Subgroups — C₃₁×#19701 #21476460

Order 7936.

854 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	C₃₁×#3227	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3206	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1721	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1720	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1717	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1715	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1452	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	—	2	2	no	—
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
17	C₃₁×#498	C₂²	1	2	yes	complemented
18	C₃₁×#495	—	2	2	no	—
19	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
20	C₃₁×#493	C₂²	1	2	yes	—
21	C₃₁×#493	C₂²	1	2	yes	—
22	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#492	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×#492	C₂²	1	1	yes	characteristic, complemented
25	C₃₁×#492	C₂²	1	2	yes	complemented
26	C₃₁×#487	C₂²	1	2	yes	complemented
27	C₃₁×#486	—	2	2	no	—
28	C₃₁×#486	C₂²	1	2	yes	—
29	C₃₁×#486	C₂²	1	2	yes	—
30	C₃₁×#486	C₂²	1	2	yes	—
31	C₃₁×#486	C₂²	1	2	yes	—
32	C₃₁×#486	C₂²	1	2	yes	complemented
33	C₃₁×#481	—	2	2	no	—
34	C₃₁×#476	—	2	2	no	—
35	C₃₁×#461	—	2	2	no	—
36	C₃₁×#459	—	2	2	no	—
37	C₃₁×#459	—	2	2	no	—
38	C₃₁×#448	—	2	2	no	—
39	C₃₁×#447	—	2	2	no	—
40	C₃₁×#447	—	2	2	no	—
41	C₃₁×#431	C₂²	1	1	yes	characteristic
42	C₃₁×#431	C₂²	1	1	yes	characteristic
43	C₂×C₃₁×#102	—	2	2	no	—
44	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#322	C₂²	1	2	yes	complemented
46	C₃₁×#322	C₂²	1	2	yes	complemented
47	C₂×#41×C₃₁	—	2	1	no	—
48	C₂×#41×C₃₁	—	2	1	no	—
49	C₂²×Q₈×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	—	2	1	no	—
53	C₃₁×#135	—	4	8	no	—
54	C₂×Q₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×SD₁₆×C₃₁	—	2	2	no	—
57	C₂×SD₁₆×C₃₁	—	2	4	no	—
58	C₂×D₈×C₃₁	—	2	2	no	—
59	C₂×D₈×C₃₁	—	2	4	no	—
60	C₂×D₈×C₃₁	—	2	4	no	—
61	C₂²×C₈×C₃₁	D₄	1	2	yes	—
62	C₃₁×C₄⋊Q₈	—	2	1	no	—
63	C₃₁×C₄⋊Q₈	—	2	1	no	—
64	C₃₁×C₄⋊Q₈	—	2	1	no	—
65	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
66	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
67	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
68	C₃₁×#127	—	2	1	no	—
69	C₃₁×#127	—	2	1	no	—
70	C₃₁×#127	—	2	1	no	—
71	C₃₁×#127	C₂³	1	2	yes	—
72	C₃₁×#127	C₂³	1	1	yes	characteristic
73	C₃₁×#127	C₂³	1	1	yes	characteristic
74	C₃₁×#124	—	2	1	no	—
75	C₃₁×#124	—	2	1	no	—
76	C₃₁×#121	—	2	1	no	—
77	C₃₁×#121	—	2	1	no	—
78	C₃₁×#121	—	2	1	no	—
79	C₃₁×#121	—	2	1	no	—
80	C₄×D₄×C₃₁	—	2	1	no	—
81	C₄×D₄×C₃₁	—	2	1	no	—
82	C₄×D₄×C₃₁	—	2	1	no	—
83	C₄×D₄×C₃₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
87	C₃₁×#106	—	2	4	no	—
88	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
89	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
90	C₃₁×#103	—	2	4	no	—
91	C₃₁×#102	—	2	4	no	—
92	C₃₁×#102	—	2	4	no	—
93	C₃₁×#102	—	2	4	no	—
94	C₃₁×#102	—	2	4	no	—
