Subgroups — C₃₁×#61269 #21518028

Order 7936.

1750 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3495	C₂	1	2	yes	maximal
2	C₃₁×#3356	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3352	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3349	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3344	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3163	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3097	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3095	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3082	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3059	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#470	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2950	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2930	C₂	1	1	yes	maximal, characteristic
14	C₃₁×#2912	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#2911	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#556	C₂²	1	4	yes	—
20	C₃₁×#554	C₂²	1	4	yes	—
21	C₃₁×#552	—	2	4	no	—
22	C₃₁×#552	C₂²	1	4	yes	—
23	C₃₁×#552	C₂²	1	4	yes	—
24	C₃₁×#551	—	2	4	no	—
25	C₃₁×#544	C₂²	1	2	yes	—
26	C₃₁×#542	—	2	2	no	—
27	C₃₁×#541	—	2	2	no	—
28	C₃₁×#536	C₂²	1	2	yes	—
29	C₃₁×#533	—	2	2	no	—
30	C₃₁×#529	—	2	2	no	—
31	C₃₁×#525	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
33	C₃₁×#520	C₂²	1	2	yes	—
34	C₃₁×#519	C₂²	1	2	yes	—
35	C₂×C₃₁×#117	C₂²	1	2	yes	—
36	C₃₁×#501	C₂²	1	4	yes	complemented
37	C₃₁×#499	C₂²	1	2	yes	complemented
38	C₃₁×#498	C₂²	1	2	yes	complemented
39	C₃₁×#485	C₂²	1	4	yes	—
40	C₃₁×#485	C₂²	1	4	yes	—
41	C₃₁×#485	C₂²	1	4	yes	complemented
42	C₃₁×#485	C₂²	1	4	yes	complemented
43	C₃₁×#484	C₂²	1	4	yes	—
44	C₃₁×#484	C₂²	1	4	yes	complemented
45	C₃₁×#483	C₂²	1	4	yes	—
46	C₃₁×#483	C₂²	1	4	yes	complemented
47	C₃₁×#479	C₂²	1	4	yes	complemented
48	C₃₁×#479	C₂²	1	4	yes	complemented
49	C₃₁×#477	C₂²	1	4	yes	complemented
50	C₃₁×#475	C₂²	1	4	yes	complemented
51	C₃₁×#470	C₂²	1	4	yes	—
52	C₃₁×#470	C₂²	1	4	yes	—
53	C₃₁×#470	C₂²	1	2	yes	complemented
54	C₃₁×#470	C₂²	1	2	yes	complemented
55	C₃₁×#467	C₂²	1	4	yes	complemented
56	C₃₁×#441	C₂²	1	2	yes	complemented
57	C₃₁×#441	C₂²	1	2	yes	complemented
58	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	complemented
59	C₃₁×#436	C₂²	1	4	yes	—
60	C₃₁×#435	C₂²	1	4	yes	—
61	C₃₁×#428	C₂²	1	2	yes	—
62	C₃₁×#428	C₂²	1	2	yes	—
63	C₃₁×#428	C₂²	1	2	yes	—
64	C₃₁×#428	C₂²	1	2	yes	—
65	C₃₁×#428	C₂²	1	2	yes	—
66	C₃₁×#428	C₂²	1	2	yes	—
67	C₃₁×#428	C₂²	1	4	yes	complemented
68	C₃₁×#427	C₂²	1	2	yes	—
69	C₃₁×#427	C₂²	1	1	yes	characteristic
70	C₃₁×#427	C₂²	1	1	yes	characteristic
71	C₃₁×#427	C₂²	1	2	yes	complemented
72	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
73	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
74	C₃₁×#424	C₂²	1	1	yes	characteristic
75	C₃₁×#419	C₂²	1	2	yes	—
76	C₃₁×#419	C₂²	1	2	yes	—
77	C₂×C₃₁×#103	C₂²	1	2	yes	—
78	C₃₁×#408	C₂²	1	1	yes	characteristic
79	C₃₁×#407	C₂²	1	1	yes	characteristic
80	C₃₁×#405	C₂²	1	1	yes	characteristic
81	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
82	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
