Subgroups — C₃₁×#17912 #21474671

Order 7936.

700 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3106	C₂	1	1	yes	maximal, characteristic
2	C₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1588	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1586	C₂	1	2	yes	maximal, complemented
6	C₈×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
7	C₈×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1574	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1489	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1484	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#482	C₄	1	2	yes	—
16	C₃₁×#477	C₄	1	1	yes	characteristic
17	C₃₁×#476	C₄	1	1	yes	characteristic
18	C₃₁×#460	C₄	1	1	yes	characteristic
19	C₃₁×#459	C₄	1	1	yes	characteristic
20	C₃₁×#449	C₄	1	2	yes	—
21	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
24	C₄×D₈×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#436	—	2	1	no	—
26	C₃₁×#436	C₂²	1	2	yes	—
27	C₈×D₄×C₃₁	—	2	1	no	—
28	C₈×D₄×C₃₁	—	2	1	no	—
29	C₈×D₄×C₃₁	—	2	1	no	—
30	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₈×D₄×C₃₁	C₂²	1	2	yes	—
32	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#431	—	2	1	no	—
34	C₃₁×#431	C₂²	1	1	yes	characteristic
35	C₃₁×#431	C₂²	1	1	yes	characteristic
36	C₃₁×#423	C₂²	1	1	yes	characteristic
37	C₃₁×#408	—	2	1	no	—
38	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₃₁	—	2	1	no	—
40	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#335	C₂²	1	2	yes	—
42	C₃₁×#334	C₂²	1	2	yes	—
43	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
45	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
46	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
47	C₃₁×#322	C₂²	1	2	yes	—
48	C₈²×C₃₁	C₂²	1	2	yes	—
49	C₂×#41×C₃₁	—	2	1	no	—
50	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
52	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
53	C₃₁×#131	—	4	4	no	—
54	C₂×M₁₆×C₃₁	—	2	1	no	—
55	C₂×M₁₆×C₃₁	—	2	1	no	—
56	C₂×M₁₆×C₃₁	—	2	2	no	—
57	C₂²×C₈×C₃₁	—	2	1	no	—
58	C₂²×C₈×C₃₁	—	2	1	no	—
59	C₂²×C₈×C₃₁	—	2	2	no	—
60	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
63	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
64	C₃₁×#124	—	2	1	no	—
65	C₃₁×#121	—	2	1	no	—
66	C₃₁×#121	C₂×C₄	1	2	yes	—
67	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₄×D₄×C₃₁	—	2	1	no	—
70	C₄×D₄×C₃₁	—	2	1	no	—
71	C₄×D₄×C₃₁	—	2	1	no	—
72	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₄×D₄×C₃₁	C₂³	1	2	yes	—
75	C₃₁×#117	—	2	1	no	—
76	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	—	2	1	no	—
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#107	C₂×C₄	1	2	yes	—
80	C₃₁×#106	C₂×C₄	1	2	yes	—
81	C₃₁×C₄⋊C₈	—	2	1	no	—
82	C₃₁×C₄⋊C₈	—	2	1	no	—
83	C₃₁×C₄⋊C₈	—	2	1	no	—
84	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
85	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
86	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
87	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
88	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
89	C₃₁×#103	C₂×C₄	1	2	yes	—
90	C₃₁×#102	C₂×C₄	1	2	yes	—
91	C₃₁×#102	C₂×C₄	1	2	yes	—
92	C₃₁×#102	C₂×C₄	1	2	yes	—
93	C₃₁×C₂²⋊C₈	—	2	1	no	—
94	C₃₁×C₂²⋊C₈	—	2	2	no	—
95	C₃₁×C₂²⋊C₈	—	2	2	no	—
96	C₃₁×C₂²⋊C₈	—	2	1	no	—
97	C₃₁×C₂²⋊C₈	—	2	1	no	—
98	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
99	C₃₁×#97	—	2	1	no	—
100	C₄×C₈×C₃₁	—	2	1	no	—
101	C₄×C₈×C₃₁	—	2	1	no	—
102	C₄×C₈×C₃₁	—	2	1	no	—
103	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
104	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
105	C₄×C₈×C₃₁	C₂³	1	2	yes	—
106	C₄×C₈×C₃₁	C₂³	1	2	yes	—
107	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
108	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
109	#41×C₃₁	—	2	2	no	—
110	#41×C₃₁	—	2	2	no	—
111	#41×C₃₁	—	4	2	no	—
112	#41×C₃₁	—	4	2	no	—
113	C₂×Q₈×C₃₁	—	2	1	no	—
