Subgroups — C₃₁×#26996 #21483755

Order 7936.

812 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3099	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3062	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3041	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2946	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2090	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2037	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2018	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2006	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1972	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1938	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1931	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1907	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1883	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1856	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1765	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
19	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
21	C₃₁×#537	C₂²	1	1	yes	characteristic
22	C₃₁×#535	C₂²	1	1	yes	characteristic
23	C₃₁×#520	C₂²	1	1	yes	characteristic
24	C₃₁×#518	C₂²	1	1	yes	characteristic
25	C₃₁×#485	—	2	1	no	—
26	C₃₁×#483	—	2	1	no	—
27	C₃₁×#482	—	2	1	no	—
28	C₃₁×#481	—	2	1	no	—
29	C₃₁×#469	—	2	1	no	—
30	C₃₁×#468	—	2	1	no	—
31	C₃₁×#466	—	2	1	no	—
32	C₃₁×#465	—	2	1	no	—
33	C₃₁×#463	—	2	1	no	—
34	C₃₁×#463	—	2	1	no	—
35	C₃₁×#459	—	2	1	no	—
36	C₃₁×#459	—	2	1	no	—
37	C₃₁×#442	—	2	1	no	—
38	C₃₁×#440	—	2	1	no	—
39	C₄×SD₁₆×C₃₁	—	2	1	no	—
40	C₄×D₈×C₃₁	—	2	1	no	—
41	C₃₁×#428	C₂²	1	1	yes	characteristic
42	C₃₁×#428	C₂²	1	1	yes	characteristic
43	C₃₁×#428	C₂²	1	1	yes	characteristic
44	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
45	C₃₁×#424	C₂²	1	1	yes	characteristic
46	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
47	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
48	C₃₁×#418	C₂²	1	1	yes	characteristic
49	C₃₁×#417	C₂²	1	1	yes	characteristic
50	C₃₁×#416	C₂²	1	1	yes	characteristic
51	C₃₁×#416	C₂²	1	1	yes	characteristic
52	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
53	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
54	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
55	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
56	C₃₁×#408	C₂²	1	1	yes	characteristic
57	C₃₁×#405	C₂²	1	1	yes	characteristic
58	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
59	C₃₁×#390	C₂²	1	1	yes	characteristic
60	C₃₁×#390	C₂²	1	1	yes	characteristic
61	C₃₁×#343	C₂²	1	1	yes	characteristic
62	C₃₁×#343	C₂²	1	1	yes	characteristic
63	C₃₁×#341	C₂²	1	1	yes	characteristic
64	C₃₁×#341	C₂²	1	1	yes	characteristic
65	C₃₁×#339	C₂²	1	1	yes	characteristic
66	C₃₁×#339	C₂²	1	1	yes	characteristic
67	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
72	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
73	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