83	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×Q₁₆×C₃₁	—	2	4	no	—
85	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
86	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
87	C₃₁×#131	—	2	4	no	—
88	C₂×M₁₆×C₃₁	—	2	4	no	—
89	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
90	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
91	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
92	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
93	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
94	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
96	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
97	C₃₁×C₄⋊Q₈	—	2	2	no	—
98	C₃₁×C₄⋊Q₈	—	2	2	no	—
99	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
100	C₃₁×#126	—	2	4	no	—
101	C₃₁×#126	—	2	4	no	—
102	C₃₁×#126	—	2	4	no	—
103	C₃₁×#126	—	2	4	no	—
104	C₃₁×#125	—	2	2	no	—
105	C₃₁×#125	—	2	4	no	—
106	C₃₁×#125	—	2	4	no	—
107	C₃₁×#125	—	2	4	no	—
108	C₃₁×#125	C₂³	1	4	yes	—
109	C₃₁×#124	—	2	2	no	—
110	C₃₁×#124	—	2	4	no	—
111	C₃₁×#123	—	2	4	no	—
112	C₃₁×#123	—	2	4	no	—
113	C₃₁×#123	—	2	4	no	—
114	C₃₁×#123	D₄	1	2	yes	—
115	C₃₁×C₂²⋊Q₈	—	2	2	no	—
116	C₃₁×C₂²⋊Q₈	—	2	2	no	—
117	C₃₁×C₂²⋊Q₈	—	2	4	no	—
118	C₃₁×C₂²⋊Q₈	—	2	4	no	—
119	C₃₁×C₂²⋊Q₈	—	2	4	no	—
120	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
121	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
122	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
123	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
124	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
125	C₃₁×#121	D₄	1	2	yes	—
126	C₃₁×#120	D₄	1	2	yes	—
127	C₄×Q₈×C₃₁	—	2	2	no	—
128	C₄×Q₈×C₃₁	—	2	2	no	—
129	C₄×Q₈×C₃₁	—	2	2	no	—
130	C₄×Q₈×C₃₁	—	2	4	no	—
131	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
132	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
133	C₄×D₄×C₃₁	—	2	2	no	—
134	C₄×D₄×C₃₁	—	2	4	no	—
135	C₄×D₄×C₃₁	—	2	4	no	—
136	C₄×D₄×C₃₁	C₂³	1	4	yes	—
137	C₃₁×#117	—	2	2	no	—
138	C₃₁×#117	—	2	2	no	—
139	C₃₁×#117	—	2	4	no	—
140	C₃₁×#117	—	2	4	no	—
141	C₃₁×#117	—	2	4	no	—
142	C₃₁×#117	C₂³	1	2	yes	—
143	C₃₁×#117	C₂³	1	2	yes	—
144	C₃₁×#117	C₂³	1	4	yes	—
145	C₃₁×#117	C₂³	1	2	yes	—
146	C₃₁×#117	C₂³	1	2	yes	—
147	C₃₁×#117	C₂³	1	1	yes	characteristic
148	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
150	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
151	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
152	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
153	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
154	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
155	C₂×C₄²×C₃₁	—	2	2	no	—
156	C₃₁×#107	C₂³	1	2	yes	—
157	C₃₁×#107	C₂³	1	2	yes	—
158	C₃₁×#107	C₂³	1	2	yes	—
159	C₃₁×#107	C₂³	1	2	yes	—
160	C₃₁×#107	C₂³	1	4	yes	—
161	C₃₁×#107	C₂³	1	4	yes	—
162	C₃₁×#106	C₂³	1	2	yes	—
163	C₃₁×#106	C₂³	1	2	yes	—
164	C₃₁×#106	C₂³	1	1	yes	characteristic
165	C₃₁×#106	C₂³	1	4	yes	—
166	C₃₁×#106	C₂³	1	1	yes	characteristic
167	C₃₁×#106	C₂³	1	1	yes	characteristic
168	C₃₁×#106	C₂³	1	1	yes	characteristic
169	C₃₁×#106	C₂³	1	2	yes	—
170	C₃₁×#106	C₂³	1	2	yes	—
171	C₃₁×#106	C₂³	1	4	yes	—
