Subgroups — C₃₁×#19166 #21475925

Order 7936.

758 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3138	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3101	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#459	C₂	1	1	yes	maximal, characteristic
4	D₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1701	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1697	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1681	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1677	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×#1545	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	—	2	1	no	—
13	C₂²×M₁₆×C₃₁	—	2	1	no	—
14	C₃₁×#533	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#490	—	2	2	no	—
18	C₃₁×#486	C₂²	1	2	yes	complemented
19	C₃₁×#486	C₂²	1	2	yes	complemented
20	C₃₁×#483	—	2	1	no	—
21	C₃₁×#476	C₂²	1	1	yes	characteristic, complemented
22	C₃₁×#476	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#461	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×#461	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×#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	complemented
33	C₃₁×#435	—	2	2	no	—
34	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
35	C₃₁×#431	C₂²	1	1	yes	characteristic
36	C₃₁×#423	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₃₁×#418	—	2	1	no	—
39	C₃₁×#418	—	2	1	no	—
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×#333	C₂²	1	2	yes	complemented
46	C₃₁×#332	C₂²	1	2	yes	complemented
47	C₃₁×#331	C₂²	1	2	yes	complemented
48	C₃₁×#331	C₂²	1	2	yes	complemented
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₃₁	—	2	1	no	—
54	C₂×SD₁₆×C₃₁	—	2	1	no	—
55	C₂×SD₁₆×C₃₁	—	2	1	no	—
56	C₂×D₈×C₃₁	—	2	2	no	—
57	C₂×D₈×C₃₁	—	2	2	no	—
58	C₂×D₈×C₃₁	—	4	4	no	—
59	C₂×M₁₆×C₃₁	—	2	1	no	—
60	C₂×M₁₆×C₃₁	—	2	2	no	—
61	C₂×M₁₆×C₃₁	—	4	4	no	—
62	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
63	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	—	2	1	no	—
65	C₂²×C₈×C₃₁	—	2	2	no	—
66	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
67	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
68	C₃₁×#127	—	2	1	no	—
69	C₃₁×#127	—	2	1	no	—
70	C₃₁×#127	—	2	1	no	—
71	C₃₁×#127	C₂³	1	2	yes	—
72	C₃₁×#125	—	2	1	no	—
73	C₃₁×C₂²⋊Q₈	—	2	1	no	—
74	C₃₁×C₂²⋊Q₈	—	2	1	no	—
75	C₄×Q₈×C₃₁	—	2	1	no	—
76	C₄×Q₈×C₃₁	—	2	1	no	—
77	C₄×D₄×C₃₁	—	2	1	no	—
78	C₄×D₄×C₃₁	—	2	1	no	—
79	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
80	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
81	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×#117	—	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×#106	—	2	1	no	—
88	C₃₁×#106	—	2	1	no	—
89	C₃₁×C₄⋊C₈	—	2	1	no	—
90	C₃₁×C₄⋊C₈	—	2	1	no	—
91	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
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×#103	—	2	1	no	—
96	C₃₁×#103	—	2	1	no	—
97	C₃₁×#102	—	2	1	no	—
98	C₃₁×#102	—	2	1	no	—
99	C₃₁×#102	—	2	2	no	—
100	C₃₁×#102	—	2	2	no	—
101	C₃₁×#102	—	2	2	no	—
102	C₃₁×#102	—	2	2	no	—
103	C₃₁×#102	—	2	2	no	—
104	C₃₁×#102	—	2	2	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	C₂⁴×C₃₁	—	4	1	no	—
113	C₂×Q₈×C₃₁	—	2	1	no	—
114	C₂×Q₈×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	—	2	2	no	—
118	C₂×D₄×C₃₁	—	2	2	no	—
