Subgroups — C₃₁×#60820 #21517579

Order 7936.

2122 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3221	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3207	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3206	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3151	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3139	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3042	C₂	1	4	yes	maximal, complemented
11	C₃₁×#3010	C₂	1	4	yes	maximal, complemented
12	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	2	yes	—
14	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	—
15	C₂²×SD₁₆×C₃₁	C₂²	1	4	yes	—
16	C₃₁×#559	—	2	2	no	—
17	C₃₁×#550	—	2	2	no	—
18	D₄×Q₈×C₃₁	—	2	2	no	—
19	C₃₁×#547	—	2	4	no	—
20	C₃₁×#547	—	2	2	no	—
21	C₃₁×#546	—	2	4	no	—
22	C₃₁×#540	C₂²	1	4	yes	—
23	C₃₁×#536	C₂²	1	2	yes	—
24	C₃₁×#536	C₂²	1	4	yes	complemented
25	C₃₁×#535	C₂²	1	2	yes	—
26	C₃₁×#533	C₂²	1	1	yes	characteristic
27	C₃₁×#532	C₂²	1	1	yes	characteristic
28	C₃₁×#532	C₂²	1	4	yes	—
29	C₃₁×#532	C₂²	1	2	yes	complemented
30	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	complemented
32	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
33	C₃₁×#518	C₂²	1	2	yes	—
34	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#496	C₂²	1	4	yes	—
36	C₃₁×#493	C₂²	1	2	yes	—
37	C₃₁×#492	C₂²	1	4	yes	—
38	C₃₁×#492	C₂²	1	4	yes	—
39	C₃₁×#492	C₂²	1	2	yes	—
40	C₃₁×#491	C₂²	1	4	yes	complemented
41	C₃₁×#488	C₂²	1	4	yes	complemented
42	C₃₁×#487	C₂²	1	2	yes	complemented
43	C₃₁×#487	C₂²	1	4	yes	complemented
44	C₃₁×#486	C₂²	1	2	yes	complemented
45	C₃₁×#463	C₂²	1	16	yes	complemented
46	C₃₁×#461	C₂²	1	16	yes	complemented
47	C₃₁×#449	C₂²	1	16	yes	complemented
48	C₃₁×#448	C₂²	1	16	yes	complemented
49	C₃₁×#433	C₂²	1	4	yes	complemented
50	C₃₁×#431	C₂²	1	4	yes	complemented
51	C₃₁×#424	C₂²	1	2	yes	—
52	C₃₁×#419	C₂²	1	4	yes	—
53	C₃₁×#416	C₂²	1	4	yes	—
54	C₃₁×#408	C₂²	1	2	yes	—
55	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
56	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
57	C₃₁×2₋¹⁺⁴	—	4	8	no	—
58	C₂×#41×C₃₁	—	2	4	no	—
59	C₂×#41×C₃₁	—	2	2	no	—
60	C₂×#41×C₃₁	—	2	4	no	—
61	C₂×#41×C₃₁	C₂³	1	2	yes	—
62	C₂×#41×C₃₁	C₂³	1	4	yes	—
63	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×Q₈×C₃₁	—	2	2	no	—
65	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
66	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
67	C₂²×D₄×C₃₁	—	2	2	no	—
68	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
69	C₃₁×AGL(1,ℤ/8)	—	4	8	no	—
70	C₂×SD₁₆×C₃₁	—	4	8	no	—
71	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
72	C₂×SD₁₆×C₃₁	C₂³	1	16	yes	—
73	C₂×D₈×C₃₁	C₂³	1	8	yes	—
74	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
75	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
76	C₃₁×C₄⋊Q₈	—	2	2	no	—
77	C₃₁×C₄⋊Q₈	—	2	4	no	—
78	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
79	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
80	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
82	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
83	C₃₁×#127	—	2	2	no	—
84	C₃₁×#127	—	2	2	no	—
85	C₃₁×#127	C₂³	1	1	yes	characteristic
86	C₃₁×#127	C₂³	1	2	yes	—
87	C₃₁×#127	C₂³	1	2	yes	—
88	C₃₁×#125	C₂³	1	2	yes	—
89	C₃₁×#124	—	2	2	no	—
