Subgroups — C₃₁×#33048 #21489807

Order 7936.

1054 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2857	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2856	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1936	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1895	C₂	1	4	yes	maximal, complemented
9	C₃₁×#1834	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
14	C₂×C₃₁×#121	—	2	1	no	—
15	C₂×C₃₁×#121	—	2	1	no	—
16	C₂×C₃₁×#121	C₂²	1	2	yes	—
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#486	—	2	4	no	—
19	C₂×C₃₁×#102	C₂²	1	4	yes	—
20	C₂×C₃₁×#102	C₂²	1	4	yes	—
21	C₄×M₁₆×C₃₁	—	2	4	no	—
22	C₂×C₃₁×#97	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#391	—	2	1	no	—
26	C₃₁×#390	—	2	1	no	—
27	C₃₁×#390	—	2	1	no	—
28	C₃₁×#390	C₂²	1	2	yes	—
29	C₃₁×#389	—	2	1	no	—
30	C₃₁×#389	C₂²	1	2	yes	—
31	C₃₁×#384	C₂²	1	1	yes	characteristic
32	C₃₁×#384	C₂²	1	1	yes	characteristic
33	C₃₁×#380	—	2	1	no	—
34	C₃₁×#380	C₂²	1	2	yes	—
35	C₃₁×#340	C₂²	1	4	yes	—
36	C₃₁×#340	C₂²	1	4	yes	—
37	C₂²×Q₈×C₃₁	—	2	1	no	—
38	C₂²×D₄×C₃₁	—	2	2	no	—
39	C₂²×D₄×C₃₁	—	2	1	no	—
40	C₂²×D₄×C₃₁	—	2	1	no	—
41	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
42	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₃₁	—	2	4	no	—
46	C₂×M₁₆×C₃₁	—	2	4	no	—
47	C₂×M₁₆×C₃₁	—	2	4	no	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₃₁×C₄⋊Q₈	—	2	2	no	—
51	C₃₁×C₄⋊Q₈	—	2	2	no	—
52	C₃₁×#127	—	2	2	no	—
53	C₃₁×#127	—	2	2	no	—
54	C₃₁×C₂²⋊Q₈	—	4	4	no	—
55	C₃₁×#121	—	2	8	no	—
56	C₃₁×#121	—	4	4	no	—
57	C₃₁×#121	—	4	4	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₄²×C₃₁	—	2	2	no	—
72	C₂×C₄²×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#102	—	2	8	no	—
78	C₃₁×#102	—	2	8	no	—
79	C₃₁×#97	—	2	4	no	—
80	C₄×C₈×C₃₁	—	2	2	no	—
81	C₄×C₈×C₃₁	D₄	1	4	yes	complemented
82	C₃₁×#95	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₂⁴×C₃₁	—	2	2	no	—
85	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂×Q₈×C₃₁	—	4	2	no	—
87	C₂×Q₈×C₃₁	—	4	2	no	—
88	C₂×Q₈×C₃₁	—	4	2	no	—
89	C₂×D₄×C₃₁	—	2	2	no	—
90	C₂×D₄×C₃₁	—	2	4	no	—
91	C₂×D₄×C₃₁	—	2	4	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	4	4	no	—
94	C₂×D₄×C₃₁	—	4	4	no	—
95	C₂×D₄×C₃₁	—	4	4	no	—
96	C₂×D₄×C₃₁	—	4	2	no	—
97	C₂×D₄×C₃₁	—	4	2	no	—
98	C₂×D₄×C₃₁	—	4	2	no	—
99	C₂×D₄×C₃₁	—	4	2	no	—
100	C₂×D₄×C₃₁	—	4	2	no	—
101	C₂×D₄×C₃₁	—	4	2	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	#41	1	4	yes	—
120	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
121	M₁₆×C₃₁	—	2	16	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
127	C₂×C₈×C₃₁	#41	1	4	yes	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	4	no	—
130	C₄⋊C₄×C₃₁	—	2	4	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	4	4	no	—
133	C₄⋊C₄×C₃₁	—	4	2	no	—
134	C₄⋊C₄×C₃₁	—	4	2	no	—
135	C₄⋊C₄×C₃₁	—	4	2	no	—
136	C₄⋊C₄×C₃₁	—	4	2	no	—
137	C₄⋊C₄×C₃₁	—	4	2	no	—
138	C₂²⋊C₄×C₃₁	—	4	4	no	—
139	C₂²⋊C₄×C₃₁	—	4	4	no	—
140	C₂²⋊C₄×C₃₁	—	4	4	no	—
141	C₂²⋊C₄×C₃₁	—	4	4	no	—
142	C₄²×C₃₁	—	2	2	no	—
143	C₄²×C₃₁	—	2	2	no	—
144	C₄²×C₃₁	—	2	1	no	—
145	C₄²×C₃₁	—	2	1	no	—
146	C₄²×C₃₁	C₂×D₄	1	2	yes	—
147	C₄²×C₃₁	#41	1	2	yes	—
148	#33048	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂³×C₃₁	—	2	1	no	—
150	C₂³×C₃₁	—	2	1	no	—
151	C₂³×C₃₁	—	2	1	no	—
152	C₂³×C₃₁	—	2	1	no	—
153	C₂³×C₃₁	—	2	1	no	—
