Subgroups — C₃₁×#24674 #21481433

Order 7936.

796 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3040	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#463	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2925	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2052	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2051	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2019	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2018	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1978	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1977	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1967	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1931	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1930	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1800	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#137	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×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
22	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#464	—	2	1	no	—
25	C₃₁×#464	—	2	1	no	—
26	C₃₁×#463	—	2	1	no	—
27	C₃₁×#463	—	2	1	no	—
28	C₃₁×#463	—	2	1	no	—
29	C₃₁×#463	—	2	1	no	—
30	C₃₁×#463	—	2	1	no	—
31	C₃₁×#463	—	2	1	no	—
32	C₃₁×#442	—	2	1	no	—
33	C₃₁×#440	—	2	1	no	—
34	C₄×SD₁₆×C₃₁	—	2	1	no	—
35	C₄×D₈×C₃₁	—	2	1	no	—
36	C₃₁×#433	—	2	1	no	—
37	C₃₁×#433	—	2	1	no	—
38	C₃₁×#432	—	2	1	no	—
39	C₃₁×#431	—	2	1	no	—
40	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
41	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
42	C₃₁×#428	C₂²	1	1	yes	characteristic
43	C₃₁×#423	C₂²	1	1	yes	characteristic
44	C₃₁×#423	C₂²	1	1	yes	characteristic
45	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₃₁×#419	C₂²	1	1	yes	characteristic
47	C₃₁×#418	C₂²	1	1	yes	characteristic
48	C₃₁×#418	C₂²	1	1	yes	characteristic
49	C₃₁×#417	C₂²	1	1	yes	characteristic
50	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
51	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
52	C₂×C₃₁×#101	C₂²	1	1	yes	characteristic
53	C₂×C₃₁×#101	C₂²	1	1	yes	characteristic
54	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
55	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
56	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
57	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
58	C₃₁×#408	C₂²	1	1	yes	characteristic
59	C₃₁×#408	C₂²	1	1	yes	characteristic
60	C₃₁×#405	C₂²	1	1	yes	characteristic
61	C₃₁×#405	C₂²	1	1	yes	characteristic
62	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
63	C₃₁×#343	C₂²	1	1	yes	characteristic
64	C₃₁×#343	C₂²	1	1	yes	characteristic
65	C₃₁×#341	C₂²	1	1	yes	characteristic
66	C₃₁×#341	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	C₂³×C₄×C₃₁	—	2	1	no	—
72	C₃₁×#137	—	4	1	no	—
73	C₃₁×#137	—	4	1	no	—
74	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
75	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
76	C₃₁×AGL(1,ℤ/8)	—	4	1	no	—
77	C₃₁×AGL(1,ℤ/8)	—	4	1	no	—
78	C₂×Q₁₆×C₃₁	—	2	1	no	—
79	C₂×SD₁₆×C₃₁	—	2	1	no	—
80	C₂×SD₁₆×C₃₁	—	2	1	no	—
81	C₂×SD₁₆×C₃₁	—	2	1	no	—
82	C₂×SD₁₆×C₃₁	—	2	2	no	—
83	C₂×SD₁₆×C₃₁	—	2	1	no	—
84	C₂×SD₁₆×C₃₁	—	2	1	no	—
85	C₂×SD₁₆×C₃₁	—	4	1	no	—
86	C₂×D₈×C₃₁	—	2	1	no	—
87	C₂×D₈×C₃₁	—	2	1	no	—
88	C₂×D₈×C₃₁	—	2	2	no	—
89	C₂×D₈×C₃₁	—	2	1	no	—
