Subgroups — C₃₁×#32789 #21489548

Order 7936.

672 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2857	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2851	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2101	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2099	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2099	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2077	C₂	1	2	yes	maximal
7	C₃₁×#2076	C₂	1	2	yes	maximal
8	C₃₁×#2059	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2058	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2057	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1817	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
15	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#107	C₂²	1	2	yes	—
17	C₂×C₃₁×#106	C₂²	1	2	yes	—
18	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₃₁×#391	—	2	1	no	—
20	C₃₁×#389	—	2	1	no	—
21	C₃₁×#389	C₂²	1	1	yes	characteristic
22	C₃₁×#389	C₂²	1	1	yes	characteristic
23	C₃₁×#389	C₂²	1	2	yes	—
24	C₃₁×#387	—	2	1	no	—
25	C₃₁×#387	—	2	1	no	—
26	C₃₁×#385	—	2	1	no	—
27	C₃₁×#385	—	2	1	no	—
28	C₃₁×#384	C₂²	1	1	yes	characteristic
29	C₃₁×#382	C₂²	1	1	yes	characteristic
30	C₃₁×#380	—	2	1	no	—
31	C₃₁×#380	C₂²	1	1	yes	characteristic
32	C₃₁×#380	C₂²	1	2	yes	—
33	C₃₁×#380	C₂²	1	1	yes	characteristic
34	C₃₁×#340	C₂²	1	2	yes	—
35	C₃₁×#340	C₂²	1	2	yes	—
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₃₁×#336	C₂²	1	1	yes	characteristic
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₃₁×C₄⋊Q₈	—	4	2	no	—
50	C₃₁×#125	—	4	2	no	—
51	C₃₁×C₂²⋊Q₈	—	4	4	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	1	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	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₄²×C₃₁	—	4	2	no	—
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	C₃₁×#107	—	4	2	no	—
75	C₃₁×#106	—	4	2	no	—
76	C₃₁×C₂²⋊C₈	—	4	2	no	—
77	C₃₁×#95	—	2	2	no	—
78	C₃₁×#95	—	2	1	no	—
79	C₃₁×#95	—	2	1	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₃₁×#95	—	2	1	no	—
82	C₃₁×#95	—	2	1	no	—
83	C₃₁×#95	—	2	1	no	—
84	C₃₁×#95	—	2	1	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	C₂²×C₄×C₃₁	—	2	2	no	—
90	C₂²×C₄×C₃₁	—	2	2	no	—
91	C₂²×C₄×C₃₁	—	2	2	no	—
92	C₂²×C₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	2	2	no	—
94	C₂²×C₄×C₃₁	—	2	1	no	—
95	C₂²×C₄×C₃₁	—	2	1	no	—
96	C₂²×C₄×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	—	4	1	no	—
103	C₂²×C₄×C₃₁	—	4	1	no	—
104	C₂²×C₄×C₃₁	—	4	1	no	—
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	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
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	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×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	—	2	1	no	—
122	C₄⋊C₄×C₃₁	—	2	1	no	—
123	C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₄⋊C₄×C₃₁	—	2	1	no	—
125	C₄⋊C₄×C₃₁	—	2	4	no	—
126	C₄⋊C₄×C₃₁	—	4	4	no	—
127	C₄⋊C₄×C₃₁	—	4	2	no	—
128	C₄⋊C₄×C₃₁	—	4	2	no	—
129	C₄⋊C₄×C₃₁	—	4	2	no	—
130	C₄⋊C₄×C₃₁	—	4	2	no	—
131	C₄⋊C₄×C₃₁	—	4	2	no	—
132	C₄⋊C₄×C₃₁	—	4	2	no	—
133	C₄⋊C₄×C₃₁	—	4	2	no	—
134	C₂²⋊C₄×C₃₁	—	4	2	no	—
135	C₂²⋊C₄×C₃₁	—	4	4	no	—
136	C₂²⋊C₄×C₃₁	—	4	2	no	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	—	2	2	no	—
139	C₄²×C₃₁	—	4	1	no	—
140	C₄²×C₃₁	—	4	1	no	—
141	#32789	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂³×C₃₁	—	2	1	no	—
143	C₂³×C₃₁	—	2	1	no	—
144	C₂³×C₃₁	—	2	1	no	—
145	C₂³×C₃₁	—	2	1	no	—
146	C₂³×C₃₁	—	2	1	no	—
147	C₂³×C₃₁	—	2	1	no	—
148	C₂³×C₃₁	#137	1	2	yes	—
149	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
150	Q₈×C₃₁	—	8	4	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	1	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	—	4	4	no	—
165	C₂×C₄×C₃₁	—	4	4	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	1	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	1	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	—	4	1	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	H(5,𝔽₂)	1	2	yes	—
