Subgroups — C₃₁×#61007 #21517766

Order 7936.

1704 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3510	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3263	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3262	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3233	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3211	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3202	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3200	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3199	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#3183	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3168	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3121	C₂	1	4	yes	maximal, complemented
12	C₃₁×#3118	C₂	1	4	yes	maximal, complemented
13	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#559	—	2	4	no	—
15	C₃₁×#556	—	2	4	no	—
16	C₃₁×#551	—	2	4	no	—
17	C₃₁×#551	—	2	4	no	—
18	C₃₁×#542	C₂²	1	4	yes	—
19	C₃₁×#542	C₂²	1	4	yes	complemented
20	C₃₁×#536	C₂²	1	2	yes	—
21	C₃₁×#535	C₂²	1	2	yes	—
22	C₃₁×#533	C₂²	1	2	yes	complemented
23	C₃₁×#529	C₂²	1	4	yes	—
24	C₃₁×#529	C₂²	1	4	yes	complemented
25	C₂×C₃₁×#124	C₂²	1	2	yes	complemented
26	C₃₁×#519	C₂²	1	2	yes	complemented
27	C₃₁×#518	C₂²	1	2	yes	complemented
28	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#499	C₂²	1	2	yes	complemented
30	C₃₁×#499	C₂²	1	2	yes	complemented
31	C₃₁×#495	C₂²	1	2	yes	—
32	C₃₁×#495	C₂²	1	2	yes	—
33	C₃₁×#488	C₂²	1	4	yes	complemented
34	C₃₁×#488	C₂²	1	4	yes	complemented
35	C₃₁×#485	C₂²	1	4	yes	complemented
36	C₃₁×#485	C₂²	1	4	yes	complemented
37	C₃₁×#482	C₂²	1	4	yes	—
38	C₃₁×#482	C₂²	1	4	yes	—
39	C₃₁×#477	C₂²	1	4	yes	complemented
40	C₃₁×#476	C₂²	1	4	yes	complemented
41	C₃₁×#475	C₂²	1	4	yes	complemented
42	C₃₁×#474	C₂²	1	4	yes	complemented
43	C₃₁×#470	C₂²	1	2	yes	complemented
44	C₃₁×#469	C₂²	1	2	yes	—
45	C₃₁×#468	C₂²	1	4	yes	complemented
46	C₃₁×#467	C₂²	1	2	yes	complemented
47	C₃₁×#466	C₂²	1	2	yes	—
48	C₃₁×#465	C₂²	1	4	yes	complemented
49	C₃₁×#464	C₂²	1	4	yes	complemented
50	C₃₁×#464	C₂²	1	4	yes	complemented
51	C₃₁×#463	C₂²	1	4	yes	—
52	C₃₁×#463	C₂²	1	4	yes	—
53	C₃₁×#451	C₂²	1	4	yes	complemented
54	C₃₁×#450	C₂²	1	4	yes	complemented
55	C₃₁×#448	C₂²	1	4	yes	—
56	C₃₁×#447	C₂²	1	4	yes	—
57	C₃₁×#442	C₂²	1	2	yes	—
58	C₃₁×#441	C₂²	1	2	yes	complemented
59	C₃₁×#440	C₂²	1	4	yes	complemented
60	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
61	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
62	C₄×D₈×C₃₁	C₂²	1	2	yes	—
63	C₃₁×#433	C₂²	1	4	yes	complemented
64	C₃₁×#431	C₂²	1	4	yes	complemented
65	C₃₁×#424	C₂²	1	1	yes	characteristic, complemented
66	C₃₁×#424	C₂²	1	1	yes	characteristic, complemented
67	C₃₁×#408	C₂²	1	1	yes	characteristic
68	C₃₁×#408	C₂²	1	1	yes	characteristic
