Subgroups — C₃₁×#19164 #21475923

Order 7936.

706 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3165	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3101	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#461	C₂	1	1	yes	maximal, characteristic
4	D₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1693	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1691	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1673	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1671	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1550	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1549	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	—	2	1	no	—
13	C₂²×M₁₆×C₃₁	—	2	1	no	—
14	C₃₁×#533	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#501	—	2	2	no	—
18	C₃₁×#500	C₂²	1	2	yes	complemented
19	C₃₁×#498	C₂²	1	2	yes	complemented
20	C₃₁×#483	—	2	1	no	—
21	C₃₁×#476	C₂²	1	1	yes	characteristic
22	C₃₁×#476	C₂²	1	1	yes	characteristic
23	C₃₁×#461	—	2	1	no	—
24	C₃₁×#461	—	2	1	no	—
25	C₃₁×#461	C₂²	1	1	yes	characteristic
26	C₃₁×#461	C₂²	1	1	yes	characteristic
27	C₃₁×#461	C₂²	1	2	yes	—
28	C₃₁×#461	C₂²	1	2	yes	—
29	C₃₁×#450	—	2	1	no	—
30	C₄×SD₁₆×C₃₁	—	2	1	no	—
31	C₄×SD₁₆×C₃₁	—	2	1	no	—
32	C₃₁×#436	C₂²	1	2	yes	—
33	C₃₁×#435	—	2	2	no	—
34	C₈×D₄×C₃₁	C₂²	1	2	yes	—
35	C₃₁×#431	C₂²	1	1	yes	characteristic
36	C₃₁×#428	—	2	1	no	—
37	C₃₁×#428	—	2	1	no	—
38	C₃₁×#423	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#103	—	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×#333	C₂²	1	2	yes	complemented
46	C₃₁×#332	C₂²	1	2	yes	complemented
47	C₃₁×#332	C₂²	1	2	yes	complemented
48	C₃₁×#332	C₂²	1	2	yes	complemented
49	C₂²×Q₈×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	—	2	1	no	—
52	C₂×SD₁₆×C₃₁	—	2	1	no	—
53	C₂×SD₁₆×C₃₁	—	2	1	no	—
54	C₂×SD₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×SD₁₆×C₃₁	—	4	4	no	—
57	C₂×M₁₆×C₃₁	—	2	1	no	—
58	C₂×M₁₆×C₃₁	—	2	2	no	—
59	C₂×M₁₆×C₃₁	—	4	4	no	—
60	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
61	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₃₁	—	2	1	no	—
63	C₂²×C₈×C₃₁	—	2	2	no	—
64	C₃₁×C₄⋊Q₈	—	2	1	no	—
65	C₃₁×C₄⋊Q₈	—	2	1	no	—
66	C₃₁×C₄⋊Q₈	—	2	1	no	—
67	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
68	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
69	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
70	C₃₁×#125	—	2	1	no	—
71	C₃₁×C₂²⋊Q₈	—	2	1	no	—
72	C₃₁×C₂²⋊Q₈	—	2	1	no	—
73	C₄×Q₈×C₃₁	—	2	1	no	—
74	C₄×Q₈×C₃₁	—	2	1	no	—
75	C₄×D₄×C₃₁	—	2	1	no	—
76	C₄×D₄×C₃₁	—	2	1	no	—
77	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
78	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
79	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₃₁×#117	—	2	1	no	—
82	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
85	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
87	C₃₁×#107	—	2	2	no	—
88	C₃₁×#107	—	2	2	no	—
89	C₃₁×#106	—	2	2	no	—
90	C₃₁×#106	—	2	2	no	—
91	C₃₁×#106	—	2	1	no	—
92	C₃₁×#106	—	2	1	no	—
93	C₃₁×C₄⋊C₈	—	2	1	no	—
94	C₃₁×C₄⋊C₈	—	2	1	no	—