74	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
75	C₃₁×AGL(1,ℤ/8)	—	4	1	no	—
76	C₃₁×AGL(1,ℤ/8)	—	4	1	no	—
77	C₂×SD₁₆×C₃₁	—	2	1	no	—
78	C₂×SD₁₆×C₃₁	—	2	1	no	—
79	C₂×SD₁₆×C₃₁	—	2	1	no	—
80	C₂×D₈×C₃₁	—	2	1	no	—
81	C₂×D₈×C₃₁	—	2	1	no	—
82	C₂×D₈×C₃₁	—	2	1	no	—
83	C₂×D₈×C₃₁	—	2	2	no	—
84	C₂×D₈×C₃₁	—	2	1	no	—
85	C₂×D₈×C₃₁	—	2	1	no	—
86	C₂×D₈×C₃₁	—	4	1	no	—
87	C₃₁×#131	—	2	2	no	—
88	C₃₁×#131	—	4	1	no	—
89	C₂×M₁₆×C₃₁	—	2	1	no	—
90	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
91	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
92	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
93	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
94	C₂²×C₈×C₃₁	—	2	1	no	—
95	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
96	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
97	C₃₁×#127	—	2	1	no	—
98	C₃₁×#124	—	2	1	no	—
99	C₃₁×C₂²⋊Q₈	—	2	1	no	—
100	C₃₁×C₂²⋊Q₈	—	2	1	no	—
101	C₃₁×#121	—	2	1	no	—
102	C₃₁×#121	—	2	1	no	—
103	C₃₁×#121	—	2	1	no	—
104	C₃₁×#121	—	2	1	no	—
105	C₃₁×#121	—	4	1	no	—
106	C₃₁×#120	—	4	1	no	—
107	C₄×Q₈×C₃₁	—	2	1	no	—
108	C₄×D₄×C₃₁	—	2	1	no	—
109	C₄×D₄×C₃₁	—	2	1	no	—
110	C₄×D₄×C₃₁	—	2	1	no	—
111	C₄×D₄×C₃₁	—	2	1	no	—
112	C₄×D₄×C₃₁	—	2	1	no	—
113	C₄×D₄×C₃₁	—	4	1	no	—
114	C₃₁×#117	—	2	1	no	—
115	C₃₁×#117	C₂³	1	1	yes	characteristic
116	C₃₁×#117	C₂³	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
118	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
120	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
121	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
122	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
123	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
124	C₃₁×#107	—	2	1	no	—
125	C₃₁×#107	—	2	1	no	—
126	C₃₁×#107	—	2	1	no	—
127	C₃₁×#106	—	2	1	no	—
128	C₃₁×#106	—	2	1	no	—
129	C₃₁×#106	—	2	1	no	—
130	C₃₁×#106	—	2	1	no	—
131	C₃₁×#106	—	2	1	no	—
132	C₃₁×C₄⋊C₈	—	2	1	no	—
133	C₃₁×C₄⋊C₈	—	2	1	no	—
134	C₃₁×#104	—	2	2	no	—
135	C₃₁×#104	—	4	1	no	—
136	C₃₁×#103	—	2	1	no	—
137	C₃₁×#103	D₄	1	1	yes	characteristic
138	C₃₁×#103	D₄	1	1	yes	characteristic
139	C₃₁×#103	D₄	1	1	yes	characteristic
140	C₃₁×#103	D₄	1	1	yes	characteristic
141	C₃₁×#102	—	2	1	no	—
142	C₃₁×#102	—	2	1	no	—
143	C₃₁×#102	—	2	1	no	—
144	C₃₁×#102	—	2	1	no	—
145	C₃₁×#102	—	2	1	no	—
146	C₃₁×#102	—	2	1	no	—
147	C₃₁×#102	—	2	1	no	—
148	C₃₁×#102	D₄	1	1	yes	characteristic
149	C₃₁×#102	D₄	1	1	yes	characteristic
150	C₃₁×#102	D₄	1	1	yes	characteristic
151	C₃₁×#102	D₄	1	1	yes	characteristic
152	C₃₁×#100	—	2	2	no	—
153	C₃₁×#100	—	2	2	no	—
154	C₃₁×C₂²⋊C₈	—	2	1	no	—
155	C₃₁×C₂²⋊C₈	—	2	1	no	—
156	C₃₁×#97	—	2	1	no	—
157	C₃₁×#97	—	4	1	no	—
158	C₄×C₈×C₃₁	—	2	1	no	—
159	C₂⁴×C₃₁	—	2	1	no	—
160	C₂⁴×C₃₁	—	2	1	no	—
161	#41×C₃₁	—	2	1	no	—
162	#41×C₃₁	—	2	1	no	—
163	#41×C₃₁	—	2	1	no	—
164	#41×C₃₁	—	2	1	no	—
165	#41×C₃₁	—	2	1	no	—
166	#41×C₃₁	—	2	1	no	—
167	#41×C₃₁	—	2	1	no	—
168	#41×C₃₁	—	2	1	no	—
169	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