69	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
70	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
71	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
73	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
74	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
75	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
76	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
77	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
78	C₂×D₈×C₃₁	C₂³	1	2	yes	—
79	C₂×D₈×C₃₁	C₂³	1	2	yes	—
80	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×C₄⋊Q₈	—	2	2	no	—
85	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
86	C₃₁×#127	—	2	2	no	—
87	C₃₁×#126	—	2	4	no	—
88	C₃₁×#126	—	2	4	no	—
89	C₃₁×#126	—	2	4	no	—
90	C₃₁×#126	—	2	4	no	—
91	C₃₁×#125	C₂³	1	4	yes	—
92	C₃₁×#124	—	2	2	no	—
93	C₃₁×#124	—	2	2	no	—
94	C₃₁×#124	—	2	4	no	—
95	C₃₁×#124	—	2	4	no	—
96	C₃₁×#124	—	2	4	no	—
97	C₃₁×#124	C₂³	1	4	yes	—
98	C₃₁×#124	C₂³	1	4	yes	—
99	C₃₁×#123	—	2	2	no	—
100	C₃₁×#123	—	2	2	no	—
101	C₃₁×#123	—	2	4	no	—
102	C₃₁×#123	—	2	4	no	—
103	C₃₁×#123	—	2	4	no	—
104	C₃₁×#123	—	2	4	no	—
105	C₃₁×C₂²⋊Q₈	—	2	4	no	—
106	C₃₁×C₂²⋊Q₈	—	2	4	no	—
107	C₃₁×C₂²⋊Q₈	—	2	4	no	—
108	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
109	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
110	C₃₁×#121	C₂³	1	4	yes	—
111	C₃₁×#121	C₂³	1	4	yes	—
112	C₃₁×#120	—	2	2	no	—
113	C₃₁×#120	—	2	2	no	—
114	C₄×Q₈×C₃₁	—	2	4	no	—
115	C₄×Q₈×C₃₁	—	2	4	no	—
116	C₄×Q₈×C₃₁	D₄	1	2	yes	—
117	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
118	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
119	C₄×D₄×C₃₁	—	2	4	no	—
120	C₄×D₄×C₃₁	—	2	4	no	—
121	C₄×D₄×C₃₁	—	2	4	no	—
122	C₄×D₄×C₃₁	D₄	1	2	yes	—
123	C₄×D₄×C₃₁	D₄	1	4	yes	—
124	C₄×D₄×C₃₁	C₂³	1	4	yes	—
125	C₄×D₄×C₃₁	C₂³	1	4	yes	—
126	C₃₁×#117	—	2	4	no	—
127	C₃₁×#117	—	2	4	no	—
128	C₃₁×#117	—	2	4	no	—
129	C₃₁×#117	C₂³	1	2	yes	—
130	C₃₁×#117	C₂³	1	2	yes	—
131	C₃₁×#117	C₂³	1	1	yes	characteristic
132	C₃₁×#117	C₂³	1	2	yes	—
133	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
134	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
135	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
137	C₃₁×#107	C₂³	1	2	yes	—
138	C₃₁×#107	C₂³	1	2	yes	—
139	C₃₁×#107	C₂³	1	4	yes	—
140	C₃₁×#106	C₂³	1	4	yes	—
141	C₃₁×#106	C₂³	1	2	yes	—
142	C₃₁×#106	C₂³	1	2	yes	—
143	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
144	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
145	C₃₁×#103	C₂³	1	4	yes	—
146	C₃₁×#103	C₂³	1	4	yes	—
147	C₃₁×#103	C₂³	1	4	yes	—
148	C₃₁×#103	C₂³	1	4	yes	—
149	C₃₁×#102	C₂³	1	4	yes	—
150	C₃₁×#102	C₂³	1	4	yes	—
151	C₃₁×#102	C₂³	1	4	yes	—
152	C₃₁×#102	C₂³	1	4	yes	—
153	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
154	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