95	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
96	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
97	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
98	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
99	C₃₁×#103	—	2	2	no	—
100	C₃₁×#103	—	2	2	no	—
101	C₃₁×#103	—	2	1	no	—
102	C₃₁×#103	—	2	1	no	—
103	C₃₁×#102	—	2	1	no	—
104	C₃₁×#102	—	2	1	no	—
105	C₃₁×C₂²⋊C₈	—	2	1	no	—
106	C₃₁×C₂²⋊C₈	—	2	2	no	—
107	C₃₁×C₂²⋊C₈	—	2	2	no	—
108	C₃₁×#97	D₄	1	2	yes	—
109	C₄×C₈×C₃₁	—	2	1	no	—
110	C₄×C₈×C₃₁	C₂³	1	2	yes	—
111	C₂⁴×C₃₁	—	2	1	no	—
112	C₂×Q₈×C₃₁	—	2	1	no	—
113	C₂×Q₈×C₃₁	—	2	2	no	—
114	C₂×Q₈×C₃₁	—	2	1	no	—
115	C₂×Q₈×C₃₁	—	4	2	no	—
116	C₂×Q₈×C₃₁	—	4	2	no	—
117	C₂×Q₈×C₃₁	—	4	1	no	—
118	C₂×D₄×C₃₁	—	2	2	no	—
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	—	4	1	no	—
121	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	2	1	no	—
128	C₂²×C₄×C₃₁	—	4	2	no	—
129	C₂²×C₄×C₃₁	—	4	2	no	—
130	C₂²×C₄×C₃₁	—	4	1	no	—
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
134	SD₁₆×C₃₁	—	4	2	no	—
135	SD₁₆×C₃₁	—	4	4	no	—
136	SD₁₆×C₃₁	—	4	2	no	—
137	SD₁₆×C₃₁	—	4	4	no	—
138	M₁₆×C₃₁	—	2	2	no	—
139	M₁₆×C₃₁	—	4	2	no	—
140	M₁₆×C₃₁	—	4	4	no	—
141	M₁₆×C₃₁	SD₁₆	1	4	yes	complemented
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	1	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	4	2	no	—
147	C₂×C₈×C₃₁	—	4	4	no	—
148	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
149	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
150	C₄⋊C₄×C₃₁	—	2	1	no	—
151	C₄⋊C₄×C₃₁	—	2	1	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	2	1	no	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	—	2	1	no	—
157	C₄⋊C₄×C₃₁	—	4	1	no	—
158	C₄⋊C₄×C₃₁	—	4	1	no	—
159	C₄⋊C₄×C₃₁	—	4	1	no	—
160	C₄⋊C₄×C₃₁	—	4	1	no	—
161	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂²⋊C₄×C₃₁	—	2	1	no	—
164	C₂²⋊C₄×C₃₁	—	2	1	no	—
165	C₂²⋊C₄×C₃₁	—	2	1	no	—
166	C₂²⋊C₄×C₃₁	—	2	1	no	—
167	C₂²⋊C₄×C₃₁	—	4	1	no	—
168	C₄²×C₃₁	—	4	1	no	—
169	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
171	C₄²×C₃₁	#41	1	1	yes	characteristic
172	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
173	#19164	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂³×C₃₁	—	2	1	no	—
175	C₂³×C₃₁	—	2	1	no	—
176	C₂³×C₃₁	—	4	2	no	—
177	C₂³×C₃₁	—	4	2	no	—
178	C₂³×C₃₁	—	8	1	no	—
179	C₂³×C₃₁	#121	1	1	yes	characteristic
180	Q₈×C₃₁	—	4	2	no	—
181	Q₈×C₃₁	—	4	1	no	—
182	Q₈×C₃₁	—	4	1	no	—
183	Q₈×C₃₁	—	4	2	no	—
184	Q₈×C₃₁	—	8	1	no	—
185	Q₈×C₃₁	—	8	2	no	—
186	D₄×C₃₁	—	2	1	no	—
187	D₄×C₃₁	—	2	1	no	—
188	D₄×C₃₁	—	2	1	no	—
189	D₄×C₃₁	—	4	2	no	—
190	D₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	1	no	—
194	C₂×C₄×C₃₁	—	2	1	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	2	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	1	no	—
199	C₂×C₄×C₃₁	—	4	1	no	—
200	C₂×C₄×C₃₁	—	4	1	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	1	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	8	2	no	—
207	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
208	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
