Subgroups — C₃₁×#24283 #21481042

Order 7936.

1304 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#473	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#471	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3005	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#3004	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2930	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2047	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2046	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2020	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2019	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1979	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1977	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1811	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1785	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1784	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
17	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#137	—	2	4	no	—
19	C₂×C₃₁×#137	—	2	4	no	—
20	C₂×C₃₁×#137	—	2	2	no	—
21	C₂×C₃₁×#137	—	2	2	no	—
22	C₂×C₃₁×#137	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#137	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#137	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
26	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
28	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
29	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
31	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#525	C₂²	1	1	yes	characteristic
33	C₃₁×#473	—	2	4	no	—
34	C₃₁×#471	—	2	4	no	—
35	C₃₁×#452	—	2	2	no	—
36	C₃₁×#451	—	2	2	no	—
37	C₃₁×#449	—	2	2	no	—
38	C₃₁×#448	—	2	2	no	—
39	C₃₁×#436	—	2	2	no	—
40	C₃₁×#435	—	2	2	no	—
41	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
43	C₃₁×#424	C₂²	1	1	yes	characteristic
44	C₃₁×#419	C₂²	1	1	yes	characteristic
45	C₃₁×#419	C₂²	1	1	yes	characteristic
46	C₃₁×#419	C₂²	1	1	yes	characteristic
47	C₃₁×#418	C₂²	1	1	yes	characteristic
48	C₂×C₃₁×#101	C₂²	1	1	yes	characteristic
49	C₂×C₃₁×#101	C₂²	1	1	yes	characteristic
50	C₂×C₃₁×#101	C₂²	1	1	yes	characteristic
51	C₂×C₃₁×#101	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	C₃₁×#408	C₂²	1	1	yes	characteristic
57	C₃₁×#407	C₂²	1	1	yes	characteristic
58	C₃₁×#407	C₂²	1	1	yes	characteristic
59	C₃₁×#407	C₂²	1	1	yes	characteristic
60	C₃₁×#405	C₂²	1	1	yes	characteristic
61	C₃₁×#341	C₂²	1	1	yes	characteristic
62	C₃₁×#341	C₂²	1	1	yes	characteristic
63	C₂×#41×C₃₁	—	2	2	no	—
64	C₂×#41×C₃₁	—	2	2	no	—
65	C₂×#41×C₃₁	—	2	2	no	—
66	C₂×#41×C₃₁	—	2	2	no	—
67	C₂×#41×C₃₁	—	4	1	no	—
68	C₂×#41×C₃₁	—	4	1	no	—
69	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₃₁	—	2	1	no	—
71	C₂²×Q₈×C₃₁	—	2	2	no	—
72	C₂²×Q₈×C₃₁	—	2	2	no	—
73	C₂²×Q₈×C₃₁	—	2	2	no	—
74	C₂²×Q₈×C₃₁	—	2	2	no	—
75	C₂²×Q₈×C₃₁	—	4	1	no	—
76	C₂²×Q₈×C₃₁	—	4	1	no	—
77	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₂²×D₄×C₃₁	—	2	1	no	—
81	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₃₁	—	2	1	no	—
83	C₂³×C₄×C₃₁	—	2	1	no	—
