Subgroups — C₃₁×#19230 #21475989

Order 7936.

790 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3245	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3151	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2935	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×#1540	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#331	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	C₂×C₃₁×#131	—	2	1	no	—
15	C₃₁×#532	C₂²	1	1	yes	characteristic
16	C₃₁×#532	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
18	C₃₁×#496	—	2	2	no	—
19	C₃₁×#493	—	2	1	no	—
20	C₃₁×#493	C₂²	1	2	yes	—
21	C₃₁×#492	C₂²	1	2	yes	—
22	C₃₁×#486	—	2	1	no	—
23	C₃₁×#474	—	2	1	no	—
24	C₃₁×#474	C₂²	1	2	yes	complemented
25	C₃₁×#466	—	2	2	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	2	yes	complemented
30	C₃₁×#447	—	2	2	no	—
31	C₃₁×#436	C₂²	1	2	yes	complemented
32	C₃₁×#435	—	2	2	no	—
33	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
34	C₃₁×#431	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₃₁×#408	—	2	1	no	—
38	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#335	C₂²	1	2	yes	complemented
40	C₃₁×#334	C₂²	1	2	yes	complemented
41	C₃₁×#332	C₂²	1	1	yes	characteristic, complemented
42	C₃₁×#332	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×#331	C₂²	1	2	yes	—
44	C₃₁×#331	C₂²	1	2	yes	—
45	C₃₁×#331	C₂²	1	1	yes	characteristic, complemented
46	C₃₁×#331	C₂²	1	1	yes	characteristic, complemented
47	C₂×#41×C₃₁	—	2	1	no	—
48	C₂×#41×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
53	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
54	C₂×SD₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×D₈×C₃₁	—	2	2	no	—
57	C₂×D₈×C₃₁	—	2	2	no	—
58	C₂×D₈×C₃₁	—	2	2	no	—
59	C₂×D₈×C₃₁	—	2	2	no	—
60	C₂×D₈×C₃₁	—	2	2	no	—
61	C₃₁×#131	—	4	4	no	—
62	C₂×M₁₆×C₃₁	—	2	2	no	—
63	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
64	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
65	C₂²×C₈×C₃₁	—	2	1	no	—
66	C₂²×C₈×C₃₁	—	2	1	no	—
67	C₂²×C₈×C₃₁	—	2	2	no	—
68	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
69	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
70	C₃₁×#127	—	2	1	no	—
71	C₃₁×#127	—	2	1	no	—
72	C₃₁×#127	—	2	1	no	—
73	C₃₁×#127	D₄	1	1	yes	characteristic
74	C₃₁×#127	C₂³	1	2	yes	—
75	C₃₁×#127	C₂³	1	1	yes	characteristic
76	C₃₁×#124	—	2	1	no	—
77	C₃₁×#124	—	2	1	no	—
78	C₃₁×#121	—	2	1	no	—
79	C₃₁×#121	—	2	1	no	—
80	C₃₁×#121	—	2	1	no	—
81	C₃₁×#121	D₄	1	2	yes	—
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	C₂³	1	2	yes	—
86	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
87	C₃₁×#107	—	2	2	no	—
88	C₃₁×C₄⋊C₈	—	2	1	no	—
89	C₃₁×C₄⋊C₈	—	2	1	no	—
90	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
91	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
92	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
93	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
94	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
95	C₃₁×#102	—	2	2	no	—
96	C₃₁×#102	—	2	2	no	—
97	C₃₁×#102	—	2	2	no	—
98	C₃₁×#102	—	2	2	no	—
99	C₃₁×C₂²⋊C₈	—	2	1	no	—
100	C₃₁×C₂²⋊C₈	—	2	2	no	—
101	C₃₁×C₂²⋊C₈	—	2	2	no	—
