Subgroups — C₃₁×#19162 #21475921

Order 7936.

844 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3151	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3100	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#459	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1687	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1685	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1667	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1665	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1547	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
13	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
14	C₂²×D₈×C₃₁	—	2	1	no	—
15	C₂²×M₁₆×C₃₁	—	2	1	no	—
16	C₃₁×#532	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#496	—	2	2	no	—
20	C₃₁×#493	C₂²	1	2	yes	complemented
21	C₃₁×#492	C₂²	1	2	yes	complemented
22	C₃₁×#480	—	2	1	no	—
23	C₃₁×#474	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×#474	C₂²	1	1	yes	characteristic, complemented
25	C₃₁×#459	—	2	1	no	—
26	C₃₁×#459	—	2	1	no	—
27	C₃₁×#459	C₂²	1	2	yes	—
28	C₃₁×#459	C₂²	1	2	yes	—
29	C₃₁×#459	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×#459	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#447	—	2	1	no	—
32	C₄×D₈×C₃₁	—	2	1	no	—
33	C₄×D₈×C₃₁	—	2	1	no	—
34	C₃₁×#436	C₂²	1	2	yes	complemented
35	C₃₁×#435	—	2	2	no	—
36	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
37	C₃₁×#431	C₂²	1	1	yes	characteristic
38	C₃₁×#423	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#102	—	2	1	no	—
41	C₃₁×#407	—	2	1	no	—
42	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
43	C₃₁×#335	C₂²	1	2	yes	complemented
44	C₃₁×#334	C₂²	1	2	yes	complemented
45	C₃₁×#332	C₂²	1	2	yes	complemented
46	C₃₁×#331	C₂²	1	2	yes	complemented
47	C₃₁×#331	C₂²	1	2	yes	complemented
48	C₃₁×#331	C₂²	1	2	yes	complemented
49	C₂×#41×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	—	2	1	no	—
53	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₃₁	—	2	1	no	—
55	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
56	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
57	C₂×SD₁₆×C₃₁	—	2	2	no	—
58	C₂×SD₁₆×C₃₁	—	2	2	no	—
59	C₂×D₈×C₃₁	—	2	1	no	—
60	C₂×D₈×C₃₁	—	2	1	no	—
61	C₂×D₈×C₃₁	—	2	2	no	—
62	C₂×D₈×C₃₁	—	2	2	no	—
63	C₂×D₈×C₃₁	—	2	2	no	—
64	C₂×D₈×C₃₁	—	2	2	no	—
65	C₂×D₈×C₃₁	—	4	4	no	—
66	C₂×M₁₆×C₃₁	—	2	1	no	—
67	C₂×M₁₆×C₃₁	—	2	2	no	—
68	C₂×M₁₆×C₃₁	—	4	4	no	—
69	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
70	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
71	C₂²×C₈×C₃₁	—	2	1	no	—
72	C₂²×C₈×C₃₁	—	2	2	no	—
73	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