172	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
173	C₃₁×#103	C₂³	1	4	yes	—
174	C₃₁×#103	C₂³	1	4	yes	—
175	C₃₁×#103	C₂³	1	4	yes	—
176	C₃₁×#103	C₂³	1	4	yes	—
177	C₃₁×#103	C₂³	1	4	yes	—
178	C₃₁×#103	C₂³	1	4	yes	—
179	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
180	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
181	C₃₁×#97	C₂³	1	2	yes	—
182	C₄×C₈×C₃₁	C₂³	1	2	yes	—
183	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
184	#41×C₃₁	—	2	2	no	—
185	#41×C₃₁	—	2	2	no	—
186	C₂×Q₈×C₃₁	—	2	2	no	—
187	C₂×Q₈×C₃₁	—	2	4	no	—
188	C₂×Q₈×C₃₁	—	2	4	no	—
189	C₂×Q₈×C₃₁	—	2	4	no	—
190	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
191	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
192	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
193	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
195	C₂²×C₄×C₃₁	—	2	2	no	—
196	C₂²×C₄×C₃₁	—	2	2	no	—
197	C₂²×C₄×C₃₁	—	2	2	no	—
198	C₂²×C₄×C₃₁	—	2	2	no	—
199	C₂²×C₄×C₃₁	—	2	2	no	—
200	C₂²×C₄×C₃₁	—	2	2	no	—
201	C₂²×C₄×C₃₁	—	2	4	no	—
202	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
203	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
204	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
205	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
206	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
207	Q₁₆×C₃₁	—	2	4	no	—
208	Q₁₆×C₃₁	—	2	4	no	—
209	M₁₆×C₃₁	—	2	2	no	—
210	M₁₆×C₃₁	—	2	2	no	—
211	M₁₆×C₃₁	—	2	4	no	—
212	M₁₆×C₃₁	—	2	4	no	—
213	M₁₆×C₃₁	—	2	4	no	—
214	C₂×C₈×C₃₁	—	2	2	no	—
215	C₂×C₈×C₃₁	—	2	4	no	—
216	C₂×C₈×C₃₁	#41	1	4	yes	—
217	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
218	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
219	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
220	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
221	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
222	C₄⋊C₄×C₃₁	—	2	4	no	—
223	C₄⋊C₄×C₃₁	—	2	4	no	—
224	C₄⋊C₄×C₃₁	—	2	2	no	—
225	C₄⋊C₄×C₃₁	—	2	2	no	—
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	4	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	4	no	—
235	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
236	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
237	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
238	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
239	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
240	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
241	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
242	C₂²⋊C₄×C₃₁	—	2	2	no	—
243	C₂²⋊C₄×C₃₁	—	2	2	no	—
244	C₂²⋊C₄×C₃₁	—	2	2	no	—
245	C₂²⋊C₄×C₃₁	—	2	2	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	4	no	—
250	C₂²⋊C₄×C₃₁	—	2	4	no	—
251	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
252	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
253	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
254	C₄²×C₃₁	—	2	2	no	—
255	C₄²×C₃₁	—	2	2	no	—
256	C₄²×C₃₁	—	2	2	no	—
257	C₄²×C₃₁	—	2	2	no	—
258	C₄²×C₃₁	—	2	4	no	—
259	C₄²×C₃₁	—	2	4	no	—
260	C₄²×C₃₁	—	2	4	no	—
261	C₄²×C₃₁	C₂⁴	1	2	yes	—
262	#61269	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
