Subgroups — C₃₁×#26649 #21483408

Order 7936.

810 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3097	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3042	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2948	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2946	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2063	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2018	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#2012	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2003	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1977	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1930	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1929	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1922	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1885	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1883	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1856	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	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#536	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
22	C₃₁×#520	C₂²	1	1	yes	characteristic
23	C₃₁×#518	C₂²	1	1	yes	characteristic
24	C₃₁×#518	C₂²	1	1	yes	characteristic
25	C₃₁×#485	—	2	1	no	—
26	C₃₁×#485	—	2	1	no	—
27	C₃₁×#484	—	2	1	no	—
28	C₃₁×#483	—	2	1	no	—
29	C₃₁×#463	—	2	1	no	—
30	C₃₁×#463	—	2	1	no	—
31	C₃₁×#461	—	2	1	no	—
32	C₃₁×#461	—	2	1	no	—
33	C₃₁×#442	—	2	1	no	—
34	C₃₁×#442	—	2	1	no	—
35	C₃₁×#442	—	2	1	no	—
36	C₃₁×#440	—	2	1	no	—
37	C₄×SD₁₆×C₃₁	—	2	1	no	—
38	C₄×D₈×C₃₁	—	2	1	no	—
39	C₄×D₈×C₃₁	—	2	1	no	—
40	C₄×D₈×C₃₁	—	2	1	no	—
41	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
42	C₃₁×#428	C₂²	1	1	yes	characteristic
43	C₃₁×#428	C₂²	1	1	yes	characteristic
44	C₃₁×#427	C₂²	1	1	yes	characteristic
45	C₃₁×#424	C₂²	1	1	yes	characteristic
46	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
47	C₃₁×#419	C₂²	1	1	yes	characteristic
48	C₃₁×#419	C₂²	1	1	yes	characteristic
49	C₃₁×#418	C₂²	1	1	yes	characteristic
50	C₃₁×#418	C₂²	1	1	yes	characteristic
51	C₃₁×#416	C₂²	1	1	yes	characteristic
52	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
53	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
54	C₂×C₃₁×#101	C₂²	1	1	yes	characteristic
55	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
56	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
57	C₃₁×#408	C₂²	1	1	yes	characteristic
58	C₃₁×#405	C₂²	1	1	yes	characteristic
59	C₃₁×#405	C₂²	1	1	yes	characteristic
60	C₃₁×#390	C₂²	1	1	yes	characteristic
61	C₃₁×#343	C₂²	1	1	yes	characteristic
62	C₃₁×#341	C₂²	1	1	yes	characteristic
63	C₃₁×#339	C₂²	1	1	yes	characteristic
64	C₃₁×#339	C₂²	1	1	yes	characteristic
65	C₃₁×#339	C₂²	1	1	yes	characteristic
66	C₃₁×#336	C₂²	1	1	yes	characteristic