74	C₃₁×#127	—	2	1	no	—
75	C₃₁×#127	—	2	1	no	—
76	C₃₁×#127	—	2	1	no	—
77	C₃₁×#127	C₂³	1	2	yes	—
78	C₃₁×#127	C₂³	1	1	yes	characteristic
79	C₃₁×#124	—	2	1	no	—
80	C₃₁×#121	—	2	1	no	—
81	C₃₁×#121	—	2	1	no	—
82	C₄×D₄×C₃₁	—	2	1	no	—
83	C₄×D₄×C₃₁	—	2	1	no	—
84	C₄×D₄×C₃₁	—	2	1	no	—
85	C₄×D₄×C₃₁	—	2	1	no	—
86	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
87	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
88	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
89	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
91	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
92	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
93	C₃₁×#107	—	2	1	no	—
94	C₃₁×#107	—	2	1	no	—
95	C₃₁×C₄⋊C₈	—	2	1	no	—
96	C₃₁×C₄⋊C₈	—	2	1	no	—
97	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
98	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
99	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
100	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
101	C₃₁×#102	—	2	1	no	—
102	C₃₁×#102	—	2	1	no	—
103	C₃₁×#102	—	2	2	no	—
104	C₃₁×#102	—	2	2	no	—
105	C₃₁×#102	—	2	1	no	—
106	C₃₁×#102	—	2	1	no	—
107	C₃₁×C₂²⋊C₈	—	2	1	no	—
108	C₃₁×C₂²⋊C₈	—	2	2	no	—
109	C₃₁×C₂²⋊C₈	—	2	2	no	—
110	C₃₁×#97	D₄	1	2	yes	—
111	C₄×C₈×C₃₁	—	2	1	no	—
112	C₄×C₈×C₃₁	C₂³	1	2	yes	—
113	C₂⁴×C₃₁	—	2	1	no	—
114	C₂⁴×C₃₁	—	4	1	no	—
115	#41×C₃₁	—	4	2	no	—
116	#41×C₃₁	—	4	2	no	—
117	C₂×Q₈×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	2	no	—
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	—	2	1	no	—
121	C₂×D₄×C₃₁	—	2	2	no	—
122	C₂×D₄×C₃₁	—	2	1	no	—
123	C₂×D₄×C₃₁	—	2	2	no	—
124	C₂×D₄×C₃₁	—	2	1	no	—
125	C₂×D₄×C₃₁	—	2	2	no	—
126	C₂×D₄×C₃₁	—	4	1	no	—
127	C₂×D₄×C₃₁	—	4	2	no	—
128	C₂×D₄×C₃₁	—	4	2	no	—
129	C₂×D₄×C₃₁	—	4	1	no	—
130	C₂×D₄×C₃₁	—	4	1	no	—
131	C₂×D₄×C₃₁	—	4	1	no	—
132	C₂×D₄×C₃₁	—	4	1	no	—
133	C₂×D₄×C₃₁	—	4	2	no	—
134	C₂×D₄×C₃₁	—	4	2	no	—
135	C₂×D₄×C₃₁	—	8	2	no	—
136	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	—	2	1	no	—
139	C₂²×C₄×C₃₁	—	2	1	no	—
140	C₂²×C₄×C₃₁	—	2	1	no	—
141	C₂²×C₄×C₃₁	—	2	1	no	—
142	C₂²×C₄×C₃₁	—	2	1	no	—
143	C₂²×C₄×C₃₁	—	4	2	no	—
144	C₂²×C₄×C₃₁	—	4	2	no	—
145	C₂²×C₄×C₃₁	—	4	1	no	—
146	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
149	SD₁₆×C₃₁	—	4	4	no	—
150	SD₁₆×C₃₁	—	4	4	no	—
151	D₈×C₃₁	—	4	4	no	—
152	D₈×C₃₁	—	4	2	no	—
153	D₈×C₃₁	—	4	4	no	—
154	D₈×C₃₁	—	4	2	no	—
155	D₈×C₃₁	—	4	4	no	—
156	D₈×C₃₁	—	4	4	no	—
157	M₁₆×C₃₁	—	2	2	no	—
158	M₁₆×C₃₁	—	4	2	no	—
159	M₁₆×C₃₁	—	4	4	no	—
160	M₁₆×C₃₁	D₈	1	4	yes	complemented
161	C₂×C₈×C₃₁	—	2	1	no	—
162	C₂×C₈×C₃₁	—	2	1	no	—
163	C₂×C₈×C₃₁	—	2	2	no	—
164	C₂×C₈×C₃₁	—	2	2	no	—
