Subgroups — C₃₁×#17515 #21474274

Order 7936.

994 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3480	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2906	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1495	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1489	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1487	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1483	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1480	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1477	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₃₁×#559	—	2	1	no	—
14	C₃₁×#559	C₄	1	1	yes	characteristic
15	C₃₁×#557	C₄	1	1	yes	characteristic
16	C₃₁×#556	—	2	1	no	—
17	C₃₁×#556	C₄	1	1	yes	characteristic
18	C₃₁×#553	—	2	2	no	—
19	C₃₁×#553	C₄	1	2	yes	—
20	C₃₁×#550	C₄	1	1	yes	characteristic
21	C₃₁×#548	C₄	1	2	yes	—
22	C₃₁×#532	—	2	1	no	—
23	C₃₁×#532	C₂²	1	1	yes	characteristic
24	C₃₁×#529	—	2	2	no	—
25	C₃₁×#529	C₂²	1	2	yes	—
26	C₄×#41×C₃₁	—	2	1	no	—
27	C₄×#41×C₃₁	C₂²	1	2	yes	—
28	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#446	—	2	1	no	—
31	C₃₁×#436	—	2	1	no	—
32	C₃₁×#435	—	2	2	no	—
33	C₃₁×#431	C₂²	1	2	yes	—
34	C₃₁×#431	C₂²	1	1	yes	characteristic
35	C₃₁×#424	—	2	1	no	—
36	C₃₁×#424	—	2	2	no	—
37	C₃₁×#423	C₂²	1	1	yes	characteristic
38	C₃₁×#423	C₂²	1	2	yes	—
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
41	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
42	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
43	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
44	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
45	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
46	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
47	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
50	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
51	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
52	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
53	C₂×#41×C₃₁	—	2	1	no	—
54	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
55	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
56	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₃₁×#131	—	4	4	no	—
58	C₂×M₁₆×C₃₁	—	2	2	no	—
59	C₂×M₁₆×C₃₁	—	2	2	no	—
60	C₂×M₁₆×C₃₁	—	2	1	no	—
61	C₂²×C₈×C₃₁	—	2	1	no	—
62	C₂²×C₈×C₃₁	—	2	2	no	—
63	C₂²×C₈×C₃₁	—	2	1	no	—
64	C₃₁×C₄⋊Q₈	—	2	1	no	—
65	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
66	C₃₁×#127	—	2	1	no	—
67	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
68	C₃₁×#126	—	2	2	no	—
69	C₃₁×#126	—	2	2	no	—
70	C₃₁×#126	—	2	2	no	—
71	C₃₁×#125	—	2	2	no	—
72	C₃₁×#125	C₂×C₄	1	2	yes	—
73	C₃₁×#124	—	2	1	no	—
74	C₃₁×#124	—	2	1	no	—
75	C₃₁×#124	—	2	2	no	—
76	C₃₁×#124	—	2	1	no	—
77	C₃₁×#124	C₂×C₄	1	2	yes	—
78	C₃₁×#123	—	2	2	no	—
79	C₃₁×#123	—	2	2	no	—
80	C₃₁×#123	—	2	2	no	—
81	C₃₁×C₂²⋊Q₈	—	2	2	no	—
82	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
83	C₃₁×#121	—	2	2	no	—
84	C₃₁×#121	—	2	2	no	—
85	C₃₁×#121	—	2	1	no	—
86	C₃₁×#121	—	2	2	no	—
87	C₃₁×#121	C₂×C₄	1	2	yes	—
88	C₃₁×#121	C₂×C₄	1	2	yes	—
89	C₄×Q₈×C₃₁	—	2	1	no	—
90	C₄×Q₈×C₃₁	—	2	1	no	—
91	C₄×Q₈×C₃₁	—	2	1	no	—
92	C₄×Q₈×C₃₁	—	2	1	no	—
93	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
95	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
96	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
97	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
98	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
99	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×D₄×C₃₁	—	2	2	no	—
101	C₄×D₄×C₃₁	—	2	2	no	—