209	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
210	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
211	C₂×C₄×C₃₁	#137	1	2	yes	—
212	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
215	C₈×C₃₁	—	2	2	no	—
216	C₈×C₃₁	—	4	2	no	—
217	C₈×C₃₁	—	4	2	no	—
218	C₈×C₃₁	—	4	1	no	—
219	C₈×C₃₁	—	4	1	no	—
220	C₈×C₃₁	C₂×SD₁₆	1	4	yes	complemented
221	#3165	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#3101	C₂×C₃₁	1	1	yes	characteristic
223	C₂×#461	C₂×C₃₁	1	1	yes	characteristic
224	D₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
225	#1693	C₂×C₃₁	1	2	yes	complemented
226	#1691	C₂×C₃₁	1	2	yes	complemented
227	#1673	C₂×C₃₁	1	2	yes	complemented
228	#1671	C₂×C₃₁	1	2	yes	complemented
229	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1550	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#1549	C₂×C₃₁	1	1	yes	characteristic, complemented
232	C₂²×C₃₁	—	2	1	no	—
233	C₂²×C₃₁	—	2	1	no	—
234	C₂²×C₃₁	—	2	2	no	—
235	C₂²×C₃₁	—	4	1	no	—
236	C₂²×C₃₁	—	4	1	no	—
237	C₂²×C₃₁	—	4	1	no	—
238	C₂²×C₃₁	—	4	1	no	—
239	C₂²×C₃₁	—	4	1	no	—
240	C₂²×C₃₁	—	8	4	no	—
241	C₂²×C₃₁	#461	1	2	yes	—
242	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
243	C₄×C₃₁	—	2	1	no	—
244	C₄×C₃₁	—	2	1	no	—
245	C₄×C₃₁	—	2	1	no	—
246	C₄×C₃₁	—	4	1	no	—
247	C₄×C₃₁	—	4	1	no	—
248	C₄×C₃₁	—	8	2	no	—
249	C₄×C₃₁	—	8	1	no	—
250	C₄×C₃₁	—	8	1	no	—
251	C₄×C₃₁	#461	1	2	yes	—
252	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
253	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
254	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
255	C₄×C₃₁	#575	1	1	yes	characteristic
256	C₂²×SD₁₆	—	2	1	no	—
257	C₂²×M₁₆	—	2	1	no	—
258	#533	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
261	#501	—	2	2	no	—
262	#500	C₂²×C₃₁	1	2	yes	complemented
263	#498	C₂²×C₃₁	1	2	yes	complemented
264	#483	—	2	1	no	—
265	#476	C₂²×C₃₁	1	1	yes	characteristic
266	#476	C₂²×C₃₁	1	1	yes	characteristic
267	#461	—	2	1	no	—
268	#461	—	2	1	no	—
269	#461	C₂²×C₃₁	1	1	yes	characteristic
270	#461	C₂²×C₃₁	1	1	yes	characteristic
271	#461	C₂²×C₃₁	1	2	yes	—
272	#461	C₂²×C₃₁	1	2	yes	—
273	#450	—	2	1	no	—
274	C₄×SD₁₆	—	2	1	no	—
275	C₄×SD₁₆	—	2	1	no	—
276	#436	C₂²×C₃₁	1	2	yes	—
277	#435	—	2	2	no	—
278	C₈×D₄	C₂²×C₃₁	1	2	yes	—
279	#431	C₂²×C₃₁	1	1	yes	characteristic
280	#428	—	2	1	no	—
281	#428	—	2	1	no	—
282	#423	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
284	C₂×#103	—	2	1	no	—
285	#407	—	2	1	no	—
286	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
287	#335	C₂²×C₃₁	1	2	yes	complemented
288	#334	C₂²×C₃₁	1	2	yes	complemented
289	#333	C₂²×C₃₁	1	2	yes	complemented
290	#332	C₂²×C₃₁	1	2	yes	complemented
291	#332	C₂²×C₃₁	1	2	yes	complemented
292	#332	C₂²×C₃₁	1	2	yes	complemented
293	C₂×C₃₁	—	2	2	no	—
294	C₂×C₃₁	—	4	2	no	—
295	C₂×C₃₁	—	8	1	no	—
296	C₂×C₃₁	C₂×#461	1	1	yes	characteristic
297	C₂×C₃₁	#3055	1	1	yes	characteristic
298	C₂×C₃₁	#3069	1	1	yes	characteristic
299	C₂²×Q₈	—	2	1	no	—
300	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
301	C₂³×C₄	—	2	1	no	—
302	C₂×SD₁₆	—	2	2	no	—