90	C₂×D₈×C₃₁	—	2	1	no	—
91	C₂×D₈×C₃₁	—	4	1	no	—
92	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
93	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
94	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
95	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
96	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
97	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
98	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
99	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₃₁×#124	—	2	1	no	—
101	C₃₁×#124	—	2	1	no	—
102	C₃₁×#124	—	2	1	no	—
103	C₃₁×#124	—	2	1	no	—
104	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
105	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
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	—	2	1	no	—
111	C₄×D₄×C₃₁	—	2	1	no	—
112	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
113	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
114	C₃₁×#117	—	2	1	no	—
115	C₃₁×#117	C₂³	1	1	yes	characteristic
116	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
117	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
118	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
119	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
120	C₂×C₄²×C₃₁	—	2	1	no	—
121	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
122	C₃₁×#108	—	4	1	no	—
123	C₃₁×#108	—	4	1	no	—
124	C₃₁×#107	—	2	1	no	—
125	C₃₁×#106	—	2	1	no	—
126	C₃₁×C₄⋊C₈	—	2	1	no	—
127	C₃₁×C₄⋊C₈	—	2	1	no	—
128	C₃₁×C₄⋊C₈	—	2	1	no	—
129	C₃₁×C₄⋊C₈	—	2	1	no	—
130	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
131	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
132	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
133	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
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×#102	—	2	1	no	—
140	C₃₁×#102	—	2	1	no	—
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×#101	—	2	2	no	—
147	C₃₁×#101	—	4	1	no	—
148	C₃₁×#100	—	2	2	no	—
149	C₃₁×#100	—	4	1	no	—
150	C₃₁×#100	—	4	1	no	—
151	C₃₁×#100	—	4	1	no	—
152	C₃₁×C₂²⋊C₈	—	2	1	no	—
153	C₃₁×C₂²⋊C₈	—	2	1	no	—
154	C₃₁×C₂²⋊C₈	—	2	1	no	—
155	C₃₁×C₂²⋊C₈	—	2	1	no	—
156	C₃₁×#97	—	2	1	no	—
157	C₃₁×#97	—	2	1	no	—
158	C₃₁×#97	—	2	1	no	—
159	C₄×C₈×C₃₁	—	2	1	no	—
160	C₄×C₈×C₃₁	—	2	1	no	—
161	C₄×C₈×C₃₁	—	2	1	no	—
162	C₂⁴×C₃₁	—	2	1	no	—
163	C₂⁴×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	C₂×Q₈×C₃₁	—	2	1	no	—
169	C₂×Q₈×C₃₁	—	2	1	no	—
170	C₂×Q₈×C₃₁	—	2	1	no	—
171	C₂×Q₈×C₃₁	—	2	1	no	—
172	C₂×Q₈×C₃₁	—	2	1	no	—
173	C₂×Q₈×C₃₁	—	2	1	no	—
174	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
185	C₂×D₄×C₃₁	—	2	1	no	—
186	C₂×D₄×C₃₁	—	2	1	no	—
187	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₂×D₄×C₃₁	#41	1	2	yes	—
189	C₂²×C₄×C₃₁	—	2	1	no	—
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	—	4	1	no	—
195	C₂²×C₄×C₃₁	—	4	1	no	—
196	C₂²×C₄×C₃₁	—	8	2	no	—
197	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
198	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
199	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
200	Q₁₆×C₃₁	—	4	2	no	—
201	SD₁₆×C₃₁	—	2	4	no	—
202	SD₁₆×C₃₁	—	4	2	no	—
203	SD₁₆×C₃₁	—	4	2	no	—
204	SD₁₆×C₃₁	—	4	2	no	—
205	SD₁₆×C₃₁	—	4	2	no	—
206	D₈×C₃₁	—	2	4	no	—
207	D₈×C₃₁	—	4	2	no	—