165	C₂×C₈×C₃₁	—	4	2	no	—
166	C₂×C₈×C₃₁	—	4	4	no	—
167	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
168	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₂²⋊C₄×C₃₁	—	2	1	no	—
175	C₂²⋊C₄×C₃₁	—	2	1	no	—
176	C₂²⋊C₄×C₃₁	—	2	1	no	—
177	C₂²⋊C₄×C₃₁	—	2	1	no	—
178	C₂²⋊C₄×C₃₁	—	4	1	no	—
179	C₂²⋊C₄×C₃₁	—	4	1	no	—
180	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₃₁	#41	1	1	yes	characteristic
183	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
184	#19162	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂³×C₃₁	—	2	1	no	—
186	C₂³×C₃₁	—	2	1	no	—
187	C₂³×C₃₁	—	4	2	no	—
188	C₂³×C₃₁	—	4	2	no	—
189	C₂³×C₃₁	—	4	1	no	—
190	C₂³×C₃₁	—	4	2	no	—
191	C₂³×C₃₁	—	8	1	no	—
192	C₂³×C₃₁	—	8	2	no	—
193	C₂³×C₃₁	—	8	2	no	—
194	C₂³×C₃₁	—	8	2	no	—
195	C₂³×C₃₁	#121	1	1	yes	characteristic
196	Q₈×C₃₁	—	4	1	no	—
197	Q₈×C₃₁	—	4	1	no	—
198	D₄×C₃₁	—	2	1	no	—
199	D₄×C₃₁	—	2	1	no	—
200	D₄×C₃₁	—	2	1	no	—
201	D₄×C₃₁	—	4	2	no	—
202	D₄×C₃₁	—	4	1	no	—
203	D₄×C₃₁	—	4	2	no	—
204	D₄×C₃₁	—	4	2	no	—
205	D₄×C₃₁	—	4	1	no	—
206	D₄×C₃₁	—	4	2	no	—
207	D₄×C₃₁	—	4	1	no	—
208	D₄×C₃₁	—	4	1	no	—
209	D₄×C₃₁	—	4	2	no	—
210	D₄×C₃₁	—	8	1	no	—
211	D₄×C₃₁	—	8	2	no	—
212	D₄×C₃₁	—	8	2	no	—
213	D₄×C₃₁	—	8	2	no	—
214	D₄×C₃₁	—	8	2	no	—
215	D₄×C₃₁	—	8	2	no	—
216	D₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	complemented
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	1	no	—
221	C₂×C₄×C₃₁	—	4	1	no	—
222	C₂×C₄×C₃₁	—	4	2	no	—
223	C₂×C₄×C₃₁	—	4	1	no	—
224	C₂×C₄×C₃₁	—	4	1	no	—
225	C₂×C₄×C₃₁	—	4	1	no	—
226	C₂×C₄×C₃₁	—	4	1	no	—
227	C₂×C₄×C₃₁	—	4	2	no	—
228	C₂×C₄×C₃₁	—	4	2	no	—
229	C₂×C₄×C₃₁	—	4	1	no	—
230	C₂×C₄×C₃₁	—	8	2	no	—
231	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
232	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
233	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
234	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
235	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
239	C₈×C₃₁	—	2	2	no	—
240	C₈×C₃₁	—	4	2	no	—
241	C₈×C₃₁	—	4	2	no	—
242	C₈×C₃₁	—	4	1	no	—
243	C₈×C₃₁	—	4	1	no	—
244	C₈×C₃₁	C₂×D₈	1	4	yes	complemented
245	#3151	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#3100	C₂×C₃₁	1	1	yes	characteristic, complemented
247	C₂×#459	C₂×C₃₁	1	1	yes	characteristic, complemented
248	D₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#1687	C₂×C₃₁	1	2	yes	complemented
250	#1685	C₂×C₃₁	1	2	yes	complemented
251	#1667	C₂×C₃₁	1	2	yes	complemented
252	#1665	C₂×C₃₁	1	2	yes	complemented
253	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#1547	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#1545	C₂×C₃₁	1	1	yes	characteristic, complemented