155	C₃₁×#97	C₂³	1	2	yes	—
156	C₄×C₈×C₃₁	C₂³	1	2	yes	—
157	C₄×C₈×C₃₁	C₂³	1	2	yes	—
158	C₄×C₈×C₃₁	C₂³	1	2	yes	—
159	C₂⁴×C₃₁	—	2	2	no	—
160	#41×C₃₁	—	4	1	no	—
161	#41×C₃₁	—	4	1	no	—
162	C₂×Q₈×C₃₁	—	2	2	no	—
163	C₂×Q₈×C₃₁	—	2	4	no	—
164	C₂×Q₈×C₃₁	—	4	2	no	—
165	C₂×Q₈×C₃₁	—	4	2	no	—
166	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
168	C₂×D₄×C₃₁	—	2	4	no	—
169	C₂×D₄×C₃₁	—	2	4	no	—
170	C₂×D₄×C₃₁	—	2	2	no	—
171	C₂×D₄×C₃₁	—	4	2	no	—
172	C₂×D₄×C₃₁	—	4	2	no	—
173	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₃₁	—	2	2	no	—
177	C₂²×C₄×C₃₁	—	2	4	no	—
178	C₂²×C₄×C₃₁	—	2	4	no	—
179	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
180	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
181	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
183	Q₁₆×C₃₁	—	4	2	no	—
184	Q₁₆×C₃₁	—	4	2	no	—
185	SD₁₆×C₃₁	—	4	4	no	—
186	SD₁₆×C₃₁	—	4	4	no	—
187	D₈×C₃₁	—	4	2	no	—
188	D₈×C₃₁	—	4	2	no	—
189	M₁₆×C₃₁	—	4	1	no	—
190	M₁₆×C₃₁	—	4	1	no	—
191	C₂×C₈×C₃₁	—	4	1	no	—
192	C₂×C₈×C₃₁	—	4	1	no	—
193	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
194	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
195	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
196	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
197	C₄⋊C₄×C₃₁	—	2	4	no	—
198	C₄⋊C₄×C₃₁	—	2	4	no	—
199	C₄⋊C₄×C₃₁	—	2	4	no	—
200	C₄⋊C₄×C₃₁	—	2	4	no	—
201	C₄⋊C₄×C₃₁	—	2	4	no	—
202	C₄⋊C₄×C₃₁	—	2	4	no	—
203	C₄⋊C₄×C₃₁	—	2	4	no	—
204	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
205	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
206	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
207	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
208	C₂²⋊C₄×C₃₁	—	2	2	no	—
209	C₂²⋊C₄×C₃₁	—	2	2	no	—
210	C₂²⋊C₄×C₃₁	—	2	4	no	—
211	C₂²⋊C₄×C₃₁	—	2	4	no	—
212	C₂²⋊C₄×C₃₁	—	2	4	no	—
213	C₂²⋊C₄×C₃₁	—	2	4	no	—
214	C₂²⋊C₄×C₃₁	—	2	4	no	—
215	C₂²⋊C₄×C₃₁	—	2	4	no	—
216	C₂²⋊C₄×C₃₁	—	2	4	no	—
217	C₂²⋊C₄×C₃₁	—	2	4	no	—
218	C₂²⋊C₄×C₃₁	—	2	4	no	—
219	C₂²⋊C₄×C₃₁	—	2	4	no	—
220	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
221	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
222	C₄²×C₃₁	—	2	4	no	—
223	C₄²×C₃₁	—	2	4	no	—
224	C₄²×C₃₁	—	2	2	no	—
225	C₄²×C₃₁	—	2	2	no	—
226	C₄²×C₃₁	C₂×D₄	1	2	yes	—
227	C₄²×C₃₁	C₂×D₄	1	4	yes	—
228	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
229	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
230	C₄²×C₃₁	C₂⁴	1	2	yes	—
231	#61007	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂³×C₃₁	—	2	4	no	—
233	C₂³×C₃₁	—	4	4	no	—
234	C₂³×C₃₁	—	8	4	no	—
235	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
236	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	Q₈×C₃₁	—	2	4	no	—
238	Q₈×C₃₁	—	2	4	no	—
239	Q₈×C₃₁	—	4	1	no	—
240	Q₈×C₃₁	—	4	4	no	—