208	D₈×C₃₁	—	4	2	no	—
209	D₈×C₃₁	—	4	2	no	—
210	M₁₆×C₃₁	—	2	1	no	—
211	M₁₆×C₃₁	—	2	1	no	—
212	M₁₆×C₃₁	—	2	1	no	—
213	M₁₆×C₃₁	—	2	1	no	—
214	M₁₆×C₃₁	—	2	1	no	—
215	M₁₆×C₃₁	—	2	1	no	—
216	M₁₆×C₃₁	—	2	1	no	—
217	M₁₆×C₃₁	—	2	1	no	—
218	M₁₆×C₃₁	—	2	1	no	—
219	M₁₆×C₃₁	—	2	1	no	—
220	M₁₆×C₃₁	—	2	1	no	—
221	M₁₆×C₃₁	#41	1	2	yes	—
222	C₂×C₈×C₃₁	—	2	1	no	—
223	C₂×C₈×C₃₁	—	2	1	no	—
224	C₂×C₈×C₃₁	—	2	1	no	—
225	C₂×C₈×C₃₁	—	2	1	no	—
226	C₂×C₈×C₃₁	—	2	1	no	—
227	C₂×C₈×C₃₁	—	2	1	no	—
228	C₂×C₈×C₃₁	—	2	1	no	—
229	C₂×C₈×C₃₁	—	2	1	no	—
230	C₂×C₈×C₃₁	—	2	1	no	—
231	C₂×C₈×C₃₁	—	2	1	no	—
232	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
233	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
234	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
235	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
236	C₄⋊C₄×C₃₁	—	2	1	no	—
237	C₄⋊C₄×C₃₁	—	2	1	no	—
238	C₄⋊C₄×C₃₁	—	2	1	no	—
239	C₄⋊C₄×C₃₁	—	2	1	no	—
240	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
241	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
242	C₂²⋊C₄×C₃₁	—	2	1	no	—
243	C₂²⋊C₄×C₃₁	—	2	1	no	—
244	C₂²⋊C₄×C₃₁	—	2	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	—	4	1	no	—
249	C₂²⋊C₄×C₃₁	—	4	1	no	—
250	C₂²⋊C₄×C₃₁	—	4	1	no	—
251	C₄²×C₃₁	—	2	1	no	—
252	C₄²×C₃₁	—	2	1	no	—
253	C₄²×C₃₁	—	2	1	no	—
254	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
255	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
256	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
257	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
258	#24674	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
259	C₂³×C₃₁	—	2	1	no	—
260	C₂³×C₃₁	—	2	2	no	—
261	C₂³×C₃₁	—	4	1	no	—
262	C₂³×C₃₁	—	4	1	no	—
263	C₂³×C₃₁	—	4	1	no	—
264	C₂³×C₃₁	—	4	1	no	—
265	C₂³×C₃₁	—	4	1	no	—
266	C₂³×C₃₁	—	8	1	no	—
267	C₂³×C₃₁	—	8	1	no	—
268	C₂³×C₃₁	—	8	1	no	—
269	C₂³×C₃₁	—	8	1	no	—
270	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
271	Q₈×C₃₁	—	2	1	no	—
272	Q₈×C₃₁	—	2	1	no	—
273	Q₈×C₃₁	—	4	1	no	—
274	Q₈×C₃₁	—	4	1	no	—
275	Q₈×C₃₁	—	4	1	no	—
276	Q₈×C₃₁	—	4	1	no	—
277	D₄×C₃₁	—	2	1	no	—
278	D₄×C₃₁	—	2	2	no	—
279	D₄×C₃₁	—	2	1	no	—
280	D₄×C₃₁	—	2	2	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	C₂×C₄×C₃₁	—	2	1	no	—
290	C₂×C₄×C₃₁	—	2	1	no	—
291	C₂×C₄×C₃₁	—	2	1	no	—
292	C₂×C₄×C₃₁	—	4	1	no	—
293	C₂×C₄×C₃₁	—	4	1	no	—
294	C₂×C₄×C₃₁	—	4	2	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	—	4	1	no	—
301	C₂×C₄×C₃₁	—	8	1	no	—
302	C₂×C₄×C₃₁	—	8	2	no	—
303	C₂×C₄×C₃₁	—	8	2	no	—
304	C₂×C₄×C₃₁	—	8	1	no	—
305	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
306	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
307	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
308	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
309	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
310	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
311	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
312	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
313	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
314	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
315	C₈×C₃₁	—	2	2	no	—
316	C₈×C₃₁	—	2	1	no	—
317	C₈×C₃₁	—	2	1	no	—
318	C₈×C₃₁	—	2	1	no	—
319	C₈×C₃₁	—	2	1	no	—
320	C₈×C₃₁	—	4	1	no	—
321	C₈×C₃₁	—	4	1	no	—