95	C₃₁×#102	—	2	4	no	—
96	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
97	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
98	C₄×C₈×C₃₁	D₄	1	2	yes	—
99	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₃₁	C₂³	1	2	yes	—
102	C₄×C₈×C₃₁	C₂³	1	2	yes	—
103	C₂⁴×C₃₁	—	4	1	no	—
104	#41×C₃₁	—	4	4	no	—
105	#41×C₃₁	—	4	4	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	—	4	1	no	—
110	C₂×Q₈×C₃₁	—	8	2	no	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	1	no	—
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	—	4	1	no	—
121	C₂×D₄×C₃₁	—	4	1	no	—
122	C₂×D₄×C₃₁	—	4	1	no	—
123	C₂×D₄×C₃₁	—	4	1	no	—
124	C₂×D₄×C₃₁	—	4	1	no	—
125	C₂×D₄×C₃₁	—	4	4	no	—
126	C₂×D₄×C₃₁	—	8	2	no	—
127	C₂×D₄×C₃₁	—	8	2	no	—
128	C₂²×C₄×C₃₁	—	4	1	no	—
129	C₂²×C₄×C₃₁	—	4	1	no	—
130	C₂²×C₄×C₃₁	—	4	1	no	—
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	Q₁₆×C₃₁	—	4	4	no	—
135	SD₁₆×C₃₁	—	4	4	no	—
136	SD₁₆×C₃₁	—	4	4	no	—
137	SD₁₆×C₃₁	—	8	4	no	—
138	D₈×C₃₁	—	4	4	no	—
139	D₈×C₃₁	—	8	4	no	—
140	D₈×C₃₁	—	8	4	no	—
141	C₂×C₈×C₃₁	—	2	4	no	—
142	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
144	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
145	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	2	1	no	—
151	C₄⋊C₄×C₃₁	—	2	1	no	—
152	C₄⋊C₄×C₃₁	—	4	1	no	—
153	C₄⋊C₄×C₃₁	—	4	1	no	—
154	C₂²⋊C₄×C₃₁	—	4	1	no	—
155	C₂²⋊C₄×C₃₁	—	4	1	no	—
156	C₂²⋊C₄×C₃₁	—	4	1	no	—
157	C₂²⋊C₄×C₃₁	—	4	1	no	—
158	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
159	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
162	#19701	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂³×C₃₁	—	4	1	no	—
164	C₂³×C₃₁	—	4	1	no	—
165	C₂³×C₃₁	—	4	2	no	—
166	C₂³×C₃₁	—	8	2	no	—
167	C₂³×C₃₁	—	8	2	no	—
168	C₂³×C₃₁	—	8	2	no	—
169	C₂³×C₃₁	#127	1	1	yes	characteristic
170	Q₈×C₃₁	—	4	2	no	—
171	Q₈×C₃₁	—	4	2	no	—
172	Q₈×C₃₁	—	4	4	no	—
173	Q₈×C₃₁	—	8	2	no	—
174	D₄×C₃₁	—	4	2	no	—
175	D₄×C₃₁	—	4	2	no	—
176	D₄×C₃₁	—	4	4	no	—
177	D₄×C₃₁	—	4	2	no	—
178	D₄×C₃₁	—	4	4	no	—
179	D₄×C₃₁	—	4	2	no	—
180	D₄×C₃₁	—	4	2	no	—
181	D₄×C₃₁	—	4	4	no	—
182	D₄×C₃₁	—	4	2	no	—
183	D₄×C₃₁	—	8	2	no	—
184	D₄×C₃₁	—	8	2	no	—
185	D₄×C₃₁	—	8	2	no	—
186	D₄×C₃₁	—	8	2	no	—
187	D₄×C₃₁	—	8	2	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	8	2	no	—
194	C₂×C₄×C₃₁	—	8	2	no	—
195	C₂×C₄×C₃₁	—	8	2	no	—
196	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
197	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
198	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
199	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
200	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
203	C₈×C₃₁	—	2	2	no	—
204	C₈×C₃₁	—	2	2	no	—
205	C₈×C₃₁	—	2	4	no	—
206	C₈×C₃₁	—	2	4	no	—
207	#3227	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#3206	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#3205	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#3205	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#3147	C₂×C₃₁	1	1	yes	characteristic, complemented