241	Q₈×C₃₁	—	8	4	no	—
242	D₄×C₃₁	—	2	4	no	—
243	D₄×C₃₁	—	2	4	no	—
244	D₄×C₃₁	—	4	2	no	—
245	D₄×C₃₁	—	4	1	no	—
246	D₄×C₃₁	—	4	4	no	—
247	D₄×C₃₁	—	8	4	no	—
248	D₄×C₃₁	—	8	4	no	—
249	C₂×C₄×C₃₁	—	2	1	no	—
250	C₂×C₄×C₃₁	—	2	1	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	2	2	no	—
253	C₂×C₄×C₃₁	—	2	4	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	4	no	—
256	C₂×C₄×C₃₁	—	4	2	no	—
257	C₂×C₄×C₃₁	—	4	4	no	—
258	C₂×C₄×C₃₁	—	4	4	no	—
259	C₂×C₄×C₃₁	—	8	4	no	—
260	C₂×C₄×C₃₁	—	8	4	no	—
261	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
262	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
263	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
264	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
267	C₈×C₃₁	—	2	2	no	—
268	C₈×C₃₁	—	2	2	no	—
269	C₈×C₃₁	—	2	2	no	—
270	C₈×C₃₁	—	2	2	no	—
271	#3510	C₂×C₃₁	1	2	yes	complemented
272	#3263	C₂×C₃₁	1	2	yes	complemented
273	#3262	C₂×C₃₁	1	2	yes	complemented
274	#3233	C₂×C₃₁	1	4	yes	complemented
275	#3211	C₂×C₃₁	1	4	yes	complemented
276	#3202	C₂×C₃₁	1	1	yes	characteristic, complemented
277	#3200	C₂×C₃₁	1	2	yes	complemented
278	#3199	C₂×C₃₁	1	1	yes	characteristic
279	#3183	C₂×C₃₁	1	2	yes	complemented
280	#3168	C₂×C₃₁	1	2	yes	complemented
281	#3121	C₂×C₃₁	1	4	yes	complemented
282	#3118	C₂×C₃₁	1	4	yes	complemented
283	#2922	C₂×C₃₁	1	1	yes	characteristic, complemented
284	C₂²×C₃₁	—	2	2	no	—
285	C₂²×C₃₁	—	2	4	no	—
286	C₂²×C₃₁	—	2	2	no	—
287	C₂²×C₃₁	—	2	1	no	—
288	C₂²×C₃₁	—	4	4	no	—
289	C₂²×C₃₁	—	4	4	no	—
290	C₂²×C₃₁	—	4	4	no	—
291	C₂²×C₃₁	—	16	4	no	—
292	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
293	C₄×C₃₁	—	2	2	no	—
294	C₄×C₃₁	—	2	2	no	—
295	C₄×C₃₁	—	2	4	no	—
296	C₄×C₃₁	—	2	2	no	—
297	C₄×C₃₁	—	2	1	no	—
298	C₄×C₃₁	—	4	4	no	—
299	C₄×C₃₁	—	4	4	no	—
300	C₄×C₃₁	—	4	4	no	—
301	C₄×C₃₁	#585	1	1	yes	characteristic
302	C₄×C₃₁	#585	1	1	yes	characteristic
303	#559	—	2	4	no	—
304	#556	—	2	4	no	—
305	#551	—	2	4	no	—
306	#551	—	2	4	no	—
307	#542	C₂²×C₃₁	1	4	yes	—
308	#542	C₂²×C₃₁	1	4	yes	complemented
309	#536	C₂²×C₃₁	1	2	yes	—
310	#535	C₂²×C₃₁	1	2	yes	—
311	#533	C₂²×C₃₁	1	2	yes	complemented
312	#529	C₂²×C₃₁	1	4	yes	—
313	#529	C₂²×C₃₁	1	4	yes	complemented
314	C₂×#124	C₂²×C₃₁	1	2	yes	complemented
315	#519	C₂²×C₃₁	1	2	yes	complemented
316	#518	C₂²×C₃₁	1	2	yes	complemented
317	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
318	#499	C₂²×C₃₁	1	2	yes	complemented
319	#499	C₂²×C₃₁	1	2	yes	complemented
320	#495	C₂²×C₃₁	1	2	yes	—
321	#495	C₂²×C₃₁	1	2	yes	—
322	#488	C₂²×C₃₁	1	4	yes	complemented
323	#488	C₂²×C₃₁	1	4	yes	complemented
324	#485	C₂²×C₃₁	1	4	yes	complemented
325	#485	C₂²×C₃₁	1	4	yes	complemented
326	#482	C₂²×C₃₁	1	4	yes	—
327	#482	C₂²×C₃₁	1	4	yes	—
328	#477	C₂²×C₃₁	1	4	yes	complemented
329	#476	C₂²×C₃₁	1	4	yes	complemented
