Subgroups — C₃₁×#33722 #21490481

Order 7936.

826 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#509	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#508	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃₁×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3156	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3155	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2251	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2250	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2249	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2248	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	—	2	1	no	—
13	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
14	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
16	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#536	—	2	1	no	—
19	C₃₁×#535	—	2	1	no	—
20	C₃₁×#510	—	2	4	no	—
21	C₃₁×#509	—	2	4	no	—
22	C₃₁×#508	—	2	4	no	—
23	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	complemented
25	C₂×C₃₁×SD₃₂	—	2	4	no	—
26	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
27	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
28	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
29	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	complemented
32	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
33	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#497	—	2	2	no	—
37	C₃₁×#496	—	2	2	no	—
38	C₃₁×#495	C₂²	1	2	yes	—
39	C₃₁×#495	C₂²	1	2	yes	complemented
40	C₃₁×#494	C₂²	1	2	yes	complemented
41	C₃₁×#493	C₂²	1	2	yes	—
42	C₃₁×#405	C₂²	1	1	yes	characteristic
43	C₃₁×#346	C₂²	1	2	yes	complemented
44	C₄×C₁₆×C₃₁	C₂²	1	2	yes	complemented
45	C₂×#41×C₃₁	—	2	1	no	—
46	C₂×#41×C₃₁	—	2	1	no	—
47	C₂²×Q₈×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	—	2	1	no	—
49	C₃₁×#137	—	4	4	no	—
50	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
51	C₃₁×#135	—	2	2	no	—
52	C₃₁×#135	—	2	2	no	—
53	C₃₁×#135	—	2	2	no	—
54	C₃₁×#135	—	2	2	no	—
55	C₂×Q₁₆×C₃₁	—	2	2	no	—
56	C₂×Q₁₆×C₃₁	—	2	2	no	—
57	C₂×Q₁₆×C₃₁	—	2	2	no	—
58	C₂×Q₁₆×C₃₁	—	2	1	no	—
59	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×SD₁₆×C₃₁	—	2	2	no	—
63	C₂×SD₁₆×C₃₁	—	2	2	no	—
64	C₂×SD₁₆×C₃₁	—	2	1	no	—
65	C₂×SD₁₆×C₃₁	—	2	1	no	—
66	C₂×D₈×C₃₁	—	2	2	no	—
67	C₂×D₈×C₃₁	—	2	2	no	—
68	C₂×D₈×C₃₁	—	2	2	no	—
69	C₂×D₈×C₃₁	—	2	1	no	—
70	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×D₈×C₃₁	C₂³	1	2	yes	—
72	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
74	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₃₁×C₄⋊Q₈	—	2	2	no	—
76	C₃₁×#127	—	2	2	no	—
77	C₃₁×#124	—	2	2	no	—
78	C₃₁×#124	—	2	2	no	—
79	C₃₁×#123	—	4	2	no	—
80	C₃₁×C₂²⋊Q₈	—	4	2	no	—
81	C₃₁×#121	—	4	2	no	—
82	C₃₁×#120	—	4	2	no	—
83	C₃₁×#117	D₄	1	1	yes	characteristic
84	C₃₁×Q₃₂	—	2	2	no	—
85	C₃₁×Q₃₂	—	2	4	no	—
86	C₃₁×SD₃₂	—	2	4	no	—
87	C₃₁×SD₃₂	—	2	4	no	—
88	C₃₁×SD₃₂	—	2	4	no	—
89	C₃₁×SD₃₂	—	2	2	no	—
90	C₃₁×SD₃₂	—	2	2	no	—
91	C₃₁×SD₃₂	—	2	4	no	—
92	C₃₁×D₁₆	—	2	4	no	—
93	C₃₁×D₁₆	—	2	2	no	—
94	C₃₁×M₃₂	D₄	1	4	yes	complemented
95	C₂×C₁₆×C₃₁	D₄	1	4	yes	complemented
