Subgroups — C₃₁×#32825 #21489584

Order 7936.

662 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2861	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2096	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2093	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2072	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2071	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#2068	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2065	C₂	1	2	yes	maximal
9	C₃₁×#2059	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2042	C₂	1	2	yes	maximal
11	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#125	—	2	1	no	—
13	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#107	C₂²	1	2	yes	—
16	C₂×C₃₁×#106	C₂²	1	2	yes	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃₁×#103	C₂²	1	2	yes	—
19	C₂×C₃₁×#103	C₂²	1	2	yes	—
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₃₁×#391	C₂²	1	1	yes	characteristic
23	C₃₁×#391	C₂²	1	1	yes	characteristic
24	C₃₁×#390	C₂²	1	1	yes	characteristic
25	C₃₁×#390	C₂²	1	1	yes	characteristic
26	C₃₁×#389	—	2	1	no	—
27	C₃₁×#389	—	2	1	no	—
28	C₃₁×#389	C₂²	1	2	yes	—
29	C₃₁×#389	C₂²	1	2	yes	—
30	C₃₁×#388	—	2	1	no	—
31	C₃₁×#388	—	2	1	no	—
32	C₃₁×#388	—	2	1	no	—
33	C₃₁×#388	—	2	1	no	—
34	C₃₁×#384	C₂²	1	1	yes	characteristic
35	C₃₁×#384	C₂²	1	1	yes	characteristic
36	C₃₁×#382	—	2	1	no	—
37	C₄³×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	1	yes	characteristic
42	C₃₁×#336	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₃₁×#125	—	4	2	no	—
48	C₃₁×#125	—	4	4	no	—
49	C₃₁×#124	—	4	2	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₃₁×#107	—	4	2	no	—
76	C₃₁×#106	—	4	2	no	—
77	C₃₁×C₄⋊C₈	—	4	2	no	—
78	C₃₁×#103	—	4	2	no	—
79	C₃₁×#103	—	4	2	no	—
80	C₃₁×#102	—	4	2	no	—
81	C₃₁×#102	—	4	2	no	—
82	C₃₁×#95	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₃₁×#95	—	2	1	no	—
85	C₃₁×#95	—	2	1	no	—
86	C₃₁×#95	—	2	1	no	—
87	C₃₁×#95	—	2	1	no	—
88	C₂⁴×C₃₁	—	2	1	no	—
89	C₂×Q₈×C₃₁	—	2	1	no	—
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	—	2	2	no	—
92	C₂×Q₈×C₃₁	—	2	1	no	—
93	C₂×D₄×C₃₁	—	2	1	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₄⋊C₄×C₃₁	—	4	2	no	—
122	C₄⋊C₄×C₃₁	—	4	2	no	—
123	C₄⋊C₄×C₃₁	—	4	2	no	—
124	C₄⋊C₄×C₃₁	—	4	4	no	—
125	C₄⋊C₄×C₃₁	—	4	4	no	—
126	C₄⋊C₄×C₃₁	—	4	2	no	—
127	C₄⋊C₄×C₃₁	—	4	2	no	—
128	C₄⋊C₄×C₃₁	—	4	2	no	—
129	C₂²⋊C₄×C₃₁	—	4	2	no	—
130	C₂²⋊C₄×C₃₁	—	4	2	no	—
131	C₂²⋊C₄×C₃₁	—	4	2	no	—
132	C₂²⋊C₄×C₃₁	—	4	2	no	—
133	C₂²⋊C₄×C₃₁	—	4	2	no	—
134	C₂²⋊C₄×C₃₁	—	4	2	no	—
135	C₄²×C₃₁	—	2	2	no	—
136	C₄²×C₃₁	—	2	2	no	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	—	2	1	no	—
139	C₄²×C₃₁	—	2	2	no	—
140	C₄²×C₃₁	—	2	2	no	—
141	C₄²×C₃₁	—	2	1	no	—
142	C₄²×C₃₁	—	4	1	no	—
143	#32825	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	4	1	no	—
145	C₂³×C₃₁	—	4	1	no	—
146	C₂³×C₃₁	—	4	1	no	—
147	C₂³×C₃₁	—	4	1	no	—
148	C₂³×C₃₁	—	4	1	no	—
149	C₂³×C₃₁	—	4	1	no	—
150	C₂³×C₃₁	—	4	1	no	—
151	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	Q₈×C₃₁	—	4	2	no	—
153	Q₈×C₃₁	—	4	2	no	—
154	D₄×C₃₁	—	4	2	no	—
155	D₄×C₃₁	—	4	2	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	1	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	1	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	1	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	#135	1	2	yes	—
187	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
188	C₂×C₄×C₃₁	#137	1	2	yes	—
189	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
190	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
191	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