212	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#1721	C₂×C₃₁	1	2	yes	complemented
214	#1720	C₂×C₃₁	1	2	yes	complemented
215	#1717	C₂×C₃₁	1	2	yes	complemented
216	#1715	C₂×C₃₁	1	2	yes	complemented
217	#1452	C₂×C₃₁	1	1	yes	characteristic, complemented
218	C₂²×C₃₁	—	2	1	no	—
219	C₂²×C₃₁	—	2	1	no	—
220	C₂²×C₃₁	—	8	1	no	—
221	C₂²×C₃₁	—	8	1	no	—
222	C₂²×C₃₁	—	8	2	no	—
223	C₂²×C₃₁	—	8	2	no	—
224	C₂²×C₃₁	—	8	1	no	—
225	C₂²×C₃₁	—	8	1	no	—
226	C₂²×C₃₁	—	8	2	no	—
227	C₂²×C₃₁	—	8	1	no	—
228	C₂²×C₃₁	—	8	1	no	—
229	C₂²×C₃₁	—	16	4	no	—
230	C₂²×C₃₁	#496	1	2	yes	—
231	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
232	C₄×C₃₁	—	2	1	no	—
233	C₄×C₃₁	—	2	1	no	—
234	C₄×C₃₁	—	8	1	no	—
235	C₄×C₃₁	—	8	1	no	—
236	C₄×C₃₁	—	8	2	no	—
237	C₄×C₃₁	#496	1	2	yes	—
238	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
239	C₄×C₃₁	#576	1	2	yes	—
240	C₄×C₃₁	#577	1	2	yes	—
241	C₂×#135	—	2	2	no	—
242	#532	C₂²×C₃₁	1	1	yes	characteristic
243	#532	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
246	#498	C₂²×C₃₁	1	2	yes	complemented
247	#495	—	2	2	no	—
248	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
249	#493	C₂²×C₃₁	1	2	yes	—
250	#493	C₂²×C₃₁	1	2	yes	—
251	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
252	#492	C₂²×C₃₁	1	1	yes	characteristic, complemented
253	#492	C₂²×C₃₁	1	1	yes	characteristic, complemented
254	#492	C₂²×C₃₁	1	2	yes	complemented
255	#487	C₂²×C₃₁	1	2	yes	complemented
256	#486	—	2	2	no	—
257	#486	C₂²×C₃₁	1	2	yes	—
258	#486	C₂²×C₃₁	1	2	yes	—
259	#486	C₂²×C₃₁	1	2	yes	—
260	#486	C₂²×C₃₁	1	2	yes	—
261	#486	C₂²×C₃₁	1	2	yes	complemented
262	#481	—	2	2	no	—
263	#476	—	2	2	no	—
264	#461	—	2	2	no	—
265	#459	—	2	2	no	—
266	#459	—	2	2	no	—
267	#448	—	2	2	no	—
268	#447	—	2	2	no	—
269	#447	—	2	2	no	—
270	#431	C₂²×C₃₁	1	1	yes	characteristic
271	#431	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×#102	—	2	2	no	—
273	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
274	#322	C₂²×C₃₁	1	2	yes	complemented
275	#322	C₂²×C₃₁	1	2	yes	complemented
276	C₂×C₃₁	—	2	2	no	—
277	C₂×C₃₁	—	16	1	no	—
278	C₂×C₃₁	—	16	1	no	—
279	C₂×C₃₁	—	16	2	no	—
280	C₂×C₃₁	C₂×#496	1	1	yes	characteristic
281	C₂×C₃₁	#3154	1	1	yes	characteristic
282	C₂×C₃₁	#3155	1	1	yes	characteristic
283	C₂×#41	—	2	1	no	—
284	C₂×#41	—	2	1	no	—
285	C₂²×Q₈	—	2	1	no	—
286	C₂²×D₄	—	2	1	no	—
287	C₂²×D₄	—	2	1	no	—
288	C₂²×D₄	—	2	1	no	—
289	#135	—	4	8	no	—
290	C₂×Q₁₆	—	2	2	no	—
291	C₂×SD₁₆	—	2	2	no	—
292	C₂×SD₁₆	—	2	2	no	—
293	C₂×SD₁₆	—	2	4	no	—
294	C₂×D₈	—	2	2	no	—
295	C₂×D₈	—	2	4	no	—
296	C₂×D₈	—	2	4	no	—
297	C₂²×C₈	D₄×C₃₁	1	2	yes	—
298	C₄⋊Q₈	—	2	1	no	—
299	C₄⋊Q₈	—	2	1	no	—
300	C₄⋊Q₈	—	2	1	no	—
301	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
302	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
303	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
304	#127	—	2	1	no	—
305	#127	—	2	1	no	—