102	C₄×D₄×C₃₁	—	2	2	no	—
103	C₄×D₄×C₃₁	—	2	1	no	—
104	C₄×D₄×C₃₁	—	2	1	no	—
105	C₄×D₄×C₃₁	—	2	2	no	—
106	C₄×D₄×C₃₁	—	2	2	no	—
107	C₄×D₄×C₃₁	—	2	1	no	—
108	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
109	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
110	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
111	C₄×D₄×C₃₁	D₄	1	2	yes	—
112	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
113	C₄×D₄×C₃₁	C₂³	1	2	yes	—
114	C₃₁×#117	—	2	2	no	—
115	C₃₁×#117	—	2	2	no	—
116	C₃₁×#117	C₂×C₄	1	2	yes	—
117	C₃₁×#117	D₄	1	1	yes	characteristic
118	C₃₁×#117	D₄	1	2	yes	—
119	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
120	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
121	C₂×C₄²×C₃₁	—	2	1	no	—
122	C₂×C₄²×C₃₁	—	2	2	no	—
123	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
125	C₃₁×C₄⋊C₈	—	2	2	no	—
126	C₃₁×C₄⋊C₈	—	2	1	no	—
127	C₃₁×C₄⋊C₈	—	2	2	no	—
128	C₃₁×C₄⋊C₈	—	2	1	no	—
129	C₃₁×C₄⋊C₈	—	2	2	no	—
130	C₃₁×C₄⋊C₈	—	2	1	no	—
131	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
132	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
133	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
134	C₃₁×C₂²⋊C₈	—	2	1	no	—
135	C₃₁×C₂²⋊C₈	—	2	2	no	—
136	C₃₁×#97	—	2	2	no	—
137	C₄×C₈×C₃₁	—	2	1	no	—
138	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₃₁	C₂³	1	2	yes	—
141	#41×C₃₁	—	2	2	no	—
142	#41×C₃₁	—	2	2	no	—
143	#41×C₃₁	—	2	4	no	—
144	#41×C₃₁	—	4	2	no	—
145	#41×C₃₁	—	4	2	no	—
146	#41×C₃₁	—	4	2	no	—
147	C₂×Q₈×C₃₁	—	2	1	no	—
148	C₂×Q₈×C₃₁	—	2	1	no	—
149	C₂×Q₈×C₃₁	—	2	2	no	—
150	C₂×Q₈×C₃₁	—	4	1	no	—
151	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
152	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
153	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
154	C₂×D₄×C₃₁	—	2	2	no	—
155	C₂×D₄×C₃₁	—	2	2	no	—
156	C₂×D₄×C₃₁	—	2	2	no	—
157	C₂×D₄×C₃₁	—	2	1	no	—
158	C₂×D₄×C₃₁	—	2	1	no	—
159	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
160	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
161	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
162	C₂²×C₄×C₃₁	—	2	1	no	—
163	C₂²×C₄×C₃₁	—	2	1	no	—
164	C₂²×C₄×C₃₁	—	2	2	no	—
165	C₂²×C₄×C₃₁	—	2	2	no	—
166	C₂²×C₄×C₃₁	—	2	2	no	—
167	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
169	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
173	M₁₆×C₃₁	—	4	4	no	—
174	M₁₆×C₃₁	—	4	4	no	—
175	M₁₆×C₃₁	—	4	2	no	—
176	C₂×C₈×C₃₁	—	2	1	no	—
177	C₂×C₈×C₃₁	—	2	1	no	—
178	C₂×C₈×C₃₁	—	2	2	no	—
179	C₂×C₈×C₃₁	—	2	1	no	—
180	C₂×C₈×C₃₁	—	2	2	no	—
181	C₂×C₈×C₃₁	—	2	1	no	—
182	C₂×C₈×C₃₁	—	4	2	no	—
183	C₂×C₈×C₃₁	—	4	4	no	—
184	C₂×C₈×C₃₁	—	4	2	no	—
185	C₄⋊C₄×C₃₁	—	2	1	no	—
186	C₄⋊C₄×C₃₁	—	2	1	no	—
187	C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₄⋊C₄×C₃₁	—	2	2	no	—
189	C₄⋊C₄×C₃₁	—	2	2	no	—
190	C₄⋊C₄×C₃₁	—	2	2	no	—
191	C₄⋊C₄×C₃₁	—	2	1	no	—
192	C₄⋊C₄×C₃₁	—	2	2	no	—
193	C₄⋊C₄×C₃₁	—	2	1	no	—
194	C₄⋊C₄×C₃₁	—	2	1	no	—
195	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
196	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
197	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
198	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
199	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
200	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
201	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
202	C₂²⋊C₄×C₃₁	—	2	2	no	—
203	C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂²⋊C₄×C₃₁	—	2	2	no	—
205	C₂²⋊C₄×C₃₁	—	2	2	no	—
206	C₂²⋊C₄×C₃₁	—	2	2	no	—
207	C₂²⋊C₄×C₃₁	—	2	2	no	—
208	C₂²⋊C₄×C₃₁	—	2	2	no	—