189	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
190	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
191	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
192	C₈×C₃₁	—	4	2	no	—
193	C₈×C₃₁	—	4	2	no	—
194	#2857	C₂×C₃₁	1	1	yes	characteristic
195	#2851	C₂×C₃₁	1	1	yes	characteristic
196	#2101	C₂×C₃₁	1	2	yes	complemented
197	#2099	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#2099	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#2077	C₂×C₃₁	1	2	yes	—
200	#2076	C₂×C₃₁	1	2	yes	—
201	#2059	C₂×C₃₁	1	2	yes	complemented
202	#2058	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#2057	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#1817	C₂×C₃₁	1	1	yes	characteristic
205	C₂²×C₃₁	—	2	2	no	—
206	C₂²×C₃₁	—	2	2	no	—
207	C₂²×C₃₁	—	2	2	no	—
208	C₂²×C₃₁	—	4	1	no	—
209	C₂²×C₃₁	—	4	1	no	—
210	C₂²×C₃₁	—	4	1	no	—
211	C₂²×C₃₁	—	4	1	no	—
212	C₂²×C₃₁	#534	1	1	yes	characteristic
213	C₂²×C₃₁	#536	1	1	yes	characteristic
214	C₂²×C₃₁	#536	1	1	yes	characteristic
215	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
216	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
218	C₂²×C₃₁	#575	1	1	yes	characteristic
219	C₄×C₃₁	—	2	2	no	—
220	C₄×C₃₁	—	2	2	no	—
221	C₄×C₃₁	—	4	1	no	—
222	C₄×C₃₁	—	4	1	no	—
223	C₄×C₃₁	—	4	1	no	—
224	C₄×C₃₁	—	8	1	no	—
225	C₄×C₃₁	—	8	1	no	—
226	C₄×C₃₁	—	8	2	no	—
227	C₄×C₃₁	—	8	2	no	—
228	C₄×C₃₁	—	8	1	no	—
229	C₄×C₃₁	—	8	1	no	—
230	C₄×C₃₁	#563	1	2	yes	—
231	C₄×C₃₁	#563	1	2	yes	—
232	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×C₂²⋊Q₈	—	2	1	no	—
235	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×#107	C₂²×C₃₁	1	2	yes	—
237	C₂×#106	C₂²×C₃₁	1	2	yes	—
238	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
239	#391	—	2	1	no	—
240	#389	—	2	1	no	—
241	#389	C₂²×C₃₁	1	1	yes	characteristic
242	#389	C₂²×C₃₁	1	1	yes	characteristic
243	#389	C₂²×C₃₁	1	2	yes	—
244	#387	—	2	1	no	—
245	#387	—	2	1	no	—
246	#385	—	2	1	no	—
247	#385	—	2	1	no	—
248	#384	C₂²×C₃₁	1	1	yes	characteristic
249	#382	C₂²×C₃₁	1	1	yes	characteristic
250	#380	—	2	1	no	—
251	#380	C₂²×C₃₁	1	1	yes	characteristic
252	#380	C₂²×C₃₁	1	2	yes	—
253	#380	C₂²×C₃₁	1	1	yes	characteristic
254	#340	C₂²×C₃₁	1	2	yes	—
255	#340	C₂²×C₃₁	1	2	yes	—
256	#340	C₂²×C₃₁	1	2	yes	—
257	#340	C₂²×C₃₁	1	2	yes	—
258	#340	C₂²×C₃₁	1	2	yes	—
259	#340	C₂²×C₃₁	1	2	yes	—
260	#336	C₂²×C₃₁	1	1	yes	characteristic
261	#336	C₂²×C₃₁	1	1	yes	characteristic
262	#336	C₂²×C₃₁	1	2	yes	—
263	C₂×C₃₁	—	4	2	no	—
264	C₂×C₃₁	#2817	1	1	yes	characteristic
265	C₂×C₃₁	#3190	1	1	yes	characteristic
266	C₂×C₃₁	#3194	1	1	yes	characteristic
267	C₂×C₃₁	#3228	1	1	yes	characteristic
268	C₂×C₃₁	#3233	1	1	yes	characteristic
269	C₂×C₃₁	#3236	1	1	yes	characteristic
270	C₂×C₃₁	#3239	1	1	yes	characteristic
271	C₂²×Q₈	—	2	1	no	—
272	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
273	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
274	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
275	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
276	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
277	C₄⋊Q₈	—	4	2	no	—
278	#125	—	4	2	no	—
279	C₂²⋊Q₈	—	4	4	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	—	2	1	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	—	2	1	no	—
288	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
289	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
291	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
292	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
293	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
294	C₂×C₂²⋊C₄	—	2	2	no	—
295	C₂×C₂²⋊C₄	—	2	1	no	—
296	C₂×C₂²⋊C₄	—	2	1	no	—
297	C₂×C₄²	—	4	2	no	—
298	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
299	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
