Subgroups — C₃₁×#18017 #21474776

Order 7936.

600 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3108	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2994	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2984	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1591	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1590	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1588	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1587	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1587	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1585	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1573	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1494	C₂	1	1	yes	maximal, characteristic
14	C₂×C₃₁×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#529	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#483	C₄	1	1	yes	characteristic
19	C₃₁×#480	C₄	1	1	yes	characteristic
20	C₃₁×#479	C₄	1	1	yes	characteristic
21	C₃₁×#479	C₄	1	1	yes	characteristic
22	C₃₁×#464	C₄	1	1	yes	characteristic
23	C₃₁×#464	C₄	1	1	yes	characteristic
24	C₃₁×#451	C₄	1	1	yes	characteristic
25	C₃₁×#448	C₄	1	1	yes	characteristic
26	C₃₁×#446	—	2	1	no	—
27	C₃₁×#446	—	2	1	no	—
28	C₃₁×#446	C₂²	1	1	yes	characteristic
29	C₃₁×#446	C₂²	1	1	yes	characteristic
30	C₃₁×#446	C₂²	1	1	yes	characteristic
31	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
36	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
37	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#436	C₂²	1	1	yes	characteristic
39	C₃₁×#435	—	2	1	no	—
40	C₈×D₄×C₃₁	—	2	1	no	—
41	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#433	—	2	1	no	—
43	C₃₁×#432	—	2	1	no	—
44	C₃₁×#432	C₂²	1	1	yes	characteristic
45	C₃₁×#431	—	2	1	no	—
46	C₃₁×#431	C₂²	1	1	yes	characteristic
47	C₃₁×#431	C₂²	1	1	yes	characteristic
48	C₃₁×#424	—	2	1	no	—
49	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
50	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
51	C₃₁×#335	C₂²	1	1	yes	characteristic
52	C₃₁×#335	C₂²	1	1	yes	characteristic
53	C₃₁×#334	C₂²	1	1	yes	characteristic
54	C₃₁×#334	C₂²	1	1	yes	characteristic
55	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
56	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
57	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
58	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
59	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
60	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
61	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
62	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
63	C₃₁×#322	C₂²	1	1	yes	characteristic
64	C₃₁×#322	C₂²	1	1	yes	characteristic
65	C₈²×C₃₁	C₂²	1	1	yes	characteristic
66	C₈²×C₃₁	C₂²	1	1	yes	characteristic
67	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₂×SD₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
71	C₂×SD₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
72	C₂×M₁₆×C₃₁	—	2	1	no	—
73	C₂²×C₈×C₃₁	—	2	1	no	—
74	C₂²×C₈×C₃₁	—	2	1	no	—
75	C₂²×C₈×C₃₁	—	2	1	no	—
76	C₃₁×#126	—	2	1	no	—
77	C₃₁×#125	C₂×C₄	1	1	yes	characteristic
78	C₃₁×#124	C₂×C₄	1	1	yes	characteristic
79	C₃₁×#123	—	2	1	no	—
80	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
81	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
82	C₄×Q₈×C₃₁	—	2	1	no	—
83	C₄×Q₈×C₃₁	—	2	1	no	—
84	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
87	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₄×D₄×C₃₁	—	2	1	no	—
90	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
91	C₃₁×#117	—	2	1	no	—
92	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
93	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₃₁	—	2	1	no	—
95	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
96	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
97	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
98	C₃₁×#106	C₂×C₄	1	1	yes	characteristic
99	C₃₁×#106	C₂×C₄	1	1	yes	characteristic
100	C₃₁×C₄⋊C₈	—	2	1	no	—
101	C₃₁×C₄⋊C₈	—	2	1	no	—
102	C₃₁×C₄⋊C₈	—	2	1	no	—
103	C₃₁×C₄⋊C₈	—	2	1	no	—
104	C₃₁×C₄⋊C₈	—	2	1	no	—
105	C₃₁×C₄⋊C₈	—	2	1	no	—