154	C₂³×C₃₁	—	2	1	no	—
155	C₂³×C₃₁	—	4	2	no	—
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	—	4	2	no	—
158	C₂³×C₃₁	—	4	2	no	—
159	C₂³×C₃₁	—	4	2	no	—
160	C₂³×C₃₁	—	4	2	no	—
161	C₂³×C₃₁	—	4	2	no	—
162	C₂³×C₃₁	#124	1	2	yes	—
163	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
164	Q₈×C₃₁	—	8	4	no	—
165	D₄×C₃₁	—	4	4	no	—
166	D₄×C₃₁	—	4	4	no	—
167	D₄×C₃₁	—	8	8	no	—
168	D₄×C₃₁	—	8	4	no	—
169	D₄×C₃₁	—	8	4	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	4	no	—
173	C₂×C₄×C₃₁	—	2	4	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	4	4	no	—
181	C₂×C₄×C₃₁	—	4	4	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	#121	1	2	yes	—
192	C₂×C₄×C₃₁	#121	1	2	yes	—
193	C₂×C₄×C₃₁	#124	1	2	yes	—
194	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
195	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
196	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
197	C₂×C₄×C₃₁	#137	1	2	yes	—
198	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
200	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
201	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
202	C₈×C₃₁	—	2	4	no	—
203	C₈×C₃₁	—	2	4	no	—
204	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
205	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#2857	C₂×C₃₁	1	1	yes	characteristic, complemented
207	#2856	C₂×C₃₁	1	1	yes	characteristic
208	#1990	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#1957	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#1936	C₂×C₃₁	1	4	yes	complemented
211	#1895	C₂×C₃₁	1	4	yes	complemented
212	#1834	C₂×C₃₁	1	1	yes	characteristic, complemented
213	C₂²×C₃₁	—	2	2	no	—
214	C₂²×C₃₁	—	2	2	no	—
215	C₂²×C₃₁	—	2	2	no	—
216	C₂²×C₃₁	—	4	1	no	—
217	C₂²×C₃₁	—	4	1	no	—
218	C₂²×C₃₁	—	4	1	no	—
219	C₂²×C₃₁	—	4	1	no	—
220	C₂²×C₃₁	—	8	2	no	—
221	C₂²×C₃₁	—	8	2	no	—
222	C₂²×C₃₁	—	8	2	no	—
223	C₂²×C₃₁	—	8	2	no	—
224	C₂²×C₃₁	—	8	2	no	—
225	C₂²×C₃₁	—	8	2	no	—
226	C₂²×C₃₁	—	8	2	no	—
227	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
228	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
229	C₂²×C₃₁	#532	1	1	yes	characteristic
230	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
231	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
232	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
233	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
234	C₄×C₃₁	—	2	2	no	—
235	C₄×C₃₁	—	4	1	no	—
236	C₄×C₃₁	—	4	1	no	—
237	C₄×C₃₁	—	8	2	no	—
238	C₄×C₃₁	—	8	2	no	—
239	C₄×C₃₁	—	8	2	no	—
240	C₄×C₃₁	#468	1	4	yes	—
241	C₄×C₃₁	#469	1	4	yes	—
242	C₄×C₃₁	#525	1	2	yes	—
243	C₄×C₃₁	#529	1	2	yes	—
244	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
245	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×C₂²⋊Q₈	—	2	1	no	—
248	C₂×#121	—	2	1	no	—
249	C₂×#121	—	2	1	no	—
250	C₂×#121	C₂²×C₃₁	1	2	yes	—
251	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
252	#486	—	2	4	no	—
253	C₂×#102	C₂²×C₃₁	1	4	yes	—
254	C₂×#102	C₂²×C₃₁	1	4	yes	—
255	C₄×M₁₆	—	2	4	no	—
256	C₂×#97	C₂²×C₃₁	1	2	yes	—
257	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
259	#391	—	2	1	no	—
260	#390	—	2	1	no	—
261	#390	—	2	1	no	—
262	#390	C₂²×C₃₁	1	2	yes	—
263	#389	—	2	1	no	—
264	#389	C₂²×C₃₁	1	2	yes	—
265	#384	C₂²×C₃₁	1	1	yes	characteristic
266	#384	C₂²×C₃₁	1	1	yes	characteristic