300	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
301	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
302	#107	—	4	2	no	—
303	#106	—	4	2	no	—
304	C₂²⋊C₈	—	4	2	no	—
305	#95	—	2	2	no	—
306	#95	—	2	1	no	—
307	#95	—	2	1	no	—
308	#95	—	2	2	no	—
309	#95	—	2	1	no	—
310	#95	—	2	1	no	—
311	#95	—	2	1	no	—
312	#95	—	2	1	no	—
313	C₃₁	#32789	1	1	yes	characteristic, direct product factor, Sylow
314	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
315	C₂×Q₈	—	4	2	no	—
316	C₂×Q₈	—	4	2	no	—
317	C₂×Q₈	—	4	2	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	4	1	no	—
332	C₂²×C₄	—	4	1	no	—
333	C₂²×C₄	—	4	1	no	—
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	C₂×D₄×C₃₁	1	1	yes	characteristic
339	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
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	C₂×C₈	—	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	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	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	C₄⋊C₄	—	4	2	no	—
360	C₄⋊C₄	—	4	2	no	—
361	C₄⋊C₄	—	4	2	no	—
362	C₄⋊C₄	—	4	2	no	—
363	C₂²⋊C₄	—	4	2	no	—
364	C₂²⋊C₄	—	4	4	no	—
365	C₂²⋊C₄	—	4	2	no	—
366	C₄²	—	2	2	no	—
367	C₄²	—	2	2	no	—
368	C₄²	—	4	1	no	—
369	C₄²	—	4	1	no	—
370	C₂³	—	2	1	no	—
371	C₂³	—	2	1	no	—
372	C₂³	—	2	1	no	—
373	C₂³	—	2	1	no	—
374	C₂³	—	2	1	no	—
375	C₂³	—	2	1	no	—
376	C₂³	C₃₁×#137	1	2	yes	—
377	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
378	Q₈	—	8	4	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	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	—	4	4	no	—
393	C₂×C₄	—	4	4	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	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	2	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	C₃₁×#135	1	2	yes	—
415	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
416	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
417	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
418	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
419	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
420	C₈	—	4	2	no	—
421	C₈	—	4	2	no	—
422	C₂²	—	2	2	no	—
423	C₂²	—	2	2	no	—
424	C₂²	—	2	2	no	—
425	C₂²	—	4	1	no	—
426	C₂²	—	4	1	no	—
427	C₂²	—	4	1	no	—
428	C₂²	—	4	1	no	—
429	C₂²	C₃₁×#534	1	1	yes	characteristic
430	C₂²	C₃₁×#536	1	1	yes	characteristic
431	C₂²	C₃₁×#536	1	1	yes	characteristic
432	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
433	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
434	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
435	C₂²	C₃₁×#575	1	1	yes	characteristic
436	C₄	—	2	2	no	—
437	C₄	—	2	2	no	—
438	C₄	—	4	1	no	—
439	C₄	—	4	1	no	—
440	C₄	—	4	1	no	—
441	C₄	—	8	1	no	—
442	C₄	—	8	2	no	—
443	C₄	—	8	1	no	—
444	C₄	—	8	2	no	—
445	C₄	—	8	1	no	—
446	C₄	—	8	1	no	—
447	C₄	C₃₁×#563	1	2	yes	—
448	C₄	C₃₁×#563	1	2	yes	—
449	C₂	—	4	2	no	—
450	C₂	C₃₁×#2817	1	1	yes	characteristic
451	C₂	C₃₁×#3190	1	1	yes	characteristic
452	C₂	C₃₁×#3194	1	1	yes	characteristic
453	C₂	C₃₁×#3228	1	1	yes	characteristic
454	C₂	C₃₁×#3233	1	1	yes	characteristic
455	C₂	C₃₁×#3236	1	1	yes	characteristic
456	C₂	C₃₁×#3239	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
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
3. C₃₁×#2101	C₂	semidirect
4. C₃₁×#2099	C₂	semidirect
5. C₃₁×#2099	C₂	semidirect
8. C₃₁×#2059	C₂	semidirect
9. C₃₁×#2058	C₂	semidirect
10. C₃₁×#2057	C₂	semidirect
141. #32789	C₃₁	direct (N × H)
196. #2101	C₂×C₃₁	semidirect
197. #2099	C₂×C₃₁	semidirect
198. #2099	C₂×C₃₁	semidirect
201. #2059	C₂×C₃₁	semidirect
202. #2058	C₂×C₃₁	semidirect
203. #2057	C₂×C₃₁	semidirect
313. C₃₁	#32789	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×#2857	2
C₃₁×#2851	2
C₃₁×#2101	2
C₃₁×#2099	2
C₃₁×#2077	2
C₃₁×#2076	2
C₃₁×#2059	2
C₃₁×#2058	2
C₃₁×#2057	2
C₃₁×#1817	2
#32789	31

Subgroups — C31×#32789 #21489548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32789 #21489548