106	C₃₁×C₄⋊C₈	—	2	1	no	—
107	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
108	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
109	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
110	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
111	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
114	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
115	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
116	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
117	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
118	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
119	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
120	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
121	C₃₁×#102	C₂×C₄	1	1	yes	characteristic
122	C₃₁×#102	C₂×C₄	1	1	yes	characteristic
123	C₃₁×C₂²⋊C₈	—	2	1	no	—
124	C₃₁×C₂²⋊C₈	—	2	1	no	—
125	C₃₁×C₂²⋊C₈	—	2	1	no	—
126	C₃₁×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
127	C₃₁×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
128	C₃₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
129	C₃₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
130	C₃₁×#97	—	2	1	no	—
131	C₃₁×#97	—	2	1	no	—
132	C₃₁×#97	—	2	1	no	—
133	C₃₁×#97	—	2	1	no	—
134	C₄×C₈×C₃₁	—	2	1	no	—
135	C₄×C₈×C₃₁	—	2	1	no	—
136	C₄×C₈×C₃₁	—	2	1	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	1	yes	characteristic
141	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
144	C₂×Q₈×C₃₁	—	2	1	no	—
145	C₂×Q₈×C₃₁	—	2	2	no	—
146	C₂×Q₈×C₃₁	—	4	1	no	—
147	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
148	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
149	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
150	C₂×D₄×C₃₁	—	2	1	no	—
151	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₃₁	—	2	1	no	—
153	C₂²×C₄×C₃₁	—	2	1	no	—
154	C₂²×C₄×C₃₁	—	2	1	no	—
155	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
158	Q₁₆×C₃₁	—	2	2	no	—
159	Q₁₆×C₃₁	—	4	1	no	—
160	SD₁₆×C₃₁	—	2	2	no	—
161	SD₁₆×C₃₁	—	4	1	no	—
162	M₁₆×C₃₁	—	8	1	no	—
163	C₂×C₈×C₃₁	—	2	4	no	—
164	C₂×C₈×C₃₁	—	2	1	no	—
165	C₂×C₈×C₃₁	—	2	1	no	—
166	C₂×C₈×C₃₁	—	2	1	no	—
167	C₂×C₈×C₃₁	—	2	1	no	—
168	C₂×C₈×C₃₁	—	2	1	no	—
169	C₂×C₈×C₃₁	—	2	1	no	—
170	C₂×C₈×C₃₁	—	8	1	no	—
171	C₂×C₈×C₃₁	—	8	1	no	—
172	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
173	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
174	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
176	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
180	C₄⋊C₄×C₃₁	—	2	1	no	—
181	C₄⋊C₄×C₃₁	—	2	1	no	—
182	C₄⋊C₄×C₃₁	—	2	1	no	—
183	C₄⋊C₄×C₃₁	—	2	1	no	—
184	C₄⋊C₄×C₃₁	—	2	1	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	1	no	—
189	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
190	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
191	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
192	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
193	C₂²⋊C₄×C₃₁	—	2	1	no	—
194	C₂²⋊C₄×C₃₁	—	2	1	no	—
195	C₂²⋊C₄×C₃₁	—	2	1	no	—
196	C₂²⋊C₄×C₃₁	—	2	1	no	—
197	C₂²⋊C₄×C₃₁	—	2	1	no	—
198	C₄²×C₃₁	—	2	1	no	—
199	C₄²×C₃₁	—	2	1	no	—
200	C₄²×C₃₁	—	2	1	no	—
201	C₄²×C₃₁	—	2	1	no	—
202	C₄²×C₃₁	—	2	1	no	—
203	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
204	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
205	C₄²×C₃₁	#41	1	1	yes	characteristic
206	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
207	#18017	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂³×C₃₁	—	2	1	no	—
209	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
210	Q₈×C₃₁	—	2	1	no	—
211	Q₈×C₃₁	—	2	1	no	—
212	Q₈×C₃₁	—	2	1	no	—
213	Q₈×C₃₁	—	2	1	no	—
214	Q₈×C₃₁	—	2	1	no	—
215	Q₈×C₃₁	—	4	2	no	—
216	Q₈×C₃₁	#131	1	2	yes	—
217	D₄×C₃₁	—	2	1	no	—
218	D₄×C₃₁	—	2	1	no	—
219	D₄×C₃₁	—	4	2	no	—
220	C₂×C₄×C₃₁	—	2	1	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	1	no	—
223	C₂×C₄×C₃₁	—	2	1	no	—
224	C₂×C₄×C₃₁	—	2	1	no	—
225	C₂×C₄×C₃₁	—	2	1	no	—
226	C₂×C₄×C₃₁	—	2	1	no	—
227	C₂×C₄×C₃₁	—	2	1	no	—
228	C₂×C₄×C₃₁	—	2	1	no	—
229	C₂×C₄×C₃₁	—	2	1	no	—
230	C₂×C₄×C₃₁	—	2	1	no	—
231	C₂×C₄×C₃₁	—	2	1	no	—