90	C₃₁×#124	—	2	4	no	—
91	C₃₁×#124	C₂³	1	4	yes	—
92	C₃₁×#124	C₂³	1	2	yes	—
93	C₃₁×#123	—	2	8	no	—
94	C₃₁×#123	—	2	4	no	—
95	C₃₁×#123	—	2	8	no	—
96	C₃₁×C₂²⋊Q₈	—	2	2	no	—
97	C₃₁×C₂²⋊Q₈	—	2	4	no	—
98	C₃₁×C₂²⋊Q₈	—	2	8	no	—
99	C₃₁×#121	—	2	4	no	—
100	C₃₁×#121	—	2	2	no	—
101	C₃₁×#121	—	2	4	no	—
102	C₃₁×#121	—	2	8	no	—
103	C₃₁×#121	C₂³	1	8	yes	—
104	C₃₁×#120	—	2	4	no	—
105	C₄×Q₈×C₃₁	—	2	2	no	—
106	C₄×Q₈×C₃₁	D₄	1	2	yes	—
107	C₄×D₄×C₃₁	—	2	4	no	—
108	C₄×D₄×C₃₁	—	2	4	no	—
109	C₄×D₄×C₃₁	—	2	2	no	—
110	C₄×D₄×C₃₁	—	2	4	no	—
111	C₄×D₄×C₃₁	D₄	1	4	yes	—
112	C₄×D₄×C₃₁	D₄	1	2	yes	—
113	C₄×D₄×C₃₁	C₂³	1	8	yes	—
114	C₃₁×#117	C₂³	1	2	yes	—
115	C₃₁×#117	C₂³	1	1	yes	characteristic
116	C₃₁×#117	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
119	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
122	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
123	C₃₁×#103	C₂³	1	8	yes	—
124	C₃₁×#103	C₂³	1	16	yes	—
125	C₃₁×#102	C₂³	1	8	yes	—
126	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
127	C₃₁×#97	C₂³	1	2	yes	—
128	C₄×C₈×C₃₁	C₂³	1	2	yes	—
129	C₄×C₈×C₃₁	C₂³	1	2	yes	—
130	C₄×C₈×C₃₁	C₂³	1	2	yes	—
131	C₂⁴×C₃₁	—	2	2	no	—
132	#41×C₃₁	—	2	4	no	—
133	#41×C₃₁	—	2	8	no	—
134	#41×C₃₁	—	4	8	no	—
135	#41×C₃₁	—	4	16	no	—
136	#41×C₃₁	—	4	8	no	—
137	#41×C₃₁	—	8	8	no	—
138	C₂×Q₈×C₃₁	—	2	2	no	—
139	C₂×Q₈×C₃₁	—	2	4	no	—
140	C₂×Q₈×C₃₁	—	2	2	no	—
141	C₂×Q₈×C₃₁	—	2	8	no	—
142	C₂×Q₈×C₃₁	—	4	8	no	—
143	C₂×Q₈×C₃₁	—	4	8	no	—
144	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
146	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
147	C₂×D₄×C₃₁	—	2	4	no	—
148	C₂×D₄×C₃₁	—	2	2	no	—
149	C₂×D₄×C₃₁	—	2	8	no	—
150	C₂×D₄×C₃₁	—	2	2	no	—
151	C₂×D₄×C₃₁	—	2	2	no	—
152	C₂×D₄×C₃₁	—	2	4	no	—
153	C₂×D₄×C₃₁	—	2	4	no	—
154	C₂×D₄×C₃₁	—	2	8	no	—
155	C₂×D₄×C₃₁	—	8	4	no	—
156	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
157	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
159	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
160	C₂²×C₄×C₃₁	—	2	4	no	—
161	C₂²×C₄×C₃₁	—	2	2	no	—
162	C₂²×C₄×C₃₁	—	2	4	no	—
163	C₂²×C₄×C₃₁	—	2	2	no	—
164	C₂²×C₄×C₃₁	—	2	8	no	—
165	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
166	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
169	SD₁₆×C₃₁	—	4	16	no	—
170	SD₁₆×C₃₁	—	4	8	no	—
171	D₈×C₃₁	—	4	8	no	—
172	M₁₆×C₃₁	—	2	4	no	—
173	C₂×C₈×C₃₁	—	2	4	no	—
174	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
175	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
176	C₄⋊C₄×C₃₁	—	2	4	no	—
177	C₄⋊C₄×C₃₁	—	2	2	no	—
178	C₄⋊C₄×C₃₁	—	2	8	no	—
179	C₄⋊C₄×C₃₁	—	2	4	no	—
180	C₄⋊C₄×C₃₁	—	2	2	no	—
181	C₄⋊C₄×C₃₁	—	2	4	no	—
182	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
183	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
184	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
185	C₂²⋊C₄×C₃₁	—	2	2	no	—
186	C₂²⋊C₄×C₃₁	—	2	8	no	—
187	C₂²⋊C₄×C₃₁	—	2	8	no	—
188	C₂²⋊C₄×C₃₁	—	2	8	no	—
189	C₂²⋊C₄×C₃₁	—	2	4	no	—
190	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
191	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
192	C₄²×C₃₁	—	2	2	no	—
193	C₄²×C₃₁	C₂×D₄	1	4	yes	—
194	C₄²×C₃₁	C₂×D₄	1	2	yes	—
195	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
196	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