306	#127	—	2	1	no	—
307	#127	C₂³×C₃₁	1	2	yes	—
308	#127	C₂³×C₃₁	1	1	yes	characteristic
309	#127	C₂³×C₃₁	1	1	yes	characteristic
310	#124	—	2	1	no	—
311	#124	—	2	1	no	—
312	#121	—	2	1	no	—
313	#121	—	2	1	no	—
314	#121	—	2	1	no	—
315	#121	—	2	1	no	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	—	2	1	no	—
319	C₄×D₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
323	#106	—	2	4	no	—
324	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
325	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
326	#103	—	2	4	no	—
327	#102	—	2	4	no	—
328	#102	—	2	4	no	—
329	#102	—	2	4	no	—
330	#102	—	2	4	no	—
331	#102	—	2	4	no	—
332	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
333	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
334	C₄×C₈	D₄×C₃₁	1	2	yes	—
335	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
336	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
337	C₄×C₈	C₂³×C₃₁	1	2	yes	—
338	C₄×C₈	C₂³×C₃₁	1	2	yes	—
339	C₃₁	#19701	1	1	yes	characteristic, direct product factor, Sylow
340	C₂⁴	—	4	1	no	—
341	#41	—	4	4	no	—
342	#41	—	4	4	no	—
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	—	2	1	no	—
345	C₂×Q₈	—	2	1	no	—
346	C₂×Q₈	—	4	1	no	—
347	C₂×Q₈	—	8	2	no	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	—	2	2	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	4	1	no	—
358	C₂×D₄	—	4	1	no	—
359	C₂×D₄	—	4	1	no	—
360	C₂×D₄	—	4	1	no	—
361	C₂×D₄	—	4	1	no	—
362	C₂×D₄	—	4	4	no	—
363	C₂×D₄	—	8	2	no	—
364	C₂×D₄	—	8	2	no	—
365	C₂²×C₄	—	4	1	no	—
366	C₂²×C₄	—	4	1	no	—
367	C₂²×C₄	—	4	1	no	—
368	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
369	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
371	Q₁₆	—	4	4	no	—
372	SD₁₆	—	4	4	no	—
373	SD₁₆	—	4	4	no	—
374	SD₁₆	—	8	4	no	—
375	D₈	—	4	4	no	—
376	D₈	—	8	4	no	—
377	D₈	—	8	4	no	—
378	C₂×C₈	—	2	4	no	—
379	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
380	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
381	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
382	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	4	1	no	—
390	C₄⋊C₄	—	4	1	no	—
391	C₂²⋊C₄	—	4	1	no	—
392	C₂²⋊C₄	—	4	1	no	—
393	C₂²⋊C₄	—	4	1	no	—
394	C₂²⋊C₄	—	4	1	no	—
395	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
396	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
397	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
398	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
399	C₂³	—	4	1	no	—
400	C₂³	—	4	1	no	—
401	C₂³	—	4	2	no	—
402	C₂³	—	8	2	no	—
403	C₂³	—	8	2	no	—
404	C₂³	—	8	2	no	—
405	C₂³	C₃₁×#127	1	1	yes	characteristic
406	Q₈	—	4	2	no	—
407	Q₈	—	4	2	no	—
408	Q₈	—	4	4	no	—
409	Q₈	—	8	2	no	—
410	D₄	—	4	2	no	—
411	D₄	—	4	2	no	—
412	D₄	—	4	4	no	—
413	D₄	—	4	2	no	—
414	D₄	—	4	4	no	—
415	D₄	—	4	2	no	—
416	D₄	—	4	2	no	—
417	D₄	—	4	4	no	—
418	D₄	—	4	2	no	—
419	D₄	—	8	2	no	—
420	D₄	—	8	2	no	—
421	D₄	—	8	2	no	—
422	D₄	—	8	2	no	—