84	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₃₁×#137	—	2	16	no	—
86	C₃₁×#137	—	2	16	no	—
87	C₃₁×#137	—	2	2	no	—
88	C₃₁×#137	—	2	2	no	—
89	C₃₁×#137	—	2	2	no	—
90	C₃₁×#137	—	4	1	no	—
91	C₃₁×#137	—	4	1	no	—
92	C₃₁×#137	—	4	1	no	—
93	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
94	C₃₁×AGL(1,ℤ/8)	—	4	1	no	—
95	C₂×Q₁₆×C₃₁	—	2	2	no	—
96	C₂×Q₁₆×C₃₁	—	2	2	no	—
97	C₂×Q₁₆×C₃₁	—	2	1	no	—
98	C₂×Q₁₆×C₃₁	—	2	1	no	—
99	C₂×Q₁₆×C₃₁	—	2	1	no	—
100	C₂×Q₁₆×C₃₁	—	2	2	no	—
101	C₂×Q₁₆×C₃₁	—	2	1	no	—
102	C₂×Q₁₆×C₃₁	—	2	1	no	—
103	C₂×Q₁₆×C₃₁	—	2	2	no	—
104	C₂×Q₁₆×C₃₁	D₄	1	4	yes	complemented
105	C₂×SD₁₆×C₃₁	—	2	2	no	—
106	C₂×SD₁₆×C₃₁	—	2	2	no	—
107	C₂×SD₁₆×C₃₁	—	2	1	no	—
108	C₂×SD₁₆×C₃₁	—	2	1	no	—
109	C₂×SD₁₆×C₃₁	—	2	1	no	—
110	C₂×SD₁₆×C₃₁	—	2	1	no	—
111	C₂×SD₁₆×C₃₁	—	2	2	no	—
112	C₂×SD₁₆×C₃₁	—	2	1	no	—
113	C₂×SD₁₆×C₃₁	—	2	1	no	—
114	C₂×SD₁₆×C₃₁	—	2	2	no	—
115	C₂×SD₁₆×C₃₁	D₄	1	4	yes	complemented
116	C₂×D₈×C₃₁	—	2	1	no	—
117	C₃₁×#131	—	2	2	no	—
118	C₃₁×#131	—	2	2	no	—
119	C₂×M₁₆×C₃₁	—	2	2	no	—
120	C₂×M₁₆×C₃₁	—	2	2	no	—
121	C₂×M₁₆×C₃₁	—	2	1	no	—
122	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
123	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
124	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
125	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
126	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
127	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
128	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
129	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
130	C₂²×C₈×C₃₁	—	2	1	no	—
131	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
132	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
133	C₃₁×#123	—	2	1	no	—
134	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
135	C₃₁×#121	D₄	1	2	yes	—
136	C₃₁×#120	—	2	1	no	—
137	C₄×D₄×C₃₁	—	2	2	no	—
138	C₃₁×#117	C₂³	1	1	yes	characteristic
139	C₃₁×#108	—	2	2	no	—
140	C₃₁×#108	—	2	2	no	—
141	C₃₁×C₄⋊C₈	—	2	2	no	—
142	C₃₁×#103	—	2	2	no	—
143	C₃₁×#103	—	2	2	no	—
144	C₃₁×#103	—	2	2	no	—
145	C₃₁×#102	—	2	2	no	—
146	C₃₁×#101	—	2	2	no	—
147	C₃₁×#101	—	2	2	no	—
148	C₃₁×#101	—	2	2	no	—
149	C₃₁×#101	—	4	1	no	—
150	C₃₁×#101	—	4	1	no	—
151	C₃₁×#101	—	4	1	no	—
152	C₃₁×#100	—	2	2	no	—
153	C₃₁×#100	—	4	1	no	—
154	C₃₁×C₂²⋊C₈	—	2	2	no	—
155	C₃₁×C₂²⋊C₈	—	2	2	no	—
156	C₃₁×C₂²⋊C₈	—	2	2	no	—
157	C₃₁×C₂²⋊C₈	—	2	2	no	—
158	C₃₁×#97	—	2	1	no	—
159	C₄×C₈×C₃₁	—	2	1	no	—
160	C₂⁴×C₃₁	—	2	1	no	—
161	C₂⁴×C₃₁	#41	1	1	yes	characteristic
162	#41×C₃₁	—	2	8	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	—	4	4	no	—
168	#41×C₃₁	—	4	4	no	—
169	#41×C₃₁	—	4	4	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	8	no	—
174	C₂×Q₈×C₃₁	—	2	1	no	—
175	C₂×Q₈×C₃₁	—	2	1	no	—
176	C₂×Q₈×C₃₁	—	2	1	no	—
177	C₂×Q₈×C₃₁	—	2	1	no	—
178	C₂×Q₈×C₃₁	—	2	1	no	—
179	C₂×Q₈×C₃₁	—	2	1	no	—
180	C₂×Q₈×C₃₁	—	2	1	no	—
181	C₂×Q₈×C₃₁	—	2	1	no	—
182	C₂×Q₈×C₃₁	—	2	1	no	—
183	C₂×Q₈×C₃₁	—	2	1	no	—
184	C₂×Q₈×C₃₁	—	2	1	no	—
185	C₂×Q₈×C₃₁	—	2	1	no	—
186	C₂×Q₈×C₃₁	—	4	2	no	—
187	C₂×Q₈×C₃₁	—	4	2	no	—
188	C₂×Q₈×C₃₁	—	4	4	no	—
189	C₂×Q₈×C₃₁	—	4	4	no	—
190	C₂×Q₈×C₃₁	—	4	4	no	—