322	C₈×C₃₁	—	4	1	no	—
323	C₈×C₃₁	—	4	1	no	—
324	C₈×C₃₁	—	4	1	no	—
325	#3040	C₂×C₃₁	1	1	yes	characteristic, complemented
326	C₂×#463	C₂×C₃₁	1	1	yes	characteristic, complemented
327	C₄×AGL(1,ℤ/8)	C₂×C₃₁	1	1	yes	characteristic, complemented
328	#2925	C₂×C₃₁	1	1	yes	characteristic, complemented
329	#2052	C₂×C₃₁	1	1	yes	characteristic, complemented
330	#2051	C₂×C₃₁	1	1	yes	characteristic, complemented
331	#2019	C₂×C₃₁	1	1	yes	characteristic, complemented
332	#2018	C₂×C₃₁	1	1	yes	characteristic
333	#1978	C₂×C₃₁	1	1	yes	characteristic, complemented
334	#1977	C₂×C₃₁	1	1	yes	characteristic, complemented
335	#1967	C₂×C₃₁	1	1	yes	characteristic, complemented
336	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
337	#1931	C₂×C₃₁	1	1	yes	characteristic, complemented
338	#1930	C₂×C₃₁	1	1	yes	characteristic, complemented
339	#1800	C₂×C₃₁	1	1	yes	characteristic, complemented
340	C₂²×C₃₁	—	2	1	no	—
341	C₂²×C₃₁	—	2	1	no	—
342	C₂²×C₃₁	—	4	1	no	—
343	C₂²×C₃₁	—	4	2	no	—
344	C₂²×C₃₁	—	4	1	no	—
345	C₂²×C₃₁	—	4	1	no	—
346	C₂²×C₃₁	—	8	2	no	—
347	C₂²×C₃₁	—	8	2	no	—
348	C₂²×C₃₁	—	8	1	no	—
349	C₂²×C₃₁	—	8	1	no	—
350	C₂²×C₃₁	—	8	1	no	—
351	C₂²×C₃₁	—	8	1	no	—
352	C₂²×C₃₁	—	8	1	no	—
353	C₂²×C₃₁	—	16	1	no	—
354	C₂²×C₃₁	—	16	1	no	—
355	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
356	C₂²×C₃₁	#553	1	1	yes	characteristic
357	C₂²×C₃₁	#555	1	1	yes	characteristic
358	C₄×C₃₁	—	2	1	no	—
359	C₄×C₃₁	—	2	1	no	—
360	C₄×C₃₁	—	2	1	no	—
361	C₄×C₃₁	—	2	1	no	—
362	C₄×C₃₁	—	4	1	no	—
363	C₄×C₃₁	—	4	1	no	—
364	C₄×C₃₁	—	4	1	no	—
365	C₄×C₃₁	—	8	1	no	—
366	C₄×C₃₁	—	8	1	no	—
367	C₄×C₃₁	#525	1	1	yes	characteristic
368	C₄×C₃₁	#535	1	1	yes	characteristic
369	C₄×C₃₁	#546	1	1	yes	characteristic
370	C₄×C₃₁	#550	1	1	yes	characteristic
371	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic
372	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
373	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
374	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
375	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
376	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
377	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
378	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
379	#464	—	2	1	no	—
380	#464	—	2	1	no	—
381	#463	—	2	1	no	—
382	#463	—	2	1	no	—
383	#463	—	2	1	no	—
384	#463	—	2	1	no	—
385	#463	—	2	1	no	—
386	#463	—	2	1	no	—
387	#442	—	2	1	no	—
388	#440	—	2	1	no	—
389	C₄×SD₁₆	—	2	1	no	—
390	C₄×D₈	—	2	1	no	—
391	#433	—	2	1	no	—
392	#433	—	2	1	no	—
393	#432	—	2	1	no	—
394	#431	—	2	1	no	—
395	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
396	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
397	#428	C₂²×C₃₁	1	1	yes	characteristic
398	#423	C₂²×C₃₁	1	1	yes	characteristic
399	#423	C₂²×C₃₁	1	1	yes	characteristic
400	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
401	#419	C₂²×C₃₁	1	1	yes	characteristic
402	#418	C₂²×C₃₁	1	1	yes	characteristic
403	#418	C₂²×C₃₁	1	1	yes	characteristic
404	#417	C₂²×C₃₁	1	1	yes	characteristic
405	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
406	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
407	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
408	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