67	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×Q₈×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)	—	4	1	no	—
75	C₂×SD₁₆×C₃₁	—	2	1	no	—
76	C₂×SD₁₆×C₃₁	—	2	1	no	—
77	C₂×SD₁₆×C₃₁	—	2	2	no	—
78	C₂×SD₁₆×C₃₁	—	2	1	no	—
79	C₂×SD₁₆×C₃₁	—	2	1	no	—
80	C₂×SD₁₆×C₃₁	—	4	1	no	—
81	C₂×D₈×C₃₁	—	2	1	no	—
82	C₂×D₈×C₃₁	—	2	2	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	—	2	1	no	—
87	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
88	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
90	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
92	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₃₁×C₄⋊Q₈	—	2	1	no	—
94	C₃₁×#124	—	2	1	no	—
95	C₃₁×#124	—	2	2	no	—
96	C₃₁×#124	—	4	1	no	—
97	C₃₁×#123	—	4	1	no	—
98	C₃₁×C₂²⋊Q₈	—	2	1	no	—
99	C₃₁×C₂²⋊Q₈	—	2	1	no	—
100	C₄×Q₈×C₃₁	—	2	1	no	—
101	C₄×D₄×C₃₁	—	2	1	no	—
102	C₄×D₄×C₃₁	—	2	1	no	—
103	C₄×D₄×C₃₁	—	2	1	no	—
104	C₄×D₄×C₃₁	—	2	1	no	—
105	C₄×D₄×C₃₁	—	2	1	no	—
106	C₄×D₄×C₃₁	—	2	1	no	—
107	C₄×D₄×C₃₁	—	2	1	no	—
108	C₄×D₄×C₃₁	—	2	1	no	—
109	C₄×D₄×C₃₁	—	2	1	no	—
110	C₄×D₄×C₃₁	—	4	1	no	—
111	C₃₁×#117	—	2	1	no	—
112	C₃₁×#117	C₂³	1	1	yes	characteristic
113	C₃₁×#117	C₂³	1	1	yes	characteristic
114	C₃₁×#117	C₂³	1	1	yes	characteristic
115	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
116	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
117	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
118	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
119	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
120	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
121	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
122	C₃₁×#108	—	4	1	no	—
123	C₃₁×#107	—	2	1	no	—
124	C₃₁×#107	—	2	1	no	—
125	C₃₁×#107	D₄	1	1	yes	characteristic
126	C₃₁×#107	D₄	1	1	yes	characteristic
127	C₃₁×#106	—	2	1	no	—
128	C₃₁×#106	—	2	1	no	—
129	C₃₁×#106	D₄	1	1	yes	characteristic
130	C₃₁×#106	D₄	1	1	yes	characteristic
131	C₃₁×C₄⋊C₈	—	2	1	no	—
132	C₃₁×C₄⋊C₈	—	2	1	no	—
133	C₃₁×#104	—	4	1	no	—
134	C₃₁×#103	—	2	1	no	—
135	C₃₁×#103	—	2	1	no	—
136	C₃₁×#103	—	2	1	no	—
137	C₃₁×#103	—	2	1	no	—
138	C₃₁×#103	—	2	1	no	—
139	C₃₁×#103	—	2	1	no	—
140	C₃₁×#103	D₄	1	1	yes	characteristic
141	C₃₁×#103	D₄	1	1	yes	characteristic
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×#101	—	2	2	no	—
151	C₃₁×#100	—	2	2	no	—
152	C₃₁×#100	—	4	1	no	—
153	C₃₁×C₂²⋊C₈	—	2	1	no	—
154	C₃₁×C₂²⋊C₈	—	2	1	no	—
155	C₃₁×#97	—	2	1	no	—
156	C₃₁×#97	—	2	1	no	—
157	C₄×C₈×C₃₁	—	2	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	C₂×Q₈×C₃₁	—	2	1	no	—
166	C₂×Q₈×C₃₁	—	2	1	no	—
167	C₂×Q₈×C₃₁	—	2	1	no	—
168	C₂×Q₈×C₃₁	—	2	1	no	—
169	C₂×Q₈×C₃₁	—	2	1	no	—
170	C₂×Q₈×C₃₁	—	2	1	no	—
171	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
184	C₂×D₄×C₃₁	#41	1	2	yes	—
185	C₂×D₄×C₃₁	#41	1	2	yes	—