96	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
97	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
98	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
99	C₃₁×#97	D₄	1	2	yes	—
100	C₄×C₈×C₃₁	C₂³	1	2	yes	—
101	C₂⁴×C₃₁	—	4	1	no	—
102	#41×C₃₁	—	4	2	no	—
103	#41×C₃₁	—	4	2	no	—
104	#41×C₃₁	—	4	2	no	—
105	#41×C₃₁	—	4	2	no	—
106	C₂×Q₈×C₃₁	—	2	1	no	—
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	—	4	1	no	—
110	C₂×Q₈×C₃₁	—	4	2	no	—
111	C₂×Q₈×C₃₁	—	4	2	no	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	4	1	no	—
116	C₂×D₄×C₃₁	—	4	1	no	—
117	C₂×D₄×C₃₁	—	4	2	no	—
118	C₂×D₄×C₃₁	—	4	2	no	—
119	C₂×D₄×C₃₁	—	4	1	no	—
120	C₂×D₄×C₃₁	—	4	4	no	—
121	C₂²×C₄×C₃₁	—	4	1	no	—
122	C₂²×C₄×C₃₁	—	4	1	no	—
123	C₂²×C₄×C₃₁	—	4	1	no	—
124	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
125	Q₁₆×C₃₁	—	2	2	no	—
126	Q₁₆×C₃₁	—	2	1	no	—
127	Q₁₆×C₃₁	—	2	2	no	—
128	Q₁₆×C₃₁	—	2	1	no	—
129	Q₁₆×C₃₁	—	2	1	no	—
130	Q₁₆×C₃₁	—	2	1	no	—
131	Q₁₆×C₃₁	—	4	2	no	—
132	Q₁₆×C₃₁	—	4	4	no	—
133	SD₁₆×C₃₁	—	4	2	no	—
134	SD₁₆×C₃₁	—	4	2	no	—
135	SD₁₆×C₃₁	—	4	4	no	—
136	SD₁₆×C₃₁	—	4	4	no	—
137	D₈×C₃₁	—	2	1	no	—
138	D₈×C₃₁	—	2	2	no	—
139	D₈×C₃₁	—	2	1	no	—
140	D₈×C₃₁	—	2	1	no	—
141	D₈×C₃₁	—	2	2	no	—
142	D₈×C₃₁	—	2	1	no	—
143	D₈×C₃₁	—	4	2	no	—
144	D₈×C₃₁	—	4	4	no	—
145	M₁₆×C₃₁	D₈	1	4	yes	—
146	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
148	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
149	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
150	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
151	C₄⋊C₄×C₃₁	—	4	4	no	—
152	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
153	C₂²⋊C₄×C₃₁	—	4	4	no	—
154	C₂²⋊C₄×C₃₁	—	4	4	no	—
155	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
156	C₄²×C₃₁	C₂×D₄	1	2	yes	—
157	C₁₆×C₃₁	C₂×D₄	1	2	yes	—
158	C₁₆×C₃₁	C₂×D₄	1	2	yes	—
159	C₁₆×C₃₁	C₂×D₄	1	4	yes	—
160	#33722	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₃₁	—	4	1	no	—
162	C₂³×C₃₁	—	4	2	no	—
163	C₂³×C₃₁	—	4	1	no	—
164	C₂³×C₃₁	—	8	2	no	—
165	C₂³×C₃₁	—	16	2	no	—
166	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
167	Q₈×C₃₁	—	4	1	no	—
168	Q₈×C₃₁	—	4	1	no	—
169	Q₈×C₃₁	—	4	2	no	—
170	Q₈×C₃₁	—	4	1	no	—
171	Q₈×C₃₁	—	4	2	no	—
172	Q₈×C₃₁	—	4	1	no	—
173	Q₈×C₃₁	—	8	2	no	—
174	D₄×C₃₁	—	4	2	no	—
175	D₄×C₃₁	—	4	1	no	—
176	D₄×C₃₁	—	4	1	no	—
177	D₄×C₃₁	—	4	1	no	—
178	D₄×C₃₁	—	4	2	no	—
179	D₄×C₃₁	—	4	1	no	—
180	D₄×C₃₁	—	8	2	no	—
181	D₄×C₃₁	—	8	2	no	—
182	D₄×C₃₁	—	8	2	no	—
183	D₄×C₃₁	—	16	4	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	8	2	no	—
188	C₂×C₄×C₃₁	—	8	2	no	—
189	C₂×C₄×C₃₁	—	8	2	no	—
190	C₂×C₄×C₃₁	#127	1	2	yes	—
191	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
192	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
193	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
195	C₈×C₃₁	#127	1	2	yes	—
196	C₈×C₃₁	C₂×D₈	1	4	yes	—
197	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
198	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
199	C₂×#510	C₂×C₃₁	1	1	yes	characteristic, complemented
200	C₂×#509	C₂×C₃₁	1	1	yes	characteristic, complemented