191	C₂×Q₈×C₃₁	—	4	2	no	—
192	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
193	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
194	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
195	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
196	C₂×D₄×C₃₁	—	2	1	no	—
197	C₂×D₄×C₃₁	—	2	2	no	—
198	C₂×D₄×C₃₁	—	2	1	no	—
199	C₂×D₄×C₃₁	—	2	1	no	—
200	C₂×D₄×C₃₁	—	2	1	no	—
201	C₂×D₄×C₃₁	—	2	1	no	—
202	C₂×D₄×C₃₁	—	2	1	no	—
203	C₂×D₄×C₃₁	—	4	2	no	—
204	C₂×D₄×C₃₁	—	4	2	no	—
205	C₂×D₄×C₃₁	—	4	2	no	—
206	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
207	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
208	C₂²×C₄×C₃₁	—	2	1	no	—
209	C₂²×C₄×C₃₁	—	2	1	no	—
210	C₂²×C₄×C₃₁	—	2	1	no	—
211	C₂²×C₄×C₃₁	—	2	1	no	—
212	C₂²×C₄×C₃₁	—	2	1	no	—
213	C₂²×C₄×C₃₁	—	2	1	no	—
214	C₂²×C₄×C₃₁	—	2	1	no	—
215	C₂²×C₄×C₃₁	—	2	1	no	—
216	C₂²×C₄×C₃₁	—	4	2	no	—
217	C₂²×C₄×C₃₁	—	4	2	no	—
218	C₂²×C₄×C₃₁	—	4	2	no	—
219	C₂²×C₄×C₃₁	—	4	2	no	—
220	C₂²×C₄×C₃₁	—	4	2	no	—
221	C₂²×C₄×C₃₁	—	4	2	no	—
222	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
223	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
224	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
225	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
226	Q₁₆×C₃₁	—	2	8	no	—
227	Q₁₆×C₃₁	—	2	4	no	—
228	Q₁₆×C₃₁	—	2	4	no	—
229	Q₁₆×C₃₁	—	4	2	no	—
230	Q₁₆×C₃₁	—	4	2	no	—
231	Q₁₆×C₃₁	—	4	2	no	—
232	SD₁₆×C₃₁	—	2	8	no	—
233	SD₁₆×C₃₁	—	2	4	no	—
234	SD₁₆×C₃₁	—	2	4	no	—
235	SD₁₆×C₃₁	—	4	2	no	—
236	SD₁₆×C₃₁	—	4	2	no	—
237	SD₁₆×C₃₁	—	4	2	no	—
238	SD₁₆×C₃₁	—	4	2	no	—
239	D₈×C₃₁	—	4	2	no	—
240	M₁₆×C₃₁	—	2	4	no	—
241	M₁₆×C₃₁	—	2	4	no	—
242	M₁₆×C₃₁	—	2	1	no	—
243	M₁₆×C₃₁	—	2	1	no	—
244	M₁₆×C₃₁	—	2	1	no	—
245	M₁₆×C₃₁	—	2	1	no	—
246	M₁₆×C₃₁	—	2	1	no	—
247	M₁₆×C₃₁	—	2	1	no	—
248	M₁₆×C₃₁	—	2	1	no	—
249	M₁₆×C₃₁	—	2	1	no	—
250	M₁₆×C₃₁	—	2	1	no	—
251	M₁₆×C₃₁	—	2	1	no	—
252	M₁₆×C₃₁	—	2	1	no	—
253	M₁₆×C₃₁	—	2	1	no	—
254	M₁₆×C₃₁	—	4	2	no	—
255	C₂×C₈×C₃₁	—	2	1	no	—
256	C₂×C₈×C₃₁	—	2	1	no	—
257	C₂×C₈×C₃₁	—	2	1	no	—
258	C₂×C₈×C₃₁	—	2	1	no	—
259	C₂×C₈×C₃₁	—	2	1	no	—
260	C₂×C₈×C₃₁	—	2	1	no	—
261	C₂×C₈×C₃₁	—	2	1	no	—
262	C₂×C₈×C₃₁	—	2	1	no	—
263	C₂×C₈×C₃₁	—	4	2	no	—
264	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
265	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
266	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
267	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
268	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
269	C₄⋊C₄×C₃₁	—	2	2	no	—
270	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
271	C₂²⋊C₄×C₃₁	—	2	2	no	—
272	C₂²⋊C₄×C₃₁	—	2	1	no	—
273	C₂²⋊C₄×C₃₁	—	2	2	no	—
274	C₄²×C₃₁	—	2	1	no	—
275	#24283	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
276	C₂³×C₃₁	—	2	1	no	—
277	C₂³×C₃₁	—	2	1	no	—
278	C₂³×C₃₁	—	2	1	no	—
279	C₂³×C₃₁	—	2	2	no	—
280	C₂³×C₃₁	—	4	2	no	—
281	C₂³×C₃₁	—	4	1	no	—
282	C₂³×C₃₁	—	4	1	no	—
283	C₂³×C₃₁	—	8	2	no	—
284	C₂³×C₃₁	#121	1	2	yes	—
285	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
286	Q₈×C₃₁	—	2	2	no	—
287	Q₈×C₃₁	—	2	2	no	—
288	Q₈×C₃₁	—	2	2	no	—