186	C₂²×C₄×C₃₁	—	2	1	no	—
187	C₂²×C₄×C₃₁	—	2	1	no	—
188	C₂²×C₄×C₃₁	—	2	1	no	—
189	C₂²×C₄×C₃₁	—	2	1	no	—
190	C₂²×C₄×C₃₁	—	4	1	no	—
191	C₂²×C₄×C₃₁	—	4	1	no	—
192	C₂²×C₄×C₃₁	—	4	1	no	—
193	C₂²×C₄×C₃₁	—	4	1	no	—
194	C₂²×C₄×C₃₁	—	4	1	no	—
195	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
196	SD₁₆×C₃₁	—	2	4	no	—
197	SD₁₆×C₃₁	—	4	2	no	—
198	SD₁₆×C₃₁	—	4	2	no	—
199	SD₁₆×C₃₁	—	4	2	no	—
200	D₈×C₃₁	—	2	4	no	—
201	D₈×C₃₁	—	2	4	no	—
202	D₈×C₃₁	—	2	4	no	—
203	D₈×C₃₁	—	4	2	no	—
204	M₁₆×C₃₁	—	2	1	no	—
205	M₁₆×C₃₁	—	2	1	no	—
206	M₁₆×C₃₁	—	2	1	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	#41	1	2	yes	—
212	C₂×C₈×C₃₁	—	2	1	no	—
213	C₂×C₈×C₃₁	—	2	1	no	—
214	C₂×C₈×C₃₁	—	2	1	no	—
215	C₂×C₈×C₃₁	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
221	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
222	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
223	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
224	C₄⋊C₄×C₃₁	—	2	1	no	—
225	C₄⋊C₄×C₃₁	—	2	1	no	—
226	C₄⋊C₄×C₃₁	—	4	1	no	—
227	C₄⋊C₄×C₃₁	—	4	1	no	—
228	C₄⋊C₄×C₃₁	—	4	1	no	—
229	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
230	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
231	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
232	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
233	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
234	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
235	C₂²⋊C₄×C₃₁	—	2	1	no	—
236	C₂²⋊C₄×C₃₁	—	2	1	no	—
237	C₂²⋊C₄×C₃₁	—	2	1	no	—
238	C₂²⋊C₄×C₃₁	—	4	1	no	—
239	C₂²⋊C₄×C₃₁	—	4	1	no	—
240	C₂²⋊C₄×C₃₁	—	4	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	4	no	—
246	C₂²⋊C₄×C₃₁	—	8	2	no	—
247	C₄²×C₃₁	—	2	1	no	—
248	C₄²×C₃₁	—	2	1	no	—
249	C₄²×C₃₁	—	2	1	no	—
250	C₄²×C₃₁	—	2	1	no	—
251	C₄²×C₃₁	—	2	1	no	—
252	C₄²×C₃₁	—	4	1	no	—
253	#26649	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
254	C₂³×C₃₁	—	2	1	no	—
255	C₂³×C₃₁	—	2	2	no	—
256	C₂³×C₃₁	—	2	2	no	—
257	C₂³×C₃₁	—	4	1	no	—
258	C₂³×C₃₁	—	4	1	no	—
259	C₂³×C₃₁	—	4	1	no	—
260	C₂³×C₃₁	—	4	1	no	—
261	C₂³×C₃₁	—	8	1	no	—
262	C₂³×C₃₁	—	8	1	no	—
263	C₂³×C₃₁	—	8	1	no	—
264	C₂³×C₃₁	—	8	1	no	—
265	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
266	Q₈×C₃₁	—	2	1	no	—
267	Q₈×C₃₁	—	2	1	no	—
268	Q₈×C₃₁	—	4	1	no	—
269	Q₈×C₃₁	—	4	1	no	—
270	Q₈×C₃₁	—	4	1	no	—
271	Q₈×C₃₁	—	4	1	no	—
272	D₄×C₃₁	—	2	2	no	—
273	D₄×C₃₁	—	2	2	no	—
274	D₄×C₃₁	—	2	1	no	—
275	D₄×C₃₁	—	2	2	no	—
276	D₄×C₃₁	—	2	1	no	—
277	D₄×C₃₁	—	2	2	no	—
278	D₄×C₃₁	—	4	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	C₂×C₄×C₃₁	—	2	1	no	—
285	C₂×C₄×C₃₁	—	2	1	no	—
286	C₂×C₄×C₃₁	—	2	1	no	—
287	C₂×C₄×C₃₁	—	2	1	no	—
288	C₂×C₄×C₃₁	—	2	1	no	—
289	C₂×C₄×C₃₁	—	2	1	no	—
290	C₂×C₄×C₃₁	—	2	1	no	—
291	C₂×C₄×C₃₁	—	4	1	no	—
292	C₂×C₄×C₃₁	—	4	1	no	—