263	C₂³×C₃₁	—	2	2	no	—
264	C₂³×C₃₁	—	4	2	no	—
265	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
266	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
267	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
268	Q₈×C₃₁	—	2	2	no	—
269	Q₈×C₃₁	—	2	4	no	—
270	Q₈×C₃₁	—	2	4	no	—
271	Q₈×C₃₁	—	4	4	no	—
272	Q₈×C₃₁	—	8	4	no	—
273	D₄×C₃₁	—	2	4	no	—
274	D₄×C₃₁	—	2	2	no	—
275	D₄×C₃₁	—	4	4	no	—
276	C₂×C₄×C₃₁	—	2	2	no	—
277	C₂×C₄×C₃₁	—	2	4	no	—
278	C₂×C₄×C₃₁	—	2	2	no	—
279	C₂×C₄×C₃₁	—	2	2	no	—
280	C₂×C₄×C₃₁	—	2	2	no	—
281	C₂×C₄×C₃₁	—	2	2	no	—
282	C₂×C₄×C₃₁	—	2	2	no	—
283	C₂×C₄×C₃₁	—	2	2	no	—
284	C₂×C₄×C₃₁	—	2	2	no	—
285	C₂×C₄×C₃₁	—	2	4	no	—
286	C₂×C₄×C₃₁	—	2	2	no	—
287	C₂×C₄×C₃₁	—	2	4	no	—
288	C₂×C₄×C₃₁	—	4	4	no	—
289	C₂×C₄×C₃₁	—	4	4	no	—
290	C₂×C₄×C₃₁	—	8	4	no	—
291	C₂×C₄×C₃₁	—	8	2	no	—
292	C₂×C₄×C₃₁	—	8	2	no	—
293	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
294	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
295	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
296	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
297	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
298	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
299	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
300	C₈×C₃₁	—	2	2	no	—
301	C₈×C₃₁	—	2	2	no	—
302	C₈×C₃₁	—	2	2	no	—
303	C₈×C₃₁	C₂×#41	1	2	yes	—
304	C₈×C₃₁	C₂×#41	1	2	yes	—
305	#3495	C₂×C₃₁	1	2	yes	—
306	#3356	C₂×C₃₁	1	4	yes	complemented
307	#3352	C₂×C₃₁	1	4	yes	complemented
308	#3349	C₂×C₃₁	1	4	yes	complemented
309	#3344	C₂×C₃₁	1	4	yes	complemented
310	#3163	C₂×C₃₁	1	1	yes	characteristic, complemented
311	#3097	C₂×C₃₁	1	2	yes	complemented
312	#3095	C₂×C₃₁	1	2	yes	complemented
313	#3082	C₂×C₃₁	1	2	yes	complemented
314	#3059	C₂×C₃₁	1	1	yes	characteristic, complemented
315	C₂×#470	C₂×C₃₁	1	1	yes	characteristic, complemented
316	#2950	C₂×C₃₁	1	1	yes	characteristic, complemented
317	#2930	C₂×C₃₁	1	1	yes	characteristic
318	#2912	C₂×C₃₁	1	1	yes	characteristic, complemented
319	#2911	C₂×C₃₁	1	1	yes	characteristic, complemented
320	C₂²×C₃₁	—	2	2	no	—
321	C₂²×C₃₁	—	2	2	no	—
322	C₂²×C₃₁	—	2	4	no	—
323	C₂²×C₃₁	—	2	2	no	—
324	C₂²×C₃₁	—	2	2	no	—
325	C₂²×C₃₁	—	2	2	no	—
326	C₂²×C₃₁	—	8	2	no	—
327	C₂²×C₃₁	#547	1	2	yes	—
328	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
329	C₄×C₃₁	—	2	4	no	—
330	C₄×C₃₁	—	2	2	no	—
331	C₄×C₃₁	—	2	2	no	—
332	C₄×C₃₁	—	4	2	no	—
333	C₄×C₃₁	—	4	2	no	—
334	C₄×C₃₁	—	4	2	no	—
335	C₄×C₃₁	—	4	4	no	—
336	C₄×C₃₁	—	4	2	no	—
337	C₄×C₃₁	—	4	4	no	—
338	C₄×C₃₁	#547	1	2	yes	—
339	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
340	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
341	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
342	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
343	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
344	#556	C₂²×C₃₁	1	4	yes	—
345	#554	C₂²×C₃₁	1	4	yes	—
346	#552	—	2	4	no	—
347	#552	C₂²×C₃₁	1	4	yes	—
348	#552	C₂²×C₃₁	1	4	yes	—
349	#551	—	2	4	no	—
350	#544	C₂²×C₃₁	1	2	yes	—
351	#542	—	2	2	no	—
352	#541	—	2	2	no	—
353	#536	C₂²×C₃₁	1	2	yes	—
354	#533	—	2	2	no	—