289	Q₈×C₃₁	—	2	1	no	—
290	Q₈×C₃₁	—	2	2	no	—
291	Q₈×C₃₁	—	2	2	no	—
292	Q₈×C₃₁	—	2	2	no	—
293	Q₈×C₃₁	—	2	1	no	—
294	Q₈×C₃₁	—	4	8	no	—
295	Q₈×C₃₁	—	4	1	no	—
296	Q₈×C₃₁	—	4	1	no	—
297	Q₈×C₃₁	—	4	1	no	—
298	Q₈×C₃₁	—	4	1	no	—
299	Q₈×C₃₁	—	4	1	no	—
300	Q₈×C₃₁	—	4	1	no	—
301	D₄×C₃₁	—	2	2	no	—
302	D₄×C₃₁	—	2	1	no	—
303	D₄×C₃₁	—	2	2	no	—
304	D₄×C₃₁	—	2	1	no	—
305	D₄×C₃₁	—	4	8	no	—
306	D₄×C₃₁	—	4	1	no	—
307	D₄×C₃₁	—	4	1	no	—
308	D₄×C₃₁	—	4	1	no	—
309	D₄×C₃₁	—	4	1	no	—
310	D₄×C₃₁	—	8	2	no	—
311	C₂×C₄×C₃₁	—	2	1	no	—
312	C₂×C₄×C₃₁	—	2	1	no	—
313	C₂×C₄×C₃₁	—	2	2	no	—
314	C₂×C₄×C₃₁	—	2	2	no	—
315	C₂×C₄×C₃₁	—	2	2	no	—
316	C₂×C₄×C₃₁	—	2	2	no	—
317	C₂×C₄×C₃₁	—	2	2	no	—
318	C₂×C₄×C₃₁	—	2	1	no	—
319	C₂×C₄×C₃₁	—	2	2	no	—
320	C₂×C₄×C₃₁	—	2	2	no	—
321	C₂×C₄×C₃₁	—	2	2	no	—
322	C₂×C₄×C₃₁	—	4	8	no	—
323	C₂×C₄×C₃₁	—	4	8	no	—
324	C₂×C₄×C₃₁	—	4	1	no	—
325	C₂×C₄×C₃₁	—	4	1	no	—
326	C₂×C₄×C₃₁	—	4	1	no	—
327	C₂×C₄×C₃₁	—	4	1	no	—
328	C₂×C₄×C₃₁	—	4	1	no	—
329	C₂×C₄×C₃₁	—	4	1	no	—
330	C₂×C₄×C₃₁	—	4	1	no	—
331	C₂×C₄×C₃₁	—	4	1	no	—
332	C₂×C₄×C₃₁	#121	1	2	yes	—
333	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
334	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
335	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
336	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
337	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
338	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
339	C₈×C₃₁	—	2	2	no	—
340	C₈×C₃₁	—	4	1	no	—
341	C₈×C₃₁	—	4	1	no	—
342	C₈×C₃₁	—	4	1	no	—
343	C₈×C₃₁	—	4	1	no	—
344	C₈×C₃₁	—	4	1	no	—
345	C₈×C₃₁	—	4	1	no	—
346	C₈×C₃₁	#120	1	2	yes	—
347	C₈×C₃₁	#120	1	2	yes	—
348	C₂²×#137	C₂×C₃₁	1	1	yes	characteristic, complemented
349	C₂×#473	C₂×C₃₁	1	1	yes	characteristic, complemented
350	C₂×#471	C₂×C₃₁	1	1	yes	characteristic, complemented
351	#3005	C₂×C₃₁	1	1	yes	characteristic
352	#3004	C₂×C₃₁	1	1	yes	characteristic, complemented
353	#2930	C₂×C₃₁	1	1	yes	characteristic, complemented
354	#2047	C₂×C₃₁	1	1	yes	characteristic, complemented
355	#2046	C₂×C₃₁	1	1	yes	characteristic, complemented
356	#2020	C₂×C₃₁	1	1	yes	characteristic, complemented
357	#2019	C₂×C₃₁	1	1	yes	characteristic, complemented
358	#1979	C₂×C₃₁	1	1	yes	characteristic, complemented
359	#1977	C₂×C₃₁	1	1	yes	characteristic, complemented
360	#1811	C₂×C₃₁	1	1	yes	characteristic, complemented
361	#1785	C₂×C₃₁	1	1	yes	characteristic, complemented
362	#1784	C₂×C₃₁	1	1	yes	characteristic, complemented
363	C₂²×C₃₁	—	2	2	no	—
364	C₂²×C₃₁	—	2	2	no	—
365	C₂²×C₃₁	—	2	2	no	—
366	C₂²×C₃₁	—	2	2	no	—
367	C₂²×C₃₁	—	4	2	no	—
368	C₂²×C₃₁	—	4	2	no	—
369	C₂²×C₃₁	—	4	2	no	—
370	C₂²×C₃₁	—	4	1	no	—
371	C₂²×C₃₁	—	4	2	no	—
372	C₂²×C₃₁	—	8	2	no	—
373	C₂²×C₃₁	—	8	2	no	—
374	C₂²×C₃₁	—	8	2	no	—
375	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
376	C₂²×C₃₁	#546	1	1	yes	characteristic
377	C₂²×C₃₁	#546	1	1	yes	characteristic
378	C₄×C₃₁	—	2	2	no	—
379	C₄×C₃₁	—	4	2	no	—
380	C₄×C₃₁	—	4	2	no	—
381	C₄×C₃₁	—	4	2	no	—
382	C₄×C₃₁	—	4	1	no	—
383	C₄×C₃₁	—	4	2	no	—
384	C₄×C₃₁	C₂×#120	1	1	yes	characteristic