293	C₂×C₄×C₃₁	—	4	1	no	—
294	C₂×C₄×C₃₁	—	4	1	no	—
295	C₂×C₄×C₃₁	—	4	1	no	—
296	C₂×C₄×C₃₁	—	4	1	no	—
297	C₂×C₄×C₃₁	—	4	1	no	—
298	C₂×C₄×C₃₁	—	4	1	no	—
299	C₂×C₄×C₃₁	—	4	1	no	—
300	C₂×C₄×C₃₁	—	8	1	no	—
301	C₂×C₄×C₃₁	—	16	1	no	—
302	C₂×C₄×C₃₁	—	16	1	no	—
303	C₂×C₄×C₃₁	—	16	1	no	—
304	C₂×C₄×C₃₁	—	16	1	no	—
305	C₂×C₄×C₃₁	—	16	1	no	—
306	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
307	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
308	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
309	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
310	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
311	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
312	C₈×C₃₁	—	2	1	no	—
313	C₈×C₃₁	—	2	1	no	—
314	C₈×C₃₁	—	2	2	no	—
315	C₈×C₃₁	—	2	1	no	—
316	C₈×C₃₁	—	2	1	no	—
317	C₈×C₃₁	—	4	1	no	—
318	C₈×C₃₁	—	4	1	no	—
319	C₈×C₃₁	—	4	1	no	—
320	#3097	C₂×C₃₁	1	1	yes	characteristic, complemented
321	#3042	C₂×C₃₁	1	1	yes	characteristic, complemented
322	#2948	C₂×C₃₁	1	1	yes	characteristic, complemented
323	#2946	C₂×C₃₁	1	1	yes	characteristic, complemented
324	#2063	C₂×C₃₁	1	1	yes	characteristic, complemented
325	#2018	C₂×C₃₁	1	1	yes	characteristic
326	#2012	C₂×C₃₁	1	1	yes	characteristic, complemented
327	#2003	C₂×C₃₁	1	1	yes	characteristic, complemented
328	#1977	C₂×C₃₁	1	1	yes	characteristic, complemented
329	#1930	C₂×C₃₁	1	1	yes	characteristic, complemented
330	#1929	C₂×C₃₁	1	1	yes	characteristic, complemented
331	#1922	C₂×C₃₁	1	1	yes	characteristic, complemented
332	#1885	C₂×C₃₁	1	1	yes	characteristic, complemented
333	#1883	C₂×C₃₁	1	1	yes	characteristic, complemented
334	#1856	C₂×C₃₁	1	1	yes	characteristic, complemented
335	C₂²×C₃₁	—	2	1	no	—
336	C₂²×C₃₁	—	2	1	no	—
337	C₂²×C₃₁	—	4	1	no	—
338	C₂²×C₃₁	—	4	2	no	—
339	C₂²×C₃₁	—	4	2	no	—
340	C₂²×C₃₁	—	4	1	no	—
341	C₂²×C₃₁	—	4	1	no	—
342	C₂²×C₃₁	—	8	2	no	—
343	C₂²×C₃₁	—	8	2	no	—
344	C₂²×C₃₁	—	8	1	no	—
345	C₂²×C₃₁	—	8	1	no	—
346	C₂²×C₃₁	—	8	1	no	—
347	C₂²×C₃₁	—	8	1	no	—
348	C₂²×C₃₁	—	16	1	no	—
349	C₂²×C₃₁	—	16	1	no	—
350	C₂²×C₃₁	#538	1	1	yes	characteristic
351	C₂²×C₃₁	D₄²	1	1	yes	characteristic
352	C₂²×C₃₁	#553	1	1	yes	characteristic
353	C₄×C₃₁	—	4	1	no	—
354	C₄×C₃₁	—	4	1	no	—
355	C₄×C₃₁	—	4	1	no	—
356	C₄×C₃₁	—	4	1	no	—
357	C₄×C₃₁	—	4	1	no	—
358	C₄×C₃₁	—	4	1	no	—
359	C₄×C₃₁	—	4	1	no	—
360	C₄×C₃₁	—	8	1	no	—
361	C₄×C₃₁	—	8	1	no	—
362	C₄×C₃₁	—	8	1	no	—
363	C₄×C₃₁	#525	1	1	yes	characteristic
364	C₄×C₃₁	#525	1	1	yes	characteristic
365	C₄×C₃₁	#546	1	1	yes	characteristic
366	C₄×C₃₁	#550	1	1	yes	characteristic
367	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
368	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
369	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
370	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