209	C₂²⋊C₄×C₃₁	—	2	2	no	—
210	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
211	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
212	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
213	C₄²×C₃₁	—	2	2	no	—
214	C₄²×C₃₁	—	2	2	no	—
215	C₄²×C₃₁	—	2	1	no	—
216	C₄²×C₃₁	—	2	1	no	—
217	C₄²×C₃₁	—	2	1	no	—
218	C₄²×C₃₁	—	2	1	no	—
219	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
220	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
221	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
222	C₄²×C₃₁	C₂×D₄	1	2	yes	—
223	C₄²×C₃₁	C₂×D₄	1	2	yes	—
224	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
225	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
226	#17515	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
227	C₂³×C₃₁	—	2	2	no	—
228	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
229	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
230	Q₈×C₃₁	—	2	1	no	—
231	Q₈×C₃₁	—	2	2	no	—
232	Q₈×C₃₁	—	2	1	no	—
233	Q₈×C₃₁	—	2	1	no	—
234	Q₈×C₃₁	—	4	2	no	—
235	Q₈×C₃₁	—	4	2	no	—
236	Q₈×C₃₁	#131	1	2	yes	complemented
237	D₄×C₃₁	—	2	4	no	—
238	D₄×C₃₁	—	2	2	no	—
239	D₄×C₃₁	—	2	2	no	—
240	D₄×C₃₁	—	2	2	no	—
241	D₄×C₃₁	—	4	4	no	—
242	D₄×C₃₁	—	4	4	no	—
243	D₄×C₃₁	—	4	2	no	—
244	D₄×C₃₁	—	8	1	no	—
245	D₄×C₃₁	—	8	2	no	—
246	C₂×C₄×C₃₁	—	2	1	no	—
247	C₂×C₄×C₃₁	—	2	1	no	—
248	C₂×C₄×C₃₁	—	2	2	no	—
249	C₂×C₄×C₃₁	—	2	1	no	—
250	C₂×C₄×C₃₁	—	2	2	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	2	2	no	—
253	C₂×C₄×C₃₁	—	2	1	no	—
254	C₂×C₄×C₃₁	—	2	1	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	1	no	—
257	C₂×C₄×C₃₁	—	4	1	no	—
258	C₂×C₄×C₃₁	—	4	2	no	—
259	C₂×C₄×C₃₁	—	4	4	no	—
260	C₂×C₄×C₃₁	—	4	2	no	—
261	C₂×C₄×C₃₁	—	4	2	no	—
262	C₂×C₄×C₃₁	—	4	2	no	—
263	C₂×C₄×C₃₁	—	8	2	no	—
264	C₂×C₄×C₃₁	—	8	1	no	—
265	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
266	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
267	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
268	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
272	C₂×C₄×C₃₁	#137	1	2	yes	—
273	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
274	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
276	C₈×C₃₁	—	2	2	no	—
277	C₈×C₃₁	—	4	1	no	—
278	C₈×C₃₁	—	4	2	no	—
279	C₈×C₃₁	—	4	1	no	—
280	C₈×C₃₁	—	4	2	no	—
281	C₈×C₃₁	—	4	1	no	—
282	#3480	C₂×C₃₁	1	1	yes	characteristic
283	#2965	C₂×C₃₁	1	1	yes	characteristic, complemented
284	#2906	C₂×C₃₁	1	1	yes	characteristic, complemented
285	#1495	C₂×C₃₁	1	2	yes	complemented
286	#1492	C₂×C₃₁	1	1	yes	characteristic, complemented
287	#1489	C₂×C₃₁	1	2	yes	complemented
288	#1487	C₂×C₃₁	1	1	yes	characteristic, complemented
289	#1483	C₂×C₃₁	1	2	yes	complemented
290	#1480	C₂×C₃₁	1	1	yes	characteristic, complemented
291	#1477	C₂×C₃₁	1	2	yes	complemented
292	C₂×Q₈⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
293	C₂²×C₃₁	—	2	1	no	—
294	C₂²×C₃₁	—	2	1	no	—
295	C₂²×C₃₁	—	2	2	no	—
296	C₂²×C₃₁	—	2	2	no	—
297	C₂²×C₃₁	—	2	2	no	—
298	C₂²×C₃₁	—	4	2	no	—
299	C₂²×C₃₁	—	4	2	no	—
300	C₂²×C₃₁	—	4	2	no	—
301	C₂²×C₃₁	#417	1	2	yes	—
302	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
303	C₄×C₃₁	—	2	2	no	—
304	C₄×C₃₁	—	2	1	no	—
305	C₄×C₃₁	—	2	2	no	—
306	C₄×C₃₁	—	2	2	no	—
307	C₄×C₃₁	—	2	1	no	—
308	C₄×C₃₁	—	2	2	no	—
309	C₄×C₃₁	—	2	1	no	—
310	C₄×C₃₁	—	4	1	no	—
311	C₄×C₃₁	—	4	2	no	—
312	C₄×C₃₁	—	4	1	no	—
313	C₄×C₃₁	—	4	1	no	—
314	C₄×C₃₁	#417	1	2	yes	—
315	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
316	C₄×C₃₁	#568	1	1	yes	characteristic
317	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