256	C₂²×C₃₁	—	2	1	no	—
257	C₂²×C₃₁	—	2	1	no	—
258	C₂²×C₃₁	—	2	2	no	—
259	C₂²×C₃₁	—	4	1	no	—
260	C₂²×C₃₁	—	4	1	no	—
261	C₂²×C₃₁	—	4	1	no	—
262	C₂²×C₃₁	—	4	1	no	—
263	C₂²×C₃₁	—	4	1	no	—
264	C₂²×C₃₁	—	8	4	no	—
265	C₂²×C₃₁	—	8	1	no	—
266	C₂²×C₃₁	—	8	2	no	—
267	C₂²×C₃₁	—	8	2	no	—
268	C₂²×C₃₁	—	8	1	no	—
269	C₂²×C₃₁	—	8	2	no	—
270	C₂²×C₃₁	—	8	1	no	—
271	C₂²×C₃₁	—	16	4	no	—
272	C₂²×C₃₁	#459	1	2	yes	—
273	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
274	C₄×C₃₁	—	2	1	no	—
275	C₄×C₃₁	—	2	1	no	—
276	C₄×C₃₁	—	2	1	no	—
277	C₄×C₃₁	—	4	1	no	—
278	C₄×C₃₁	—	4	1	no	—
279	C₄×C₃₁	—	8	1	no	—
280	C₄×C₃₁	#459	1	2	yes	—
281	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
282	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
283	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
284	C₄×C₃₁	#575	1	1	yes	characteristic
285	C₂×AGL(1,ℤ/8)	—	2	1	no	—
286	C₂×AGL(1,ℤ/8)	—	2	1	no	—
287	C₂²×D₈	—	2	1	no	—
288	C₂²×M₁₆	—	2	1	no	—
289	#532	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
292	#496	—	2	2	no	—
293	#493	C₂²×C₃₁	1	2	yes	complemented
294	#492	C₂²×C₃₁	1	2	yes	complemented
295	#480	—	2	1	no	—
296	#474	C₂²×C₃₁	1	1	yes	characteristic, complemented
297	#474	C₂²×C₃₁	1	1	yes	characteristic, complemented
298	#459	—	2	1	no	—
299	#459	—	2	1	no	—
300	#459	C₂²×C₃₁	1	2	yes	—
301	#459	C₂²×C₃₁	1	2	yes	—
302	#459	C₂²×C₃₁	1	1	yes	characteristic, complemented
303	#459	C₂²×C₃₁	1	1	yes	characteristic, complemented
304	#447	—	2	1	no	—
305	C₄×D₈	—	2	1	no	—
306	C₄×D₈	—	2	1	no	—
307	#436	C₂²×C₃₁	1	2	yes	complemented
308	#435	—	2	2	no	—
309	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
310	#431	C₂²×C₃₁	1	1	yes	characteristic
311	#423	C₂²×C₃₁	1	1	yes	characteristic
312	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
313	C₂×#102	—	2	1	no	—
314	#407	—	2	1	no	—
315	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
316	#335	C₂²×C₃₁	1	2	yes	complemented
317	#334	C₂²×C₃₁	1	2	yes	complemented
318	#332	C₂²×C₃₁	1	2	yes	complemented
319	#331	C₂²×C₃₁	1	2	yes	complemented
320	#331	C₂²×C₃₁	1	2	yes	complemented
321	#331	C₂²×C₃₁	1	2	yes	complemented
322	C₂×C₃₁	—	2	2	no	—
323	C₂×C₃₁	—	4	2	no	—
324	C₂×C₃₁	—	8	1	no	—
325	C₂×C₃₁	—	16	1	no	—
326	C₂×C₃₁	—	16	2	no	—
327	C₂×C₃₁	C₂×#459	1	1	yes	characteristic
328	C₂×C₃₁	#3055	1	1	yes	characteristic
329	C₂×C₃₁	#3068	1	1	yes	characteristic
330	C₂×#41	—	2	1	no	—
331	C₂²×D₄	—	2	1	no	—
332	C₂²×D₄	—	2	1	no	—
333	C₂²×D₄	—	2	1	no	—
334	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
335	C₂³×C₄	—	2	1	no	—
336	AGL(1,ℤ/8)	—	4	4	no	—
337	AGL(1,ℤ/8)	—	4	4	no	—
338	C₂×SD₁₆	—	2	2	no	—