171	C₂×D₄×C₃₁	—	2	1	no	—
172	C₂×D₄×C₃₁	—	2	1	no	—
173	C₂×D₄×C₃₁	—	2	1	no	—
174	C₂×D₄×C₃₁	—	2	1	no	—
175	C₂×D₄×C₃₁	—	2	1	no	—
176	C₂×D₄×C₃₁	—	2	1	no	—
177	C₂×D₄×C₃₁	—	2	1	no	—
178	C₂×D₄×C₃₁	—	2	1	no	—
179	C₂×D₄×C₃₁	—	2	1	no	—
180	C₂×D₄×C₃₁	—	2	1	no	—
181	C₂×D₄×C₃₁	—	2	1	no	—
182	C₂×D₄×C₃₁	—	2	1	no	—
183	C₂×D₄×C₃₁	—	2	1	no	—
184	C₂×D₄×C₃₁	—	4	1	no	—
185	C₂×D₄×C₃₁	—	4	1	no	—
186	C₂×D₄×C₃₁	—	4	1	no	—
187	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
189	C₂×D₄×C₃₁	#41	1	2	yes	—
190	C₂²×C₄×C₃₁	—	2	1	no	—
191	C₂²×C₄×C₃₁	—	2	1	no	—
192	C₂²×C₄×C₃₁	—	2	1	no	—
193	C₂²×C₄×C₃₁	—	2	1	no	—
194	C₂²×C₄×C₃₁	—	2	1	no	—
195	C₂²×C₄×C₃₁	—	4	1	no	—
196	C₂²×C₄×C₃₁	—	4	1	no	—
197	C₂²×C₄×C₃₁	—	4	1	no	—
198	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
199	SD₁₆×C₃₁	—	2	4	no	—
200	SD₁₆×C₃₁	—	4	2	no	—
201	SD₁₆×C₃₁	—	4	2	no	—
202	D₈×C₃₁	—	2	4	no	—
203	D₈×C₃₁	—	4	2	no	—
204	D₈×C₃₁	—	4	2	no	—
205	D₈×C₃₁	—	4	2	no	—
206	D₈×C₃₁	—	4	2	no	—
207	M₁₆×C₃₁	—	2	1	no	—
208	M₁₆×C₃₁	—	2	1	no	—
209	M₁₆×C₃₁	—	2	1	no	—
210	M₁₆×C₃₁	—	2	1	no	—
211	M₁₆×C₃₁	—	2	1	no	—
212	M₁₆×C₃₁	—	2	1	no	—
213	M₁₆×C₃₁	—	4	2	no	—
214	M₁₆×C₃₁	#41	1	2	yes	—
215	M₁₆×C₃₁	#41	1	2	yes	—
216	C₂×C₈×C₃₁	—	2	1	no	—
217	C₂×C₈×C₃₁	—	2	1	no	—
218	C₂×C₈×C₃₁	—	2	1	no	—
219	C₂×C₈×C₃₁	—	2	1	no	—
220	C₂×C₈×C₃₁	—	2	1	no	—
221	C₂×C₈×C₃₁	—	2	1	no	—
222	C₂×C₈×C₃₁	—	2	1	no	—
223	C₂×C₈×C₃₁	—	2	1	no	—
224	C₂×C₈×C₃₁	—	4	2	no	—
225	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
226	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
227	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
228	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
229	C₄⋊C₄×C₃₁	—	2	1	no	—
230	C₄⋊C₄×C₃₁	—	2	1	no	—
231	C₄⋊C₄×C₃₁	—	2	1	no	—
232	C₄⋊C₄×C₃₁	—	2	1	no	—
233	C₄⋊C₄×C₃₁	—	4	1	no	—
234	C₄⋊C₄×C₃₁	—	4	1	no	—
235	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
236	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
237	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
238	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
239	C₂²⋊C₄×C₃₁	—	2	1	no	—
240	C₂²⋊C₄×C₃₁	—	2	1	no	—
241	C₂²⋊C₄×C₃₁	—	4	1	no	—
242	C₂²⋊C₄×C₃₁	—	4	1	no	—
243	C₂²⋊C₄×C₃₁	—	4	1	no	—
244	C₂²⋊C₄×C₃₁	—	4	1	no	—
245	C₂²⋊C₄×C₃₁	—	4	1	no	—
246	C₂²⋊C₄×C₃₁	—	4	1	no	—
247	C₂²⋊C₄×C₃₁	—	4	1	no	—
248	C₂²⋊C₄×C₃₁	—	8	2	no	—
249	C₂²⋊C₄×C₃₁	—	8	2	no	—
250	C₄²×C₃₁	—	2	1	no	—
251	C₄²×C₃₁	—	2	1	no	—
252	C₄²×C₃₁	—	2	1	no	—
253	C₄²×C₃₁	—	2	1	no	—
254	C₄²×C₃₁	—	4	1	no	—
255	#26996	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
256	C₂³×C₃₁	—	2	1	no	—
257	C₂³×C₃₁	—	2	1	no	—
258	C₂³×C₃₁	—	2	2	no	—
259	C₂³×C₃₁	—	4	1	no	—
260	C₂³×C₃₁	—	4	1	no	—
261	C₂³×C₃₁	—	4	1	no	—
262	C₂³×C₃₁	—	4	1	no	—
263	C₂³×C₃₁	—	4	1	no	—
264	C₂³×C₃₁	—	8	1	no	—
265	C₂³×C₃₁	—	8	1	no	—
266	C₂³×C₃₁	—	8	1	no	—
267	C₂³×C₃₁	—	8	1	no	—
268	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
269	Q₈×C₃₁	—	2	1	no	—
270	Q₈×C₃₁	—	2	1	no	—
271	Q₈×C₃₁	—	2	1	no	—