114	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	1	no	—
119	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
120	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
128	SD₁₆×C₃₁	—	2	4	no	—
129	D₈×C₃₁	—	2	4	no	—
130	M₁₆×C₃₁	—	2	4	no	—
131	M₁₆×C₃₁	—	4	4	no	—
132	M₁₆×C₃₁	—	4	2	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	—	4	1	no	—
139	C₂×C₈×C₃₁	—	4	2	no	—
140	C₂×C₈×C₃₁	—	4	2	no	—
141	C₂×C₈×C₃₁	—	4	4	no	—
142	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
143	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
144	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₄⋊C₄×C₃₁	—	2	1	no	—
151	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
152	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
153	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
154	C₂²⋊C₄×C₃₁	—	2	1	no	—
155	C₂²⋊C₄×C₃₁	—	2	1	no	—
156	C₂²⋊C₄×C₃₁	—	2	2	no	—
157	C₂²⋊C₄×C₃₁	—	2	1	no	—
158	C₂²⋊C₄×C₃₁	—	2	2	no	—
159	C₄²×C₃₁	—	2	1	no	—
160	C₄²×C₃₁	—	2	1	no	—
161	C₄²×C₃₁	—	2	1	no	—
162	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
163	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₄²×C₃₁	#41	1	1	yes	characteristic
165	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
166	#17912	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₃₁	—	2	1	no	—
168	C₂³×C₃₁	—	2	2	no	—
169	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
170	Q₈×C₃₁	—	2	1	no	—
171	Q₈×C₃₁	—	4	2	no	—
172	Q₈×C₃₁	#131	1	2	yes	complemented
173	D₄×C₃₁	—	2	4	no	—
174	D₄×C₃₁	—	2	1	no	—
175	D₄×C₃₁	—	2	2	no	—
176	D₄×C₃₁	—	2	2	no	—
177	D₄×C₃₁	—	4	2	no	—
178	D₄×C₃₁	—	4	4	no	—
179	D₄×C₃₁	—	4	2	no	—
180	D₄×C₃₁	#131	1	2	yes	complemented
181	C₂×C₄×C₃₁	—	2	4	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	2	no	—
196	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
202	C₈×C₃₁	—	2	4	no	—
203	C₈×C₃₁	—	2	2	no	—
204	C₈×C₃₁	—	2	2	no	—
205	C₈×C₃₁	—	2	1	no	—
206	C₈×C₃₁	—	2	1	no	—
207	C₈×C₃₁	—	2	1	no	—
208	C₈×C₃₁	—	4	1	no	—
209	C₈×C₃₁	—	4	2	no	—
210	C₈×C₃₁	—	4	1	no	—
211	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
212	#3106	C₂×C₃₁	1	1	yes	characteristic
213	C₈×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
214	#2931	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#1588	C₂×C₃₁	1	2	yes	complemented
216	#1586	C₂×C₃₁	1	2	yes	complemented
217	C₈×SD₁₆	C₂×C₃₁	1	2	yes	complemented
218	C₈×D₈	C₂×C₃₁	1	2	yes	complemented
219	#1574	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#1489	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#1484	C₂×C₃₁	1	1	yes	characteristic, complemented
222	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
223	C₂²×C₃₁	—	2	1	no	—
224	C₂²×C₃₁	—	2	2	no	—
225	C₂²×C₃₁	—	2	1	no	—
226	C₂²×C₃₁	—	2	1	no	—
227	C₂²×C₃₁	—	4	1	no	—
228	C₂²×C₃₁	—	4	1	no	—
229	C₂²×C₃₁	—	4	2	no	—
230	C₂²×C₃₁	—	4	2	no	—
231	C₂²×C₃₁	—	4	2	no	—
232	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
233	C₄×C₃₁	—	2	1	no	—
234	C₄×C₃₁	—	2	2	no	—
235	C₄×C₃₁	—	2	1	no	—
236	C₄×C₃₁	—	2	1	no	—
237	C₄×C₃₁	—	4	1	no	—
238	C₄×C₃₁	—	4	2	no	—
239	C₄×C₃₁	—	4	1	no	—
240	C₄×C₃₁	#443	1	2	yes	—
241	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
242	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
243	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
244	C₄×C₃₁	#575	1	1	yes	characteristic
245	C₂×#131	—	2	1	no	—
246	#532	C₂²×C₃₁	1	1	yes	characteristic
247	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
248	#482	C₄×C₃₁	1	2	yes	—
249	#477	C₄×C₃₁	1	1	yes	characteristic
250	#476	C₄×C₃₁	1	1	yes	characteristic
251	#460	C₄×C₃₁	1	1	yes	characteristic
252	#459	C₄×C₃₁	1	1	yes	characteristic
253	#449	C₄×C₃₁	1	2	yes	—
254	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
255	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
256	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
257	C₄×D₈	C₂²×C₃₁	1	2	yes	—
258	#436	—	2	1	no	—
259	#436	C₂²×C₃₁	1	2	yes	—