339	C₂×SD₁₆	—	2	2	no	—
340	C₂×D₈	—	2	2	no	—
341	C₂×D₈	—	2	2	no	—
342	C₂×D₈	—	2	1	no	—
343	C₂×D₈	—	2	1	no	—
344	C₂×D₈	—	2	2	no	—
345	C₂×D₈	—	2	2	no	—
346	C₂×D₈	—	4	4	no	—
347	C₂×M₁₆	—	2	1	no	—
348	C₂×M₁₆	—	2	2	no	—
349	C₂×M₁₆	—	4	4	no	—
350	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
351	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
352	C₂²×C₈	—	2	1	no	—
353	C₂²×C₈	—	2	2	no	—
354	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
355	#127	—	2	1	no	—
356	#127	—	2	1	no	—
357	#127	—	2	1	no	—
358	#127	C₂³×C₃₁	1	2	yes	—
359	#127	C₂³×C₃₁	1	1	yes	characteristic
360	#124	—	2	1	no	—
361	#121	—	2	1	no	—
362	#121	—	2	1	no	—
363	C₄×D₄	—	2	1	no	—
364	C₄×D₄	—	2	1	no	—
365	C₄×D₄	—	2	1	no	—
366	C₄×D₄	—	2	1	no	—
367	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
368	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
369	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
370	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
371	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
372	C₂×C₂²⋊C₄	—	2	1	no	—
373	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
374	#107	—	2	1	no	—
375	#107	—	2	1	no	—
376	C₄⋊C₈	—	2	1	no	—
377	C₄⋊C₈	—	2	1	no	—
378	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
379	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
380	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
381	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
382	#102	—	2	1	no	—
383	#102	—	2	1	no	—
384	#102	—	2	2	no	—
385	#102	—	2	2	no	—
386	#102	—	2	1	no	—
387	#102	—	2	1	no	—
388	C₂²⋊C₈	—	2	1	no	—
389	C₂²⋊C₈	—	2	2	no	—
390	C₂²⋊C₈	—	2	2	no	—
391	#97	D₄×C₃₁	1	2	yes	—
392	C₄×C₈	—	2	1	no	—
393	C₄×C₈	C₂³×C₃₁	1	2	yes	—
394	C₃₁	#19162	1	1	yes	characteristic, direct product factor, Sylow
395	C₂⁴	—	2	1	no	—
396	C₂⁴	—	4	1	no	—
397	#41	—	4	2	no	—
398	#41	—	4	2	no	—
399	C₂×Q₈	—	2	1	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	2	1	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	1	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	4	1	no	—
409	C₂×D₄	—	4	2	no	—
410	C₂×D₄	—	4	2	no	—
411	C₂×D₄	—	4	1	no	—
412	C₂×D₄	—	4	1	no	—
413	C₂×D₄	—	4	1	no	—
414	C₂×D₄	—	4	1	no	—
415	C₂×D₄	—	4	2	no	—
416	C₂×D₄	—	4	2	no	—
417	C₂×D₄	—	8	2	no	—
418	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
419	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	4	2	no	—
426	C₂²×C₄	—	4	2	no	—
427	C₂²×C₄	—	4	1	no	—
428	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
429	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
430	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
431	SD₁₆	—	4	4	no	—