371	#536	C₂²×C₃₁	1	1	yes	characteristic
372	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
373	#520	C₂²×C₃₁	1	1	yes	characteristic
374	#518	C₂²×C₃₁	1	1	yes	characteristic
375	#518	C₂²×C₃₁	1	1	yes	characteristic
376	#485	—	2	1	no	—
377	#485	—	2	1	no	—
378	#484	—	2	1	no	—
379	#483	—	2	1	no	—
380	#463	—	2	1	no	—
381	#463	—	2	1	no	—
382	#461	—	2	1	no	—
383	#461	—	2	1	no	—
384	#442	—	2	1	no	—
385	#442	—	2	1	no	—
386	#442	—	2	1	no	—
387	#440	—	2	1	no	—
388	C₄×SD₁₆	—	2	1	no	—
389	C₄×D₈	—	2	1	no	—
390	C₄×D₈	—	2	1	no	—
391	C₄×D₈	—	2	1	no	—
392	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
393	#428	C₂²×C₃₁	1	1	yes	characteristic
394	#428	C₂²×C₃₁	1	1	yes	characteristic
395	#427	C₂²×C₃₁	1	1	yes	characteristic
396	#424	C₂²×C₃₁	1	1	yes	characteristic
397	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
398	#419	C₂²×C₃₁	1	1	yes	characteristic
399	#419	C₂²×C₃₁	1	1	yes	characteristic
400	#418	C₂²×C₃₁	1	1	yes	characteristic
401	#418	C₂²×C₃₁	1	1	yes	characteristic
402	#416	C₂²×C₃₁	1	1	yes	characteristic
403	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
404	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
405	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
406	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
407	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
408	#408	C₂²×C₃₁	1	1	yes	characteristic
409	#405	C₂²×C₃₁	1	1	yes	characteristic
410	#405	C₂²×C₃₁	1	1	yes	characteristic
411	#390	C₂²×C₃₁	1	1	yes	characteristic
412	#343	C₂²×C₃₁	1	1	yes	characteristic
413	#341	C₂²×C₃₁	1	1	yes	characteristic
414	#339	C₂²×C₃₁	1	1	yes	characteristic
415	#339	C₂²×C₃₁	1	1	yes	characteristic
416	#339	C₂²×C₃₁	1	1	yes	characteristic
417	#336	C₂²×C₃₁	1	1	yes	characteristic
418	C₂×C₃₁	—	2	1	no	—
419	C₂×C₃₁	—	2	1	no	—
420	C₂×C₃₁	—	8	1	no	—
421	C₂×C₃₁	—	8	2	no	—
422	C₂×C₃₁	—	8	2	no	—
423	C₂×C₃₁	#2456	1	1	yes	characteristic
424	C₂×C₃₁	#3288	1	1	yes	characteristic
425	C₂×C₃₁	#3290	1	1	yes	characteristic
426	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
427	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
428	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
429	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
430	AGL(1,ℤ/8)	—	2	2	no	—
431	AGL(1,ℤ/8)	—	2	2	no	—
432	AGL(1,ℤ/8)	—	2	2	no	—
433	AGL(1,ℤ/8)	—	4	1	no	—
434	C₂×SD₁₆	—	2	1	no	—
435	C₂×SD₁₆	—	2	1	no	—
436	C₂×SD₁₆	—	2	2	no	—
437	C₂×SD₁₆	—	2	1	no	—
438	C₂×SD₁₆	—	2	1	no	—
439	C₂×SD₁₆	—	4	1	no	—
440	C₂×D₈	—	2	1	no	—
441	C₂×D₈	—	2	2	no	—
442	C₂×D₈	—	2	2	no	—
443	C₂×D₈	—	2	1	no	—
444	C₂×D₈	—	2	1	no	—
445	C₂×D₈	—	2	1	no	—
446	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
447	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