267	#380	—	2	1	no	—
268	#380	C₂²×C₃₁	1	2	yes	—
269	#340	C₂²×C₃₁	1	4	yes	—
270	#340	C₂²×C₃₁	1	4	yes	—
271	C₂×C₃₁	—	4	2	no	—
272	C₂×C₃₁	—	16	2	no	—
273	C₂×C₃₁	#2850	1	1	yes	characteristic
274	C₂×C₃₁	C₂×#468	1	1	yes	characteristic
275	C₂×C₃₁	C₂×#469	1	1	yes	characteristic
276	C₂×C₃₁	#3138	1	1	yes	characteristic
277	C₂×C₃₁	#3141	1	1	yes	characteristic
278	C₂×C₃₁	#3182	1	1	yes	characteristic
279	C₂×C₃₁	#3184	1	1	yes	characteristic
280	C₂²×Q₈	—	2	1	no	—
281	C₂²×D₄	—	2	2	no	—
282	C₂²×D₄	—	2	1	no	—
283	C₂²×D₄	—	2	1	no	—
284	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
285	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
286	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
287	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
288	C₂×M₁₆	—	2	4	no	—
289	C₂×M₁₆	—	2	4	no	—
290	C₂×M₁₆	—	2	4	no	—
291	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
292	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
293	C₄⋊Q₈	—	2	2	no	—
294	C₄⋊Q₈	—	2	2	no	—
295	#127	—	2	2	no	—
296	#127	—	2	2	no	—
297	C₂²⋊Q₈	—	4	4	no	—
298	#121	—	2	8	no	—
299	#121	—	4	4	no	—
300	#121	—	4	4	no	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	2	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
308	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
309	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
310	C₂×C₂²⋊C₄	—	2	2	no	—
311	C₂×C₂²⋊C₄	—	2	2	no	—
312	C₂×C₂²⋊C₄	—	2	2	no	—
313	C₂×C₂²⋊C₄	—	2	2	no	—
314	C₂×C₄²	—	2	2	no	—
315	C₂×C₄²	—	2	2	no	—
316	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
317	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
318	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
319	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
320	#102	—	2	8	no	—
321	#102	—	2	8	no	—
322	#97	—	2	4	no	—
323	C₄×C₈	—	2	2	no	—
324	C₄×C₈	D₄×C₃₁	1	4	yes	complemented
325	#95	—	2	2	no	—
326	#95	—	2	2	no	—
327	C₃₁	#33048	1	1	yes	characteristic, direct product factor, Sylow
328	C₂⁴	—	2	2	no	—
329	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
330	C₂×Q₈	—	4	2	no	—
331	C₂×Q₈	—	4	2	no	—
332	C₂×Q₈	—	4	2	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	4	4	no	—
338	C₂×D₄	—	4	4	no	—
339	C₂×D₄	—	4	4	no	—
340	C₂×D₄	—	4	2	no	—
341	C₂×D₄	—	4	2	no	—
342	C₂×D₄	—	4	2	no	—
343	C₂×D₄	—	4	2	no	—
344	C₂×D₄	—	4	2	no	—
345	C₂×D₄	—	4	2	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
360	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
361	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
363	C₂²×C₄	#41×C₃₁	1	4	yes	—
364	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
365	M₁₆	—	2	16	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
371	C₂×C₈	#41×C₃₁	1	4	yes	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	4	no	—
374	C₄⋊C₄	—	2	4	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	4	4	no	—
377	C₄⋊C₄	—	4	2	no	—
378	C₄⋊C₄	—	4	2	no	—
379	C₄⋊C₄	—	4	2	no	—
380	C₄⋊C₄	—	4	2	no	—
381	C₄⋊C₄	—	4	2	no	—
382	C₂²⋊C₄	—	4	4	no	—
383	C₂²⋊C₄	—	4	4	no	—
384	C₂²⋊C₄	—	4	4	no	—
385	C₂²⋊C₄	—	4	4	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	C₂×D₄×C₃₁	1	2	yes	—
391	C₄²	#41×C₃₁	1	2	yes	—
392	C₂³	—	2	1	no	—
393	C₂³	—	2	1	no	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	1	no	—
396	C₂³	—	2	1	no	—
397	C₂³	—	2	1	no	—