119	C₂×D₄×C₃₁	—	2	2	no	—
120	C₂×D₄×C₃₁	—	4	1	no	—
121	C₂×D₄×C₃₁	—	4	2	no	—
122	C₂×D₄×C₃₁	—	4	2	no	—
123	C₂×D₄×C₃₁	—	4	1	no	—
124	C₂×D₄×C₃₁	—	4	1	no	—
125	C₂×D₄×C₃₁	—	4	1	no	—
126	C₂×D₄×C₃₁	—	4	2	no	—
127	C₂×D₄×C₃₁	—	4	2	no	—
128	C₂×D₄×C₃₁	—	8	2	no	—
129	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	2	1	no	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	2	1	no	—
136	C₂²×C₄×C₃₁	—	4	2	no	—
137	C₂²×C₄×C₃₁	—	4	2	no	—
138	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
141	SD₁₆×C₃₁	—	4	2	no	—
142	SD₁₆×C₃₁	—	4	2	no	—
143	D₈×C₃₁	—	4	4	no	—
144	D₈×C₃₁	—	4	4	no	—
145	M₁₆×C₃₁	—	2	2	no	—
146	M₁₆×C₃₁	—	2	2	no	—
147	M₁₆×C₃₁	—	4	4	no	—
148	M₁₆×C₃₁	—	4	2	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	—	2	1	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₂×C₈×C₃₁	—	4	4	no	—
154	C₂×C₈×C₃₁	—	4	2	no	—
155	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
156	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
157	C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₄⋊C₄×C₃₁	—	2	1	no	—
160	C₄⋊C₄×C₃₁	—	2	1	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	—	4	1	no	—
163	C₄⋊C₄×C₃₁	—	4	1	no	—
164	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂²⋊C₄×C₃₁	—	2	1	no	—
167	C₂²⋊C₄×C₃₁	—	2	1	no	—
168	C₂²⋊C₄×C₃₁	—	2	1	no	—
169	C₂²⋊C₄×C₃₁	—	2	1	no	—
170	C₂²⋊C₄×C₃₁	—	4	1	no	—
171	C₄²×C₃₁	—	4	1	no	—
172	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₃₁	#41	1	1	yes	characteristic
175	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
176	#19166	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂³×C₃₁	—	2	1	no	—
178	C₂³×C₃₁	—	2	1	no	—
179	C₂³×C₃₁	—	4	2	no	—
180	C₂³×C₃₁	—	4	2	no	—
181	C₂³×C₃₁	—	4	2	no	—
182	C₂³×C₃₁	—	8	1	no	—
183	C₂³×C₃₁	—	8	2	no	—
184	C₂³×C₃₁	—	8	2	no	—
185	C₂³×C₃₁	—	8	2	no	—
186	C₂³×C₃₁	#121	1	1	yes	characteristic
187	Q₈×C₃₁	—	4	1	no	—
188	Q₈×C₃₁	—	4	1	no	—
189	Q₈×C₃₁	—	8	1	no	—
190	D₄×C₃₁	—	2	1	no	—
191	D₄×C₃₁	—	2	1	no	—
192	D₄×C₃₁	—	2	1	no	—
193	D₄×C₃₁	—	4	2	no	—
194	D₄×C₃₁	—	4	2	no	—
195	D₄×C₃₁	—	4	2	no	—
196	D₄×C₃₁	—	4	2	no	—
197	D₄×C₃₁	—	4	2	no	—
198	D₄×C₃₁	—	8	2	no	—
199	D₄×C₃₁	—	8	2	no	—
200	D₄×C₃₁	—	8	2	no	—
201	D₄×C₃₁	—	8	2	no	—
202	D₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	1	no	—
207	C₂×C₄×C₃₁	—	4	1	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	1	no	—
210	C₂×C₄×C₃₁	—	4	1	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	1	no	—
216	C₂×C₄×C₃₁	—	4	1	no	—
217	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
218	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
219	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
220	C₂×C₄×C₃₁	#135	1	2	yes	—
221	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
222	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
225	C₈×C₃₁	—	2	2	no	—
226	C₈×C₃₁	—	2	2	no	—
227	C₈×C₃₁	—	4	2	no	—
228	C₈×C₃₁	—	4	2	no	—
229	C₈×C₃₁	—	4	1	no	—
230	C₈×C₃₁	—	4	1	no	—
231	#3138	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#3101	C₂×C₃₁	1	1	yes	characteristic, complemented
233	C₂×#459	C₂×C₃₁	1	1	yes	characteristic
234	D₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#1701	C₂×C₃₁	1	2	yes	complemented
236	#1697	C₂×C₃₁	1	2	yes	complemented
237	#1681	C₂×C₃₁	1	2	yes	complemented
238	#1677	C₂×C₃₁	1	2	yes	complemented