448	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
449	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
450	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
451	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
452	C₄⋊Q₈	—	2	1	no	—
453	#124	—	2	1	no	—
454	#124	—	2	2	no	—
455	#124	—	4	1	no	—
456	#123	—	4	1	no	—
457	C₂²⋊Q₈	—	2	1	no	—
458	C₂²⋊Q₈	—	2	1	no	—
459	C₄×Q₈	—	2	1	no	—
460	C₄×D₄	—	2	1	no	—
461	C₄×D₄	—	2	1	no	—
462	C₄×D₄	—	2	1	no	—
463	C₄×D₄	—	2	1	no	—
464	C₄×D₄	—	2	1	no	—
465	C₄×D₄	—	2	1	no	—
466	C₄×D₄	—	2	1	no	—
467	C₄×D₄	—	2	1	no	—
468	C₄×D₄	—	2	1	no	—
469	C₄×D₄	—	4	1	no	—
470	#117	—	2	1	no	—
471	#117	C₂³×C₃₁	1	1	yes	characteristic
472	#117	C₂³×C₃₁	1	1	yes	characteristic
473	#117	C₂³×C₃₁	1	1	yes	characteristic
474	C₂×C₄⋊C₄	—	2	1	no	—
475	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
476	C₂×C₂²⋊C₄	—	4	1	no	—
477	C₂×C₂²⋊C₄	—	4	1	no	—
478	C₂×C₂²⋊C₄	—	4	1	no	—
479	C₂×C₂²⋊C₄	—	4	1	no	—
480	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
481	#108	—	4	1	no	—
482	#107	—	2	1	no	—
483	#107	—	2	1	no	—
484	#107	D₄×C₃₁	1	1	yes	characteristic
485	#107	D₄×C₃₁	1	1	yes	characteristic
486	#106	—	2	1	no	—
487	#106	—	2	1	no	—
488	#106	D₄×C₃₁	1	1	yes	characteristic
489	#106	D₄×C₃₁	1	1	yes	characteristic
490	C₄⋊C₈	—	2	1	no	—
491	C₄⋊C₈	—	2	1	no	—
492	#104	—	4	1	no	—
493	#103	—	2	1	no	—
494	#103	—	2	1	no	—
495	#103	—	2	1	no	—
496	#103	—	2	1	no	—
497	#103	—	2	1	no	—
498	#103	—	2	1	no	—
499	#103	D₄×C₃₁	1	1	yes	characteristic
500	#103	D₄×C₃₁	1	1	yes	characteristic
501	#102	—	2	1	no	—
502	#102	—	2	1	no	—
503	#102	—	2	1	no	—
504	#102	—	2	1	no	—
505	#102	—	2	1	no	—
506	#102	—	2	1	no	—
507	#102	D₄×C₃₁	1	1	yes	characteristic
508	#102	D₄×C₃₁	1	1	yes	characteristic
509	#101	—	2	2	no	—
510	#100	—	2	2	no	—
511	#100	—	4	1	no	—
512	C₂²⋊C₈	—	2	1	no	—
513	C₂²⋊C₈	—	2	1	no	—
514	#97	—	2	1	no	—
515	#97	—	2	1	no	—
516	C₄×C₈	—	2	1	no	—
517	C₄×C₈	—	2	1	no	—
518	C₃₁	#26649	1	1	yes	characteristic, direct product factor, Sylow
519	C₂⁴	—	2	1	no	—
520	C₂⁴	—	2	1	no	—
521	#41	—	2	1	no	—
522	#41	—	2	1	no	—
523	#41	—	2	1	no	—
524	#41	—	2	1	no	—
525	C₂×Q₈	—	2	1	no	—
526	C₂×Q₈	—	2	1	no	—
527	C₂×Q₈	—	2	1	no	—
528	C₂×Q₈	—	2	1	no	—
529	C₂×Q₈	—	2	1	no	—
530	C₂×Q₈	—	2	1	no	—
531	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
532	C₂×D₄	—	2	1	no	—
533	C₂×D₄	—	2	1	no	—
534	C₂×D₄	—	2	1	no	—
535	C₂×D₄	—	2	1	no	—
536	C₂×D₄	—	2	1	no	—
537	C₂×D₄	—	2	1	no	—
538	C₂×D₄	—	2	1	no	—
539	C₂×D₄	—	2	1	no	—
540	C₂×D₄	—	2	1	no	—
541	C₂×D₄	—	2	1	no	—
542	C₂×D₄	—	2	1	no	—
543	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
544	C₂×D₄	#41×C₃₁	1	2	yes	—
545	C₂×D₄	#41×C₃₁	1	2	yes	—
546	C₂²×C₄	—	2	1	no	—
547	C₂²×C₄	—	2	1	no	—
548	C₂²×C₄	—	2	1	no	—
549	C₂²×C₄	—	2	1	no	—
550	C₂²×C₄	—	4	1	no	—
551	C₂²×C₄	—	4	1	no	—