355	#529	—	2	2	no	—
356	#525	C₂²×C₃₁	1	1	yes	characteristic
357	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
358	#520	C₂²×C₃₁	1	2	yes	—
359	#519	C₂²×C₃₁	1	2	yes	—
360	C₂×#117	C₂²×C₃₁	1	2	yes	—
361	#501	C₂²×C₃₁	1	4	yes	complemented
362	#499	C₂²×C₃₁	1	2	yes	complemented
363	#498	C₂²×C₃₁	1	2	yes	complemented
364	#485	C₂²×C₃₁	1	4	yes	—
365	#485	C₂²×C₃₁	1	4	yes	—
366	#485	C₂²×C₃₁	1	4	yes	complemented
367	#485	C₂²×C₃₁	1	4	yes	complemented
368	#484	C₂²×C₃₁	1	4	yes	—
369	#484	C₂²×C₃₁	1	4	yes	complemented
370	#483	C₂²×C₃₁	1	4	yes	—
371	#483	C₂²×C₃₁	1	4	yes	complemented
372	#479	C₂²×C₃₁	1	4	yes	complemented
373	#479	C₂²×C₃₁	1	4	yes	complemented
374	#477	C₂²×C₃₁	1	4	yes	complemented
375	#475	C₂²×C₃₁	1	4	yes	complemented
376	#470	C₂²×C₃₁	1	4	yes	—
377	#470	C₂²×C₃₁	1	4	yes	—
378	#470	C₂²×C₃₁	1	2	yes	complemented
379	#470	C₂²×C₃₁	1	2	yes	complemented
380	#467	C₂²×C₃₁	1	4	yes	complemented
381	#441	C₂²×C₃₁	1	2	yes	complemented
382	#441	C₂²×C₃₁	1	2	yes	complemented
383	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	complemented
384	#436	C₂²×C₃₁	1	4	yes	—
385	#435	C₂²×C₃₁	1	4	yes	—
386	#428	C₂²×C₃₁	1	2	yes	—
387	#428	C₂²×C₃₁	1	2	yes	—
388	#428	C₂²×C₃₁	1	2	yes	—
389	#428	C₂²×C₃₁	1	2	yes	—
390	#428	C₂²×C₃₁	1	2	yes	—
391	#428	C₂²×C₃₁	1	2	yes	—
392	#428	C₂²×C₃₁	1	4	yes	complemented
393	#427	C₂²×C₃₁	1	2	yes	—
394	#427	C₂²×C₃₁	1	1	yes	characteristic
395	#427	C₂²×C₃₁	1	1	yes	characteristic
396	#427	C₂²×C₃₁	1	2	yes	complemented
397	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
398	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
399	#424	C₂²×C₃₁	1	1	yes	characteristic
400	#419	C₂²×C₃₁	1	2	yes	—
401	#419	C₂²×C₃₁	1	2	yes	—
402	C₂×#103	C₂²×C₃₁	1	2	yes	—
403	#408	C₂²×C₃₁	1	1	yes	characteristic
404	#407	C₂²×C₃₁	1	1	yes	characteristic
405	#405	C₂²×C₃₁	1	1	yes	characteristic
406	C₂×C₃₁	—	2	2	no	—
407	C₂×C₃₁	—	4	2	no	—
408	C₂×C₃₁	—	4	2	no	—
409	C₂×C₃₁	C₂×#547	1	1	yes	characteristic
410	C₂×C₃₁	#3586	1	2	yes	—
411	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
412	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
413	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
414	C₂×Q₁₆	—	2	4	no	—
415	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
416	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
417	#131	—	2	4	no	—
418	C₂×M₁₆	—	2	4	no	—
419	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
420	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
421	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
422	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
423	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
424	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
425	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
426	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
427	C₄⋊Q₈	—	2	2	no	—
428	C₄⋊Q₈	—	2	2	no	—
429	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
430	#126	—	2	4	no	—
431	#126	—	2	4	no	—
432	#126	—	2	4	no	—
433	#126	—	2	4	no	—
434	#125	—	2	2	no	—
435	#125	—	2	4	no	—
436	#125	—	2	4	no	—
437	#125	—	2	4	no	—
438	#125	C₂³×C₃₁	1	4	yes	—