330	#475	C₂²×C₃₁	1	4	yes	complemented
331	#474	C₂²×C₃₁	1	4	yes	complemented
332	#470	C₂²×C₃₁	1	2	yes	complemented
333	#469	C₂²×C₃₁	1	2	yes	—
334	#468	C₂²×C₃₁	1	4	yes	complemented
335	#467	C₂²×C₃₁	1	2	yes	complemented
336	#466	C₂²×C₃₁	1	2	yes	—
337	#465	C₂²×C₃₁	1	4	yes	complemented
338	#464	C₂²×C₃₁	1	4	yes	complemented
339	#464	C₂²×C₃₁	1	4	yes	complemented
340	#463	C₂²×C₃₁	1	4	yes	—
341	#463	C₂²×C₃₁	1	4	yes	—
342	#451	C₂²×C₃₁	1	4	yes	complemented
343	#450	C₂²×C₃₁	1	4	yes	complemented
344	#448	C₂²×C₃₁	1	4	yes	—
345	#447	C₂²×C₃₁	1	4	yes	—
346	#442	C₂²×C₃₁	1	2	yes	—
347	#441	C₂²×C₃₁	1	2	yes	complemented
348	#440	C₂²×C₃₁	1	4	yes	complemented
349	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
350	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
351	C₄×D₈	C₂²×C₃₁	1	2	yes	—
352	#433	C₂²×C₃₁	1	4	yes	complemented
353	#431	C₂²×C₃₁	1	4	yes	complemented
354	#424	C₂²×C₃₁	1	1	yes	characteristic, complemented
355	#424	C₂²×C₃₁	1	1	yes	characteristic, complemented
356	#408	C₂²×C₃₁	1	1	yes	characteristic
357	#408	C₂²×C₃₁	1	1	yes	characteristic
358	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
359	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
360	C₂×C₃₁	—	4	1	no	—
361	C₂×C₃₁	—	4	2	no	—
362	C₂×C₃₁	—	8	4	no	—
363	C₂×C₃₁	#3449	1	1	yes	characteristic
364	C₂×C₃₁	#3584	1	2	yes	—
365	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
366	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
367	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
368	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
369	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
370	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
371	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
372	C₂×D₈	C₂³×C₃₁	1	2	yes	—
373	C₂×D₈	C₂³×C₃₁	1	2	yes	—
374	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
375	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
376	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
377	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
378	C₄⋊Q₈	—	2	2	no	—
379	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
380	#127	—	2	2	no	—
381	#126	—	2	4	no	—
382	#126	—	2	4	no	—
383	#126	—	2	4	no	—
384	#126	—	2	4	no	—
385	#125	C₂³×C₃₁	1	4	yes	—
386	#124	—	2	2	no	—
387	#124	—	2	2	no	—
388	#124	—	2	4	no	—
389	#124	—	2	4	no	—
390	#124	—	2	4	no	—
391	#124	C₂³×C₃₁	1	4	yes	—
392	#124	C₂³×C₃₁	1	4	yes	—
393	#123	—	2	2	no	—
394	#123	—	2	2	no	—
395	#123	—	2	4	no	—
396	#123	—	2	4	no	—
397	#123	—	2	4	no	—
398	#123	—	2	4	no	—
399	C₂²⋊Q₈	—	2	4	no	—
400	C₂²⋊Q₈	—	2	4	no	—
401	C₂²⋊Q₈	—	2	4	no	—
402	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
403	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
404	#121	C₂³×C₃₁	1	4	yes	—
405	#121	C₂³×C₃₁	1	4	yes	—