260	C₈×D₄	—	2	1	no	—
261	C₈×D₄	—	2	1	no	—
262	C₈×D₄	—	2	1	no	—
263	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
264	C₈×D₄	C₂²×C₃₁	1	2	yes	—
265	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
266	#431	—	2	1	no	—
267	#431	C₂²×C₃₁	1	1	yes	characteristic
268	#431	C₂²×C₃₁	1	1	yes	characteristic
269	#423	C₂²×C₃₁	1	1	yes	characteristic
270	#408	—	2	1	no	—
271	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×C₄×C₈	—	2	1	no	—
273	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
274	#335	C₂²×C₃₁	1	2	yes	—
275	#334	C₂²×C₃₁	1	2	yes	—
276	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
277	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
278	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
279	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
280	#322	C₂²×C₃₁	1	2	yes	—
281	C₈²	C₂²×C₃₁	1	2	yes	—
282	C₂×C₃₁	—	4	1	no	—
283	C₂×C₃₁	—	4	2	no	—
284	C₂×C₃₁	—	8	2	no	—
285	C₂×C₃₁	#2929	1	1	yes	characteristic
286	C₂×C₃₁	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
287	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
288	C₂×#41	—	2	1	no	—
289	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
290	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
291	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
292	#131	—	4	4	no	—
293	C₂×M₁₆	—	2	1	no	—
294	C₂×M₁₆	—	2	1	no	—
295	C₂×M₁₆	—	2	2	no	—
296	C₂²×C₈	—	2	2	no	—
297	C₂²×C₈	—	2	1	no	—
298	C₂²×C₈	—	2	1	no	—
299	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
300	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
301	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
302	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
303	#124	—	2	1	no	—
304	#121	—	2	1	no	—
305	#121	C₂×C₄×C₃₁	1	2	yes	—
306	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
307	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
308	C₄×D₄	—	2	1	no	—
309	C₄×D₄	—	2	1	no	—
310	C₄×D₄	—	2	1	no	—
311	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
312	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
313	C₄×D₄	C₂³×C₃₁	1	2	yes	—
314	#117	—	2	1	no	—
315	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
316	C₂×C₄²	—	2	1	no	—
317	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
318	#107	C₂×C₄×C₃₁	1	2	yes	—
319	#106	C₂×C₄×C₃₁	1	2	yes	—
320	C₄⋊C₈	—	2	1	no	—
321	C₄⋊C₈	—	2	1	no	—
322	C₄⋊C₈	—	2	1	no	—
323	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
324	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
325	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
326	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
327	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
328	#103	C₂×C₄×C₃₁	1	2	yes	—
329	#102	C₂×C₄×C₃₁	1	2	yes	—
330	#102	C₂×C₄×C₃₁	1	2	yes	—
331	#102	C₂×C₄×C₃₁	1	2	yes	—
332	C₂²⋊C₈	—	2	1	no	—
333	C₂²⋊C₈	—	2	2	no	—
334	C₂²⋊C₈	—	2	2	no	—
335	C₂²⋊C₈	—	2	1	no	—
336	C₂²⋊C₈	—	2	1	no	—
337	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
338	#97	—	2	1	no	—
339	C₄×C₈	—	2	1	no	—
340	C₄×C₈	—	2	1	no	—
341	C₄×C₈	—	2	1	no	—
342	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
343	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
344	C₄×C₈	C₂³×C₃₁	1	2	yes	—
345	C₄×C₈	C₂³×C₃₁	1	2	yes	—
346	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
347	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
348	C₃₁	#17912	1	1	yes	characteristic, direct product factor, Sylow
349	#41	—	2	2	no	—
350	#41	—	2	2	no	—
351	#41	—	4	2	no	—
352	#41	—	4	2	no	—
353	C₂×Q₈	—	2	1	no	—
354	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	1	no	—
358	C₂×D₄	—	2	1	no	—
359	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
360	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
366	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
367	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
368	SD₁₆	—	2	4	no	—
369	D₈	—	2	4	no	—