409	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
410	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
411	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
412	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
413	#408	C₂²×C₃₁	1	1	yes	characteristic
414	#408	C₂²×C₃₁	1	1	yes	characteristic
415	#405	C₂²×C₃₁	1	1	yes	characteristic
416	#405	C₂²×C₃₁	1	1	yes	characteristic
417	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
418	#343	C₂²×C₃₁	1	1	yes	characteristic
419	#343	C₂²×C₃₁	1	1	yes	characteristic
420	#341	C₂²×C₃₁	1	1	yes	characteristic
421	#341	C₂²×C₃₁	1	1	yes	characteristic
422	C₂×C₃₁	—	2	1	no	—
423	C₂×C₃₁	—	2	1	no	—
424	C₂×C₃₁	—	8	1	no	—
425	C₂×C₃₁	—	8	2	no	—
426	C₂×C₃₁	—	16	1	no	—
427	C₂×C₃₁	#2414	1	1	yes	characteristic
428	C₂×C₃₁	#3067	1	1	yes	characteristic
429	C₂×C₃₁	#3068	1	1	yes	characteristic
430	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
431	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
432	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
433	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
434	C₂³×C₄	—	2	1	no	—
435	#137	—	4	1	no	—
436	#137	—	4	1	no	—
437	AGL(1,ℤ/8)	—	2	2	no	—
438	AGL(1,ℤ/8)	—	2	2	no	—
439	AGL(1,ℤ/8)	—	4	1	no	—
440	AGL(1,ℤ/8)	—	4	1	no	—
441	C₂×Q₁₆	—	2	1	no	—
442	C₂×SD₁₆	—	2	1	no	—
443	C₂×SD₁₆	—	2	1	no	—
444	C₂×SD₁₆	—	2	1	no	—
445	C₂×SD₁₆	—	2	2	no	—
446	C₂×SD₁₆	—	2	1	no	—
447	C₂×SD₁₆	—	2	1	no	—
448	C₂×SD₁₆	—	4	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	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
456	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
457	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
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	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
462	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
463	#124	—	2	1	no	—
464	#124	—	2	1	no	—
465	#124	—	2	1	no	—
466	#124	—	2	1	no	—
467	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
468	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
469	C₄×D₄	—	2	1	no	—
470	C₄×D₄	—	2	1	no	—
471	C₄×D₄	—	2	1	no	—
472	C₄×D₄	—	2	1	no	—
473	C₄×D₄	—	2	1	no	—
474	C₄×D₄	—	2	1	no	—
475	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
476	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
477	#117	—	2	1	no	—
478	#117	C₂³×C₃₁	1	1	yes	characteristic
479	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
480	C₂×C₂²⋊C₄	—	2	1	no	—
481	C₂×C₂²⋊C₄	—	2	1	no	—
482	C₂×C₂²⋊C₄	—	2	1	no	—
483	C₂×C₄²	—	2	1	no	—
484	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
485	#108	—	4	1	no	—
486	#108	—	4	1	no	—
487	#107	—	2	1	no	—
488	#106	—	2	1	no	—
489	C₄⋊C₈	—	2	1	no	—
490	C₄⋊C₈	—	2	1	no	—
491	C₄⋊C₈	—	2	1	no	—
492	C₄⋊C₈	—	2	1	no	—
493	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
494	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
495	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
496	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
497	#103	—	2	1	no	—
498	#103	—	2	1	no	—
499	#103	—	2	1	no	—
500	#103	—	2	1	no	—
501	#103	—	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	—	2	1	no	—
508	#102	—	2	1	no	—
509	#101	—	2	2	no	—
510	#101	—	4	1	no	—
511	#100	—	2	2	no	—
512	#100	—	4	1	no	—
513	#100	—	4	1	no	—
514	#100	—	4	1	no	—