439	#124	—	2	2	no	—
440	#124	—	2	4	no	—
441	#123	—	2	4	no	—
442	#123	—	2	4	no	—
443	#123	—	2	4	no	—
444	#123	D₄×C₃₁	1	2	yes	—
445	C₂²⋊Q₈	—	2	2	no	—
446	C₂²⋊Q₈	—	2	2	no	—
447	C₂²⋊Q₈	—	2	4	no	—
448	C₂²⋊Q₈	—	2	4	no	—
449	C₂²⋊Q₈	—	2	4	no	—
450	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
451	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
452	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
453	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
454	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
455	#121	D₄×C₃₁	1	2	yes	—
456	#120	D₄×C₃₁	1	2	yes	—
457	C₄×Q₈	—	2	2	no	—
458	C₄×Q₈	—	2	2	no	—
459	C₄×Q₈	—	2	2	no	—
460	C₄×Q₈	—	2	4	no	—
461	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
462	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
463	C₄×D₄	—	2	2	no	—
464	C₄×D₄	—	2	4	no	—
465	C₄×D₄	—	2	4	no	—
466	C₄×D₄	C₂³×C₃₁	1	4	yes	—
467	#117	—	2	2	no	—
468	#117	—	2	2	no	—
469	#117	—	2	4	no	—
470	#117	—	2	4	no	—
471	#117	—	2	4	no	—
472	#117	C₂³×C₃₁	1	2	yes	—
473	#117	C₂³×C₃₁	1	2	yes	—
474	#117	C₂³×C₃₁	1	2	yes	—
475	#117	C₂³×C₃₁	1	2	yes	—
476	#117	C₂³×C₃₁	1	4	yes	—
477	#117	C₂³×C₃₁	1	1	yes	characteristic
478	C₂×C₄⋊C₄	—	2	2	no	—
479	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
480	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
481	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
482	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
483	C₂×C₂²⋊C₄	—	2	2	no	—
484	C₂×C₂²⋊C₄	—	2	2	no	—
485	C₂×C₄²	—	2	2	no	—
486	#107	C₂³×C₃₁	1	2	yes	—
487	#107	C₂³×C₃₁	1	2	yes	—
488	#107	C₂³×C₃₁	1	2	yes	—
489	#107	C₂³×C₃₁	1	2	yes	—
490	#107	C₂³×C₃₁	1	4	yes	—
491	#107	C₂³×C₃₁	1	4	yes	—
492	#106	C₂³×C₃₁	1	2	yes	—
493	#106	C₂³×C₃₁	1	2	yes	—
494	#106	C₂³×C₃₁	1	1	yes	characteristic
495	#106	C₂³×C₃₁	1	4	yes	—
496	#106	C₂³×C₃₁	1	1	yes	characteristic
497	#106	C₂³×C₃₁	1	1	yes	characteristic
498	#106	C₂³×C₃₁	1	1	yes	characteristic
499	#106	C₂³×C₃₁	1	2	yes	—
500	#106	C₂³×C₃₁	1	2	yes	—
501	#106	C₂³×C₃₁	1	4	yes	—
502	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
503	#103	C₂³×C₃₁	1	4	yes	—
504	#103	C₂³×C₃₁	1	4	yes	—
505	#103	C₂³×C₃₁	1	4	yes	—
506	#103	C₂³×C₃₁	1	4	yes	—
507	#103	C₂³×C₃₁	1	4	yes	—
508	#103	C₂³×C₃₁	1	4	yes	—
509	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
510	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
511	#97	C₂³×C₃₁	1	2	yes	—
512	C₄×C₈	C₂³×C₃₁	1	2	yes	—
513	C₃₁	#61269	1	1	yes	characteristic, direct product factor, Sylow
514	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
515	#41	—	2	2	no	—
516	#41	—	2	2	no	—
517	C₂×Q₈	—	2	2	no	—
518	C₂×Q₈	—	2	4	no	—
519	C₂×Q₈	—	2	4	no	—
520	C₂×Q₈	—	2	4	no	—
521	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
522	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
523	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
524	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
525	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
526	C₂²×C₄	—	2	2	no	—
527	C₂²×C₄	—	2	2	no	—
528	C₂²×C₄	—	2	2	no	—
529	C₂²×C₄	—	2	2	no	—
530	C₂²×C₄	—	2	2	no	—
531	C₂²×C₄	—	2	2	no	—
532	C₂²×C₄	—	2	4	no	—