272	Q₈×C₃₁	—	2	1	no	—
273	D₄×C₃₁	—	2	1	no	—
274	D₄×C₃₁	—	2	2	no	—
275	D₄×C₃₁	—	2	1	no	—
276	D₄×C₃₁	—	2	2	no	—
277	D₄×C₃₁	—	2	1	no	—
278	D₄×C₃₁	—	2	1	no	—
279	D₄×C₃₁	—	4	1	no	—
280	D₄×C₃₁	—	4	1	no	—
281	D₄×C₃₁	—	4	1	no	—
282	D₄×C₃₁	—	4	1	no	—
283	D₄×C₃₁	—	4	1	no	—
284	D₄×C₃₁	—	4	1	no	—
285	D₄×C₃₁	—	4	1	no	—
286	D₄×C₃₁	—	4	1	no	—
287	D₄×C₃₁	—	4	1	no	—
288	D₄×C₃₁	—	4	1	no	—
289	D₄×C₃₁	—	16	1	no	—
290	D₄×C₃₁	—	16	1	no	—
291	D₄×C₃₁	—	16	1	no	—
292	C₂×C₄×C₃₁	—	2	1	no	—
293	C₂×C₄×C₃₁	—	2	1	no	—
294	C₂×C₄×C₃₁	—	2	1	no	—
295	C₂×C₄×C₃₁	—	2	1	no	—
296	C₂×C₄×C₃₁	—	2	1	no	—
297	C₂×C₄×C₃₁	—	2	1	no	—
298	C₂×C₄×C₃₁	—	4	1	no	—
299	C₂×C₄×C₃₁	—	4	1	no	—
300	C₂×C₄×C₃₁	—	4	1	no	—
301	C₂×C₄×C₃₁	—	4	1	no	—
302	C₂×C₄×C₃₁	—	4	1	no	—
303	C₂×C₄×C₃₁	—	4	1	no	—
304	C₂×C₄×C₃₁	—	4	1	no	—
305	C₂×C₄×C₃₁	—	4	1	no	—
306	C₂×C₄×C₃₁	—	4	1	no	—
307	C₂×C₄×C₃₁	—	8	1	no	—
308	C₂×C₄×C₃₁	—	8	1	no	—
309	C₂×C₄×C₃₁	—	16	1	no	—
310	C₂×C₄×C₃₁	—	16	1	no	—
311	C₂×C₄×C₃₁	—	16	1	no	—
312	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
313	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
314	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
315	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
316	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
317	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
318	C₈×C₃₁	—	2	2	no	—
319	C₈×C₃₁	—	2	1	no	—
320	C₈×C₃₁	—	2	1	no	—
321	C₈×C₃₁	—	2	1	no	—
322	C₈×C₃₁	—	2	1	no	—
323	C₈×C₃₁	—	2	2	no	—
324	C₈×C₃₁	—	4	1	no	—
325	C₈×C₃₁	—	4	1	no	—
326	#3099	C₂×C₃₁	1	1	yes	characteristic, complemented
327	#3062	C₂×C₃₁	1	1	yes	characteristic, complemented
328	#3041	C₂×C₃₁	1	1	yes	characteristic, complemented
329	#2946	C₂×C₃₁	1	1	yes	characteristic, complemented
330	#2090	C₂×C₃₁	1	1	yes	characteristic, complemented
331	#2037	C₂×C₃₁	1	1	yes	characteristic, complemented
332	#2018	C₂×C₃₁	1	1	yes	characteristic, complemented
333	#2006	C₂×C₃₁	1	1	yes	characteristic, complemented
334	#1972	C₂×C₃₁	1	1	yes	characteristic, complemented
335	#1938	C₂×C₃₁	1	1	yes	characteristic, complemented
336	#1931	C₂×C₃₁	1	1	yes	characteristic, complemented
337	#1907	C₂×C₃₁	1	1	yes	characteristic, complemented
338	#1883	C₂×C₃₁	1	1	yes	characteristic, complemented
339	#1856	C₂×C₃₁	1	1	yes	characteristic, complemented
340	#1765	C₂×C₃₁	1	1	yes	characteristic, complemented
341	C₂²×C₃₁	—	2	1	no	—
342	C₂²×C₃₁	—	2	1	no	—
343	C₂²×C₃₁	—	4	1	no	—
344	C₂²×C₃₁	—	4	1	no	—
345	C₂²×C₃₁	—	4	2	no	—
346	C₂²×C₃₁	—	4	1	no	—
347	C₂²×C₃₁	—	4	1	no	—
348	C₂²×C₃₁	—	4	1	no	—
349	C₂²×C₃₁	—	4	1	no	—
350	C₂²×C₃₁	—	8	2	no	—
351	C₂²×C₃₁	—	8	2	no	—
352	C₂²×C₃₁	—	8	1	no	—
353	C₂²×C₃₁	—	8	1	no	—
354	C₂²×C₃₁	—	8	1	no	—
355	C₂²×C₃₁	—	8	1	no	—
356	C₂²×C₃₁	—	8	1	no	—
357	C₂²×C₃₁	—	16	1	no	—
358	C₂²×C₃₁	—	16	1	no	—
359	C₂²×C₃₁	#540	1	1	yes	characteristic
360	C₂²×C₃₁	D₄²	1	1	yes	characteristic
361	C₂²×C₃₁	#553	1	1	yes	characteristic
362	C₄×C₃₁	—	4	1	no	—
363	C₄×C₃₁	—	4	1	no	—
364	C₄×C₃₁	—	4	1	no	—