515	C₂²⋊C₈	—	2	1	no	—
516	C₂²⋊C₈	—	2	1	no	—
517	C₂²⋊C₈	—	2	1	no	—
518	C₂²⋊C₈	—	2	1	no	—
519	#97	—	2	1	no	—
520	#97	—	2	1	no	—
521	#97	—	2	1	no	—
522	C₄×C₈	—	2	1	no	—
523	C₄×C₈	—	2	1	no	—
524	C₄×C₈	—	2	1	no	—
525	C₃₁	#24674	1	1	yes	characteristic, direct product factor, Sylow
526	C₂⁴	—	2	1	no	—
527	C₂⁴	—	2	1	no	—
528	#41	—	2	1	no	—
529	#41	—	2	1	no	—
530	#41	—	2	1	no	—
531	#41	—	2	1	no	—
532	C₂×Q₈	—	2	1	no	—
533	C₂×Q₈	—	2	1	no	—
534	C₂×Q₈	—	2	1	no	—
535	C₂×Q₈	—	2	1	no	—
536	C₂×Q₈	—	2	1	no	—
537	C₂×Q₈	—	2	1	no	—
538	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
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	—	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	C₂×D₄×C₃₁	1	1	yes	characteristic
552	C₂×D₄	#41×C₃₁	1	2	yes	—
553	C₂²×C₄	—	2	1	no	—
554	C₂²×C₄	—	2	1	no	—
555	C₂²×C₄	—	2	1	no	—
556	C₂²×C₄	—	2	1	no	—
557	C₂²×C₄	—	2	1	no	—
558	C₂²×C₄	—	4	1	no	—
559	C₂²×C₄	—	4	1	no	—
560	C₂²×C₄	—	8	2	no	—
561	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
562	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
563	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
564	Q₁₆	—	4	2	no	—
565	SD₁₆	—	2	4	no	—
566	SD₁₆	—	4	2	no	—
567	SD₁₆	—	4	2	no	—
568	SD₁₆	—	4	2	no	—
569	SD₁₆	—	4	2	no	—
570	D₈	—	2	4	no	—
571	D₈	—	4	2	no	—
572	D₈	—	4	2	no	—
573	D₈	—	4	2	no	—
574	M₁₆	—	2	1	no	—
575	M₁₆	—	2	1	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	—	2	1	no	—
583	M₁₆	—	2	1	no	—
584	M₁₆	—	2	1	no	—
585	M₁₆	#41×C₃₁	1	2	yes	—
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	—	2	1	no	—
594	C₂×C₈	—	2	1	no	—
595	C₂×C₈	—	2	1	no	—
596	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
597	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
598	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
599	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
600	C₄⋊C₄	—	2	1	no	—
601	C₄⋊C₄	—	2	1	no	—
602	C₄⋊C₄	—	2	1	no	—
603	C₄⋊C₄	—	2	1	no	—
604	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
605	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
606	C₂²⋊C₄	—	2	1	no	—
607	C₂²⋊C₄	—	2	1	no	—
608	C₂²⋊C₄	—	2	1	no	—
609	C₂²⋊C₄	—	4	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	C₄²	—	2	1	no	—
616	C₄²	—	2	1	no	—
617	C₄²	—	2	1	no	—
618	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
619	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
620	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
621	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
622	C₂³	—	2	1	no	—
623	C₂³	—	2	2	no	—
624	C₂³	—	4	1	no	—
625	C₂³	—	4	1	no	—
626	C₂³	—	4	1	no	—
627	C₂³	—	4	1	no	—
628	C₂³	—	4	1	no	—
629	C₂³	—	8	1	no	—
630	C₂³	—	8	1	no	—
631	C₂³	—	8	1	no	—
632	C₂³	—	8	1	no	—
633	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
634	Q₈	—	2	1	no	—
635	Q₈	—	2	1	no	—
636	Q₈	—	4	1	no	—
637	Q₈	—	4	1	no	—
638	Q₈	—	4	1	no	—
639	Q₈	—	4	1	no	—
640	D₄	—	2	1	no	—
641	D₄	—	2	2	no	—
642	D₄	—	2	1	no	—
643	D₄	—	2	2	no	—
644	D₄	—	4	1	no	—
645	D₄	—	4	1	no	—
646	D₄	—	4	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	C₂×C₄	—	2	1	no	—
653	C₂×C₄	—	2	1	no	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	4	1	no	—
656	C₂×C₄	—	4	1	no	—
657	C₂×C₄	—	4	2	no	—
658	C₂×C₄	—	4	1	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	—	4	1	no	—