432	SD₁₆	—	4	4	no	—
433	D₈	—	4	4	no	—
434	D₈	—	4	2	no	—
435	D₈	—	4	4	no	—
436	D₈	—	4	2	no	—
437	D₈	—	4	4	no	—
438	D₈	—	4	4	no	—
439	M₁₆	—	2	2	no	—
440	M₁₆	—	4	2	no	—
441	M₁₆	—	4	4	no	—
442	M₁₆	D₈×C₃₁	1	4	yes	complemented
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	—	2	2	no	—
447	C₂×C₈	—	4	2	no	—
448	C₂×C₈	—	4	4	no	—
449	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
450	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
455	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
456	C₂²⋊C₄	—	2	1	no	—
457	C₂²⋊C₄	—	2	1	no	—
458	C₂²⋊C₄	—	2	1	no	—
459	C₂²⋊C₄	—	2	1	no	—
460	C₂²⋊C₄	—	4	1	no	—
461	C₂²⋊C₄	—	4	1	no	—
462	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
464	C₄²	#41×C₃₁	1	1	yes	characteristic
465	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
466	C₂³	—	2	1	no	—
467	C₂³	—	2	1	no	—
468	C₂³	—	4	2	no	—
469	C₂³	—	4	2	no	—
470	C₂³	—	4	1	no	—
471	C₂³	—	4	2	no	—
472	C₂³	—	8	1	no	—
473	C₂³	—	8	2	no	—
474	C₂³	—	8	2	no	—
475	C₂³	—	8	2	no	—
476	C₂³	C₃₁×#121	1	1	yes	characteristic
477	Q₈	—	4	1	no	—
478	Q₈	—	4	1	no	—
479	D₄	—	2	1	no	—
480	D₄	—	2	1	no	—
481	D₄	—	2	1	no	—
482	D₄	—	4	2	no	—
483	D₄	—	4	1	no	—
484	D₄	—	4	2	no	—
485	D₄	—	4	2	no	—
486	D₄	—	4	1	no	—
487	D₄	—	4	2	no	—
488	D₄	—	4	1	no	—
489	D₄	—	4	1	no	—
490	D₄	—	4	2	no	—
491	D₄	—	8	1	no	—
492	D₄	—	8	2	no	—
493	D₄	—	8	2	no	—
494	D₄	—	8	2	no	—
495	D₄	—	8	2	no	—
496	D₄	—	8	2	no	—
497	D₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	complemented
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	1	no	—
507	C₂×C₄	—	4	1	no	—
508	C₂×C₄	—	4	1	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	8	2	no	—
512	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
513	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
514	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
515	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
516	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
517	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
518	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
519	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
520	C₈	—	2	2	no	—
521	C₈	—	4	2	no	—
522	C₈	—	4	2	no	—
523	C₈	—	4	1	no	—
524	C₈	—	4	1	no	—
525	C₈	C₂×D₈×C₃₁	1	4	yes	complemented
526	C₂²	—	2	1	no	—
527	C₂²	—	2	1	no	—
528	C₂²	—	2	2	no	—
529	C₂²	—	4	1	no	—
530	C₂²	—	4	1	no	—
531	C₂²	—	4	1	no	—
532	C₂²	—	4	1	no	—
533	C₂²	—	4	1	no	—
534	C₂²	—	8	4	no	—
535	C₂²	—	8	1	no	—
536	C₂²	—	8	2	no	—
537	C₂²	—	8	2	no	—
538	C₂²	—	8	1	no	—
539	C₂²	—	8	2	no	—
540	C₂²	—	8	1	no	—
541	C₂²	—	16	4	no	—
542	C₂²	C₃₁×#459	1	2	yes	—