102	C₃₁×#97	D₄	1	2	yes	—
103	C₄×C₈×C₃₁	—	2	1	no	—
104	C₄×C₈×C₃₁	C₂³	1	2	yes	—
105	C₂⁴×C₃₁	—	4	1	no	—
106	#41×C₃₁	—	4	2	no	—
107	#41×C₃₁	—	4	2	no	—
108	#41×C₃₁	—	4	2	no	—
109	#41×C₃₁	—	4	2	no	—
110	C₂×Q₈×C₃₁	—	2	1	no	—
111	C₂×Q₈×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
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	2	no	—
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	—	2	2	no	—
121	C₂×D₄×C₃₁	—	4	1	no	—
122	C₂×D₄×C₃₁	—	4	1	no	—
123	C₂×D₄×C₃₁	—	4	1	no	—
124	C₂×D₄×C₃₁	—	4	1	no	—
125	C₂×D₄×C₃₁	—	4	2	no	—
126	C₂×D₄×C₃₁	—	4	2	no	—
127	C₂×D₄×C₃₁	—	8	1	no	—
128	C₂×D₄×C₃₁	—	8	1	no	—
129	C₂×D₄×C₃₁	—	8	2	no	—
130	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₃₁	—	2	2	no	—
132	C₂²×C₄×C₃₁	—	4	1	no	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
136	SD₁₆×C₃₁	—	4	4	no	—
137	SD₁₆×C₃₁	—	4	4	no	—
138	D₈×C₃₁	—	4	4	no	—
139	D₈×C₃₁	—	4	4	no	—
140	D₈×C₃₁	—	8	2	no	—
141	D₈×C₃₁	—	8	2	no	—
142	D₈×C₃₁	—	8	2	no	—
143	M₁₆×C₃₁	—	2	2	no	—
144	M₁₆×C₃₁	—	4	4	no	—
145	M₁₆×C₃₁	D₈	1	4	yes	complemented
146	C₂×C₈×C₃₁	—	2	1	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	2	1	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	4	2	no	—
152	C₂×C₈×C₃₁	—	4	4	no	—
153	C₂×C₈×C₃₁	—	8	1	no	—
154	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
155	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
156	C₄⋊C₄×C₃₁	—	2	1	no	—
157	C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
160	C₂²⋊C₄×C₃₁	—	2	2	no	—
161	C₂²⋊C₄×C₃₁	—	2	2	no	—
162	C₂²⋊C₄×C₃₁	—	4	1	no	—
163	C₂²⋊C₄×C₃₁	—	4	1	no	—
164	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₄²×C₃₁	#41	1	1	yes	characteristic
167	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
168	#19230	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂³×C₃₁	—	2	2	no	—
170	C₂³×C₃₁	—	4	1	no	—
171	C₂³×C₃₁	—	4	2	no	—
172	C₂³×C₃₁	—	8	2	no	—
173	C₂³×C₃₁	—	8	2	no	—
174	C₂³×C₃₁	—	8	2	no	—
175	C₂³×C₃₁	#121	1	1	yes	characteristic
176	Q₈×C₃₁	—	4	2	no	—
177	Q₈×C₃₁	—	4	1	no	—
178	Q₈×C₃₁	—	4	1	no	—
179	D₄×C₃₁	—	2	2	no	—
180	D₄×C₃₁	—	2	2	no	—
181	D₄×C₃₁	—	4	4	no	—
182	D₄×C₃₁	—	4	1	no	—
183	D₄×C₃₁	—	4	2	no	—
184	D₄×C₃₁	—	4	1	no	—
185	D₄×C₃₁	—	4	2	no	—
186	D₄×C₃₁	—	4	1	no	—
187	D₄×C₃₁	—	4	2	no	—
188	D₄×C₃₁	—	4	2	no	—
189	D₄×C₃₁	—	4	1	no	—
190	D₄×C₃₁	—	4	2	no	—
191	D₄×C₃₁	—	8	1	no	—
192	D₄×C₃₁	—	8	1	no	—
193	D₄×C₃₁	—	8	2	no	—
194	D₄×C₃₁	—	8	2	no	—
195	D₄×C₃₁	—	8	2	no	—
196	D₄×C₃₁	—	8	2	no	—
197	D₄×C₃₁	—	8	2	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	4	1	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	1	no	—
205	C₂×C₄×C₃₁	—	8	2	no	—
206	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
207	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
208	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
209	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
210	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
211	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
214	C₈×C₃₁	—	2	2	no	—
215	C₈×C₃₁	—	4	1	no	—
216	C₈×C₃₁	—	4	1	no	—
217	C₈×C₃₁	—	4	2	no	—
218	C₈×C₃₁	—	4	1	no	—
219	C₈×C₃₁	—	4	1	no	—
220	C₈×C₃₁	C₂×D₈	1	4	yes	—