385	C₄×C₃₁	#535	1	1	yes	characteristic
386	C₄×C₃₁	D₄²	1	1	yes	characteristic
387	C₄×C₃₁	D₄²	1	1	yes	characteristic
388	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
389	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
390	C₂×#137	—	2	4	no	—
391	C₂×#137	—	2	4	no	—
392	C₂×#137	—	2	2	no	—
393	C₂×#137	—	2	2	no	—
394	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic
395	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic
396	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic
397	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
398	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
399	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
400	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
401	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
402	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
403	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
404	#525	C₂²×C₃₁	1	1	yes	characteristic
405	#473	—	2	4	no	—
406	#471	—	2	4	no	—
407	#452	—	2	2	no	—
408	#451	—	2	2	no	—
409	#449	—	2	2	no	—
410	#448	—	2	2	no	—
411	#436	—	2	2	no	—
412	#435	—	2	2	no	—
413	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
414	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
415	#424	C₂²×C₃₁	1	1	yes	characteristic
416	#419	C₂²×C₃₁	1	1	yes	characteristic
417	#419	C₂²×C₃₁	1	1	yes	characteristic
418	#419	C₂²×C₃₁	1	1	yes	characteristic
419	#418	C₂²×C₃₁	1	1	yes	characteristic
420	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
421	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
422	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
423	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
424	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
425	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
426	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
427	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
428	#408	C₂²×C₃₁	1	1	yes	characteristic
429	#407	C₂²×C₃₁	1	1	yes	characteristic
430	#407	C₂²×C₃₁	1	1	yes	characteristic
431	#407	C₂²×C₃₁	1	1	yes	characteristic
432	#405	C₂²×C₃₁	1	1	yes	characteristic
433	#341	C₂²×C₃₁	1	1	yes	characteristic
434	#341	C₂²×C₃₁	1	1	yes	characteristic
435	C₂×C₃₁	—	2	1	no	—
436	C₂×C₃₁	—	2	1	no	—
437	C₂×C₃₁	—	4	2	no	—
438	C₂×C₃₁	—	8	1	no	—
439	C₂×C₃₁	—	8	2	no	—
440	C₂×C₃₁	#2408	1	1	yes	characteristic
441	C₂×C₃₁	#3069	1	2	yes	—
442	C₂×#41	—	2	2	no	—
443	C₂×#41	—	2	2	no	—
444	C₂×#41	—	2	2	no	—
445	C₂×#41	—	2	2	no	—
446	C₂×#41	—	4	1	no	—
447	C₂×#41	—	4	1	no	—
448	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
449	C₂²×Q₈	—	2	1	no	—
450	C₂²×Q₈	—	2	2	no	—
451	C₂²×Q₈	—	2	2	no	—
452	C₂²×Q₈	—	2	2	no	—
453	C₂²×Q₈	—	2	2	no	—
454	C₂²×Q₈	—	4	1	no	—
455	C₂²×Q₈	—	4	1	no	—
456	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
457	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
458	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
459	C₂²×D₄	—	2	1	no	—
460	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
461	C₂³×C₄	—	2	1	no	—
462	C₂³×C₄	—	2	1	no	—