197	C₄²×C₃₁	C₂⁴	1	2	yes	—
198	#60820	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₂³×C₃₁	—	2	2	no	—
200	C₂³×C₃₁	—	2	4	no	—
201	C₂³×C₃₁	—	4	4	no	—
202	C₂³×C₃₁	—	8	4	no	—
203	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
204	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
205	Q₈×C₃₁	—	2	2	no	—
206	Q₈×C₃₁	—	2	8	no	—
207	Q₈×C₃₁	—	2	4	no	—
208	Q₈×C₃₁	—	4	8	no	—
209	Q₈×C₃₁	—	4	4	no	—
210	Q₈×C₃₁	—	4	4	no	—
211	D₄×C₃₁	—	2	4	no	—
212	D₄×C₃₁	—	2	2	no	—
213	D₄×C₃₁	—	2	4	no	—
214	D₄×C₃₁	—	2	8	no	—
215	D₄×C₃₁	—	4	8	no	—
216	D₄×C₃₁	—	4	4	no	—
217	D₄×C₃₁	—	4	4	no	—
218	D₄×C₃₁	—	4	4	no	—
219	D₄×C₃₁	—	4	8	no	—
220	D₄×C₃₁	—	8	4	no	—
221	D₄×C₃₁	—	8	8	no	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	2	8	no	—
224	C₂×C₄×C₃₁	—	2	4	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	4	8	no	—
229	C₂×C₄×C₃₁	—	4	8	no	—
230	C₂×C₄×C₃₁	—	4	4	no	—
231	C₂×C₄×C₃₁	—	4	4	no	—
232	C₂×C₄×C₃₁	—	8	8	no	—
233	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
234	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
235	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
238	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
239	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
240	C₈×C₃₁	—	2	4	no	—
241	C₈×C₃₁	—	2	4	no	—
242	#3508	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#3499	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#3221	C₂×C₃₁	1	8	yes	complemented
245	#3207	C₂×C₃₁	1	4	yes	complemented
246	#3206	C₂×C₃₁	1	4	yes	complemented
247	#3151	C₂×C₃₁	1	1	yes	characteristic, complemented
248	C₂×#492	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#3139	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#3131	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#3042	C₂×C₃₁	1	4	yes	complemented
252	#3010	C₂×C₃₁	1	4	yes	complemented
253	#2922	C₂×C₃₁	1	1	yes	characteristic, complemented
254	C₂²×C₃₁	—	2	2	no	—
255	C₂²×C₃₁	—	2	2	no	—
256	C₂²×C₃₁	—	2	4	no	—
257	C₂²×C₃₁	—	4	2	no	—
258	C₂²×C₃₁	—	4	4	no	—
259	C₂²×C₃₁	—	4	8	no	—
260	C₂²×C₃₁	—	4	4	no	—
261	C₂²×C₃₁	—	16	4	no	—
262	C₂²×C₃₁	#535	1	2	yes	—
263	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
264	C₄×C₃₁	—	2	2	no	—
265	C₄×C₃₁	—	2	4	no	—
266	C₄×C₃₁	—	2	2	no	—
267	C₄×C₃₁	—	4	4	no	—
268	C₄×C₃₁	—	4	2	no	—
269	C₄×C₃₁	—	4	4	no	—
270	C₄×C₃₁	#535	1	2	yes	—
271	C₄×C₃₁	#577	1	4	yes	—
272	C₄×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
273	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	2	yes	—
274	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	—
275	C₂²×SD₁₆	C₂²×C₃₁	1	4	yes	—
276	#559	—	2	2	no	—
277	#550	—	2	2	no	—
278	D₄×Q₈	—	2	2	no	—
279	#547	—	2	4	no	—
280	#547	—	2	2	no	—
281	#546	—	2	4	no	—
282	#540	C₂²×C₃₁	1	4	yes	—
283	#536	C₂²×C₃₁	1	2	yes	—
284	#536	C₂²×C₃₁	1	4	yes	complemented
285	#535	C₂²×C₃₁	1	2	yes	—
286	#533	C₂²×C₃₁	1	1	yes	characteristic
287	#532	C₂²×C₃₁	1	1	yes	characteristic
288	#532	C₂²×C₃₁	1	4	yes	—
289	#532	C₂²×C₃₁	1	2	yes	complemented
290	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
292	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
293	#518	C₂²×C₃₁	1	2	yes	—
294	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
295	#496	C₂²×C₃₁	1	4	yes	—