365	C₄×C₃₁	—	4	1	no	—
366	C₄×C₃₁	—	4	1	no	—
367	C₄×C₃₁	—	4	1	no	—
368	C₄×C₃₁	—	8	1	no	—
369	C₄×C₃₁	—	8	1	no	—
370	C₄×C₃₁	—	8	1	no	—
371	C₄×C₃₁	#525	1	1	yes	characteristic
372	C₄×C₃₁	#532	1	1	yes	characteristic
373	C₄×C₃₁	#546	1	1	yes	characteristic
374	C₄×C₃₁	#546	1	1	yes	characteristic
375	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
376	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
377	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
378	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
379	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
380	#537	C₂²×C₃₁	1	1	yes	characteristic
381	#535	C₂²×C₃₁	1	1	yes	characteristic
382	#520	C₂²×C₃₁	1	1	yes	characteristic
383	#518	C₂²×C₃₁	1	1	yes	characteristic
384	#485	—	2	1	no	—
385	#483	—	2	1	no	—
386	#482	—	2	1	no	—
387	#481	—	2	1	no	—
388	#469	—	2	1	no	—
389	#468	—	2	1	no	—
390	#466	—	2	1	no	—
391	#465	—	2	1	no	—
392	#463	—	2	1	no	—
393	#463	—	2	1	no	—
394	#459	—	2	1	no	—
395	#459	—	2	1	no	—
396	#442	—	2	1	no	—
397	#440	—	2	1	no	—
398	C₄×SD₁₆	—	2	1	no	—
399	C₄×D₈	—	2	1	no	—
400	#428	C₂²×C₃₁	1	1	yes	characteristic
401	#428	C₂²×C₃₁	1	1	yes	characteristic
402	#428	C₂²×C₃₁	1	1	yes	characteristic
403	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
404	#424	C₂²×C₃₁	1	1	yes	characteristic
405	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
406	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
407	#418	C₂²×C₃₁	1	1	yes	characteristic
408	#417	C₂²×C₃₁	1	1	yes	characteristic
409	#416	C₂²×C₃₁	1	1	yes	characteristic
410	#416	C₂²×C₃₁	1	1	yes	characteristic
411	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
412	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
413	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
414	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
415	#408	C₂²×C₃₁	1	1	yes	characteristic
416	#405	C₂²×C₃₁	1	1	yes	characteristic
417	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
418	#390	C₂²×C₃₁	1	1	yes	characteristic
419	#390	C₂²×C₃₁	1	1	yes	characteristic
420	#343	C₂²×C₃₁	1	1	yes	characteristic
421	#343	C₂²×C₃₁	1	1	yes	characteristic
422	#341	C₂²×C₃₁	1	1	yes	characteristic
423	#341	C₂²×C₃₁	1	1	yes	characteristic
424	#339	C₂²×C₃₁	1	1	yes	characteristic
425	#339	C₂²×C₃₁	1	1	yes	characteristic
426	C₂×C₃₁	—	2	1	no	—
427	C₂×C₃₁	—	2	1	no	—
428	C₂×C₃₁	—	8	1	no	—
429	C₂×C₃₁	—	8	1	no	—
430	C₂×C₃₁	—	8	2	no	—
431	C₂×C₃₁	—	16	1	no	—
432	C₂×C₃₁	#2464	1	1	yes	characteristic
433	C₂×C₃₁	#3288	1	1	yes	characteristic
434	C₂×C₃₁	#3288	1	1	yes	characteristic
435	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
436	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
437	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
438	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
439	AGL(1,ℤ/8)	—	2	2	no	—
440	AGL(1,ℤ/8)	—	2	2	no	—
441	AGL(1,ℤ/8)	—	2	2	no	—
442	AGL(1,ℤ/8)	—	2	2	no	—
443	AGL(1,ℤ/8)	—	4	1	no	—
444	AGL(1,ℤ/8)	—	4	1	no	—