463	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
464	#137	—	2	16	no	—
465	#137	—	2	16	no	—
466	#137	—	2	2	no	—
467	#137	—	2	2	no	—
468	#137	—	2	2	no	—
469	#137	—	4	1	no	—
470	#137	—	4	1	no	—
471	#137	—	4	1	no	—
472	AGL(1,ℤ/8)	—	2	2	no	—
473	AGL(1,ℤ/8)	—	4	1	no	—
474	C₂×Q₁₆	—	2	2	no	—
475	C₂×Q₁₆	—	2	2	no	—
476	C₂×Q₁₆	—	2	1	no	—
477	C₂×Q₁₆	—	2	1	no	—
478	C₂×Q₁₆	—	2	1	no	—
479	C₂×Q₁₆	—	2	2	no	—
480	C₂×Q₁₆	—	2	1	no	—
481	C₂×Q₁₆	—	2	1	no	—
482	C₂×Q₁₆	—	2	2	no	—
483	C₂×Q₁₆	D₄×C₃₁	1	4	yes	complemented
484	C₂×SD₁₆	—	2	2	no	—
485	C₂×SD₁₆	—	2	2	no	—
486	C₂×SD₁₆	—	2	1	no	—
487	C₂×SD₁₆	—	2	1	no	—
488	C₂×SD₁₆	—	2	1	no	—
489	C₂×SD₁₆	—	2	1	no	—
490	C₂×SD₁₆	—	2	2	no	—
491	C₂×SD₁₆	—	2	1	no	—
492	C₂×SD₁₆	—	2	1	no	—
493	C₂×SD₁₆	—	2	2	no	—
494	C₂×SD₁₆	D₄×C₃₁	1	4	yes	complemented
495	C₂×D₈	—	2	1	no	—
496	#131	—	2	2	no	—
497	#131	—	2	2	no	—
498	C₂×M₁₆	—	2	2	no	—
499	C₂×M₁₆	—	2	2	no	—
500	C₂×M₁₆	—	2	1	no	—
501	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
502	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
503	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
504	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
505	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
506	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
507	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
508	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
509	C₂²×C₈	—	2	1	no	—
510	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
511	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
512	#123	—	2	1	no	—
513	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
514	#121	D₄×C₃₁	1	2	yes	—
515	#120	—	2	1	no	—
516	C₄×D₄	—	2	2	no	—
517	#117	C₂³×C₃₁	1	1	yes	characteristic
518	#108	—	2	2	no	—
519	#108	—	2	2	no	—
520	C₄⋊C₈	—	2	2	no	—
521	#103	—	2	2	no	—
522	#103	—	2	2	no	—
523	#103	—	2	2	no	—
524	#102	—	2	2	no	—
525	#101	—	2	2	no	—
526	#101	—	2	2	no	—
527	#101	—	2	2	no	—
528	#101	—	4	1	no	—
529	#101	—	4	1	no	—
530	#101	—	4	1	no	—
531	#100	—	2	2	no	—
532	#100	—	4	1	no	—
533	C₂²⋊C₈	—	2	2	no	—
534	C₂²⋊C₈	—	2	2	no	—
535	C₂²⋊C₈	—	2	2	no	—
536	C₂²⋊C₈	—	2	2	no	—
537	#97	—	2	1	no	—
538	C₄×C₈	—	2	1	no	—
539	C₃₁	#24283	1	1	yes	characteristic, direct product factor, Sylow
540	C₂⁴	—	2	1	no	—
541	C₂⁴	#41×C₃₁	1	1	yes	characteristic
542	#41	—	2	8	no	—
543	#41	—	2	1	no	—
544	#41	—	2	1	no	—
545	#41	—	2	1	no	—
546	#41	—	2	1	no	—
547	#41	—	4	4	no	—
548	#41	—	4	4	no	—
549	#41	—	4	4	no	—
550	C₂×Q₈	—	2	1	no	—
551	C₂×Q₈	—	2	1	no	—
552	C₂×Q₈	—	2	1	no	—
553	C₂×Q₈	—	2	8	no	—
554	C₂×Q₈	—	2	1	no	—
555	C₂×Q₈	—	2	1	no	—
556	C₂×Q₈	—	2	1	no	—
557	C₂×Q₈	—	2	1	no	—
558	C₂×Q₈	—	2	1	no	—
559	C₂×Q₈	—	2	1	no	—
560	C₂×Q₈	—	2	1	no	—
561	C₂×Q₈	—	2	1	no	—
562	C₂×Q₈	—	2	1	no	—
563	C₂×Q₈	—	2	1	no	—
564	C₂×Q₈	—	2	1	no	—
565	C₂×Q₈	—	2	1	no	—
566	C₂×Q₈	—	4	2	no	—
567	C₂×Q₈	—	4	2	no	—
568	C₂×Q₈	—	4	4	no	—
569	C₂×Q₈	—	4	4	no	—
570	C₂×Q₈	—	4	4	no	—
571	C₂×Q₈	—	4	2	no	—
572	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
573	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