232	C₂×C₄×C₃₁	—	4	2	no	—
233	C₂×C₄×C₃₁	—	4	2	no	—
234	C₂×C₄×C₃₁	—	4	1	no	—
235	C₂×C₄×C₃₁	—	4	1	no	—
236	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
239	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
240	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
242	C₈×C₃₁	—	2	2	no	—
243	C₈×C₃₁	—	2	2	no	—
244	C₈×C₃₁	—	2	1	no	—
245	C₈×C₃₁	—	2	1	no	—
246	C₈×C₃₁	—	2	1	no	—
247	C₈×C₃₁	—	2	1	no	—
248	C₈×C₃₁	—	2	1	no	—
249	C₈×C₃₁	—	2	1	no	—
250	C₈×C₃₁	—	2	1	no	—
251	C₈×C₃₁	—	2	1	no	—
252	C₈×C₃₁	—	4	1	no	—
253	C₈×C₃₁	—	4	1	no	—
254	C₈×C₃₁	—	4	1	no	—
255	C₈×C₃₁	—	4	1	no	—
256	#3108	C₂×C₃₁	1	1	yes	characteristic
257	#2994	C₂×C₃₁	1	1	yes	characteristic
258	#2984	C₂×C₃₁	1	1	yes	characteristic, complemented
259	#1591	C₂×C₃₁	1	1	yes	characteristic, complemented
260	#1590	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#1588	C₂×C₃₁	1	1	yes	characteristic, complemented
262	#1587	C₂×C₃₁	1	1	yes	characteristic, complemented
263	#1587	C₂×C₃₁	1	1	yes	characteristic, complemented
264	#1585	C₂×C₃₁	1	1	yes	characteristic, complemented
265	C₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
266	C₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#1573	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#1494	C₂×C₃₁	1	1	yes	characteristic
269	C₂×Q₈⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#1450	C₂×C₃₁	1	1	yes	characteristic, complemented
271	C₂²×C₃₁	—	2	1	no	—
272	C₂²×C₃₁	—	2	1	no	—
273	C₂²×C₃₁	—	4	1	no	—
274	C₂²×C₃₁	—	4	1	no	—
275	C₂²×C₃₁	—	4	1	no	—
276	C₂²×C₃₁	#443	1	2	yes	—
277	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
278	C₄×C₃₁	—	2	1	no	—
279	C₄×C₃₁	—	2	2	no	—
280	C₄×C₃₁	—	2	1	no	—
281	C₄×C₃₁	—	2	1	no	—
282	C₄×C₃₁	—	2	1	no	—
283	C₄×C₃₁	—	4	1	no	—
284	C₄×C₃₁	—	4	1	no	—
285	C₄×C₃₁	—	4	1	no	—
286	C₄×C₃₁	—	4	1	no	—
287	C₄×C₃₁	—	4	1	no	—
288	C₄×C₃₁	—	4	1	no	—
289	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
290	C₄×C₃₁	#568	1	1	yes	characteristic
291	C₄×C₃₁	#577	1	1	yes	characteristic
292	C₄×C₃₁	#578	1	1	yes	characteristic
293	#529	C₂²×C₃₁	1	1	yes	characteristic
294	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
295	#483	C₄×C₃₁	1	1	yes	characteristic
296	#480	C₄×C₃₁	1	1	yes	characteristic
297	#479	C₄×C₃₁	1	1	yes	characteristic
298	#479	C₄×C₃₁	1	1	yes	characteristic
299	#464	C₄×C₃₁	1	1	yes	characteristic
300	#464	C₄×C₃₁	1	1	yes	characteristic
301	#451	C₄×C₃₁	1	1	yes	characteristic
302	#448	C₄×C₃₁	1	1	yes	characteristic
303	#446	—	2	1	no	—
304	#446	—	2	1	no	—
305	#446	C₂²×C₃₁	1	1	yes	characteristic
306	#446	C₂²×C₃₁	1	1	yes	characteristic
307	#446	C₂²×C₃₁	1	1	yes	characteristic
308	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
309	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
310	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
311	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
312	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
313	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
314	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
315	#436	C₂²×C₃₁	1	1	yes	characteristic
316	#435	—	2	1	no	—
317	C₈×D₄	—	2	1	no	—
318	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
319	#433	—	2	1	no	—
320	#432	—	2	1	no	—
321	#432	C₂²×C₃₁	1	1	yes	characteristic
322	#431	—	2	1	no	—
323	#431	C₂²×C₃₁	1	1	yes	characteristic
324	#431	C₂²×C₃₁	1	1	yes	characteristic
325	#424	—	2	1	no	—
326	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
327	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
328	#335	C₂²×C₃₁	1	1	yes	characteristic
329	#335	C₂²×C₃₁	1	1	yes	characteristic
330	#334	C₂²×C₃₁	1	1	yes	characteristic
331	#334	C₂²×C₃₁	1	1	yes	characteristic
332	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
333	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
334	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
335	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
336	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
337	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
338	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