552	C₂²×C₄	—	4	1	no	—
553	C₂²×C₄	—	4	1	no	—
554	C₂²×C₄	—	4	1	no	—
555	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
556	SD₁₆	—	2	4	no	—
557	SD₁₆	—	4	2	no	—
558	SD₁₆	—	4	2	no	—
559	SD₁₆	—	4	2	no	—
560	D₈	—	2	4	no	—
561	D₈	—	2	4	no	—
562	D₈	—	2	4	no	—
563	D₈	—	4	2	no	—
564	M₁₆	—	2	1	no	—
565	M₁₆	—	2	1	no	—
566	M₁₆	—	2	1	no	—
567	M₁₆	—	2	1	no	—
568	M₁₆	—	2	1	no	—
569	M₁₆	—	2	1	no	—
570	M₁₆	—	2	1	no	—
571	M₁₆	#41×C₃₁	1	2	yes	—
572	C₂×C₈	—	2	1	no	—
573	C₂×C₈	—	2	1	no	—
574	C₂×C₈	—	2	1	no	—
575	C₂×C₈	—	2	1	no	—
576	C₂×C₈	—	2	1	no	—
577	C₂×C₈	—	2	1	no	—
578	C₂×C₈	—	2	1	no	—
579	C₂×C₈	—	2	1	no	—
580	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
581	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
582	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
583	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
584	C₄⋊C₄	—	2	1	no	—
585	C₄⋊C₄	—	2	1	no	—
586	C₄⋊C₄	—	4	1	no	—
587	C₄⋊C₄	—	4	1	no	—
588	C₄⋊C₄	—	4	1	no	—
589	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
590	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
591	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
592	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
593	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
594	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
595	C₂²⋊C₄	—	2	1	no	—
596	C₂²⋊C₄	—	2	1	no	—
597	C₂²⋊C₄	—	2	1	no	—
598	C₂²⋊C₄	—	4	1	no	—
599	C₂²⋊C₄	—	4	1	no	—
600	C₂²⋊C₄	—	4	1	no	—
601	C₂²⋊C₄	—	4	1	no	—
602	C₂²⋊C₄	—	4	1	no	—
603	C₂²⋊C₄	—	4	1	no	—
604	C₂²⋊C₄	—	4	1	no	—
605	C₂²⋊C₄	—	4	4	no	—
606	C₂²⋊C₄	—	8	2	no	—
607	C₄²	—	2	1	no	—
608	C₄²	—	2	1	no	—
609	C₄²	—	2	1	no	—
610	C₄²	—	2	1	no	—
611	C₄²	—	2	1	no	—
612	C₄²	—	4	1	no	—
613	C₂³	—	2	1	no	—
614	C₂³	—	2	2	no	—
615	C₂³	—	2	2	no	—
616	C₂³	—	4	1	no	—
617	C₂³	—	4	1	no	—
618	C₂³	—	4	1	no	—
619	C₂³	—	4	1	no	—
620	C₂³	—	8	1	no	—
621	C₂³	—	8	1	no	—
622	C₂³	—	8	1	no	—
623	C₂³	—	8	1	no	—
624	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
625	Q₈	—	2	1	no	—
626	Q₈	—	2	1	no	—
627	Q₈	—	4	1	no	—
628	Q₈	—	4	1	no	—
629	Q₈	—	4	1	no	—
630	Q₈	—	4	1	no	—
631	D₄	—	2	2	no	—
632	D₄	—	2	2	no	—
633	D₄	—	2	1	no	—
634	D₄	—	2	2	no	—
635	D₄	—	2	1	no	—
636	D₄	—	2	2	no	—
637	D₄	—	4	1	no	—
638	D₄	—	4	1	no	—
639	D₄	—	4	1	no	—
640	D₄	—	4	1	no	—
641	D₄	—	4	1	no	—
642	D₄	—	4	1	no	—
643	C₂×C₄	—	2	1	no	—
644	C₂×C₄	—	2	1	no	—
645	C₂×C₄	—	2	1	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	1	no	—
649	C₂×C₄	—	2	1	no	—
650	C₂×C₄	—	4	1	no	—
651	C₂×C₄	—	4	1	no	—
652	C₂×C₄	—	4	1	no	—
653	C₂×C₄	—	4	1	no	—
654	C₂×C₄	—	4	1	no	—
655	C₂×C₄	—	4	1	no	—
656	C₂×C₄	—	4	1	no	—
657	C₂×C₄	—	4	1	no	—
658	C₂×C₄	—	4	1	no	—
659	C₂×C₄	—	8	1	no	—
660	C₂×C₄	—	16	1	no	—
661	C₂×C₄	—	16	1	no	—
662	C₂×C₄	—	16	1	no	—
663	C₂×C₄	—	16	1	no	—
664	C₂×C₄	—	16	1	no	—
