Subgroups — C₃₁×#33143 #21489902

Order 7936.

982 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#469	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2854	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1972	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1970	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1943	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1941	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1936	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1934	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1895	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1809	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#123	—	2	2	no	—
16	C₂×C₃₁×#121	C₂²	1	2	yes	—
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#469	—	2	4	no	—
19	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₄×M₁₆×C₃₁	—	2	4	no	—
26	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#390	C₂²	1	2	yes	—
31	C₃₁×#390	C₂²	1	2	yes	—
32	C₃₁×#389	C₂²	1	2	yes	—
33	C₃₁×#389	C₂²	1	2	yes	—
34	C₃₁×#388	—	2	2	no	—
35	C₃₁×#388	—	2	2	no	—
36	C₃₁×#386	—	2	2	no	—
37	C₃₁×#380	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₂²×D₄×C₃₁	—	2	2	no	—
42	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
43	C₂³×C₄×C₃₁	D₄	1	2	yes	—
44	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×D₈×C₃₁	—	2	2	no	—
46	C₂×D₈×C₃₁	—	2	2	no	—
47	C₂×D₈×C₃₁	—	2	2	no	—
48	C₂×M₁₆×C₃₁	—	2	2	no	—
49	C₂×M₁₆×C₃₁	—	2	2	no	—
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂×M₁₆×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×#123	—	4	8	no	—
59	C₃₁×#121	—	2	4	no	—
60	C₃₁×#121	—	2	4	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
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×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄²×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	—	2	2	no	—
74	C₂×C₄²×C₃₁	D₄	1	2	yes	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#107	—	2	2	no	—
78	C₃₁×#106	—	2	2	no	—
79	C₃₁×#106	—	2	2	no	—
80	C₃₁×#102	—	2	4	no	—
81	C₃₁×#102	—	2	4	no	—
82	C₃₁×#102	—	2	4	no	—
83	C₃₁×#97	—	2	2	no	—
84	C₃₁×#97	—	2	2	no	—
85	C₄×C₈×C₃₁	—	2	2	no	—
86	C₄×C₈×C₃₁	—	2	2	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₃₁×#95	—	2	2	no	—
90	C₂⁴×C₃₁	—	2	2	no	—
91	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	—	4	4	no	—
99	C₂×D₄×C₃₁	—	4	4	no	—
100	C₂×D₄×C₃₁	—	4	4	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	2	4	no	—
106	C₂²×C₄×C₃₁	—	2	4	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	—	4	2	no	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₃₁	#41	1	2	yes	—
116	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
117	D₈×C₃₁	—	4	4	no	—
118	M₁₆×C₃₁	—	2	8	no	—
119	M₁₆×C₃₁	—	2	8	no	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
127	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₃₁	#41	1	2	yes	—
129	C₂×C₈×C₃₁	#41	1	2	yes	—
130	C₂×C₈×C₃₁	#41	1	2	yes	—
131	C₂×C₈×C₃₁	#41	1	2	yes	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	4	4	no	—
139	C₄⋊C₄×C₃₁	—	4	4	no	—
140	C₂²⋊C₄×C₃₁	—	4	4	no	—
141	C₂²⋊C₄×C₃₁	—	4	4	no	—
142	C₂²⋊C₄×C₃₁	—	4	4	no	—
143	C₂²⋊C₄×C₃₁	—	4	4	no	—
144	C₂²⋊C₄×C₃₁	—	4	4	no	—
145	C₂²⋊C₄×C₃₁	—	4	4	no	—
146	C₄²×C₃₁	—	2	4	no	—
147	C₄²×C₃₁	—	2	1	no	—
148	C₄²×C₃₁	—	2	1	no	—
149	C₄²×C₃₁	—	2	1	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	#33143	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂³×C₃₁	—	2	1	no	—
153	C₂³×C₃₁	—	2	1	no	—
154	C₂³×C₃₁	—	2	2	no	—
155	C₂³×C₃₁	—	2	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	—	4	2	no	—
163	C₂³×C₃₁	#121	1	2	yes	—
164	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	4	2	no	—
167	D₄×C₃₁	—	4	2	no	—
168	D₄×C₃₁	—	4	2	no	—
169	D₄×C₃₁	—	8	8	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	4	8	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	AGL(1,ℤ/8)	1	4	yes	—
193	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
199	C₈×C₃₁	—	2	2	no	—
200	C₈×C₃₁	—	2	2	no	—
201	C₈×C₃₁	—	2	2	no	—