318	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
319	C₂×#131	—	2	1	no	—
320	#559	—	2	1	no	—
321	#559	C₄×C₃₁	1	1	yes	characteristic
322	#557	C₄×C₃₁	1	1	yes	characteristic
323	#556	—	2	1	no	—
324	#556	C₄×C₃₁	1	1	yes	characteristic
325	#553	—	2	2	no	—
326	#553	C₄×C₃₁	1	2	yes	—
327	#550	C₄×C₃₁	1	1	yes	characteristic
328	#548	C₄×C₃₁	1	2	yes	—
329	#532	—	2	1	no	—
330	#532	C₂²×C₃₁	1	1	yes	characteristic
331	#529	—	2	2	no	—
332	#529	C₂²×C₃₁	1	2	yes	—
333	C₄×#41	—	2	1	no	—
334	C₄×#41	C₂²×C₃₁	1	2	yes	—
335	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
336	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
337	#446	—	2	1	no	—
338	#436	—	2	1	no	—
339	#435	—	2	2	no	—
340	#431	C₂²×C₃₁	1	2	yes	—
341	#431	C₂²×C₃₁	1	1	yes	characteristic
342	#424	—	2	1	no	—
343	#424	—	2	2	no	—
344	#423	C₂²×C₃₁	1	1	yes	characteristic
345	#423	C₂²×C₃₁	1	2	yes	—
346	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
347	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
348	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
349	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
350	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
351	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
352	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
353	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
354	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
355	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
356	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
357	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
358	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
359	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
360	C₂×C₃₁	—	2	2	no	—
361	C₂×C₃₁	—	4	2	no	—
362	C₂×C₃₁	—	8	2	no	—
363	C₂×C₃₁	#2880	1	1	yes	characteristic
364	C₂×C₃₁	C₂×#417	1	1	yes	characteristic
365	C₂×C₃₁	#2901	1	1	yes	characteristic
366	C₂×#41	—	2	1	no	—
367	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
368	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
369	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
370	#131	—	4	4	no	—
371	C₂×M₁₆	—	2	2	no	—
372	C₂×M₁₆	—	2	1	no	—
373	C₂×M₁₆	—	2	2	no	—
374	C₂²×C₈	—	2	2	no	—
375	C₂²×C₈	—	2	1	no	—
376	C₂²×C₈	—	2	1	no	—
377	C₄⋊Q₈	—	2	1	no	—
378	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
379	#127	—	2	1	no	—
380	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
381	#126	—	2	2	no	—
382	#126	—	2	2	no	—
383	#126	—	2	2	no	—
384	#125	—	2	2	no	—
385	#125	C₂×C₄×C₃₁	1	2	yes	—
386	#124	—	2	1	no	—
387	#124	—	2	1	no	—
388	#124	—	2	2	no	—
389	#124	—	2	1	no	—
390	#124	C₂×C₄×C₃₁	1	2	yes	—
391	#123	—	2	2	no	—
392	#123	—	2	2	no	—
393	#123	—	2	2	no	—
394	C₂²⋊Q₈	—	2	2	no	—
395	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
396	#121	—	2	2	no	—
397	#121	—	2	2	no	—
398	#121	—	2	1	no	—
399	#121	—	2	2	no	—
400	#121	C₂×C₄×C₃₁	1	2	yes	—
401	#121	C₂×C₄×C₃₁	1	2	yes	—
402	C₄×Q₈	—	2	1	no	—
403	C₄×Q₈	—	2	1	no	—
404	C₄×Q₈	—	2	1	no	—
405	C₄×Q₈	—	2	1	no	—
406	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
407	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
408	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
409	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
410	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
411	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
412	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	—	2	2	no	—
415	C₄×D₄	—	2	2	no	—
416	C₄×D₄	—	2	1	no	—
417	C₄×D₄	—	2	1	no	—
418	C₄×D₄	—	2	2	no	—