239	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#1550	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#1545	C₂×C₃₁	1	1	yes	characteristic, complemented
242	C₂²×C₃₁	—	2	1	no	—
243	C₂²×C₃₁	—	2	1	no	—
244	C₂²×C₃₁	—	2	2	no	—
245	C₂²×C₃₁	—	4	1	no	—
246	C₂²×C₃₁	—	4	1	no	—
247	C₂²×C₃₁	—	4	1	no	—
248	C₂²×C₃₁	—	4	1	no	—
249	C₂²×C₃₁	—	4	1	no	—
250	C₂²×C₃₁	—	8	4	no	—
251	C₂²×C₃₁	—	8	2	no	—
252	C₂²×C₃₁	—	8	2	no	—
253	C₂²×C₃₁	—	8	2	no	—
254	C₂²×C₃₁	—	16	4	no	—
255	C₂²×C₃₁	#463	1	2	yes	—
256	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
257	C₄×C₃₁	—	2	1	no	—
258	C₄×C₃₁	—	2	1	no	—
259	C₄×C₃₁	—	2	1	no	—
260	C₄×C₃₁	—	4	1	no	—
261	C₄×C₃₁	—	4	1	no	—
262	C₄×C₃₁	—	8	1	no	—
263	C₄×C₃₁	—	8	1	no	—
264	C₄×C₃₁	#463	1	2	yes	—
265	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
266	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
267	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
268	C₄×C₃₁	#575	1	1	yes	characteristic
269	C₂²×D₈	—	2	1	no	—
270	C₂²×M₁₆	—	2	1	no	—
271	#533	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
274	#490	—	2	2	no	—
275	#486	C₂²×C₃₁	1	2	yes	complemented
276	#486	C₂²×C₃₁	1	2	yes	complemented
277	#483	—	2	1	no	—
278	#476	C₂²×C₃₁	1	1	yes	characteristic, complemented
279	#476	C₂²×C₃₁	1	1	yes	characteristic, complemented
280	#461	C₂²×C₃₁	1	1	yes	characteristic, complemented
281	#461	C₂²×C₃₁	1	1	yes	characteristic, complemented
282	#459	—	2	1	no	—
283	#459	—	2	1	no	—
284	#459	C₂²×C₃₁	1	2	yes	—
285	#459	C₂²×C₃₁	1	2	yes	—
286	#450	—	2	1	no	—
287	C₄×SD₁₆	—	2	1	no	—
288	C₄×SD₁₆	—	2	1	no	—
289	#436	C₂²×C₃₁	1	2	yes	complemented
290	#435	—	2	2	no	—
291	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
292	#431	C₂²×C₃₁	1	1	yes	characteristic
293	#423	C₂²×C₃₁	1	1	yes	characteristic
294	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
295	#418	—	2	1	no	—
296	#418	—	2	1	no	—
297	C₂×#102	—	2	1	no	—
298	#407	—	2	1	no	—
299	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
300	#335	C₂²×C₃₁	1	2	yes	complemented
301	#334	C₂²×C₃₁	1	2	yes	complemented
302	#333	C₂²×C₃₁	1	2	yes	complemented
303	#332	C₂²×C₃₁	1	2	yes	complemented
304	#331	C₂²×C₃₁	1	2	yes	complemented
305	#331	C₂²×C₃₁	1	2	yes	complemented
306	C₂×C₃₁	—	2	2	no	—
307	C₂×C₃₁	—	4	2	no	—
308	C₂×C₃₁	—	8	1	no	—
309	C₂×C₃₁	—	16	2	no	—
310	C₂×C₃₁	C₂×#463	1	1	yes	characteristic
311	C₂×C₃₁	#3055	1	1	yes	characteristic
312	C₂×C₃₁	#3068	1	1	yes	characteristic
313	C₂²×D₄	—	2	1	no	—
314	C₂²×D₄	—	2	1	no	—
315	C₂²×D₄	—	2	1	no	—
316	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
317	C₂³×C₄	—	2	1	no	—
318	C₂×SD₁₆	—	2	1	no	—
319	C₂×SD₁₆	—	2	1	no	—
320	C₂×D₈	—	2	2	no	—
321	C₂×D₈	—	2	2	no	—
322	C₂×D₈	—	4	4	no	—
323	C₂×M₁₆	—	2	1	no	—
324	C₂×M₁₆	—	2	2	no	—
325	C₂×M₁₆	—	4	4	no	—
326	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
327	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
328	C₂²×C₈	—	2	1	no	—
329	C₂²×C₈	—	2	2	no	—
330	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
331	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
332	#127	—	2	1	no	—
333	#127	—	2	1	no	—
334	#127	—	2	1	no	—
335	#127	C₂³×C₃₁	1	2	yes	—
336	#125	—	2	1	no	—
337	C₂²⋊Q₈	—	2	1	no	—
338	C₂²⋊Q₈	—	2	1	no	—
339	C₄×Q₈	—	2	1	no	—
340	C₄×Q₈	—	2	1	no	—
341	C₄×D₄	—	2	1	no	—