202	C₈×C₃₁	#123	1	4	yes	complemented
203	C₂×#469	C₂×C₃₁	1	1	yes	characteristic
204	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#2854	C₂×C₃₁	1	2	yes	complemented
206	#1972	C₂×C₃₁	1	1	yes	characteristic, complemented
207	#1970	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#1943	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#1941	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#1936	C₂×C₃₁	1	2	yes	complemented
211	#1934	C₂×C₃₁	1	2	yes	complemented
212	#1895	C₂×C₃₁	1	2	yes	complemented
213	#1809	C₂×C₃₁	1	1	yes	characteristic, complemented
214	C₂²×C₃₁	—	2	2	no	—
215	C₂²×C₃₁	—	2	4	no	—
216	C₂²×C₃₁	—	4	2	no	—
217	C₂²×C₃₁	—	4	1	no	—
218	C₂²×C₃₁	—	4	1	no	—
219	C₂²×C₃₁	—	8	2	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	C₂×#121	1	1	yes	characteristic
227	C₂²×C₃₁	#532	1	1	yes	characteristic
228	C₂²×C₃₁	#532	1	1	yes	characteristic
229	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
230	C₂²×C₃₁	#575	1	2	yes	—
231	C₄×C₃₁	—	2	2	no	—
232	C₄×C₃₁	—	2	4	no	—
233	C₄×C₃₁	—	4	2	no	—
234	C₄×C₃₁	—	8	2	no	—
235	C₄×C₃₁	—	8	2	no	—
236	C₄×C₃₁	—	8	2	no	—
237	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
238	C₄×C₃₁	#525	1	1	yes	characteristic
239	C₄×C₃₁	#529	1	1	yes	characteristic
240	C₄×C₃₁	#529	1	1	yes	characteristic
241	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
243	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×#123	—	2	2	no	—
245	C₂×#121	C₂²×C₃₁	1	2	yes	—
246	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
247	#469	—	2	4	no	—
248	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×#102	C₂²×C₃₁	1	2	yes	—
252	C₂×#102	C₂²×C₃₁	1	2	yes	—
253	C₂×#102	C₂²×C₃₁	1	2	yes	—
254	C₄×M₁₆	—	2	4	no	—
255	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
259	#390	C₂²×C₃₁	1	2	yes	—
260	#390	C₂²×C₃₁	1	2	yes	—
261	#389	C₂²×C₃₁	1	2	yes	—
262	#389	C₂²×C₃₁	1	2	yes	—
263	#388	—	2	2	no	—
264	#388	—	2	2	no	—
265	#386	—	2	2	no	—
266	#380	C₂²×C₃₁	1	2	yes	—
267	#340	C₂²×C₃₁	1	2	yes	—
268	#340	C₂²×C₃₁	1	2	yes	—
269	#340	C₂²×C₃₁	1	2	yes	—
270	C₂×C₃₁	—	4	2	no	—
271	C₂×C₃₁	—	16	2	no	—
272	C₂×C₃₁	#2854	1	1	yes	characteristic
273	C₂×C₃₁	#3055	1	2	yes	—
274	C₂×C₃₁	#3171	1	2	yes	—
275	C₂×C₃₁	#3183	1	1	yes	characteristic
276	C₂×C₃₁	#3184	1	1	yes	characteristic
277	C₂²×D₄	—	2	2	no	—
278	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
279	C₂³×C₄	D₄×C₃₁	1	2	yes	—
280	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
281	C₂×D₈	—	2	2	no	—
282	C₂×D₈	—	2	2	no	—
283	C₂×D₈	—	2	2	no	—
284	C₂×M₁₆	—	2	2	no	—
285	C₂×M₁₆	—	2	2	no	—
286	C₂×M₁₆	—	2	2	no	—
287	C₂×M₁₆	—	2	2	no	—
288	C₂×M₁₆	—	2	2	no	—
289	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
290	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
292	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
293	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
294	#123	—	4	8	no	—
295	#121	—	2	4	no	—
296	#121	—	2	4	no	—
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
300	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
301	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
302	C₂×C₂²⋊C₄	—	2	2	no	—
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	C₂×C₂²⋊C₄	—	2	2	no	—
305	C₂×C₂²⋊C₄	—	2	2	no	—
306	C₂×C₂²⋊C₄	—	2	2	no	—
307	C₂×C₂²⋊C₄	—	2	2	no	—
308	C₂×C₄²	—	2	2	no	—
309	C₂×C₄²	—	2	2	no	—
310	C₂×C₄²	D₄×C₃₁	1	2	yes	—
311	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
312	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
313	#107	—	2	2	no	—
314	#106	—	2	2	no	—
315	#106	—	2	2	no	—
316	#102	—	2	4	no	—
317	#102	—	2	4	no	—
318	#102	—	2	4	no	—
319	#97	—	2	2	no	—
320	#97	—	2	2	no	—
321	C₄×C₈	—	2	2	no	—
322	C₄×C₈	—	2	2	no	—
323	#95	—	2	2	no	—
324	#95	—	2	2	no	—
325	#95	—	2	2	no	—
326	C₃₁	#33143	1	1	yes	characteristic, direct product factor, Sylow
327	C₂⁴	—	2	2	no	—
328	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	4	4	no	—
336	C₂×D₄	—	4	4	no	—
337	C₂×D₄	—	4	4	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	2	4	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	4	2	no	—