221	#3245	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#3205	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#3151	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#2935	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#1687	C₂×C₃₁	1	2	yes	complemented
226	#1685	C₂×C₃₁	1	2	yes	complemented
227	#1667	C₂×C₃₁	1	2	yes	complemented
228	#1665	C₂×C₃₁	1	2	yes	complemented
229	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1540	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂×#331	C₂×C₃₁	1	1	yes	characteristic, complemented
232	C₂²×C₃₁	—	2	1	no	—
233	C₂²×C₃₁	—	2	1	no	—
234	C₂²×C₃₁	—	4	2	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	8	1	no	—
238	C₂²×C₃₁	—	8	2	no	—
239	C₂²×C₃₁	—	8	2	no	—
240	C₂²×C₃₁	—	8	1	no	—
241	C₂²×C₃₁	—	8	2	no	—
242	C₂²×C₃₁	—	8	1	no	—
243	C₂²×C₃₁	—	16	4	no	—
244	C₂²×C₃₁	#459	1	2	yes	—
245	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
246	C₄×C₃₁	—	2	1	no	—
247	C₄×C₃₁	—	2	1	no	—
248	C₄×C₃₁	—	2	1	no	—
249	C₄×C₃₁	—	4	2	no	—
250	C₄×C₃₁	—	8	1	no	—
251	C₄×C₃₁	#459	1	2	yes	—
252	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
253	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
254	C₄×C₃₁	#576	1	1	yes	characteristic
255	C₄×C₃₁	#577	1	1	yes	characteristic
256	C₂×AGL(1,ℤ/8)	—	2	1	no	—
257	C₂×AGL(1,ℤ/8)	—	2	1	no	—
258	C₂×#131	—	2	1	no	—
259	#532	C₂²×C₃₁	1	1	yes	characteristic
260	#532	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
262	#496	—	2	2	no	—
263	#493	—	2	1	no	—
264	#493	C₂²×C₃₁	1	2	yes	—
265	#492	C₂²×C₃₁	1	2	yes	—
266	#486	—	2	1	no	—
267	#474	—	2	1	no	—
268	#474	C₂²×C₃₁	1	2	yes	complemented
269	#466	—	2	2	no	—
270	#459	—	2	1	no	—
271	#459	C₂²×C₃₁	1	2	yes	—
272	#459	C₂²×C₃₁	1	2	yes	—
273	#459	C₂²×C₃₁	1	2	yes	complemented
274	#447	—	2	2	no	—
275	#436	C₂²×C₃₁	1	2	yes	complemented
276	#435	—	2	2	no	—
277	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
278	#431	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
281	#408	—	2	1	no	—
282	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
283	#335	C₂²×C₃₁	1	2	yes	complemented
284	#334	C₂²×C₃₁	1	2	yes	complemented
285	#332	C₂²×C₃₁	1	1	yes	characteristic, complemented
286	#332	C₂²×C₃₁	1	1	yes	characteristic, complemented
287	#331	C₂²×C₃₁	1	2	yes	—
288	#331	C₂²×C₃₁	1	2	yes	—
289	#331	C₂²×C₃₁	1	1	yes	characteristic, complemented
290	#331	C₂²×C₃₁	1	1	yes	characteristic, complemented
291	C₂×C₃₁	—	2	2	no	—
292	C₂×C₃₁	—	8	2	no	—
293	C₂×C₃₁	—	16	1	no	—
294	C₂×C₃₁	—	16	2	no	—
295	C₂×C₃₁	C₂×#459	1	1	yes	characteristic
296	C₂×C₃₁	#3062	1	1	yes	characteristic
297	C₂×C₃₁	#3068	1	1	yes	characteristic
298	C₂×#41	—	2	1	no	—
299	C₂×#41	—	2	1	no	—
300	C₂²×D₄	—	2	1	no	—
301	C₂²×D₄	—	2	1	no	—
302	C₂²×D₄	—	2	1	no	—
303	AGL(1,ℤ/8)	—	4	4	no	—
304	AGL(1,ℤ/8)	—	4	4	no	—
305	C₂×SD₁₆	—	2	2	no	—
306	C₂×SD₁₆	—	2	2	no	—
307	C₂×D₈	—	2	2	no	—
308	C₂×D₈	—	2	2	no	—
309	C₂×D₈	—	2	2	no	—
310	C₂×D₈	—	2	2	no	—
311	C₂×D₈	—	2	2	no	—
312	#131	—	4	4	no	—
313	C₂×M₁₆	—	2	2	no	—
314	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
315	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
316	C₂²×C₈	—	2	1	no	—