574	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
575	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
576	C₂×D₄	—	2	1	no	—
577	C₂×D₄	—	2	2	no	—
578	C₂×D₄	—	2	1	no	—
579	C₂×D₄	—	2	1	no	—
580	C₂×D₄	—	2	1	no	—
581	C₂×D₄	—	2	1	no	—
582	C₂×D₄	—	2	1	no	—
583	C₂×D₄	—	4	2	no	—
584	C₂×D₄	—	4	2	no	—
585	C₂×D₄	—	4	2	no	—
586	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
587	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
588	C₂²×C₄	—	2	1	no	—
589	C₂²×C₄	—	2	1	no	—
590	C₂²×C₄	—	2	1	no	—
591	C₂²×C₄	—	2	1	no	—
592	C₂²×C₄	—	2	1	no	—
593	C₂²×C₄	—	2	1	no	—
594	C₂²×C₄	—	2	1	no	—
595	C₂²×C₄	—	2	1	no	—
596	C₂²×C₄	—	4	2	no	—
597	C₂²×C₄	—	4	2	no	—
598	C₂²×C₄	—	4	2	no	—
599	C₂²×C₄	—	4	2	no	—
600	C₂²×C₄	—	4	2	no	—
601	C₂²×C₄	—	4	2	no	—
602	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
603	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
604	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
605	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
606	Q₁₆	—	2	8	no	—
607	Q₁₆	—	2	4	no	—
608	Q₁₆	—	2	4	no	—
609	Q₁₆	—	4	2	no	—
610	Q₁₆	—	4	2	no	—
611	Q₁₆	—	4	2	no	—
612	SD₁₆	—	2	8	no	—
613	SD₁₆	—	2	4	no	—
614	SD₁₆	—	2	4	no	—
615	SD₁₆	—	4	2	no	—
616	SD₁₆	—	4	2	no	—
617	SD₁₆	—	4	2	no	—
618	SD₁₆	—	4	2	no	—
619	D₈	—	4	2	no	—
620	M₁₆	—	2	4	no	—
621	M₁₆	—	2	4	no	—
622	M₁₆	—	2	1	no	—
623	M₁₆	—	2	1	no	—
624	M₁₆	—	2	1	no	—
625	M₁₆	—	2	1	no	—
626	M₁₆	—	2	1	no	—
627	M₁₆	—	2	1	no	—
628	M₁₆	—	2	1	no	—
629	M₁₆	—	2	1	no	—
630	M₁₆	—	2	1	no	—
631	M₁₆	—	2	1	no	—
632	M₁₆	—	2	1	no	—
633	M₁₆	—	2	1	no	—
634	M₁₆	—	4	2	no	—
635	C₂×C₈	—	2	1	no	—
636	C₂×C₈	—	2	1	no	—
637	C₂×C₈	—	2	1	no	—
638	C₂×C₈	—	2	1	no	—
639	C₂×C₈	—	2	1	no	—
640	C₂×C₈	—	2	1	no	—
641	C₂×C₈	—	2	1	no	—
642	C₂×C₈	—	2	1	no	—
643	C₂×C₈	—	4	2	no	—
644	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
645	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
646	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
647	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
648	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
649	C₄⋊C₄	—	2	2	no	—
650	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
651	C₂²⋊C₄	—	2	2	no	—
652	C₂²⋊C₄	—	2	1	no	—
653	C₂²⋊C₄	—	2	2	no	—
654	C₄²	—	2	1	no	—
655	C₂³	—	2	1	no	—
656	C₂³	—	2	1	no	—
657	C₂³	—	2	1	no	—
658	C₂³	—	2	2	no	—
659	C₂³	—	4	2	no	—
660	C₂³	—	4	1	no	—
661	C₂³	—	4	1	no	—
662	C₂³	—	8	2	no	—
663	C₂³	C₃₁×#121	1	2	yes	—
664	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
665	Q₈	—	2	2	no	—
666	Q₈	—	2	2	no	—
667	Q₈	—	2	2	no	—
668	Q₈	—	2	1	no	—
669	Q₈	—	2	2	no	—
670	Q₈	—	2	2	no	—
671	Q₈	—	2	2	no	—
672	Q₈	—	2	1	no	—
673	Q₈	—	4	8	no	—
674	Q₈	—	4	1	no	—
675	Q₈	—	4	1	no	—
676	Q₈	—	4	1	no	—
677	Q₈	—	4	1	no	—
678	Q₈	—	4	1	no	—
679	Q₈	—	4	1	no	—
680	D₄	—	2	2	no	—
681	D₄	—	2	1	no	—
682	D₄	—	2	2	no	—
683	D₄	—	2	1	no	—
684	D₄	—	4	8	no	—
685	D₄	—	4	1	no	—
686	D₄	—	4	1	no	—
687	D₄	—	4	1	no	—
688	D₄	—	4	1	no	—
689	D₄	—	8	2	no	—
690	C₂×C₄	—	2	1	no	—
691	C₂×C₄	—	2	1	no	—
692	C₂×C₄	—	2	2	no	—
693	C₂×C₄	—	2	2	no	—
694	C₂×C₄	—	2	2	no	—
695	C₂×C₄	—	2	2	no	—
696	C₂×C₄	—	2	2	no	—