665	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
666	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
667	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
668	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
669	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
670	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
671	C₈	—	2	1	no	—
672	C₈	—	2	1	no	—
673	C₈	—	2	2	no	—
674	C₈	—	2	1	no	—
675	C₈	—	2	1	no	—
676	C₈	—	4	1	no	—
677	C₈	—	4	1	no	—
678	C₈	—	4	1	no	—
679	C₂²	—	2	1	no	—
680	C₂²	—	2	1	no	—
681	C₂²	—	4	1	no	—
682	C₂²	—	4	2	no	—
683	C₂²	—	4	2	no	—
684	C₂²	—	4	1	no	—
685	C₂²	—	4	1	no	—
686	C₂²	—	8	2	no	—
687	C₂²	—	8	2	no	—
688	C₂²	—	8	1	no	—
689	C₂²	—	8	1	no	—
690	C₂²	—	8	1	no	—
691	C₂²	—	8	1	no	—
692	C₂²	—	16	1	no	—
693	C₂²	—	16	1	no	—
694	C₂²	C₃₁×#538	1	1	yes	characteristic
695	C₂²	D₄²×C₃₁	1	1	yes	characteristic
696	C₂²	C₃₁×#553	1	1	yes	characteristic
697	C₄	—	4	1	no	—
698	C₄	—	4	1	no	—
699	C₄	—	4	1	no	—
700	C₄	—	4	1	no	—
701	C₄	—	4	1	no	—
702	C₄	—	4	1	no	—
703	C₄	—	4	1	no	—
704	C₄	—	8	1	no	—
705	C₄	—	8	1	no	—
706	C₄	—	8	1	no	—
707	C₄	C₃₁×#525	1	1	yes	characteristic
708	C₄	C₃₁×#525	1	1	yes	characteristic
709	C₄	C₃₁×#546	1	1	yes	characteristic
710	C₄	C₃₁×#550	1	1	yes	characteristic
711	C₂	—	2	1	no	—
712	C₂	—	2	1	no	—
713	C₂	—	8	1	no	—
714	C₂	—	8	2	no	—
715	C₂	—	8	2	no	—
716	C₂	C₃₁×#2456	1	1	yes	characteristic
717	C₂	C₃₁×#3288	1	1	yes	characteristic
718	C₂	C₃₁×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#2018	C₂
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×#3097	C₂	semidirect
2. C₃₁×#3042	C₂	semidirect
3. C₃₁×#2948	C₂	semidirect
4. C₃₁×#2946	C₂	semidirect
5. C₃₁×#2063	C₂	semidirect
7. C₃₁×#2012	C₂	semidirect
8. C₃₁×#2003	C₂	semidirect
9. C₃₁×#1977	C₂	semidirect
10. C₃₁×#1930	C₂	semidirect
11. C₃₁×#1929	C₂	semidirect
12. C₃₁×#1922	C₂	semidirect
13. C₃₁×#1885	C₂	semidirect
14. C₃₁×#1883	C₂	semidirect
15. C₃₁×#1856	C₂	semidirect
253. #26649	C₃₁	direct (N × H)
320. #3097	C₂×C₃₁	semidirect
321. #3042	C₂×C₃₁	semidirect
322. #2948	C₂×C₃₁	semidirect
323. #2946	C₂×C₃₁	semidirect
324. #2063	C₂×C₃₁	semidirect
326. #2012	C₂×C₃₁	semidirect
327. #2003	C₂×C₃₁	semidirect
328. #1977	C₂×C₃₁	semidirect
329. #1930	C₂×C₃₁	semidirect
330. #1929	C₂×C₃₁	semidirect
331. #1922	C₂×C₃₁	semidirect
332. #1885	C₂×C₃₁	semidirect
333. #1883	C₂×C₃₁	semidirect
334. #1856	C₂×C₃₁	semidirect
518. C₃₁	#26649	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×#3097	2
C₃₁×#3042	2
C₃₁×#2948	2
C₃₁×#2946	2
C₃₁×#2063	2
C₃₁×#2018	2
C₃₁×#2012	2
C₃₁×#2003	2
C₃₁×#1977	2
C₃₁×#1930	2
C₃₁×#1929	2
C₃₁×#1922	2
C₃₁×#1885	2
C₃₁×#1883	2
C₃₁×#1856	2
#26649	31

Subgroups — C31×#26649 #21483408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26649 #21483408