317	C₂²×C₈	—	2	1	no	—
318	C₂²×C₈	—	2	2	no	—
319	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
320	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
321	#127	—	2	1	no	—
322	#127	—	2	1	no	—
323	#127	—	2	1	no	—
324	#127	D₄×C₃₁	1	1	yes	characteristic
325	#127	C₂³×C₃₁	1	2	yes	—
326	#127	C₂³×C₃₁	1	1	yes	characteristic
327	#124	—	2	1	no	—
328	#124	—	2	1	no	—
329	#121	—	2	1	no	—
330	#121	—	2	1	no	—
331	#121	—	2	1	no	—
332	#121	D₄×C₃₁	1	2	yes	—
333	C₄×D₄	—	2	1	no	—
334	C₄×D₄	—	2	1	no	—
335	C₄×D₄	—	2	1	no	—
336	C₄×D₄	C₂³×C₃₁	1	2	yes	—
337	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
338	#107	—	2	2	no	—
339	C₄⋊C₈	—	2	1	no	—
340	C₄⋊C₈	—	2	1	no	—
341	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
342	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
343	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
344	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
345	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
346	#102	—	2	2	no	—
347	#102	—	2	2	no	—
348	#102	—	2	2	no	—
349	#102	—	2	2	no	—
350	C₂²⋊C₈	—	2	1	no	—
351	C₂²⋊C₈	—	2	2	no	—
352	C₂²⋊C₈	—	2	2	no	—
353	#97	D₄×C₃₁	1	2	yes	—
354	C₄×C₈	—	2	1	no	—
355	C₄×C₈	C₂³×C₃₁	1	2	yes	—
356	C₃₁	#19230	1	1	yes	characteristic, direct product factor, Sylow
357	C₂⁴	—	4	1	no	—
358	#41	—	4	2	no	—
359	#41	—	4	2	no	—
360	#41	—	4	2	no	—
361	#41	—	4	2	no	—
362	C₂×Q₈	—	2	1	no	—
363	C₂×Q₈	—	2	1	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	4	1	no	—
374	C₂×D₄	—	4	1	no	—
375	C₂×D₄	—	4	1	no	—
376	C₂×D₄	—	4	1	no	—
377	C₂×D₄	—	4	2	no	—
378	C₂×D₄	—	4	2	no	—
379	C₂×D₄	—	8	1	no	—
380	C₂×D₄	—	8	1	no	—
381	C₂×D₄	—	8	2	no	—
382	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	4	1	no	—
385	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
386	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
387	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
388	SD₁₆	—	4	4	no	—
389	SD₁₆	—	4	4	no	—
390	D₈	—	4	4	no	—
391	D₈	—	4	4	no	—
392	D₈	—	8	2	no	—
393	D₈	—	8	2	no	—
394	D₈	—	8	2	no	—
395	M₁₆	—	2	2	no	—
396	M₁₆	—	4	4	no	—
397	M₁₆	D₈×C₃₁	1	4	yes	complemented
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	4	2	no	—
404	C₂×C₈	—	4	4	no	—
405	C₂×C₈	—	8	1	no	—
406	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
407	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₂²⋊C₄	—	2	2	no	—
414	C₂²⋊C₄	—	4	1	no	—
415	C₂²⋊C₄	—	4	1	no	—
416	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
417	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
418	C₄²	#41×C₃₁	1	1	yes	characteristic
419	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
420	C₂³	—	2	2	no	—
421	C₂³	—	4	1	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	8	2	no	—
424	C₂³	—	8	2	no	—
425	C₂³	—	8	2	no	—
426	C₂³	C₃₁×#121	1	1	yes	characteristic
427	Q₈	—	4	2	no	—
428	Q₈	—	4	1	no	—
429	Q₈	—	4	1	no	—
430	D₄	—	2	2	no	—
431	D₄	—	2	2	no	—
432	D₄	—	4	4	no	—
433	D₄	—	4	2	no	—
434	D₄	—	4	1	no	—
435	D₄	—	4	2	no	—
436	D₄	—	4	1	no	—
437	D₄	—	4	2	no	—
438	D₄	—	4	1	no	—
439	D₄	—	4	2	no	—
440	D₄	—	4	2	no	—
441	D₄	—	4	1	no	—
442	D₄	—	8	1	no	—
443	D₄	—	8	1	no	—
444	D₄	—	8	2	no	—