201	C₂×#508	C₂×C₃₁	1	1	yes	characteristic, complemented
202	C₂²×SD₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#3156	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#3155	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#2251	C₂×C₃₁	1	2	yes	complemented
206	#2250	C₂×C₃₁	1	2	yes	complemented
207	#2249	C₂×C₃₁	1	2	yes	complemented
208	#2248	C₂×C₃₁	1	2	yes	complemented
209	#2108	C₂×C₃₁	1	1	yes	characteristic, complemented
210	C₂²×C₃₁	—	2	2	no	—
211	C₂²×C₃₁	—	8	1	no	—
212	C₂²×C₃₁	—	8	1	no	—
213	C₂²×C₃₁	—	8	2	no	—
214	C₂²×C₃₁	—	8	2	no	—
215	C₂²×C₃₁	—	8	4	no	—
216	C₂²×C₃₁	—	8	2	no	—
217	C₂²×C₃₁	—	16	4	no	—
218	C₂²×C₃₁	#493	1	2	yes	—
219	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
220	C₄×C₃₁	—	2	4	no	—
221	C₄×C₃₁	—	8	1	no	—
222	C₄×C₃₁	—	8	1	no	—
223	C₄×C₃₁	—	8	2	no	—
224	C₄×C₃₁	#493	1	2	yes	—
225	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
226	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
227	C₂×#137	—	2	1	no	—
228	C₂×AGL(1,ℤ/8)	—	2	1	no	—
229	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
231	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
232	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
233	#536	—	2	1	no	—
234	#535	—	2	1	no	—
235	#510	—	2	4	no	—
236	#509	—	2	4	no	—
237	#508	—	2	4	no	—
238	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic, complemented
239	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	complemented
240	C₂×SD₃₂	—	2	4	no	—
241	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
242	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
243	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
244	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic, complemented
245	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic, complemented
246	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	complemented
247	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
248	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
249	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
250	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
251	#497	—	2	2	no	—
252	#496	—	2	2	no	—
253	#495	C₂²×C₃₁	1	2	yes	—
254	#495	C₂²×C₃₁	1	2	yes	complemented
255	#494	C₂²×C₃₁	1	2	yes	complemented
256	#493	C₂²×C₃₁	1	2	yes	—
257	#405	C₂²×C₃₁	1	1	yes	characteristic
258	#346	C₂²×C₃₁	1	2	yes	complemented
259	C₄×C₁₆	C₂²×C₃₁	1	2	yes	complemented
260	C₂×C₃₁	—	2	2	no	—
261	C₂×C₃₁	—	16	1	no	—
262	C₂×C₃₁	—	16	1	no	—
263	C₂×C₃₁	—	16	2	no	—
264	C₂×C₃₁	C₂×#493	1	1	yes	characteristic
265	C₂×C₃₁	#3416	1	2	yes	—
266	C₂×#41	—	2	1	no	—
267	C₂×#41	—	2	1	no	—
268	C₂²×Q₈	—	2	1	no	—
269	C₂²×D₄	—	2	1	no	—
270	#137	—	4	4	no	—
271	AGL(1,ℤ/8)	—	4	4	no	—
272	#135	—	2	2	no	—
273	#135	—	2	2	no	—
274	#135	—	2	2	no	—
275	#135	—	2	2	no	—
276	C₂×Q₁₆	—	2	2	no	—
277	C₂×Q₁₆	—	2	2	no	—
278	C₂×Q₁₆	—	2	1	no	—
279	C₂×Q₁₆	—	2	2	no	—
280	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
281	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
282	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
283	C₂×SD₁₆	—	2	1	no	—
284	C₂×SD₁₆	—	2	1	no	—