303	C₂×SD₁₆	—	2	2	no	—
304	C₂×SD₁₆	—	2	1	no	—
305	C₂×SD₁₆	—	2	1	no	—
306	C₂×SD₁₆	—	4	4	no	—
307	C₂×M₁₆	—	2	1	no	—
308	C₂×M₁₆	—	2	2	no	—
309	C₂×M₁₆	—	4	4	no	—
310	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
311	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
312	C₂²×C₈	—	2	1	no	—
313	C₂²×C₈	—	2	2	no	—
314	C₄⋊Q₈	—	2	1	no	—
315	C₄⋊Q₈	—	2	1	no	—
316	C₄⋊Q₈	—	2	1	no	—
317	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
318	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
319	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
320	#125	—	2	1	no	—
321	C₂²⋊Q₈	—	2	1	no	—
322	C₂²⋊Q₈	—	2	1	no	—
323	C₄×Q₈	—	2	1	no	—
324	C₄×Q₈	—	2	1	no	—
325	C₄×D₄	—	2	1	no	—
326	C₄×D₄	—	2	1	no	—
327	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
328	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
329	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
330	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
331	#117	—	2	1	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
335	C₂×C₂²⋊C₄	—	2	1	no	—
336	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
337	#107	—	2	2	no	—
338	#107	—	2	2	no	—
339	#106	—	2	2	no	—
340	#106	—	2	2	no	—
341	#106	—	2	1	no	—
342	#106	—	2	1	no	—
343	C₄⋊C₈	—	2	1	no	—
344	C₄⋊C₈	—	2	1	no	—
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	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
349	#103	—	2	2	no	—
350	#103	—	2	2	no	—
351	#103	—	2	1	no	—
352	#103	—	2	1	no	—
353	#102	—	2	1	no	—
354	#102	—	2	1	no	—
355	C₂²⋊C₈	—	2	1	no	—
356	C₂²⋊C₈	—	2	2	no	—
357	C₂²⋊C₈	—	2	2	no	—
358	#97	D₄×C₃₁	1	2	yes	—
359	C₄×C₈	—	2	1	no	—
360	C₄×C₈	C₂³×C₃₁	1	2	yes	—
361	C₃₁	#19164	1	1	yes	characteristic, direct product factor, Sylow
362	C₂⁴	—	2	1	no	—
363	C₂×Q₈	—	2	1	no	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	—	4	2	no	—
367	C₂×Q₈	—	4	2	no	—
368	C₂×Q₈	—	4	1	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	—	4	1	no	—
372	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
373	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	4	2	no	—
380	C₂²×C₄	—	4	2	no	—
381	C₂²×C₄	—	4	1	no	—
382	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
383	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
385	SD₁₆	—	4	2	no	—
386	SD₁₆	—	4	4	no	—
387	SD₁₆	—	4	2	no	—
388	SD₁₆	—	4	4	no	—
389	M₁₆	—	2	2	no	—
390	M₁₆	—	4	2	no	—
391	M₁₆	—	4	4	no	—
392	M₁₆	SD₁₆×C₃₁	1	4	yes	complemented
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	4	2	no	—
398	C₂×C₈	—	4	4	no	—
399	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
400	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	4	1	no	—
409	C₄⋊C₄	—	4	1	no	—
410	C₄⋊C₄	—	4	1	no	—
411	C₄⋊C₄	—	4	1	no	—
412	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
413	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
414	C₂²⋊C₄	—	2	1	no	—
415	C₂²⋊C₄	—	2	1	no	—
416	C₂²⋊C₄	—	2	1	no	—
417	C₂²⋊C₄	—	2	1	no	—
418	C₂²⋊C₄	—	4	1	no	—
419	C₄²	—	4	1	no	—
420	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
421	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