697	C₂×C₄	—	2	1	no	—
698	C₂×C₄	—	2	2	no	—
699	C₂×C₄	—	2	2	no	—
700	C₂×C₄	—	2	2	no	—
701	C₂×C₄	—	4	8	no	—
702	C₂×C₄	—	4	8	no	—
703	C₂×C₄	—	4	1	no	—
704	C₂×C₄	—	4	1	no	—
705	C₂×C₄	—	4	1	no	—
706	C₂×C₄	—	4	1	no	—
707	C₂×C₄	—	4	1	no	—
708	C₂×C₄	—	4	1	no	—
709	C₂×C₄	—	4	1	no	—
710	C₂×C₄	—	4	1	no	—
711	C₂×C₄	C₃₁×#121	1	2	yes	—
712	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
713	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
714	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
715	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
716	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
717	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
718	C₈	—	2	2	no	—
719	C₈	—	4	1	no	—
720	C₈	—	4	1	no	—
721	C₈	—	4	1	no	—
722	C₈	—	4	1	no	—
723	C₈	—	4	1	no	—
724	C₈	—	4	1	no	—
725	C₈	C₃₁×#120	1	2	yes	—
726	C₈	C₃₁×#120	1	2	yes	—
727	C₂²	—	2	2	no	—
728	C₂²	—	2	2	no	—
729	C₂²	—	2	2	no	—
730	C₂²	—	2	2	no	—
731	C₂²	—	4	2	no	—
732	C₂²	—	4	2	no	—
733	C₂²	—	4	2	no	—
734	C₂²	—	4	1	no	—
735	C₂²	—	4	2	no	—
736	C₂²	—	8	2	no	—
737	C₂²	—	8	2	no	—
738	C₂²	—	8	2	no	—
739	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
740	C₂²	C₃₁×#546	1	1	yes	characteristic
741	C₂²	C₃₁×#546	1	1	yes	characteristic
742	C₄	—	2	2	no	—
743	C₄	—	4	2	no	—
744	C₄	—	4	2	no	—
745	C₄	—	4	2	no	—
746	C₄	—	4	1	no	—
747	C₄	—	4	2	no	—
748	C₄	C₂×C₃₁×#120	1	1	yes	characteristic
749	C₄	C₃₁×#535	1	1	yes	characteristic
750	C₄	D₄²×C₃₁	1	1	yes	characteristic
751	C₄	D₄²×C₃₁	1	1	yes	characteristic
752	C₂	—	2	1	no	—
753	C₂	—	2	1	no	—
754	C₂	—	4	2	no	—
755	C₂	—	8	1	no	—
756	C₂	—	8	2	no	—
757	C₂	C₃₁×#2408	1	1	yes	characteristic
758	C₂	C₃₁×#3069	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×#137	C₂	semidirect
2. C₂×C₃₁×#473	C₂	semidirect
3. C₂×C₃₁×#471	C₂	semidirect
5. C₃₁×#3004	C₂	semidirect
6. C₃₁×#2930	C₂	semidirect
7. C₃₁×#2047	C₂	semidirect
8. C₃₁×#2046	C₂	semidirect
9. C₃₁×#2020	C₂	semidirect
10. C₃₁×#2019	C₂	semidirect
11. C₃₁×#1979	C₂	semidirect
12. C₃₁×#1977	C₂	semidirect
13. C₃₁×#1811	C₂	semidirect
14. C₃₁×#1785	C₂	semidirect
15. C₃₁×#1784	C₂	semidirect
104. C₂×Q₁₆×C₃₁	D₄	semidirect
115. C₂×SD₁₆×C₃₁	D₄	semidirect
275. #24283	C₃₁	direct (N × H)
348. C₂²×#137	C₂×C₃₁	semidirect
349. C₂×#473	C₂×C₃₁	semidirect
350. C₂×#471	C₂×C₃₁	semidirect
352. #3004	C₂×C₃₁	semidirect
353. #2930	C₂×C₃₁	semidirect
354. #2047	C₂×C₃₁	semidirect
355. #2046	C₂×C₃₁	semidirect
356. #2020	C₂×C₃₁	semidirect
357. #2019	C₂×C₃₁	semidirect
358. #1979	C₂×C₃₁	semidirect
359. #1977	C₂×C₃₁	semidirect
360. #1811	C₂×C₃₁	semidirect
361. #1785	C₂×C₃₁	semidirect
362. #1784	C₂×C₃₁	semidirect
483. C₂×Q₁₆	D₄×C₃₁	semidirect
494. C₂×SD₁₆	D₄×C₃₁	semidirect
539. C₃₁	#24283	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃₁×#137	2
C₂×C₃₁×#473	2
C₂×C₃₁×#471	2
C₃₁×#3005	2
C₃₁×#3004	2
C₃₁×#2930	2
C₃₁×#2047	2
C₃₁×#2046	2
C₃₁×#2020	2
C₃₁×#2019	2
C₃₁×#1979	2
C₃₁×#1977	2
C₃₁×#1811	2
C₃₁×#1785	2
C₃₁×#1784	2
#24283	31

Subgroups — C31×#24283 #21481042

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24283 #21481042