285	C₂×SD₁₆	—	2	2	no	—
286	C₂×SD₁₆	—	2	2	no	—
287	C₂×D₈	—	2	2	no	—
288	C₂×D₈	—	2	2	no	—
289	C₂×D₈	—	2	1	no	—
290	C₂×D₈	—	2	2	no	—
291	C₂×D₈	C₂³×C₃₁	1	2	yes	—
292	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
293	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
294	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
295	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
296	C₄⋊Q₈	—	2	2	no	—
297	#127	—	2	2	no	—
298	#124	—	2	2	no	—
299	#124	—	2	2	no	—
300	#123	—	4	2	no	—
301	C₂²⋊Q₈	—	4	2	no	—
302	#121	—	4	2	no	—
303	#120	—	4	2	no	—
304	#117	D₄×C₃₁	1	1	yes	characteristic
305	Q₃₂	—	2	2	no	—
306	Q₃₂	—	2	4	no	—
307	SD₃₂	—	2	4	no	—
308	SD₃₂	—	2	4	no	—
309	SD₃₂	—	2	4	no	—
310	SD₃₂	—	2	2	no	—
311	SD₃₂	—	2	2	no	—
312	SD₃₂	—	2	4	no	—
313	D₁₆	—	2	4	no	—
314	D₁₆	—	2	2	no	—
315	M₃₂	D₄×C₃₁	1	4	yes	complemented
316	C₂×C₁₆	D₄×C₃₁	1	4	yes	complemented
317	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
318	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
319	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
320	#97	D₄×C₃₁	1	2	yes	—
321	C₄×C₈	C₂³×C₃₁	1	2	yes	—
322	C₃₁	#33722	1	1	yes	characteristic, direct product factor, Sylow
323	C₂⁴	—	4	1	no	—
324	#41	—	4	2	no	—
325	#41	—	4	2	no	—
326	#41	—	4	2	no	—
327	#41	—	4	2	no	—
328	C₂×Q₈	—	2	1	no	—
329	C₂×Q₈	—	2	2	no	—
330	C₂×Q₈	—	2	1	no	—
331	C₂×Q₈	—	4	1	no	—
332	C₂×Q₈	—	4	2	no	—
333	C₂×Q₈	—	4	2	no	—
334	C₂×D₄	—	2	1	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	1	no	—
337	C₂×D₄	—	4	1	no	—
338	C₂×D₄	—	4	1	no	—
339	C₂×D₄	—	4	2	no	—
340	C₂×D₄	—	4	2	no	—
341	C₂×D₄	—	4	1	no	—
342	C₂×D₄	—	4	4	no	—
343	C₂²×C₄	—	4	1	no	—
344	C₂²×C₄	—	4	1	no	—
345	C₂²×C₄	—	4	1	no	—
346	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
347	Q₁₆	—	2	2	no	—
348	Q₁₆	—	2	1	no	—
349	Q₁₆	—	2	2	no	—
350	Q₁₆	—	2	1	no	—
351	Q₁₆	—	2	1	no	—
352	Q₁₆	—	2	1	no	—
353	Q₁₆	—	4	2	no	—
354	Q₁₆	—	4	4	no	—
355	SD₁₆	—	4	2	no	—
356	SD₁₆	—	4	2	no	—
357	SD₁₆	—	4	4	no	—
358	SD₁₆	—	4	4	no	—
359	D₈	—	2	1	no	—
360	D₈	—	2	2	no	—
361	D₈	—	2	1	no	—
362	D₈	—	2	1	no	—
363	D₈	—	2	2	no	—
364	D₈	—	2	1	no	—
365	D₈	—	4	2	no	—
366	D₈	—	4	4	no	—
367	M₁₆	D₈×C₃₁	1	4	yes	—
368	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
369	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
370	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
371	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
372	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
373	C₄⋊C₄	—	4	4	no	—
374	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
375	C₂²⋊C₄	—	4	4	no	—
376	C₂²⋊C₄	—	4	4	no	—
377	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
378	C₄²	C₂×D₄×C₃₁	1	2	yes	—
379	C₁₆	C₂×D₄×C₃₁	1	2	yes	—
380	C₁₆	C₂×D₄×C₃₁	1	2	yes	—
381	C₁₆	C₂×D₄×C₃₁	1	4	yes	—
382	C₂³	—	4	1	no	—
383	C₂³	—	4	2	no	—
384	C₂³	—	4	1	no	—
385	C₂³	—	8	2	no	—
386	C₂³	—	16	2	no	—
387	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
388	Q₈	—	4	1	no	—
389	Q₈	—	4	1	no	—
390	Q₈	—	4	2	no	—
391	Q₈	—	4	1	no	—
392	Q₈	—	4	2	no	—
393	Q₈	—	4	1	no	—
394	Q₈	—	8	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	1	no	—