445	C₂×SD₁₆	—	2	1	no	—
446	C₂×SD₁₆	—	2	1	no	—
447	C₂×SD₁₆	—	2	1	no	—
448	C₂×D₈	—	2	1	no	—
449	C₂×D₈	—	2	1	no	—
450	C₂×D₈	—	2	1	no	—
451	C₂×D₈	—	2	2	no	—
452	C₂×D₈	—	2	1	no	—
453	C₂×D₈	—	2	1	no	—
454	C₂×D₈	—	4	1	no	—
455	#131	—	2	2	no	—
456	#131	—	4	1	no	—
457	C₂×M₁₆	—	2	1	no	—
458	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
459	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
460	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
461	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
462	C₂²×C₈	—	2	1	no	—
463	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
464	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
465	#127	—	2	1	no	—
466	#124	—	2	1	no	—
467	C₂²⋊Q₈	—	2	1	no	—
468	C₂²⋊Q₈	—	2	1	no	—
469	#121	—	2	1	no	—
470	#121	—	2	1	no	—
471	#121	—	2	1	no	—
472	#121	—	2	1	no	—
473	#121	—	4	1	no	—
474	#120	—	4	1	no	—
475	C₄×Q₈	—	2	1	no	—
476	C₄×D₄	—	2	1	no	—
477	C₄×D₄	—	2	1	no	—
478	C₄×D₄	—	2	1	no	—
479	C₄×D₄	—	2	1	no	—
480	C₄×D₄	—	2	1	no	—
481	C₄×D₄	—	4	1	no	—
482	#117	—	2	1	no	—
483	#117	C₂³×C₃₁	1	1	yes	characteristic
484	#117	C₂³×C₃₁	1	1	yes	characteristic
485	C₂×C₄⋊C₄	—	2	1	no	—
486	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
487	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
488	C₂×C₂²⋊C₄	—	4	1	no	—
489	C₂×C₂²⋊C₄	—	4	1	no	—
490	C₂×C₂²⋊C₄	—	4	1	no	—
491	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
492	#107	—	2	1	no	—
493	#107	—	2	1	no	—
494	#107	—	2	1	no	—
495	#106	—	2	1	no	—
496	#106	—	2	1	no	—
497	#106	—	2	1	no	—
498	#106	—	2	1	no	—
499	#106	—	2	1	no	—
500	C₄⋊C₈	—	2	1	no	—
501	C₄⋊C₈	—	2	1	no	—
502	#104	—	2	2	no	—
503	#104	—	4	1	no	—
504	#103	—	2	1	no	—
505	#103	D₄×C₃₁	1	1	yes	characteristic
506	#103	D₄×C₃₁	1	1	yes	characteristic
507	#103	D₄×C₃₁	1	1	yes	characteristic
508	#103	D₄×C₃₁	1	1	yes	characteristic
509	#102	—	2	1	no	—
510	#102	—	2	1	no	—
511	#102	—	2	1	no	—
512	#102	—	2	1	no	—
513	#102	—	2	1	no	—
514	#102	—	2	1	no	—
515	#102	—	2	1	no	—
516	#102	D₄×C₃₁	1	1	yes	characteristic
517	#102	D₄×C₃₁	1	1	yes	characteristic
518	#102	D₄×C₃₁	1	1	yes	characteristic
519	#102	D₄×C₃₁	1	1	yes	characteristic
520	#100	—	2	2	no	—
521	#100	—	2	2	no	—
522	C₂²⋊C₈	—	2	1	no	—
523	C₂²⋊C₈	—	2	1	no	—
524	#97	—	2	1	no	—
525	#97	—	4	1	no	—
526	C₄×C₈	—	2	1	no	—
527	C₃₁	#26996	1	1	yes	characteristic, direct product factor, Sylow
528	C₂⁴	—	2	1	no	—
529	C₂⁴	—	2	1	no	—
530	#41	—	2	1	no	—
531	#41	—	2	1	no	—
532	#41	—	2	1	no	—
533	#41	—	2	1	no	—
534	#41	—	2	1	no	—
535	#41	—	2	1	no	—
536	#41	—	2	1	no	—
537	#41	—	2	1	no	—
538	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
539	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
540	C₂×D₄	—	2	1	no	—
541	C₂×D₄	—	2	1	no	—
542	C₂×D₄	—	2	1	no	—
543	C₂×D₄	—	2	1	no	—
544	C₂×D₄	—	2	1	no	—
545	C₂×D₄	—	2	1	no	—
546	C₂×D₄	—	2	1	no	—
547	C₂×D₄	—	2	1	no	—
548	C₂×D₄	—	2	1	no	—
549	C₂×D₄	—	2	1	no	—
550	C₂×D₄	—	2	1	no	—
551	C₂×D₄	—	2	1	no	—
552	C₂×D₄	—	2	1	no	—