342	C₄×D₄	—	2	1	no	—
343	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
344	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
345	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
346	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
347	#117	—	2	1	no	—
348	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
349	C₂×C₂²⋊C₄	—	2	1	no	—
350	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
351	#106	—	2	1	no	—
352	#106	—	2	1	no	—
353	C₄⋊C₈	—	2	1	no	—
354	C₄⋊C₈	—	2	1	no	—
355	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
356	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
357	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
358	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
359	#103	—	2	1	no	—
360	#103	—	2	1	no	—
361	#102	—	2	1	no	—
362	#102	—	2	1	no	—
363	#102	—	2	2	no	—
364	#102	—	2	2	no	—
365	#102	—	2	2	no	—
366	#102	—	2	2	no	—
367	#102	—	2	2	no	—
368	#102	—	2	2	no	—
369	C₂²⋊C₈	—	2	1	no	—
370	C₂²⋊C₈	—	2	2	no	—
371	C₂²⋊C₈	—	2	2	no	—
372	#97	D₄×C₃₁	1	2	yes	—
373	C₄×C₈	—	2	1	no	—
374	C₄×C₈	C₂³×C₃₁	1	2	yes	—
375	C₃₁	#19166	1	1	yes	characteristic, direct product factor, Sylow
376	C₂⁴	—	2	1	no	—
377	C₂⁴	—	4	1	no	—
378	C₂×Q₈	—	2	1	no	—
379	C₂×Q₈	—	2	1	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	4	1	no	—
386	C₂×D₄	—	4	2	no	—
387	C₂×D₄	—	4	2	no	—
388	C₂×D₄	—	4	1	no	—
389	C₂×D₄	—	4	1	no	—
390	C₂×D₄	—	4	1	no	—
391	C₂×D₄	—	4	2	no	—
392	C₂×D₄	—	4	2	no	—
393	C₂×D₄	—	8	2	no	—
394	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
395	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	4	2	no	—
402	C₂²×C₄	—	4	2	no	—
403	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
404	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
405	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
406	SD₁₆	—	4	2	no	—
407	SD₁₆	—	4	2	no	—
408	D₈	—	4	4	no	—
409	D₈	—	4	4	no	—
410	M₁₆	—	2	2	no	—
411	M₁₆	—	2	2	no	—
412	M₁₆	—	4	4	no	—
413	M₁₆	—	4	2	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	1	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	4	4	no	—
419	C₂×C₈	—	4	2	no	—
420	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
421	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	4	1	no	—
428	C₄⋊C₄	—	4	1	no	—
429	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
430	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
431	C₂²⋊C₄	—	2	1	no	—
432	C₂²⋊C₄	—	2	1	no	—
433	C₂²⋊C₄	—	2	1	no	—
434	C₂²⋊C₄	—	2	1	no	—
435	C₂²⋊C₄	—	4	1	no	—
436	C₄²	—	4	1	no	—
437	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
438	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
439	C₄²	#41×C₃₁	1	1	yes	characteristic
440	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
441	C₂³	—	2	1	no	—
442	C₂³	—	2	1	no	—
443	C₂³	—	4	2	no	—
444	C₂³	—	4	2	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	8	1	no	—
447	C₂³	—	8	2	no	—
448	C₂³	—	8	2	no	—
449	C₂³	—	8	2	no	—
450	C₂³	C₃₁×#121	1	1	yes	characteristic
451	Q₈	—	4	1	no	—
452	Q₈	—	4	1	no	—
453	Q₈	—	8	1	no	—
454	D₄	—	2	1	no	—
455	D₄	—	2	1	no	—
456	D₄	—	2	1	no	—
457	D₄	—	4	2	no	—
458	D₄	—	4	2	no	—
459	D₄	—	4	2	no	—
460	D₄	—	4	2	no	—
461	D₄	—	4	2	no	—
462	D₄	—	8	2	no	—
463	D₄	—	8	2	no	—
464	D₄	—	8	2	no	—
465	D₄	—	8	2	no	—
466	D₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