397	D₄	—	4	1	no	—
398	D₄	—	4	1	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	1	no	—
401	D₄	—	8	2	no	—
402	D₄	—	8	2	no	—
403	D₄	—	8	2	no	—
404	D₄	—	16	4	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	8	2	no	—
409	C₂×C₄	—	8	2	no	—
410	C₂×C₄	—	8	2	no	—
411	C₂×C₄	C₃₁×#127	1	2	yes	—
412	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
413	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
414	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
415	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
416	C₈	C₃₁×#127	1	2	yes	—
417	C₈	C₂×D₈×C₃₁	1	4	yes	—
418	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
419	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
420	C₂²	—	2	2	no	—
421	C₂²	—	8	1	no	—
422	C₂²	—	8	1	no	—
423	C₂²	—	8	2	no	—
424	C₂²	—	8	2	no	—
425	C₂²	—	8	4	no	—
426	C₂²	—	8	2	no	—
427	C₂²	—	16	4	no	—
428	C₂²	C₃₁×#493	1	2	yes	—
429	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
430	C₄	—	2	4	no	—
431	C₄	—	8	1	no	—
432	C₄	—	8	1	no	—
433	C₄	—	8	2	no	—
434	C₄	C₃₁×#493	1	2	yes	—
435	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
436	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
437	C₂	—	2	2	no	—
438	C₂	—	16	2	no	—
439	C₂	—	16	1	no	—
440	C₂	—	16	1	no	—
441	C₂	C₂×C₃₁×#493	1	1	yes	characteristic
442	C₂	C₃₁×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#510	C₂	semidirect
2. C₂×C₃₁×#509	C₂	semidirect
3. C₂×C₃₁×#508	C₂	semidirect
4. C₂²×C₃₁×SD₃₂	C₂	semidirect
5. C₃₁×#3156	C₂	semidirect
6. C₃₁×#3155	C₂	semidirect
7. C₃₁×#2251	C₂	semidirect
8. C₃₁×#2250	C₂	semidirect
9. C₃₁×#2249	C₂	semidirect
10. C₃₁×#2248	C₂	semidirect
11. C₃₁×#2108	C₂	semidirect
23. C₂×C₃₁×Q₃₂	C₂²	semidirect
24. C₂×C₃₁×Q₃₂	C₂²	semidirect
29. C₂×C₃₁×SD₃₂	C₂²	semidirect
30. C₂×C₃₁×SD₃₂	C₂²	semidirect
31. C₂×C₃₁×SD₃₂	C₂²	semidirect
33. C₂×C₃₁×D₁₆	C₂²	semidirect
39. C₃₁×#495	C₂²	semidirect
40. C₃₁×#494	C₂²	semidirect
43. C₃₁×#346	C₂²	semidirect
44. C₄×C₁₆×C₃₁	C₂²	semidirect
94. C₃₁×M₃₂	D₄	semidirect
95. C₂×C₁₆×C₃₁	D₄	semidirect
160. #33722	C₃₁	direct (N × H)
199. C₂×#510	C₂×C₃₁	semidirect
200. C₂×#509	C₂×C₃₁	semidirect
201. C₂×#508	C₂×C₃₁	semidirect
202. C₂²×SD₃₂	C₂×C₃₁	semidirect
203. #3156	C₂×C₃₁	semidirect
204. #3155	C₂×C₃₁	semidirect
205. #2251	C₂×C₃₁	semidirect
206. #2250	C₂×C₃₁	semidirect
207. #2249	C₂×C₃₁	semidirect
208. #2248	C₂×C₃₁	semidirect
209. #2108	C₂×C₃₁	semidirect
238. C₂×Q₃₂	C₂²×C₃₁	semidirect
239. C₂×Q₃₂	C₂²×C₃₁	semidirect
244. C₂×SD₃₂	C₂²×C₃₁	semidirect
245. C₂×SD₃₂	C₂²×C₃₁	semidirect
246. C₂×SD₃₂	C₂²×C₃₁	semidirect
248. C₂×D₁₆	C₂²×C₃₁	semidirect
254. #495	C₂²×C₃₁	semidirect
255. #494	C₂²×C₃₁	semidirect
258. #346	C₂²×C₃₁	semidirect
259. C₄×C₁₆	C₂²×C₃₁	semidirect
315. M₃₂	D₄×C₃₁	semidirect
316. C₂×C₁₆	D₄×C₃₁	semidirect
322. C₃₁	#33722	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×#510	2
C₂×C₃₁×#509	2
C₂×C₃₁×#508	2
C₂²×C₃₁×SD₃₂	2
C₃₁×#3156	2
C₃₁×#3155	2
C₃₁×#2251	2
C₃₁×#2250	2
C₃₁×#2249	2
C₃₁×#2248	2
C₃₁×#2108	2
#33722	31

Subgroups — C31×#33722 #21490481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33722 #21490481