370	M₁₆	—	2	4	no	—
371	M₁₆	—	4	4	no	—
372	M₁₆	—	4	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	4	1	no	—
379	C₂×C₈	—	4	2	no	—
380	C₂×C₈	—	4	2	no	—
381	C₂×C₈	—	4	4	no	—
382	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
383	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
384	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
386	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
387	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
392	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
393	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
394	C₂²⋊C₄	—	2	1	no	—
395	C₂²⋊C₄	—	2	1	no	—
396	C₂²⋊C₄	—	2	2	no	—
397	C₂²⋊C₄	—	2	1	no	—
398	C₂²⋊C₄	—	2	2	no	—
399	C₄²	—	2	1	no	—
400	C₄²	—	2	1	no	—
401	C₄²	—	2	1	no	—
402	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
403	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
404	C₄²	#41×C₃₁	1	1	yes	characteristic
405	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
406	C₂³	—	2	1	no	—
407	C₂³	—	2	2	no	—
408	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
409	Q₈	—	2	1	no	—
410	Q₈	—	4	2	no	—
411	Q₈	C₃₁×#131	1	2	yes	complemented
412	D₄	—	2	4	no	—
413	D₄	—	2	1	no	—
414	D₄	—	2	2	no	—
415	D₄	—	2	2	no	—
416	D₄	—	4	2	no	—
417	D₄	—	4	4	no	—
418	D₄	—	4	2	no	—
419	D₄	C₃₁×#131	1	2	yes	complemented
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
436	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
437	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
438	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
439	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
440	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
441	C₈	—	2	4	no	—
442	C₈	—	2	2	no	—
443	C₈	—	2	2	no	—
444	C₈	—	2	1	no	—
445	C₈	—	2	1	no	—
446	C₈	—	2	1	no	—
447	C₈	—	4	1	no	—
448	C₈	—	4	2	no	—
449	C₈	—	4	1	no	—
450	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
451	C₂²	—	2	1	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	1	no	—
454	C₂²	—	2	1	no	—
455	C₂²	—	4	1	no	—
456	C₂²	—	4	1	no	—
457	C₂²	—	4	2	no	—
458	C₂²	—	4	2	no	—
459	C₂²	—	4	2	no	—
460	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
461	C₄	—	2	1	no	—
462	C₄	—	2	2	no	—
463	C₄	—	2	1	no	—
464	C₄	—	2	1	no	—
465	C₄	—	4	2	no	—
466	C₄	—	4	1	no	—
467	C₄	—	4	1	no	—
468	C₄	C₃₁×#443	1	2	yes	—
469	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
470	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
471	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
472	C₄	C₃₁×#575	1	1	yes	characteristic
473	C₂	—	4	1	no	—
474	C₂	—	4	2	no	—
475	C₂	—	8	2	no	—
476	C₂	C₃₁×#2929	1	1	yes	characteristic
477	C₂	C₄×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
478	C₂	C₂×C₃₁×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#3106	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×#41×C₃₁	C₂	semidirect
3. C₃₁×#2931	C₂	semidirect
4. C₃₁×#1588	C₂	semidirect
5. C₃₁×#1586	C₂	semidirect
6. C₈×SD₁₆×C₃₁	C₂	semidirect
7. C₈×D₈×C₃₁	C₂	semidirect
8. C₃₁×#1574	C₂	semidirect
9. C₃₁×#1489	C₂	semidirect
10. C₃₁×#1484	C₂	semidirect
11. C₈×M₁₆×C₃₁	C₂	semidirect
166. #17912	C₃₁	direct (N × H)
172. Q₈×C₃₁	#131	semidirect
180. D₄×C₃₁	#131	semidirect
213. C₈×#41	C₂×C₃₁	semidirect
214. #2931	C₂×C₃₁	semidirect
215. #1588	C₂×C₃₁	semidirect
216. #1586	C₂×C₃₁	semidirect
217. C₈×SD₁₆	C₂×C₃₁	semidirect
218. C₈×D₈	C₂×C₃₁	semidirect
219. #1574	C₂×C₃₁	semidirect
220. #1489	C₂×C₃₁	semidirect
221. #1484	C₂×C₃₁	semidirect
222. C₈×M₁₆	C₂×C₃₁	semidirect
348. C₃₁	#17912	direct (N × H)
411. Q₈	C₃₁×#131	semidirect
419. D₄	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×#3106	2
C₈×#41×C₃₁	2
C₃₁×#2931	2
C₃₁×#1588	2
C₃₁×#1586	2
C₈×SD₁₆×C₃₁	2
C₈×D₈×C₃₁	2
C₃₁×#1574	2
C₃₁×#1489	2
C₃₁×#1484	2
C₈×M₁₆×C₃₁	2
#17912	31

Subgroups — C31×#17912 #21474671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17912 #21474671