398	C₂³	—	4	2	no	—
399	C₂³	—	4	2	no	—
400	C₂³	—	4	2	no	—
401	C₂³	—	4	2	no	—
402	C₂³	—	4	2	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	4	2	no	—
405	C₂³	C₃₁×#124	1	2	yes	—
406	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
407	Q₈	—	8	4	no	—
408	D₄	—	4	4	no	—
409	D₄	—	4	4	no	—
410	D₄	—	8	8	no	—
411	D₄	—	8	4	no	—
412	D₄	—	8	4	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	4	4	no	—
424	C₂×C₄	—	4	4	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	C₃₁×#121	1	2	yes	—
435	C₂×C₄	C₃₁×#121	1	2	yes	—
436	C₂×C₄	C₃₁×#124	1	2	yes	—
437	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
438	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
439	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
440	C₂×C₄	C₃₁×#137	1	2	yes	—
441	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
442	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
443	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
444	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
445	C₈	—	2	4	no	—
446	C₈	—	2	4	no	—
447	C₂²	—	2	2	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	4	1	no	—
451	C₂²	—	4	1	no	—
452	C₂²	—	4	1	no	—
453	C₂²	—	4	1	no	—
454	C₂²	—	8	2	no	—
455	C₂²	—	8	2	no	—
456	C₂²	—	8	2	no	—
457	C₂²	—	8	2	no	—
458	C₂²	—	8	2	no	—
459	C₂²	—	8	2	no	—
460	C₂²	—	8	2	no	—
461	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
462	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
463	C₂²	C₃₁×#532	1	1	yes	characteristic
464	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
465	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
466	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
467	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
468	C₄	—	2	2	no	—
469	C₄	—	4	1	no	—
470	C₄	—	4	1	no	—
471	C₄	—	8	2	no	—
472	C₄	—	8	2	no	—
473	C₄	—	8	2	no	—
474	C₄	C₃₁×#468	1	4	yes	—
475	C₄	C₃₁×#469	1	4	yes	—
476	C₄	C₃₁×#525	1	2	yes	—
477	C₄	C₃₁×#529	1	2	yes	—
478	C₂	—	4	2	no	—
479	C₂	—	16	2	no	—
480	C₂	C₃₁×#2850	1	1	yes	characteristic
481	C₂	C₂×C₃₁×#468	1	1	yes	characteristic
482	C₂	C₂×C₃₁×#469	1	1	yes	characteristic
483	C₂	C₃₁×#3138	1	1	yes	characteristic
484	C₂	C₃₁×#3141	1	1	yes	characteristic
485	C₂	C₃₁×#3182	1	1	yes	characteristic
486	C₂	C₃₁×#3184	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#2856	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#486	C₂	semidirect
2. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
3. C₃₁×#2857	C₂	semidirect
5. C₃₁×#1990	C₂	semidirect
6. C₃₁×#1957	C₂	semidirect
7. C₃₁×#1936	C₂	semidirect
8. C₃₁×#1895	C₂	semidirect
9. C₃₁×#1834	C₂	semidirect
81. C₄×C₈×C₃₁	D₄	semidirect
148. #33048	C₃₁	direct (N × H)
204. C₂×#486	C₂×C₃₁	semidirect
205. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
206. #2857	C₂×C₃₁	semidirect
208. #1990	C₂×C₃₁	semidirect
209. #1957	C₂×C₃₁	semidirect
210. #1936	C₂×C₃₁	semidirect
211. #1895	C₂×C₃₁	semidirect
212. #1834	C₂×C₃₁	semidirect
324. C₄×C₈	D₄×C₃₁	semidirect
327. C₃₁	#33048	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#486	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#2857	2
C₃₁×#2856	2
C₃₁×#1990	2
C₃₁×#1957	2
C₃₁×#1936	2
C₃₁×#1895	2
C₃₁×#1834	2
#33048	31

Subgroups — C31×#33048 #21489807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33048 #21489807