348	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
349	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
351	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
352	C₂²×C₄	#41×C₃₁	1	2	yes	—
353	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
354	D₈	—	4	4	no	—
355	M₁₆	—	2	8	no	—
356	M₁₆	—	2	8	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
364	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
365	C₂×C₈	#41×C₃₁	1	2	yes	—
366	C₂×C₈	#41×C₃₁	1	2	yes	—
367	C₂×C₈	#41×C₃₁	1	2	yes	—
368	C₂×C₈	#41×C₃₁	1	2	yes	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	4	4	no	—
376	C₄⋊C₄	—	4	4	no	—
377	C₂²⋊C₄	—	4	4	no	—
378	C₂²⋊C₄	—	4	4	no	—
379	C₂²⋊C₄	—	4	4	no	—
380	C₂²⋊C₄	—	4	4	no	—
381	C₂²⋊C₄	—	4	4	no	—
382	C₂²⋊C₄	—	4	4	no	—
383	C₄²	—	2	4	no	—
384	C₄²	—	2	1	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	1	no	—
387	C₄²	—	2	1	no	—
388	C₂³	—	2	1	no	—
389	C₂³	—	2	1	no	—
390	C₂³	—	2	2	no	—
391	C₂³	—	2	2	no	—
392	C₂³	—	4	2	no	—
393	C₂³	—	4	2	no	—
394	C₂³	—	4	2	no	—
395	C₂³	—	4	2	no	—
396	C₂³	—	4	2	no	—
397	C₂³	—	4	2	no	—
398	C₂³	—	4	2	no	—
399	C₂³	C₃₁×#121	1	2	yes	—
400	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
401	D₄	—	4	2	no	—
402	D₄	—	4	2	no	—
403	D₄	—	4	2	no	—
404	D₄	—	4	2	no	—
405	D₄	—	8	8	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	4	8	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	C₃₁×#121	1	2	yes	—
428	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
429	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
431	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
433	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
434	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
435	C₈	—	2	2	no	—
436	C₈	—	2	2	no	—
437	C₈	—	2	2	no	—
438	C₈	C₃₁×#123	1	4	yes	complemented
439	C₂²	—	2	2	no	—
440	C₂²	—	2	4	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	1	no	—
443	C₂²	—	4	1	no	—
444	C₂²	—	8	2	no	—
445	C₂²	—	8	2	no	—
446	C₂²	—	8	2	no	—
447	C₂²	—	8	2	no	—
448	C₂²	—	8	2	no	—
449	C₂²	—	8	2	no	—
450	C₂²	—	8	2	no	—
451	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
452	C₂²	C₃₁×#532	1	1	yes	characteristic
453	C₂²	C₃₁×#532	1	1	yes	characteristic
454	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
455	C₂²	C₃₁×#575	1	2	yes	—
456	C₄	—	2	2	no	—
457	C₄	—	2	4	no	—
458	C₄	—	4	2	no	—
459	C₄	—	8	2	no	—
460	C₄	—	8	2	no	—
461	C₄	—	8	2	no	—
462	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
463	C₄	C₃₁×#525	1	1	yes	characteristic
464	C₄	C₃₁×#529	1	1	yes	characteristic
465	C₄	C₃₁×#529	1	1	yes	characteristic
466	C₂	—	4	2	no	—
467	C₂	—	16	2	no	—
468	C₂	C₃₁×#2854	1	1	yes	characteristic
469	C₂	C₃₁×#3055	1	2	yes	—
470	C₂	C₃₁×#3171	1	2	yes	—
471	C₂	C₃₁×#3183	1	1	yes	characteristic
472	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	C₂×C₃₁×#469	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
2. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
3. C₃₁×#2854	C₂	semidirect
4. C₃₁×#1972	C₂	semidirect
5. C₃₁×#1970	C₂	semidirect
6. C₃₁×#1943	C₂	semidirect
7. C₃₁×#1941	C₂	semidirect
8. C₃₁×#1936	C₂	semidirect
9. C₃₁×#1934	C₂	semidirect
10. C₃₁×#1895	C₂	semidirect
11. C₃₁×#1809	C₂	semidirect
151. #33143	C₃₁	direct (N × H)
202. C₈×C₃₁	#123	semidirect
204. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
205. #2854	C₂×C₃₁	semidirect
206. #1972	C₂×C₃₁	semidirect
207. #1970	C₂×C₃₁	semidirect
208. #1943	C₂×C₃₁	semidirect
209. #1941	C₂×C₃₁	semidirect
210. #1936	C₂×C₃₁	semidirect
211. #1934	C₂×C₃₁	semidirect
212. #1895	C₂×C₃₁	semidirect
213. #1809	C₂×C₃₁	semidirect
326. C₃₁	#33143	direct (N × H)
438. C₈	C₃₁×#123	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#469	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#2854	2
C₃₁×#1972	2
C₃₁×#1970	2
C₃₁×#1943	2
C₃₁×#1941	2
C₃₁×#1936	2
C₃₁×#1934	2
C₃₁×#1895	2
C₃₁×#1809	2
#33143	31

Subgroups — C31×#33143 #21489902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33143 #21489902