406	#120	—	2	2	no	—
407	#120	—	2	2	no	—
408	C₄×Q₈	—	2	4	no	—
409	C₄×Q₈	—	2	4	no	—
410	C₄×Q₈	D₄×C₃₁	1	2	yes	—
411	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
412	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
413	C₄×D₄	—	2	4	no	—
414	C₄×D₄	—	2	4	no	—
415	C₄×D₄	—	2	4	no	—
416	C₄×D₄	D₄×C₃₁	1	2	yes	—
417	C₄×D₄	D₄×C₃₁	1	4	yes	—
418	C₄×D₄	C₂³×C₃₁	1	4	yes	—
419	C₄×D₄	C₂³×C₃₁	1	4	yes	—
420	#117	—	2	4	no	—
421	#117	—	2	4	no	—
422	#117	—	2	4	no	—
423	#117	C₂³×C₃₁	1	2	yes	—
424	#117	C₂³×C₃₁	1	2	yes	—
425	#117	C₂³×C₃₁	1	2	yes	—
426	#117	C₂³×C₃₁	1	1	yes	characteristic
427	C₂×C₂²⋊C₄	—	2	4	no	—
428	C₂×C₂²⋊C₄	—	2	4	no	—
429	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
430	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
431	#107	C₂³×C₃₁	1	2	yes	—
432	#107	C₂³×C₃₁	1	2	yes	—
433	#107	C₂³×C₃₁	1	4	yes	—
434	#106	C₂³×C₃₁	1	4	yes	—
435	#106	C₂³×C₃₁	1	2	yes	—
436	#106	C₂³×C₃₁	1	2	yes	—
437	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
438	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
439	#103	C₂³×C₃₁	1	4	yes	—
440	#103	C₂³×C₃₁	1	4	yes	—
441	#103	C₂³×C₃₁	1	4	yes	—
442	#103	C₂³×C₃₁	1	4	yes	—
443	#102	C₂³×C₃₁	1	4	yes	—
444	#102	C₂³×C₃₁	1	4	yes	—
445	#102	C₂³×C₃₁	1	4	yes	—
446	#102	C₂³×C₃₁	1	4	yes	—
447	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
448	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
449	#97	C₂³×C₃₁	1	2	yes	—
450	C₄×C₈	C₂³×C₃₁	1	2	yes	—
451	C₄×C₈	C₂³×C₃₁	1	2	yes	—
452	C₄×C₈	C₂³×C₃₁	1	2	yes	—
453	C₃₁	#61007	1	1	yes	characteristic, direct product factor, Sylow
454	C₂⁴	—	2	2	no	—
455	#41	—	4	1	no	—
456	#41	—	4	1	no	—
457	C₂×Q₈	—	2	2	no	—
458	C₂×Q₈	—	2	4	no	—
459	C₂×Q₈	—	4	2	no	—
460	C₂×Q₈	—	4	2	no	—
461	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
462	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
463	C₂×D₄	—	2	4	no	—
464	C₂×D₄	—	2	4	no	—
465	C₂×D₄	—	2	2	no	—
466	C₂×D₄	—	4	2	no	—
467	C₂×D₄	—	4	2	no	—
468	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
469	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
470	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	4	no	—
473	C₂²×C₄	—	2	4	no	—
474	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
475	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
476	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
477	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
478	Q₁₆	—	4	2	no	—
479	Q₁₆	—	4	2	no	—
480	SD₁₆	—	4	4	no	—
481	SD₁₆	—	4	4	no	—
482	D₈	—	4	2	no	—
483	D₈	—	4	2	no	—
484	M₁₆	—	4	1	no	—
485	M₁₆	—	4	1	no	—
486	C₂×C₈	—	4	1	no	—
487	C₂×C₈	—	4	1	no	—
488	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
489	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
490	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