422	C₄²	#41×C₃₁	1	1	yes	characteristic
423	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
424	C₂³	—	2	1	no	—
425	C₂³	—	2	1	no	—
426	C₂³	—	4	2	no	—
427	C₂³	—	4	2	no	—
428	C₂³	—	8	1	no	—
429	C₂³	C₃₁×#121	1	1	yes	characteristic
430	Q₈	—	4	2	no	—
431	Q₈	—	4	1	no	—
432	Q₈	—	4	1	no	—
433	Q₈	—	4	2	no	—
434	Q₈	—	8	1	no	—
435	Q₈	—	8	2	no	—
436	D₄	—	2	1	no	—
437	D₄	—	2	1	no	—
438	D₄	—	2	1	no	—
439	D₄	—	4	2	no	—
440	D₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	2	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×SD₁₆×C₃₁	1	2	yes	—
461	C₂×C₄	C₃₁×#137	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	2	no	—
467	C₈	—	4	2	no	—
468	C₈	—	4	1	no	—
469	C₈	—	4	1	no	—
470	C₈	C₂×SD₁₆×C₃₁	1	4	yes	complemented
471	C₂²	—	2	1	no	—
472	C₂²	—	2	1	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	4	1	no	—
475	C₂²	—	4	1	no	—
476	C₂²	—	4	1	no	—
477	C₂²	—	4	1	no	—
478	C₂²	—	4	1	no	—
479	C₂²	—	8	4	no	—
480	C₂²	C₃₁×#461	1	2	yes	—
481	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
482	C₄	—	2	1	no	—
483	C₄	—	2	1	no	—
484	C₄	—	2	1	no	—
485	C₄	—	4	1	no	—
486	C₄	—	4	1	no	—
487	C₄	—	8	2	no	—
488	C₄	—	8	1	no	—
489	C₄	—	8	1	no	—
490	C₄	C₃₁×#461	1	2	yes	—
491	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
492	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
493	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
494	C₄	C₃₁×#575	1	1	yes	characteristic
495	C₂	—	2	2	no	—
496	C₂	—	4	2	no	—
497	C₂	—	8	1	no	—
498	C₂	C₂×C₃₁×#461	1	1	yes	characteristic
499	C₂	C₃₁×#3055	1	1	yes	characteristic
500	C₂	C₃₁×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	D₄
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×#3165	C₂	semidirect
5. C₃₁×#1693	C₂	semidirect
6. C₃₁×#1691	C₂	semidirect
7. C₃₁×#1673	C₂	semidirect
8. C₃₁×#1671	C₂	semidirect
9. C₃₁×#1565	C₂	semidirect
10. C₃₁×#1550	C₂	semidirect
11. C₃₁×#1549	C₂	semidirect
18. C₃₁×#500	C₂²	semidirect
19. C₃₁×#498	C₂²	semidirect
43. C₃₁×#335	C₂²	semidirect
44. C₃₁×#334	C₂²	semidirect
45. C₃₁×#333	C₂²	semidirect
46. C₃₁×#332	C₂²	semidirect
47. C₃₁×#332	C₂²	semidirect
48. C₃₁×#332	C₂²	semidirect
141. M₁₆×C₃₁	SD₁₆	semidirect
173. #19164	C₃₁	direct (N × H)
220. C₈×C₃₁	C₂×SD₁₆	semidirect
221. #3165	C₂×C₃₁	semidirect
225. #1693	C₂×C₃₁	semidirect
226. #1691	C₂×C₃₁	semidirect
227. #1673	C₂×C₃₁	semidirect
228. #1671	C₂×C₃₁	semidirect
229. #1565	C₂×C₃₁	semidirect
230. #1550	C₂×C₃₁	semidirect
231. #1549	C₂×C₃₁	semidirect
262. #500	C₂²×C₃₁	semidirect
263. #498	C₂²×C₃₁	semidirect
287. #335	C₂²×C₃₁	semidirect
288. #334	C₂²×C₃₁	semidirect
289. #333	C₂²×C₃₁	semidirect
290. #332	C₂²×C₃₁	semidirect
291. #332	C₂²×C₃₁	semidirect
292. #332	C₂²×C₃₁	semidirect
361. C₃₁	#19164	direct (N × H)
392. M₁₆	SD₁₆×C₃₁	semidirect
470. C₈	C₂×SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3165	2
C₃₁×#3101	2
C₂×C₃₁×#461	2
D₄×M₁₆×C₃₁	2
C₃₁×#1693	2
C₃₁×#1691	2
C₃₁×#1673	2
C₃₁×#1671	2
C₃₁×#1565	2
C₃₁×#1550	2
C₃₁×#1549	2
#19164	31

Subgroups — C31×#19164 #21475923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19164 #21475923