339	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
340	#322	C₂²×C₃₁	1	1	yes	characteristic
341	#322	C₂²×C₃₁	1	1	yes	characteristic
342	C₈²	C₂²×C₃₁	1	1	yes	characteristic
343	C₈²	C₂²×C₃₁	1	1	yes	characteristic
344	C₂×C₃₁	—	2	2	no	—
345	C₂×C₃₁	—	8	1	no	—
346	C₂×C₃₁	#2933	1	1	yes	characteristic
347	C₂×C₃₁	#2947	1	1	yes	characteristic
348	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
349	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
350	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
351	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
352	C₂×SD₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
353	C₂×SD₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
354	C₂×M₁₆	—	2	1	no	—
355	C₂²×C₈	—	2	1	no	—
356	C₂²×C₈	—	2	1	no	—
357	C₂²×C₈	—	2	1	no	—
358	#126	—	2	1	no	—
359	#125	C₂×C₄×C₃₁	1	1	yes	characteristic
360	#124	C₂×C₄×C₃₁	1	1	yes	characteristic
361	#123	—	2	1	no	—
362	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
363	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
364	C₄×Q₈	—	2	1	no	—
365	C₄×Q₈	—	2	1	no	—
366	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
367	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
368	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
369	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
370	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
371	C₄×D₄	—	2	1	no	—
372	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
373	#117	—	2	1	no	—
374	C₂×C₄⋊C₄	—	2	1	no	—
375	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
376	C₂×C₄²	—	2	1	no	—
377	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
378	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
379	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
380	#106	C₂×C₄×C₃₁	1	1	yes	characteristic
381	#106	C₂×C₄×C₃₁	1	1	yes	characteristic
382	C₄⋊C₈	—	2	1	no	—
383	C₄⋊C₈	—	2	1	no	—
384	C₄⋊C₈	—	2	1	no	—
385	C₄⋊C₈	—	2	1	no	—
386	C₄⋊C₈	—	2	1	no	—
387	C₄⋊C₈	—	2	1	no	—
388	C₄⋊C₈	—	2	1	no	—
389	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
390	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
391	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
392	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
393	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
394	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
395	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
396	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
397	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
398	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
399	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
400	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
401	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
402	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
403	#102	C₂×C₄×C₃₁	1	1	yes	characteristic
404	#102	C₂×C₄×C₃₁	1	1	yes	characteristic
405	C₂²⋊C₈	—	2	1	no	—
406	C₂²⋊C₈	—	2	1	no	—
407	C₂²⋊C₈	—	2	1	no	—
408	C₂²⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
409	C₂²⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
410	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
411	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
412	#97	—	2	1	no	—
413	#97	—	2	1	no	—
414	#97	—	2	1	no	—
415	#97	—	2	1	no	—
416	C₄×C₈	—	2	1	no	—
417	C₄×C₈	—	2	1	no	—
418	C₄×C₈	—	2	1	no	—
419	C₄×C₈	—	2	1	no	—
420	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
421	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
422	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
423	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
424	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
425	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
426	C₃₁	#18017	1	1	yes	characteristic, direct product factor, Sylow
427	C₂×Q₈	—	2	1	no	—
428	C₂×Q₈	—	2	2	no	—
429	C₂×Q₈	—	4	1	no	—
430	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
431	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
432	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
439	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