419	C₄×D₄	—	2	2	no	—
420	C₄×D₄	—	2	1	no	—
421	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
422	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
423	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
424	C₄×D₄	D₄×C₃₁	1	2	yes	—
425	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
426	C₄×D₄	C₂³×C₃₁	1	2	yes	—
427	#117	—	2	2	no	—
428	#117	—	2	2	no	—
429	#117	C₂×C₄×C₃₁	1	2	yes	—
430	#117	D₄×C₃₁	1	1	yes	characteristic
431	#117	D₄×C₃₁	1	2	yes	—
432	C₂×C₄⋊C₄	—	2	1	no	—
433	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
434	C₂×C₄²	—	2	1	no	—
435	C₂×C₄²	—	2	2	no	—
436	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
437	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
438	C₄⋊C₈	—	2	2	no	—
439	C₄⋊C₈	—	2	1	no	—
440	C₄⋊C₈	—	2	2	no	—
441	C₄⋊C₈	—	2	1	no	—
442	C₄⋊C₈	—	2	2	no	—
443	C₄⋊C₈	—	2	1	no	—
444	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
445	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
446	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
447	C₂²⋊C₈	—	2	1	no	—
448	C₂²⋊C₈	—	2	2	no	—
449	#97	—	2	2	no	—
450	C₄×C₈	—	2	1	no	—
451	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
452	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
453	C₄×C₈	C₂³×C₃₁	1	2	yes	—
454	C₃₁	#17515	1	1	yes	characteristic, direct product factor, Sylow
455	#41	—	2	2	no	—
456	#41	—	2	2	no	—
457	#41	—	2	4	no	—
458	#41	—	4	2	no	—
459	#41	—	4	2	no	—
460	#41	—	4	2	no	—
461	C₂×Q₈	—	2	1	no	—
462	C₂×Q₈	—	2	1	no	—
463	C₂×Q₈	—	2	2	no	—
464	C₂×Q₈	—	4	1	no	—
465	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
466	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
467	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
468	C₂×D₄	—	2	2	no	—
469	C₂×D₄	—	2	2	no	—
470	C₂×D₄	—	2	2	no	—
471	C₂×D₄	—	2	1	no	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
474	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
475	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
482	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
483	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
484	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
485	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
486	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
487	M₁₆	—	4	4	no	—
488	M₁₆	—	4	4	no	—
489	M₁₆	—	4	2	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	2	no	—
493	C₂×C₈	—	2	1	no	—
494	C₂×C₈	—	2	2	no	—
495	C₂×C₈	—	2	1	no	—
496	C₂×C₈	—	4	2	no	—
497	C₂×C₈	—	4	4	no	—
498	C₂×C₈	—	4	2	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
510	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
511	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
512	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
513	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
514	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
515	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
516	C₂²⋊C₄	—	2	2	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	2	2	no	—
519	C₂²⋊C₄	—	2	2	no	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	2	no	—
523	C₂²⋊C₄	—	2	2	no	—
524	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
525	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
526	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
527	C₄²	—	2	2	no	—
528	C₄²	—	2	2	no	—
529	C₄²	—	2	1	no	—
530	C₄²	—	2	1	no	—
531	C₄²	—	2	1	no	—
532	C₄²	—	2	1	no	—
533	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
534	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