533	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
534	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
535	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
536	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
537	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
538	Q₁₆	—	2	4	no	—
539	Q₁₆	—	2	4	no	—
540	M₁₆	—	2	2	no	—
541	M₁₆	—	2	2	no	—
542	M₁₆	—	2	4	no	—
543	M₁₆	—	2	4	no	—
544	M₁₆	—	2	4	no	—
545	C₂×C₈	—	2	2	no	—
546	C₂×C₈	—	2	4	no	—
547	C₂×C₈	#41×C₃₁	1	4	yes	—
548	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
549	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
550	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
551	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
552	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
553	C₄⋊C₄	—	2	4	no	—
554	C₄⋊C₄	—	2	4	no	—
555	C₄⋊C₄	—	2	2	no	—
556	C₄⋊C₄	—	2	2	no	—
557	C₄⋊C₄	—	2	2	no	—
558	C₄⋊C₄	—	2	4	no	—
559	C₄⋊C₄	—	2	4	no	—
560	C₄⋊C₄	—	2	4	no	—
561	C₄⋊C₄	—	2	4	no	—
562	C₄⋊C₄	—	2	4	no	—
563	C₄⋊C₄	—	2	4	no	—
564	C₄⋊C₄	—	2	4	no	—
565	C₄⋊C₄	—	2	4	no	—
566	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
567	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
568	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
569	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
570	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
571	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
572	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
573	C₂²⋊C₄	—	2	2	no	—
574	C₂²⋊C₄	—	2	2	no	—
575	C₂²⋊C₄	—	2	2	no	—
576	C₂²⋊C₄	—	2	2	no	—
577	C₂²⋊C₄	—	2	4	no	—
578	C₂²⋊C₄	—	2	4	no	—
579	C₂²⋊C₄	—	2	4	no	—
580	C₂²⋊C₄	—	2	4	no	—
581	C₂²⋊C₄	—	2	4	no	—
582	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
583	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
584	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
585	C₄²	—	2	2	no	—
586	C₄²	—	2	2	no	—
587	C₄²	—	2	2	no	—
588	C₄²	—	2	2	no	—
589	C₄²	—	2	4	no	—
590	C₄²	—	2	4	no	—
591	C₄²	—	2	4	no	—
592	C₄²	C₂⁴×C₃₁	1	2	yes	—
593	C₂³	—	2	2	no	—
594	C₂³	—	4	2	no	—
595	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
596	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
597	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
598	Q₈	—	2	2	no	—
599	Q₈	—	2	4	no	—
600	Q₈	—	2	4	no	—
601	Q₈	—	4	4	no	—
602	Q₈	—	8	4	no	—
603	D₄	—	2	4	no	—
604	D₄	—	2	2	no	—
605	D₄	—	4	4	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	4	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	2	4	no	—
618	C₂×C₄	—	4	4	no	—
619	C₂×C₄	—	4	4	no	—
620	C₂×C₄	—	8	4	no	—
621	C₂×C₄	—	8	2	no	—
622	C₂×C₄	—	8	2	no	—
623	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
624	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
625	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
626	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
627	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
628	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
629	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
630	C₈	—	2	2	no	—
631	C₈	—	2	2	no	—
632	C₈	—	2	2	no	—
633	C₈	C₂×#41×C₃₁	1	2	yes	—