491	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	4	no	—
499	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
500	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
501	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
502	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	2	4	no	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₂²⋊C₄	—	2	4	no	—
511	C₂²⋊C₄	—	2	4	no	—
512	C₂²⋊C₄	—	2	4	no	—
513	C₂²⋊C₄	—	2	4	no	—
514	C₂²⋊C₄	—	2	4	no	—
515	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
516	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
517	C₄²	—	2	4	no	—
518	C₄²	—	2	4	no	—
519	C₄²	—	2	2	no	—
520	C₄²	—	2	2	no	—
521	C₄²	C₂×D₄×C₃₁	1	2	yes	—
522	C₄²	C₂×D₄×C₃₁	1	4	yes	—
523	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
524	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
525	C₄²	C₂⁴×C₃₁	1	2	yes	—
526	C₂³	—	2	4	no	—
527	C₂³	—	4	4	no	—
528	C₂³	—	8	4	no	—
529	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
530	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
531	Q₈	—	2	4	no	—
532	Q₈	—	2	4	no	—
533	Q₈	—	4	1	no	—
534	Q₈	—	4	4	no	—
535	Q₈	—	8	4	no	—
536	D₄	—	2	4	no	—
537	D₄	—	2	4	no	—
538	D₄	—	4	2	no	—
539	D₄	—	4	1	no	—
540	D₄	—	4	4	no	—
541	D₄	—	8	4	no	—
542	D₄	—	8	4	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	4	4	no	—
552	C₂×C₄	—	4	4	no	—
553	C₂×C₄	—	8	4	no	—
554	C₂×C₄	—	8	4	no	—
555	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
556	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
557	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
558	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
559	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
560	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
561	C₈	—	2	2	no	—
562	C₈	—	2	2	no	—
563	C₈	—	2	2	no	—
564	C₈	—	2	2	no	—
565	C₂²	—	2	2	no	—
566	C₂²	—	2	1	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	2	4	no	—
569	C₂²	—	4	4	no	—
570	C₂²	—	4	4	no	—
571	C₂²	—	4	4	no	—
572	C₂²	—	16	4	no	—
573	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
574	C₄	—	2	2	no	—
575	C₄	—	2	1	no	—
576	C₄	—	2	2	no	—
577	C₄	—	2	2	no	—
578	C₄	—	2	4	no	—
579	C₄	—	4	4	no	—
580	C₄	—	4	4	no	—
581	C₄	—	4	4	no	—
582	C₄	C₃₁×#585	1	1	yes	characteristic
583	C₄	C₃₁×#585	1	1	yes	characteristic
584	C₂	—	4	2	no	—
585	C₂	—	4	1	no	—
586	C₂	—	8	4	no	—
587	C₂	C₃₁×#3449	1	1	yes	characteristic
588	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3199	C₂
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×#3510	C₂	semidirect
2. C₃₁×#3263	C₂	semidirect
3. C₃₁×#3262	C₂	semidirect
4. C₃₁×#3233	C₂	semidirect
5. C₃₁×#3211	C₂	semidirect
6. C₃₁×#3202	C₂	semidirect
7. C₃₁×#3200	C₂	semidirect
9. C₃₁×#3183	C₂	semidirect
10. C₃₁×#3168	C₂	semidirect
11. C₃₁×#3121	C₂	semidirect
12. C₃₁×#3118	C₂	semidirect