553	C₂×D₄	—	4	1	no	—
554	C₂×D₄	—	4	1	no	—
555	C₂×D₄	—	4	1	no	—
556	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
557	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
558	C₂×D₄	#41×C₃₁	1	2	yes	—
559	C₂²×C₄	—	2	1	no	—
560	C₂²×C₄	—	2	1	no	—
561	C₂²×C₄	—	2	1	no	—
562	C₂²×C₄	—	2	1	no	—
563	C₂²×C₄	—	2	1	no	—
564	C₂²×C₄	—	4	1	no	—
565	C₂²×C₄	—	4	1	no	—
566	C₂²×C₄	—	4	1	no	—
567	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
568	SD₁₆	—	2	4	no	—
569	SD₁₆	—	4	2	no	—
570	SD₁₆	—	4	2	no	—
571	D₈	—	2	4	no	—
572	D₈	—	4	2	no	—
573	D₈	—	4	2	no	—
574	D₈	—	4	2	no	—
575	D₈	—	4	2	no	—
576	M₁₆	—	2	1	no	—
577	M₁₆	—	2	1	no	—
578	M₁₆	—	2	1	no	—
579	M₁₆	—	2	1	no	—
580	M₁₆	—	2	1	no	—
581	M₁₆	—	2	1	no	—
582	M₁₆	—	4	2	no	—
583	M₁₆	#41×C₃₁	1	2	yes	—
584	M₁₆	#41×C₃₁	1	2	yes	—
585	C₂×C₈	—	2	1	no	—
586	C₂×C₈	—	2	1	no	—
587	C₂×C₈	—	2	1	no	—
588	C₂×C₈	—	2	1	no	—
589	C₂×C₈	—	2	1	no	—
590	C₂×C₈	—	2	1	no	—
591	C₂×C₈	—	2	1	no	—
592	C₂×C₈	—	2	1	no	—
593	C₂×C₈	—	4	2	no	—
594	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
595	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
596	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
597	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
598	C₄⋊C₄	—	2	1	no	—
599	C₄⋊C₄	—	2	1	no	—
600	C₄⋊C₄	—	2	1	no	—
601	C₄⋊C₄	—	2	1	no	—
602	C₄⋊C₄	—	4	1	no	—
603	C₄⋊C₄	—	4	1	no	—
604	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
605	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
606	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
607	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
608	C₂²⋊C₄	—	2	1	no	—
609	C₂²⋊C₄	—	2	1	no	—
610	C₂²⋊C₄	—	4	1	no	—
611	C₂²⋊C₄	—	4	1	no	—
612	C₂²⋊C₄	—	4	1	no	—
613	C₂²⋊C₄	—	4	1	no	—
614	C₂²⋊C₄	—	4	1	no	—
615	C₂²⋊C₄	—	4	1	no	—
616	C₂²⋊C₄	—	4	1	no	—
617	C₂²⋊C₄	—	8	2	no	—
618	C₂²⋊C₄	—	8	2	no	—
619	C₄²	—	2	1	no	—
620	C₄²	—	2	1	no	—
621	C₄²	—	2	1	no	—
622	C₄²	—	2	1	no	—
623	C₄²	—	4	1	no	—
624	C₂³	—	2	1	no	—
625	C₂³	—	2	1	no	—
626	C₂³	—	2	2	no	—
627	C₂³	—	4	1	no	—
628	C₂³	—	4	1	no	—
629	C₂³	—	4	1	no	—
630	C₂³	—	4	1	no	—
631	C₂³	—	4	1	no	—
632	C₂³	—	8	1	no	—
633	C₂³	—	8	1	no	—
634	C₂³	—	8	1	no	—
635	C₂³	—	8	1	no	—
636	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
637	Q₈	—	2	1	no	—
638	Q₈	—	2	1	no	—
639	Q₈	—	2	1	no	—
640	Q₈	—	2	1	no	—
641	D₄	—	2	1	no	—
642	D₄	—	2	2	no	—
643	D₄	—	2	1	no	—
644	D₄	—	2	2	no	—
645	D₄	—	2	1	no	—
646	D₄	—	2	1	no	—
647	D₄	—	4	1	no	—
648	D₄	—	4	1	no	—
649	D₄	—	4	1	no	—
650	D₄	—	4	1	no	—
651	D₄	—	4	1	no	—
652	D₄	—	4	1	no	—
653	D₄	—	4	1	no	—
654	D₄	—	4	1	no	—
655	D₄	—	4	1	no	—
656	D₄	—	4	1	no	—
657	D₄	—	16	1	no	—
658	D₄	—	16	1	no	—
659	D₄	—	16	1	no	—
660	C₂×C₄	—	2	1	no	—
661	C₂×C₄	—	2	1	no	—
662	C₂×C₄	—	2	1	no	—
663	C₂×C₄	—	2	1	no	—
664	C₂×C₄	—	2	1	no	—
665	C₂×C₄	—	2	1	no	—
666	C₂×C₄	—	4	1	no	—
667	C₂×C₄	—	4	1	no	—
668	C₂×C₄	—	4	1	no	—
669	C₂×C₄	—	4	1	no	—
670	C₂×C₄	—	4	1	no	—
671	C₂×C₄	—	4	1	no	—
672	C₂×C₄	—	4	1	no	—
673	C₂×C₄	—	4	1	no	—