634	C₈	C₂×#41×C₃₁	1	2	yes	—
635	C₂²	—	2	2	no	—
636	C₂²	—	2	2	no	—
637	C₂²	—	2	2	no	—
638	C₂²	—	2	2	no	—
639	C₂²	—	2	4	no	—
640	C₂²	—	2	2	no	—
641	C₂²	—	8	2	no	—
642	C₂²	C₃₁×#547	1	2	yes	—
643	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
644	C₄	—	2	2	no	—
645	C₄	—	2	2	no	—
646	C₄	—	2	4	no	—
647	C₄	—	4	4	no	—
648	C₄	—	4	2	no	—
649	C₄	—	4	2	no	—
650	C₄	—	4	2	no	—
651	C₄	—	4	4	no	—
652	C₄	—	4	2	no	—
653	C₄	C₃₁×#547	1	2	yes	—
654	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
655	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
656	C₂	—	2	2	no	—
657	C₂	—	4	2	no	—
658	C₂	—	4	2	no	—
659	C₂	C₂×C₃₁×#547	1	1	yes	characteristic
660	C₂	C₃₁×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#525 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3356	C₂	semidirect
3. C₃₁×#3352	C₂	semidirect
4. C₃₁×#3349	C₂	semidirect
5. C₃₁×#3344	C₂	semidirect
6. C₃₁×#3163	C₂	semidirect
7. C₃₁×#3097	C₂	semidirect
8. C₃₁×#3095	C₂	semidirect
9. C₃₁×#3082	C₂	semidirect
10. C₃₁×#3059	C₂	semidirect
11. C₂×C₃₁×#470	C₂	semidirect
12. C₃₁×#2950	C₂	semidirect
14. C₃₁×#2912	C₂	semidirect
15. C₃₁×#2911	C₂	semidirect
36. C₃₁×#501	C₂²	semidirect
37. C₃₁×#499	C₂²	semidirect
38. C₃₁×#498	C₂²	semidirect
41. C₃₁×#485	C₂²	semidirect
42. C₃₁×#485	C₂²	semidirect
44. C₃₁×#484	C₂²	semidirect
46. C₃₁×#483	C₂²	semidirect
47. C₃₁×#479	C₂²	semidirect
48. C₃₁×#479	C₂²	semidirect
49. C₃₁×#477	C₂²	semidirect
50. C₃₁×#475	C₂²	semidirect
53. C₃₁×#470	C₂²	semidirect
54. C₃₁×#470	C₂²	semidirect
55. C₃₁×#467	C₂²	semidirect
56. C₃₁×#441	C₂²	semidirect
57. C₃₁×#441	C₂²	semidirect
58. C₄×Q₁₆×C₃₁	C₂²	semidirect
67. C₃₁×#428	C₂²	semidirect
71. C₃₁×#427	C₂²	semidirect
73. C₂×C₃₁×#106	C₂²	semidirect
262. #61269	C₃₁	direct (N × H)
306. #3356	C₂×C₃₁	semidirect
307. #3352	C₂×C₃₁	semidirect
308. #3349	C₂×C₃₁	semidirect
309. #3344	C₂×C₃₁	semidirect
310. #3163	C₂×C₃₁	semidirect
311. #3097	C₂×C₃₁	semidirect
312. #3095	C₂×C₃₁	semidirect
313. #3082	C₂×C₃₁	semidirect
314. #3059	C₂×C₃₁	semidirect
315. C₂×#470	C₂×C₃₁	semidirect
316. #2950	C₂×C₃₁	semidirect
318. #2912	C₂×C₃₁	semidirect
319. #2911	C₂×C₃₁	semidirect
361. #501	C₂²×C₃₁	semidirect
362. #499	C₂²×C₃₁	semidirect
363. #498	C₂²×C₃₁	semidirect
366. #485	C₂²×C₃₁	semidirect
367. #485	C₂²×C₃₁	semidirect
369. #484	C₂²×C₃₁	semidirect
371. #483	C₂²×C₃₁	semidirect
372. #479	C₂²×C₃₁	semidirect
373. #479	C₂²×C₃₁	semidirect
374. #477	C₂²×C₃₁	semidirect
375. #475	C₂²×C₃₁	semidirect
378. #470	C₂²×C₃₁	semidirect
379. #470	C₂²×C₃₁	semidirect
380. #467	C₂²×C₃₁	semidirect
381. #441	C₂²×C₃₁	semidirect
382. #441	C₂²×C₃₁	semidirect
383. C₄×Q₁₆	C₂²×C₃₁	semidirect
392. #428	C₂²×C₃₁	semidirect
396. #427	C₂²×C₃₁	semidirect
398. C₂×#106	C₂²×C₃₁	semidirect
513. C₃₁	#61269	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×#3495	2
C₃₁×#3356	2
C₃₁×#3352	2
C₃₁×#3349	2
C₃₁×#3344	2
C₃₁×#3163	2
C₃₁×#3097	2
C₃₁×#3095	2
C₃₁×#3082	2
C₃₁×#3059	2
C₂×C₃₁×#470	2
C₃₁×#2950	2
C₃₁×#2930	2
C₃₁×#2912	2
C₃₁×#2911	2
#61269	31

Subgroups — C31×#61269 #21518028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61269 #21518028