192	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
193	C₈×C₃₁	—	4	2	no	—
194	C₈×C₃₁	—	4	2	no	—
195	#2861	C₂×C₃₁	1	1	yes	characteristic
196	#2859	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#2096	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#2093	C₂×C₃₁	1	1	yes	characteristic
199	#2072	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#2071	C₂×C₃₁	1	1	yes	characteristic
201	#2068	C₂×C₃₁	1	2	yes	complemented
202	#2065	C₂×C₃₁	1	2	yes	—
203	#2059	C₂×C₃₁	1	2	yes	complemented
204	#2042	C₂×C₃₁	1	2	yes	—
205	#1831	C₂×C₃₁	1	1	yes	characteristic, complemented
206	C₂²×C₃₁	—	8	1	no	—
207	C₂²×C₃₁	—	8	1	no	—
208	C₂²×C₃₁	—	8	1	no	—
209	C₂²×C₃₁	—	8	1	no	—
210	C₂²×C₃₁	—	8	1	no	—
211	C₂²×C₃₁	—	8	1	no	—
212	C₂²×C₃₁	—	8	1	no	—
213	C₂²×C₃₁	#535	1	1	yes	characteristic
214	C₂²×C₃₁	#536	1	1	yes	characteristic
215	C₂²×C₃₁	#537	1	1	yes	characteristic
216	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
217	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
219	C₂²×C₃₁	#575	1	1	yes	characteristic
220	C₄×C₃₁	—	2	2	no	—
221	C₄×C₃₁	—	2	2	no	—
222	C₄×C₃₁	—	2	2	no	—
223	C₄×C₃₁	—	4	1	no	—
224	C₄×C₃₁	—	4	1	no	—
225	C₄×C₃₁	—	4	1	no	—
226	C₄×C₃₁	—	8	1	no	—
227	C₄×C₃₁	—	8	1	no	—
228	C₄×C₃₁	—	8	2	no	—
229	C₄×C₃₁	—	8	1	no	—
230	C₄×C₃₁	—	8	2	no	—
231	C₄×C₃₁	#562	1	2	yes	—
232	C₄×C₃₁	#563	1	2	yes	—
233	C₂×#125	—	2	1	no	—
234	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×#107	C₂²×C₃₁	1	2	yes	—
237	C₂×#106	C₂²×C₃₁	1	2	yes	—
238	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
239	C₂×#103	C₂²×C₃₁	1	2	yes	—
240	C₂×#103	C₂²×C₃₁	1	2	yes	—
241	C₂×#102	C₂²×C₃₁	1	2	yes	—
242	C₂×#102	C₂²×C₃₁	1	2	yes	—
243	#391	C₂²×C₃₁	1	1	yes	characteristic
244	#391	C₂²×C₃₁	1	1	yes	characteristic
245	#390	C₂²×C₃₁	1	1	yes	characteristic
246	#390	C₂²×C₃₁	1	1	yes	characteristic
247	#389	—	2	1	no	—
248	#389	—	2	1	no	—
249	#389	C₂²×C₃₁	1	2	yes	—
250	#389	C₂²×C₃₁	1	2	yes	—
251	#388	—	2	1	no	—
252	#388	—	2	1	no	—
253	#388	—	2	1	no	—
254	#388	—	2	1	no	—
255	#384	C₂²×C₃₁	1	1	yes	characteristic
256	#384	C₂²×C₃₁	1	1	yes	characteristic
257	#382	—	2	1	no	—
258	C₄³	C₂²×C₃₁	1	1	yes	characteristic
259	#340	C₂²×C₃₁	1	2	yes	—
260	#340	C₂²×C₃₁	1	2	yes	—
261	#336	C₂²×C₃₁	1	2	yes	—
262	#336	C₂²×C₃₁	1	1	yes	characteristic
263	#336	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×C₃₁	—	16	1	no	—
265	C₂×C₃₁	#2820	1	1	yes	characteristic
266	C₂×C₃₁	#3207	1	1	yes	characteristic
267	C₂×C₃₁	#3216	1	1	yes	characteristic
268	C₂×C₃₁	#3232	1	1	yes	characteristic
269	C₂×C₃₁	#3238	1	1	yes	characteristic
270	C₂×C₃₁	#3248	1	1	yes	characteristic
271	C₂×C₃₁	#3251	1	1	yes	characteristic
272	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
273	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
274	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
275	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
276	#125	—	4	2	no	—
277	#125	—	4	4	no	—
278	#124	—	4	2	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
289	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
290	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
291	C₂×C₂²⋊C₄	—	2	1	no	—
292	C₂×C₂²⋊C₄	—	2	1	no	—
293	C₂×C₂²⋊C₄	—	2	1	no	—
294	C₂×C₂²⋊C₄	—	2	1	no	—
295	C₂×C₂²⋊C₄	—	2	1	no	—
296	C₂×C₂²⋊C₄	—	2	1	no	—
297	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
298	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
299	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
300	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
301	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
302	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