482	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
483	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
484	C₂×C₄	C₃₁×#135	1	2	yes	—
485	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
486	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
487	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
488	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
489	C₈	—	2	2	no	—
490	C₈	—	2	2	no	—
491	C₈	—	4	2	no	—
492	C₈	—	4	2	no	—
493	C₈	—	4	1	no	—
494	C₈	—	4	1	no	—
495	C₂²	—	2	1	no	—
496	C₂²	—	2	1	no	—
497	C₂²	—	2	2	no	—
498	C₂²	—	4	1	no	—
499	C₂²	—	4	1	no	—
500	C₂²	—	4	1	no	—
501	C₂²	—	4	1	no	—
502	C₂²	—	4	1	no	—
503	C₂²	—	8	4	no	—
504	C₂²	—	8	2	no	—
505	C₂²	—	8	2	no	—
506	C₂²	—	8	2	no	—
507	C₂²	—	16	4	no	—
508	C₂²	C₃₁×#463	1	2	yes	—
509	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
510	C₄	—	2	1	no	—
511	C₄	—	2	1	no	—
512	C₄	—	2	1	no	—
513	C₄	—	4	1	no	—
514	C₄	—	4	1	no	—
515	C₄	—	8	1	no	—
516	C₄	—	8	1	no	—
517	C₄	C₃₁×#463	1	2	yes	—
518	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
519	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
520	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
521	C₄	C₃₁×#575	1	1	yes	characteristic
522	C₂	—	2	2	no	—
523	C₂	—	4	2	no	—
524	C₂	—	8	1	no	—
525	C₂	—	16	2	no	—
526	C₂	C₂×C₃₁×#463	1	1	yes	characteristic
527	C₂	C₃₁×#3055	1	1	yes	characteristic
528	C₂	C₃₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#459	C₂
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×#3138	C₂	semidirect
2. C₃₁×#3101	C₂	semidirect
4. D₄×M₁₆×C₃₁	C₂	semidirect
5. C₃₁×#1701	C₂	semidirect
6. C₃₁×#1697	C₂	semidirect
7. C₃₁×#1681	C₂	semidirect
8. C₃₁×#1677	C₂	semidirect
9. C₃₁×#1565	C₂	semidirect
10. C₃₁×#1550	C₂	semidirect
11. C₃₁×#1545	C₂	semidirect
18. C₃₁×#486	C₂²	semidirect
19. C₃₁×#486	C₂²	semidirect
21. C₃₁×#476	C₂²	semidirect
22. C₃₁×#476	C₂²	semidirect
23. C₃₁×#461	C₂²	semidirect
24. C₃₁×#461	C₂²	semidirect
32. C₃₁×#436	C₂²	semidirect
34. C₈×D₄×C₃₁	C₂²	semidirect
43. C₃₁×#335	C₂²	semidirect
44. C₃₁×#334	C₂²	semidirect
45. C₃₁×#333	C₂²	semidirect
46. C₃₁×#332	C₂²	semidirect
47. C₃₁×#331	C₂²	semidirect
48. C₃₁×#331	C₂²	semidirect
176. #19166	C₃₁	direct (N × H)
231. #3138	C₂×C₃₁	semidirect
232. #3101	C₂×C₃₁	semidirect
234. D₄×M₁₆	C₂×C₃₁	semidirect
235. #1701	C₂×C₃₁	semidirect
236. #1697	C₂×C₃₁	semidirect
237. #1681	C₂×C₃₁	semidirect
238. #1677	C₂×C₃₁	semidirect
239. #1565	C₂×C₃₁	semidirect
240. #1550	C₂×C₃₁	semidirect
241. #1545	C₂×C₃₁	semidirect
275. #486	C₂²×C₃₁	semidirect
276. #486	C₂²×C₃₁	semidirect
278. #476	C₂²×C₃₁	semidirect
279. #476	C₂²×C₃₁	semidirect
280. #461	C₂²×C₃₁	semidirect
281. #461	C₂²×C₃₁	semidirect
289. #436	C₂²×C₃₁	semidirect
291. C₈×D₄	C₂²×C₃₁	semidirect
300. #335	C₂²×C₃₁	semidirect
301. #334	C₂²×C₃₁	semidirect
302. #333	C₂²×C₃₁	semidirect
303. #332	C₂²×C₃₁	semidirect
304. #331	C₂²×C₃₁	semidirect
305. #331	C₂²×C₃₁	semidirect
375. C₃₁	#19166	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×#3138	2
C₃₁×#3101	2
C₂×C₃₁×#459	2
D₄×M₁₆×C₃₁	2
C₃₁×#1701	2
C₃₁×#1697	2
C₃₁×#1681	2
C₃₁×#1677	2
C₃₁×#1565	2
C₃₁×#1550	2
C₃₁×#1545	2
#19166	31

Subgroups — C31×#19166 #21475925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19166 #21475925