674	C₂×C₄	—	4	1	no	—
675	C₂×C₄	—	8	1	no	—
676	C₂×C₄	—	8	1	no	—
677	C₂×C₄	—	16	1	no	—
678	C₂×C₄	—	16	1	no	—
679	C₂×C₄	—	16	1	no	—
680	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
681	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
682	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
683	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
684	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
685	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
686	C₈	—	2	2	no	—
687	C₈	—	2	1	no	—
688	C₈	—	2	1	no	—
689	C₈	—	2	1	no	—
690	C₈	—	2	1	no	—
691	C₈	—	2	2	no	—
692	C₈	—	4	1	no	—
693	C₈	—	4	1	no	—
694	C₂²	—	2	1	no	—
695	C₂²	—	2	1	no	—
696	C₂²	—	4	1	no	—
697	C₂²	—	4	1	no	—
698	C₂²	—	4	2	no	—
699	C₂²	—	4	1	no	—
700	C₂²	—	4	1	no	—
701	C₂²	—	4	1	no	—
702	C₂²	—	4	1	no	—
703	C₂²	—	8	2	no	—
704	C₂²	—	8	2	no	—
705	C₂²	—	8	1	no	—
706	C₂²	—	8	1	no	—
707	C₂²	—	8	1	no	—
708	C₂²	—	8	1	no	—
709	C₂²	—	8	1	no	—
710	C₂²	—	16	1	no	—
711	C₂²	—	16	1	no	—
712	C₂²	C₃₁×#540	1	1	yes	characteristic
713	C₂²	D₄²×C₃₁	1	1	yes	characteristic
714	C₂²	C₃₁×#553	1	1	yes	characteristic
715	C₄	—	4	1	no	—
716	C₄	—	4	1	no	—
717	C₄	—	4	1	no	—
718	C₄	—	4	1	no	—
719	C₄	—	4	1	no	—
720	C₄	—	4	1	no	—
721	C₄	—	8	1	no	—
722	C₄	—	8	1	no	—
723	C₄	—	8	1	no	—
724	C₄	C₃₁×#525	1	1	yes	characteristic
725	C₄	C₃₁×#532	1	1	yes	characteristic
726	C₄	C₃₁×#546	1	1	yes	characteristic
727	C₄	C₃₁×#546	1	1	yes	characteristic
728	C₂	—	2	1	no	—
729	C₂	—	2	1	no	—
730	C₂	—	8	1	no	—
731	C₂	—	8	1	no	—
732	C₂	—	8	2	no	—
733	C₂	—	16	1	no	—
734	C₂	C₃₁×#2464	1	1	yes	characteristic
735	C₂	C₃₁×#3288	1	1	yes	characteristic
736	C₂	C₃₁×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3099	C₂	semidirect
2. C₃₁×#3062	C₂	semidirect
3. C₃₁×#3041	C₂	semidirect
4. C₃₁×#2946	C₂	semidirect
5. C₃₁×#2090	C₂	semidirect
6. C₃₁×#2037	C₂	semidirect
7. C₃₁×#2018	C₂	semidirect
8. C₃₁×#2006	C₂	semidirect
9. C₃₁×#1972	C₂	semidirect
10. C₃₁×#1938	C₂	semidirect
11. C₃₁×#1931	C₂	semidirect
12. C₃₁×#1907	C₂	semidirect
13. C₃₁×#1883	C₂	semidirect
14. C₃₁×#1856	C₂	semidirect
15. C₃₁×#1765	C₂	semidirect
255. #26996	C₃₁	direct (N × H)
326. #3099	C₂×C₃₁	semidirect
327. #3062	C₂×C₃₁	semidirect
328. #3041	C₂×C₃₁	semidirect
329. #2946	C₂×C₃₁	semidirect
330. #2090	C₂×C₃₁	semidirect
331. #2037	C₂×C₃₁	semidirect
332. #2018	C₂×C₃₁	semidirect
333. #2006	C₂×C₃₁	semidirect
334. #1972	C₂×C₃₁	semidirect
335. #1938	C₂×C₃₁	semidirect
336. #1931	C₂×C₃₁	semidirect
337. #1907	C₂×C₃₁	semidirect
338. #1883	C₂×C₃₁	semidirect
339. #1856	C₂×C₃₁	semidirect
340. #1765	C₂×C₃₁	semidirect
527. C₃₁	#26996	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×#3099	2
C₃₁×#3062	2
C₃₁×#3041	2
C₃₁×#2946	2
C₃₁×#2090	2
C₃₁×#2037	2
C₃₁×#2018	2
C₃₁×#2006	2
C₃₁×#1972	2
C₃₁×#1938	2
C₃₁×#1931	2
C₃₁×#1907	2
C₃₁×#1883	2
C₃₁×#1856	2
C₃₁×#1765	2
#26996	31

Subgroups — C31×#26996 #21483755

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26996 #21483755