303	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
304	#107	—	4	2	no	—
305	#106	—	4	2	no	—
306	C₄⋊C₈	—	4	2	no	—
307	#103	—	4	2	no	—
308	#103	—	4	2	no	—
309	#102	—	4	2	no	—
310	#102	—	4	2	no	—
311	#95	—	2	2	no	—
312	#95	—	2	2	no	—
313	#95	—	2	1	no	—
314	#95	—	2	1	no	—
315	#95	—	2	1	no	—
316	#95	—	2	1	no	—
317	C₃₁	#32825	1	1	yes	characteristic, direct product factor, Sylow
318	C₂⁴	—	2	1	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	2	2	no	—
322	C₂×Q₈	—	2	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
335	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
337	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	4	2	no	—
352	C₄⋊C₄	—	4	2	no	—
353	C₄⋊C₄	—	4	2	no	—
354	C₄⋊C₄	—	4	4	no	—
355	C₄⋊C₄	—	4	4	no	—
356	C₄⋊C₄	—	4	2	no	—
357	C₄⋊C₄	—	4	2	no	—
358	C₄⋊C₄	—	4	2	no	—
359	C₂²⋊C₄	—	4	2	no	—
360	C₂²⋊C₄	—	4	2	no	—
361	C₂²⋊C₄	—	4	2	no	—
362	C₂²⋊C₄	—	4	2	no	—
363	C₂²⋊C₄	—	4	2	no	—
364	C₂²⋊C₄	—	4	2	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	2	2	no	—
367	C₄²	—	2	2	no	—
368	C₄²	—	2	1	no	—
369	C₄²	—	2	2	no	—
370	C₄²	—	2	2	no	—
371	C₄²	—	2	1	no	—
372	C₄²	—	4	1	no	—
373	C₂³	—	4	1	no	—
374	C₂³	—	4	1	no	—
375	C₂³	—	4	1	no	—
376	C₂³	—	4	1	no	—
377	C₂³	—	4	1	no	—
378	C₂³	—	4	1	no	—
379	C₂³	—	4	1	no	—
380	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
381	Q₈	—	4	2	no	—
382	Q₈	—	4	2	no	—
383	D₄	—	4	2	no	—
384	D₄	—	4	2	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	C₃₁×#135	1	2	yes	—
416	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
417	C₂×C₄	C₃₁×#137	1	2	yes	—
418	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
419	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
420	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
421	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
422	C₈	—	4	2	no	—
423	C₈	—	4	2	no	—
424	C₂²	—	8	1	no	—
425	C₂²	—	8	1	no	—
426	C₂²	—	8	1	no	—
427	C₂²	—	8	1	no	—
428	C₂²	—	8	1	no	—
429	C₂²	—	8	1	no	—
430	C₂²	—	8	1	no	—
431	C₂²	C₃₁×#535	1	1	yes	characteristic
432	C₂²	C₃₁×#536	1	1	yes	characteristic
433	C₂²	C₃₁×#537	1	1	yes	characteristic
434	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
435	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
436	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
437	C₂²	C₃₁×#575	1	1	yes	characteristic
438	C₄	—	2	2	no	—
439	C₄	—	2	2	no	—
440	C₄	—	2	2	no	—
441	C₄	—	4	1	no	—
442	C₄	—	4	1	no	—
443	C₄	—	4	1	no	—
444	C₄	—	8	1	no	—
445	C₄	—	8	1	no	—
446	C₄	—	8	2	no	—
447	C₄	—	8	1	no	—
448	C₄	—	8	2	no	—
449	C₄	C₃₁×#562	1	2	yes	—
450	C₄	C₃₁×#563	1	2	yes	—
451	C₂	—	16	1	no	—
452	C₂	C₃₁×#2820	1	1	yes	characteristic
453	C₂	C₃₁×#3207	1	1	yes	characteristic
454	C₂	C₃₁×#3216	1	1	yes	characteristic
455	C₂	C₃₁×#3232	1	1	yes	characteristic
456	C₂	C₃₁×#3238	1	1	yes	characteristic
457	C₂	C₃₁×#3248	1	1	yes	characteristic
458	C₂	C₃₁×#3251	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2859	C₂	semidirect
3. C₃₁×#2096	C₂	semidirect
5. C₃₁×#2072	C₂	semidirect
7. C₃₁×#2068	C₂	semidirect
9. C₃₁×#2059	C₂	semidirect
11. C₃₁×#1831	C₂	semidirect
143. #32825	C₃₁	direct (N × H)
196. #2859	C₂×C₃₁	semidirect
197. #2096	C₂×C₃₁	semidirect
199. #2072	C₂×C₃₁	semidirect
201. #2068	C₂×C₃₁	semidirect
203. #2059	C₂×C₃₁	semidirect
205. #1831	C₂×C₃₁	semidirect
317. C₃₁	#32825	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×#2861	2
C₃₁×#2859	2
C₃₁×#2096	2
C₃₁×#2093	2
C₃₁×#2072	2
C₃₁×#2071	2
C₃₁×#2068	2
C₃₁×#2065	2
C₃₁×#2059	2
C₃₁×#2042	2
C₃₁×#1831	2
#32825	31

Subgroups — C31×#32825 #21489584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32825 #21489584