661	C₂×C₄	—	4	1	no	—
662	C₂×C₄	—	4	1	no	—
663	C₂×C₄	—	4	1	no	—
664	C₂×C₄	—	8	1	no	—
665	C₂×C₄	—	8	2	no	—
666	C₂×C₄	—	8	2	no	—
667	C₂×C₄	—	8	1	no	—
668	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
669	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
670	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
671	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
672	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
673	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
674	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
675	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
676	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
677	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
678	C₈	—	2	2	no	—
679	C₈	—	2	1	no	—
680	C₈	—	2	1	no	—
681	C₈	—	2	1	no	—
682	C₈	—	2	1	no	—
683	C₈	—	4	1	no	—
684	C₈	—	4	1	no	—
685	C₈	—	4	1	no	—
686	C₈	—	4	1	no	—
687	C₈	—	4	1	no	—
688	C₂²	—	2	1	no	—
689	C₂²	—	2	1	no	—
690	C₂²	—	4	1	no	—
691	C₂²	—	4	2	no	—
692	C₂²	—	4	1	no	—
693	C₂²	—	4	1	no	—
694	C₂²	—	8	2	no	—
695	C₂²	—	8	2	no	—
696	C₂²	—	8	1	no	—
697	C₂²	—	8	1	no	—
698	C₂²	—	8	1	no	—
699	C₂²	—	8	1	no	—
700	C₂²	—	8	1	no	—
701	C₂²	—	16	1	no	—
702	C₂²	—	16	1	no	—
703	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
704	C₂²	C₃₁×#553	1	1	yes	characteristic
705	C₂²	C₃₁×#555	1	1	yes	characteristic
706	C₄	—	2	1	no	—
707	C₄	—	2	1	no	—
708	C₄	—	2	1	no	—
709	C₄	—	2	1	no	—
710	C₄	—	4	1	no	—
711	C₄	—	4	1	no	—
712	C₄	—	4	1	no	—
713	C₄	—	8	1	no	—
714	C₄	—	8	1	no	—
715	C₄	C₃₁×#525	1	1	yes	characteristic
716	C₄	C₃₁×#535	1	1	yes	characteristic
717	C₄	C₃₁×#546	1	1	yes	characteristic
718	C₄	C₃₁×#550	1	1	yes	characteristic
719	C₂	—	2	1	no	—
720	C₂	—	2	1	no	—
721	C₂	—	8	1	no	—
722	C₂	—	8	2	no	—
723	C₂	—	16	1	no	—
724	C₂	C₃₁×#2414	1	1	yes	characteristic
725	C₂	C₃₁×#3067	1	1	yes	characteristic
726	C₂	C₃₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#2018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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×#3040	C₂	semidirect
2. C₂×C₃₁×#463	C₂	semidirect
3. C₄×C₃₁×AGL(1,ℤ/8)	C₂	semidirect
4. C₃₁×#2925	C₂	semidirect
5. C₃₁×#2052	C₂	semidirect
6. C₃₁×#2051	C₂	semidirect
7. C₃₁×#2019	C₂	semidirect
9. C₃₁×#1978	C₂	semidirect
10. C₃₁×#1977	C₂	semidirect
11. C₃₁×#1967	C₂	semidirect
12. C₃₁×#1950	C₂	semidirect
13. C₃₁×#1931	C₂	semidirect
14. C₃₁×#1930	C₂	semidirect
15. C₃₁×#1800	C₂	semidirect
258. #24674	C₃₁	direct (N × H)
325. #3040	C₂×C₃₁	semidirect
326. C₂×#463	C₂×C₃₁	semidirect
327. C₄×AGL(1,ℤ/8)	C₂×C₃₁	semidirect
328. #2925	C₂×C₃₁	semidirect
329. #2052	C₂×C₃₁	semidirect
330. #2051	C₂×C₃₁	semidirect
331. #2019	C₂×C₃₁	semidirect
333. #1978	C₂×C₃₁	semidirect
334. #1977	C₂×C₃₁	semidirect
335. #1967	C₂×C₃₁	semidirect
336. #1950	C₂×C₃₁	semidirect
337. #1931	C₂×C₃₁	semidirect
338. #1930	C₂×C₃₁	semidirect
339. #1800	C₂×C₃₁	semidirect
525. C₃₁	#24674	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×#3040	2
C₂×C₃₁×#463	2
C₄×C₃₁×AGL(1,ℤ/8)	2
C₃₁×#2925	2
C₃₁×#2052	2
C₃₁×#2051	2
C₃₁×#2019	2
C₃₁×#2018	2
C₃₁×#1978	2
C₃₁×#1977	2
C₃₁×#1967	2
C₃₁×#1950	2
C₃₁×#1931	2
C₃₁×#1930	2
C₃₁×#1800	2
#24674	31

Subgroups — C31×#24674 #21481433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24674 #21481433