13. C₃₁×#2922	C₂	semidirect
19. C₃₁×#542	C₂²	semidirect
22. C₃₁×#533	C₂²	semidirect
24. C₃₁×#529	C₂²	semidirect
25. C₂×C₃₁×#124	C₂²	semidirect
26. C₃₁×#519	C₂²	semidirect
27. C₃₁×#518	C₂²	semidirect
29. C₃₁×#499	C₂²	semidirect
30. C₃₁×#499	C₂²	semidirect
33. C₃₁×#488	C₂²	semidirect
34. C₃₁×#488	C₂²	semidirect
35. C₃₁×#485	C₂²	semidirect
36. C₃₁×#485	C₂²	semidirect
39. C₃₁×#477	C₂²	semidirect
40. C₃₁×#476	C₂²	semidirect
41. C₃₁×#475	C₂²	semidirect
42. C₃₁×#474	C₂²	semidirect
43. C₃₁×#470	C₂²	semidirect
45. C₃₁×#468	C₂²	semidirect
46. C₃₁×#467	C₂²	semidirect
48. C₃₁×#465	C₂²	semidirect
49. C₃₁×#464	C₂²	semidirect
50. C₃₁×#464	C₂²	semidirect
53. C₃₁×#451	C₂²	semidirect
54. C₃₁×#450	C₂²	semidirect
58. C₃₁×#441	C₂²	semidirect
59. C₃₁×#440	C₂²	semidirect
60. C₄×Q₁₆×C₃₁	C₂²	semidirect
61. C₄×SD₁₆×C₃₁	C₂²	semidirect
63. C₃₁×#433	C₂²	semidirect
64. C₃₁×#431	C₂²	semidirect
65. C₃₁×#424	C₂²	semidirect
66. C₃₁×#424	C₂²	semidirect
69. C₄×M₁₆×C₃₁	C₂²	semidirect
70. C₂×C₄×C₈×C₃₁	C₂²	semidirect
231. #61007	C₃₁	direct (N × H)
271. #3510	C₂×C₃₁	semidirect
272. #3263	C₂×C₃₁	semidirect
273. #3262	C₂×C₃₁	semidirect
274. #3233	C₂×C₃₁	semidirect
275. #3211	C₂×C₃₁	semidirect
276. #3202	C₂×C₃₁	semidirect
277. #3200	C₂×C₃₁	semidirect
279. #3183	C₂×C₃₁	semidirect
280. #3168	C₂×C₃₁	semidirect
281. #3121	C₂×C₃₁	semidirect
282. #3118	C₂×C₃₁	semidirect
283. #2922	C₂×C₃₁	semidirect
308. #542	C₂²×C₃₁	semidirect
311. #533	C₂²×C₃₁	semidirect
313. #529	C₂²×C₃₁	semidirect
314. C₂×#124	C₂²×C₃₁	semidirect
315. #519	C₂²×C₃₁	semidirect
316. #518	C₂²×C₃₁	semidirect
318. #499	C₂²×C₃₁	semidirect
319. #499	C₂²×C₃₁	semidirect
322. #488	C₂²×C₃₁	semidirect
323. #488	C₂²×C₃₁	semidirect
324. #485	C₂²×C₃₁	semidirect
325. #485	C₂²×C₃₁	semidirect
328. #477	C₂²×C₃₁	semidirect
329. #476	C₂²×C₃₁	semidirect
330. #475	C₂²×C₃₁	semidirect
331. #474	C₂²×C₃₁	semidirect
332. #470	C₂²×C₃₁	semidirect
334. #468	C₂²×C₃₁	semidirect
335. #467	C₂²×C₃₁	semidirect
337. #465	C₂²×C₃₁	semidirect
338. #464	C₂²×C₃₁	semidirect
339. #464	C₂²×C₃₁	semidirect
342. #451	C₂²×C₃₁	semidirect
343. #450	C₂²×C₃₁	semidirect
347. #441	C₂²×C₃₁	semidirect
348. #440	C₂²×C₃₁	semidirect
349. C₄×Q₁₆	C₂²×C₃₁	semidirect
350. C₄×SD₁₆	C₂²×C₃₁	semidirect
352. #433	C₂²×C₃₁	semidirect
353. #431	C₂²×C₃₁	semidirect
354. #424	C₂²×C₃₁	semidirect
355. #424	C₂²×C₃₁	semidirect
358. C₄×M₁₆	C₂²×C₃₁	semidirect
359. C₂×C₄×C₈	C₂²×C₃₁	semidirect
453. C₃₁	#61007	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×#3510	2
C₃₁×#3263	2
C₃₁×#3262	2
C₃₁×#3233	2
C₃₁×#3211	2
C₃₁×#3202	2
C₃₁×#3200	2
C₃₁×#3199	2
C₃₁×#3183	2
C₃₁×#3168	2
C₃₁×#3121	2
C₃₁×#3118	2
C₃₁×#2922	2
#61007	31

Subgroups — C31×#61007 #21517766

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61007 #21517766