440	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
441	Q₁₆	—	2	2	no	—
442	Q₁₆	—	4	1	no	—
443	SD₁₆	—	2	2	no	—
444	SD₁₆	—	4	1	no	—
445	M₁₆	—	8	1	no	—
446	C₂×C₈	—	2	4	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	8	1	no	—
454	C₂×C₈	—	8	1	no	—
455	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
456	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
457	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
458	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
459	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
462	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
473	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
474	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
475	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
476	C₂²⋊C₄	—	2	1	no	—
477	C₂²⋊C₄	—	2	1	no	—
478	C₂²⋊C₄	—	2	1	no	—
479	C₂²⋊C₄	—	2	1	no	—
480	C₂²⋊C₄	—	2	1	no	—
481	C₄²	—	2	1	no	—
482	C₄²	—	2	1	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
487	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
488	C₄²	#41×C₃₁	1	1	yes	characteristic
489	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
490	C₂³	—	2	1	no	—
491	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
492	Q₈	—	2	1	no	—
493	Q₈	—	2	1	no	—
494	Q₈	—	2	1	no	—
495	Q₈	—	2	1	no	—
496	Q₈	—	2	1	no	—
497	Q₈	—	4	2	no	—
498	Q₈	C₃₁×#131	1	2	yes	—
499	D₄	—	2	1	no	—
500	D₄	—	2	1	no	—
501	D₄	—	4	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	4	2	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	4	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
519	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
520	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
521	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
522	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
523	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
524	C₈	—	2	2	no	—
525	C₈	—	2	2	no	—
526	C₈	—	2	1	no	—
527	C₈	—	2	1	no	—
528	C₈	—	2	1	no	—
529	C₈	—	2	1	no	—
530	C₈	—	2	1	no	—
531	C₈	—	2	1	no	—
532	C₈	—	2	1	no	—
533	C₈	—	2	1	no	—
534	C₈	—	4	1	no	—
535	C₈	—	4	1	no	—
536	C₈	—	4	1	no	—
537	C₈	—	4	1	no	—
538	C₂²	—	2	1	no	—
539	C₂²	—	2	1	no	—
540	C₂²	—	4	1	no	—
541	C₂²	—	4	1	no	—
542	C₂²	—	4	1	no	—
543	C₂²	C₃₁×#443	1	2	yes	—
544	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
545	C₄	—	2	1	no	—
546	C₄	—	2	2	no	—
547	C₄	—	2	1	no	—
548	C₄	—	2	1	no	—
549	C₄	—	2	1	no	—
550	C₄	—	4	1	no	—
551	C₄	—	4	1	no	—
552	C₄	—	4	1	no	—
553	C₄	—	4	1	no	—
554	C₄	—	4	1	no	—
555	C₄	—	4	1	no	—
556	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
557	C₄	C₃₁×#568	1	1	yes	characteristic
558	C₄	C₃₁×#577	1	1	yes	characteristic
559	C₄	C₃₁×#578	1	1	yes	characteristic
560	C₂	—	2	2	no	—
561	C₂	—	8	1	no	—
562	C₂	C₃₁×#2933	1	1	yes	characteristic
563	C₂	C₃₁×#2947	1	1	yes	characteristic
564	C₂	C₂×C₃₁×#443	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	C₂×C₄×D₄
prime radical	C₃₁×#121	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2984	C₂	semidirect
4. C₃₁×#1591	C₂	semidirect
5. C₃₁×#1590	C₂	semidirect
6. C₃₁×#1588	C₂	semidirect
7. C₃₁×#1587	C₂	semidirect
8. C₃₁×#1587	C₂	semidirect
9. C₃₁×#1585	C₂	semidirect
10. C₈×Q₁₆×C₃₁	C₂	semidirect
11. C₈×SD₁₆×C₃₁	C₂	semidirect
12. C₃₁×#1573	C₂	semidirect
14. C₂×C₃₁×Q₈⋊C₈	C₂	semidirect
15. C₃₁×#1450	C₂	semidirect
207. #18017	C₃₁	direct (N × H)
258. #2984	C₂×C₃₁	semidirect
259. #1591	C₂×C₃₁	semidirect
260. #1590	C₂×C₃₁	semidirect
261. #1588	C₂×C₃₁	semidirect
262. #1587	C₂×C₃₁	semidirect
263. #1587	C₂×C₃₁	semidirect
264. #1585	C₂×C₃₁	semidirect
265. C₈×Q₁₆	C₂×C₃₁	semidirect
266. C₈×SD₁₆	C₂×C₃₁	semidirect
267. #1573	C₂×C₃₁	semidirect
269. C₂×Q₈⋊C₈	C₂×C₃₁	semidirect
270. #1450	C₂×C₃₁	semidirect
426. C₃₁	#18017	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×#3108	2
C₃₁×#2994	2
C₃₁×#2984	2
C₃₁×#1591	2
C₃₁×#1590	2
C₃₁×#1588	2
C₃₁×#1587	2
C₃₁×#1585	2
C₈×Q₁₆×C₃₁	2
C₈×SD₁₆×C₃₁	2
C₃₁×#1573	2
C₃₁×#1494	2
C₂×C₃₁×Q₈⋊C₈	2
C₃₁×#1450	2
#18017	31

Subgroups — C31×#18017 #21474776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18017 #21474776