543	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
544	C₄	—	2	1	no	—
545	C₄	—	2	1	no	—
546	C₄	—	2	1	no	—
547	C₄	—	4	1	no	—
548	C₄	—	4	1	no	—
549	C₄	—	8	1	no	—
550	C₄	C₃₁×#459	1	2	yes	—
551	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
552	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
553	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
554	C₄	C₃₁×#575	1	1	yes	characteristic
555	C₂	—	2	2	no	—
556	C₂	—	4	2	no	—
557	C₂	—	8	1	no	—
558	C₂	—	16	2	no	—
559	C₂	—	16	1	no	—
560	C₂	C₂×C₃₁×#459	1	1	yes	characteristic
561	C₂	C₃₁×#3055	1	1	yes	characteristic
562	C₂	C₃₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3151	C₂	semidirect
2. C₃₁×#3100	C₂	semidirect
3. C₂×C₃₁×#459	C₂	semidirect
4. D₄×M₁₆×C₃₁	C₂	semidirect
5. C₃₁×#1687	C₂	semidirect
6. C₃₁×#1685	C₂	semidirect
7. C₃₁×#1667	C₂	semidirect
8. C₃₁×#1665	C₂	semidirect
9. C₃₁×#1565	C₂	semidirect
10. C₃₁×#1547	C₂	semidirect
11. C₃₁×#1545	C₂	semidirect
20. C₃₁×#493	C₂²	semidirect
21. C₃₁×#492	C₂²	semidirect
23. C₃₁×#474	C₂²	semidirect
24. C₃₁×#474	C₂²	semidirect
29. C₃₁×#459	C₂²	semidirect
30. C₃₁×#459	C₂²	semidirect
34. C₃₁×#436	C₂²	semidirect
36. C₈×D₄×C₃₁	C₂²	semidirect
43. C₃₁×#335	C₂²	semidirect
44. C₃₁×#334	C₂²	semidirect
45. C₃₁×#332	C₂²	semidirect
46. C₃₁×#331	C₂²	semidirect
47. C₃₁×#331	C₂²	semidirect
48. C₃₁×#331	C₂²	semidirect
160. M₁₆×C₃₁	D₈	semidirect
184. #19162	C₃₁	direct (N × H)
216. D₄×C₃₁	AGL(1,ℤ/8)	semidirect
244. C₈×C₃₁	C₂×D₈	semidirect
245. #3151	C₂×C₃₁	semidirect
246. #3100	C₂×C₃₁	semidirect
247. C₂×#459	C₂×C₃₁	semidirect
248. D₄×M₁₆	C₂×C₃₁	semidirect
249. #1687	C₂×C₃₁	semidirect
250. #1685	C₂×C₃₁	semidirect
251. #1667	C₂×C₃₁	semidirect
252. #1665	C₂×C₃₁	semidirect
253. #1565	C₂×C₃₁	semidirect
254. #1547	C₂×C₃₁	semidirect
255. #1545	C₂×C₃₁	semidirect
293. #493	C₂²×C₃₁	semidirect
294. #492	C₂²×C₃₁	semidirect
296. #474	C₂²×C₃₁	semidirect
297. #474	C₂²×C₃₁	semidirect
302. #459	C₂²×C₃₁	semidirect
303. #459	C₂²×C₃₁	semidirect
307. #436	C₂²×C₃₁	semidirect
309. C₈×D₄	C₂²×C₃₁	semidirect
316. #335	C₂²×C₃₁	semidirect
317. #334	C₂²×C₃₁	semidirect
318. #332	C₂²×C₃₁	semidirect
319. #331	C₂²×C₃₁	semidirect
320. #331	C₂²×C₃₁	semidirect
321. #331	C₂²×C₃₁	semidirect
394. C₃₁	#19162	direct (N × H)
442. M₁₆	D₈×C₃₁	semidirect
497. D₄	C₃₁×AGL(1,ℤ/8)	semidirect
525. C₈	C₂×D₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3151	2
C₃₁×#3100	2
C₂×C₃₁×#459	2
D₄×M₁₆×C₃₁	2
C₃₁×#1687	2
C₃₁×#1685	2
C₃₁×#1667	2
C₃₁×#1665	2
C₃₁×#1565	2
C₃₁×#1547	2
C₃₁×#1545	2
#19162	31

Subgroups — C31×#19162 #21475921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19162 #21475921