445	D₄	—	8	2	no	—
446	D₄	—	8	2	no	—
447	D₄	—	8	2	no	—
448	D₄	—	8	2	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	8	2	no	—
457	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
458	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
459	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
460	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
461	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
462	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
463	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
464	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
465	C₈	—	2	2	no	—
466	C₈	—	4	1	no	—
467	C₈	—	4	1	no	—
468	C₈	—	4	2	no	—
469	C₈	—	4	1	no	—
470	C₈	—	4	1	no	—
471	C₈	C₂×D₈×C₃₁	1	4	yes	—
472	C₂²	—	2	1	no	—
473	C₂²	—	2	1	no	—
474	C₂²	—	4	2	no	—
475	C₂²	—	4	2	no	—
476	C₂²	—	4	2	no	—
477	C₂²	—	8	1	no	—
478	C₂²	—	8	2	no	—
479	C₂²	—	8	2	no	—
480	C₂²	—	8	1	no	—
481	C₂²	—	8	2	no	—
482	C₂²	—	8	1	no	—
483	C₂²	—	16	4	no	—
484	C₂²	C₃₁×#459	1	2	yes	—
485	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
486	C₄	—	2	1	no	—
487	C₄	—	2	1	no	—
488	C₄	—	2	1	no	—
489	C₄	—	4	2	no	—
490	C₄	—	8	1	no	—
491	C₄	C₃₁×#459	1	2	yes	—
492	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
493	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
494	C₄	C₃₁×#576	1	1	yes	characteristic
495	C₄	C₃₁×#577	1	1	yes	characteristic
496	C₂	—	2	2	no	—
497	C₂	—	8	2	no	—
498	C₂	—	16	2	no	—
499	C₂	—	16	1	no	—
500	C₂	C₂×C₃₁×#459	1	1	yes	characteristic
501	C₂	C₃₁×#3062	1	1	yes	characteristic
502	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×#3245	C₂	semidirect
2. C₃₁×#3205	C₂	semidirect
3. C₃₁×#3151	C₂	semidirect
4. C₃₁×#2935	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×#1540	C₂	semidirect
11. C₂×C₃₁×#331	C₂	semidirect
24. C₃₁×#474	C₂²	semidirect
29. C₃₁×#459	C₂²	semidirect
31. C₃₁×#436	C₂²	semidirect
33. C₈×D₄×C₃₁	C₂²	semidirect
39. C₃₁×#335	C₂²	semidirect
40. C₃₁×#334	C₂²	semidirect
41. C₃₁×#332	C₂²	semidirect
42. C₃₁×#332	C₂²	semidirect
45. C₃₁×#331	C₂²	semidirect
46. C₃₁×#331	C₂²	semidirect
145. M₁₆×C₃₁	D₈	semidirect
168. #19230	C₃₁	direct (N × H)
221. #3245	C₂×C₃₁	semidirect
222. #3205	C₂×C₃₁	semidirect
223. #3151	C₂×C₃₁	semidirect
224. #2935	C₂×C₃₁	semidirect
225. #1687	C₂×C₃₁	semidirect
226. #1685	C₂×C₃₁	semidirect
227. #1667	C₂×C₃₁	semidirect
228. #1665	C₂×C₃₁	semidirect
229. #1565	C₂×C₃₁	semidirect
230. #1540	C₂×C₃₁	semidirect
231. C₂×#331	C₂×C₃₁	semidirect
268. #474	C₂²×C₃₁	semidirect
273. #459	C₂²×C₃₁	semidirect
275. #436	C₂²×C₃₁	semidirect
277. C₈×D₄	C₂²×C₃₁	semidirect
283. #335	C₂²×C₃₁	semidirect
284. #334	C₂²×C₃₁	semidirect
285. #332	C₂²×C₃₁	semidirect
286. #332	C₂²×C₃₁	semidirect
289. #331	C₂²×C₃₁	semidirect
290. #331	C₂²×C₃₁	semidirect
356. C₃₁	#19230	direct (N × H)
397. M₁₆	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×#3245	2
C₃₁×#3205	2
C₃₁×#3151	2
C₃₁×#2935	2
C₃₁×#1687	2
C₃₁×#1685	2
C₃₁×#1667	2
C₃₁×#1665	2
C₃₁×#1565	2
C₃₁×#1540	2
C₂×C₃₁×#331	2
#19230	31

Subgroups — C31×#19230 #21475989

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19230 #21475989