535	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
536	C₄²	C₂×D₄×C₃₁	1	2	yes	—
537	C₄²	C₂×D₄×C₃₁	1	2	yes	—
538	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
539	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
540	C₂³	—	2	2	no	—
541	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
542	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
543	Q₈	—	2	2	no	—
544	Q₈	—	2	1	no	—
545	Q₈	—	2	1	no	—
546	Q₈	—	2	1	no	—
547	Q₈	—	4	2	no	—
548	Q₈	—	4	2	no	—
549	Q₈	C₃₁×#131	1	2	yes	complemented
550	D₄	—	2	4	no	—
551	D₄	—	2	2	no	—
552	D₄	—	2	2	no	—
553	D₄	—	2	2	no	—
554	D₄	—	4	4	no	—
555	D₄	—	4	4	no	—
556	D₄	—	4	2	no	—
557	D₄	—	8	1	no	—
558	D₄	—	8	2	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	2	no	—
572	C₂×C₄	—	4	4	no	—
573	C₂×C₄	—	4	2	no	—
574	C₂×C₄	—	4	2	no	—
575	C₂×C₄	—	4	2	no	—
576	C₂×C₄	—	8	2	no	—
577	C₂×C₄	—	8	1	no	—
578	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
579	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
580	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
581	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
582	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
583	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
584	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
585	C₂×C₄	C₃₁×#137	1	2	yes	—
586	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
587	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
588	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
589	C₈	—	2	2	no	—
590	C₈	—	4	1	no	—
591	C₈	—	4	2	no	—
592	C₈	—	4	1	no	—
593	C₈	—	4	2	no	—
594	C₈	—	4	1	no	—
595	C₂²	—	2	1	no	—
596	C₂²	—	2	1	no	—
597	C₂²	—	2	2	no	—
598	C₂²	—	2	2	no	—
599	C₂²	—	2	2	no	—
600	C₂²	—	4	2	no	—
601	C₂²	—	4	2	no	—
602	C₂²	—	4	2	no	—
603	C₂²	C₃₁×#417	1	2	yes	—
604	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
605	C₄	—	2	2	no	—
606	C₄	—	2	1	no	—
607	C₄	—	2	2	no	—
608	C₄	—	2	2	no	—
609	C₄	—	2	1	no	—
610	C₄	—	2	2	no	—
611	C₄	—	2	1	no	—
612	C₄	—	4	1	no	—
613	C₄	—	4	2	no	—
614	C₄	—	4	1	no	—
615	C₄	—	4	1	no	—
616	C₄	C₃₁×#417	1	2	yes	—
617	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
618	C₄	C₃₁×#568	1	1	yes	characteristic
619	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
620	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
621	C₂	—	2	2	no	—
622	C₂	—	4	2	no	—
623	C₂	—	8	2	no	—
624	C₂	C₃₁×#2880	1	1	yes	characteristic
625	C₂	C₂×C₃₁×#417	1	1	yes	characteristic
626	C₂	C₃₁×#2901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₃₁×#3480	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2965	C₂	semidirect
3. C₃₁×#2906	C₂	semidirect
4. C₃₁×#1495	C₂	semidirect
5. C₃₁×#1492	C₂	semidirect
6. C₃₁×#1489	C₂	semidirect
7. C₃₁×#1487	C₂	semidirect
8. C₃₁×#1483	C₂	semidirect
9. C₃₁×#1480	C₂	semidirect
10. C₃₁×#1477	C₂	semidirect
11. C₂×C₃₁×Q₈⋊C₈	C₂	semidirect
226. #17515	C₃₁	direct (N × H)
236. Q₈×C₃₁	#131	semidirect
283. #2965	C₂×C₃₁	semidirect
284. #2906	C₂×C₃₁	semidirect
285. #1495	C₂×C₃₁	semidirect
286. #1492	C₂×C₃₁	semidirect
287. #1489	C₂×C₃₁	semidirect
288. #1487	C₂×C₃₁	semidirect
289. #1483	C₂×C₃₁	semidirect
290. #1480	C₂×C₃₁	semidirect
291. #1477	C₂×C₃₁	semidirect
292. C₂×Q₈⋊C₈	C₂×C₃₁	semidirect
454. C₃₁	#17515	direct (N × H)
549. Q₈	C₃₁×#131	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3480	2
C₃₁×#2965	2
C₃₁×#2906	2
C₃₁×#1495	2
C₃₁×#1492	2
C₃₁×#1489	2
C₃₁×#1487	2
C₃₁×#1483	2
C₃₁×#1480	2
C₃₁×#1477	2
C₂×C₃₁×Q₈⋊C₈	2
#17515	31

Subgroups — C31×#17515 #21474274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17515 #21474274