423	D₄	—	8	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	8	2	no	—
430	C₂×C₄	—	8	2	no	—
431	C₂×C₄	—	8	2	no	—
432	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
433	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
434	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
435	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
436	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
437	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
438	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
439	C₈	—	2	2	no	—
440	C₈	—	2	2	no	—
441	C₈	—	2	4	no	—
442	C₈	—	2	4	no	—
443	C₂²	—	2	1	no	—
444	C₂²	—	2	1	no	—
445	C₂²	—	8	1	no	—
446	C₂²	—	8	1	no	—
447	C₂²	—	8	2	no	—
448	C₂²	—	8	2	no	—
449	C₂²	—	8	1	no	—
450	C₂²	—	8	1	no	—
451	C₂²	—	8	2	no	—
452	C₂²	—	8	1	no	—
453	C₂²	—	8	1	no	—
454	C₂²	—	16	4	no	—
455	C₂²	C₃₁×#496	1	2	yes	—
456	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
457	C₄	—	2	1	no	—
458	C₄	—	2	1	no	—
459	C₄	—	8	1	no	—
460	C₄	—	8	1	no	—
461	C₄	—	8	2	no	—
462	C₄	C₃₁×#496	1	2	yes	—
463	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
464	C₄	C₃₁×#576	1	2	yes	—
465	C₄	C₃₁×#577	1	2	yes	—
466	C₂	—	2	2	no	—
467	C₂	—	16	2	no	—
468	C₂	—	16	1	no	—
469	C₂	—	16	1	no	—
470	C₂	C₂×C₃₁×#496	1	1	yes	characteristic
471	C₂	C₃₁×#3154	1	1	yes	characteristic
472	C₂	C₃₁×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3227	C₂	semidirect
2. C₃₁×#3206	C₂	semidirect
3. C₃₁×#3205	C₂	semidirect
4. C₃₁×#3205	C₂	semidirect
5. C₃₁×#3147	C₂	semidirect
6. C₂×C₃₁×#486	C₂	semidirect
7. C₃₁×#1721	C₂	semidirect
8. C₃₁×#1720	C₂	semidirect
9. C₃₁×#1717	C₂	semidirect
10. C₃₁×#1715	C₂	semidirect
11. C₃₁×#1452	C₂	semidirect
17. C₃₁×#498	C₂²	semidirect
19. C₃₁×#494	C₂²	semidirect
22. C₃₁×#493	C₂²	semidirect
23. C₃₁×#492	C₂²	semidirect
24. C₃₁×#492	C₂²	semidirect
25. C₃₁×#492	C₂²	semidirect
26. C₃₁×#487	C₂²	semidirect
32. C₃₁×#486	C₂²	semidirect
45. C₃₁×#322	C₂²	semidirect
46. C₃₁×#322	C₂²	semidirect
162. #19701	C₃₁	direct (N × H)
207. #3227	C₂×C₃₁	semidirect
208. #3206	C₂×C₃₁	semidirect
209. #3205	C₂×C₃₁	semidirect
210. #3205	C₂×C₃₁	semidirect
211. #3147	C₂×C₃₁	semidirect
212. C₂×#486	C₂×C₃₁	semidirect
213. #1721	C₂×C₃₁	semidirect
214. #1720	C₂×C₃₁	semidirect
215. #1717	C₂×C₃₁	semidirect
216. #1715	C₂×C₃₁	semidirect
217. #1452	C₂×C₃₁	semidirect
246. #498	C₂²×C₃₁	semidirect
248. #494	C₂²×C₃₁	semidirect
251. #493	C₂²×C₃₁	semidirect
252. #492	C₂²×C₃₁	semidirect
253. #492	C₂²×C₃₁	semidirect
254. #492	C₂²×C₃₁	semidirect
255. #487	C₂²×C₃₁	semidirect
261. #486	C₂²×C₃₁	semidirect
274. #322	C₂²×C₃₁	semidirect
275. #322	C₂²×C₃₁	semidirect
339. C₃₁	#19701	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×#3227	2
C₃₁×#3206	2
C₃₁×#3205	2
C₃₁×#3147	2
C₂×C₃₁×#486	2
C₃₁×#1721	2
C₃₁×#1720	2
C₃₁×#1717	2
C₃₁×#1715	2
C₃₁×#1452	2
#19701	31

Subgroups — C31×#19701 #21476460

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19701 #21476460