁	semidirect
312. #3510	C₂×C₃₁	semidirect
313. #3183	C₂×C₃₁	semidirect
314. #3179	C₂×C₃₁	semidirect
315. #3178	C₂×C₃₁	semidirect
316. #3168	C₂×C₃₁	semidirect
317. #3166	C₂×C₃₁	semidirect
318. #3153	C₂×C₃₁	semidirect
319. #3131	C₂×C₃₁	semidirect
320. C₂×#488	C₂×C₃₁	semidirect
321. #3063	C₂×C₃₁	semidirect
322. #3062	C₂×C₃₁	semidirect
323. #2947	C₂×C₃₁	semidirect
324. #2946	C₂×C₃₁	semidirect
325. C₄×#131	C₂×C₃₁	semidirect
358. #541	C₂²×C₃₁	semidirect
359. #540	C₂²×C₃₁	semidirect
366. #529	C₂²×C₃₁	semidirect
373. #501	C₂²×C₃₁	semidirect
374. #499	C₂²×C₃₁	semidirect
381. #488	C₂²×C₃₁	semidirect
382. #488	C₂²×C₃₁	semidirect
383. #487	C₂²×C₃₁	semidirect
384. #486	C₂²×C₃₁	semidirect
385. #470	C₂²×C₃₁	semidirect
388. #468	C₂²×C₃₁	semidirect
389. #467	C₂²×C₃₁	semidirect
392. #465	C₂²×C₃₁	semidirect
394. #441	C₂²×C₃₁	semidirect
396. #440	C₂²×C₃₁	semidirect
397. C₄×Q₁₆	C₂²×C₃₁	semidirect
399. C₄×SD₁₆	C₂²×C₃₁	semidirect
408. #405	C₂²×C₃₁	semidirect
410. C₄×M₁₆	C₂²×C₃₁	semidirect
412. C₂×C₄×C₈	C₂²×C₃₁	semidirect
524. C₃₁	#60724	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3512	2
C₃₁×#3510	2
C₃₁×#3183	2
C₃₁×#3179	2
C₃₁×#3178	2
C₃₁×#3168	2
C₃₁×#3166	2
C₃₁×#3153	2
C₃₁×#3131	2
C₂×C₃₁×#488	2
C₃₁×#3063	2
C₃₁×#3062	2
C₃₁×#2947	2
C₃₁×#2946	2
C₄×C₃₁×#131	2
#60724	31

Subgroups — C31×#60724 #21517483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60724 #21517483