296	#493	C₂²×C₃₁	1	2	yes	—
297	#492	C₂²×C₃₁	1	4	yes	—
298	#492	C₂²×C₃₁	1	4	yes	—
299	#492	C₂²×C₃₁	1	2	yes	—
300	#491	C₂²×C₃₁	1	4	yes	complemented
301	#488	C₂²×C₃₁	1	4	yes	complemented
302	#487	C₂²×C₃₁	1	2	yes	complemented
303	#487	C₂²×C₃₁	1	4	yes	complemented
304	#486	C₂²×C₃₁	1	2	yes	complemented
305	#463	C₂²×C₃₁	1	16	yes	complemented
306	#461	C₂²×C₃₁	1	16	yes	complemented
307	#449	C₂²×C₃₁	1	16	yes	complemented
308	#448	C₂²×C₃₁	1	16	yes	complemented
309	#433	C₂²×C₃₁	1	4	yes	complemented
310	#431	C₂²×C₃₁	1	4	yes	complemented
311	#424	C₂²×C₃₁	1	2	yes	—
312	#419	C₂²×C₃₁	1	4	yes	—
313	#416	C₂²×C₃₁	1	4	yes	—
314	#408	C₂²×C₃₁	1	2	yes	—
315	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
316	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
317	C₂×C₃₁	—	2	2	no	—
318	C₂×C₃₁	—	4	2	no	—
319	C₂×C₃₁	—	8	2	no	—
320	C₂×C₃₁	—	8	4	no	—
321	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
322	C₂×C₃₁	C₂×#577	1	2	yes	—
323	2₋¹⁺⁴	—	4	8	no	—
324	C₂×#41	—	2	4	no	—
325	C₂×#41	—	2	2	no	—
326	C₂×#41	—	2	4	no	—
327	C₂×#41	C₂³×C₃₁	1	2	yes	—
328	C₂×#41	C₂³×C₃₁	1	4	yes	—
329	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
330	C₂²×Q₈	—	2	2	no	—
331	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
332	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
333	C₂²×D₄	—	2	2	no	—
334	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
335	AGL(1,ℤ/8)	—	4	8	no	—
336	C₂×SD₁₆	—	4	8	no	—
337	C₂×SD₁₆	C₂³×C₃₁	1	16	yes	—
338	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
339	C₂×D₈	C₂³×C₃₁	1	8	yes	—
340	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
341	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
342	C₄⋊Q₈	—	2	2	no	—
343	C₄⋊Q₈	—	2	4	no	—
344	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
345	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
346	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
347	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
348	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
349	#127	—	2	2	no	—
350	#127	—	2	2	no	—
351	#127	C₂³×C₃₁	1	2	yes	—
352	#127	C₂³×C₃₁	1	2	yes	—
353	#127	C₂³×C₃₁	1	1	yes	characteristic
354	#125	C₂³×C₃₁	1	2	yes	—
355	#124	—	2	2	no	—
356	#124	—	2	4	no	—
357	#124	C₂³×C₃₁	1	2	yes	—
358	#124	C₂³×C₃₁	1	4	yes	—
359	#123	—	2	8	no	—
360	#123	—	2	4	no	—
361	#123	—	2	8	no	—
362	C₂²⋊Q₈	—	2	2	no	—
363	C₂²⋊Q₈	—	2	4	no	—
364	C₂²⋊Q₈	—	2	8	no	—
365	#121	—	2	4	no	—
366	#121	—	2	2	no	—
367	#121	—	2	4	no	—
368	#121	—	2	8	no	—
369	#121	C₂³×C₃₁	1	8	yes	—
370	#120	—	2	4	no	—
371	C₄×Q₈	—	2	2	no	—
372	C₄×Q₈	D₄×C₃₁	1	2	yes	—
373	C₄×D₄	—	2	4	no	—
374	C₄×D₄	—	2	4	no	—
375	C₄×D₄	—	2	2	no	—
376	C₄×D₄	—	2	4	no	—
377	C₄×D₄	D₄×C₃₁	1	4	yes	—
378	C₄×D₄	D₄×C₃₁	1	2	yes	—
379	C₄×D₄	C₂³×C₃₁	1	8	yes	—
380	#117	C₂³×C₃₁	1	2	yes	—
381	#117	C₂³×C₃₁	1	1	yes	characteristic
382	#117	C₂³×C₃₁	1	2	yes	—
383	C₂×C₄⋊C₄	—	2	2	no	—
384	C₂×C₄⋊C₄	—	2	4	no	—
385	C₂×C₂²⋊C₄	—	2	4	no	—
386	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
387	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
388	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
389	#103	C₂³×C₃₁	1	8	yes	—
390	#103	C₂³×C₃₁	1	16	yes	—
391	#102	C₂³×C₃₁	1	8	yes	—
392	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
393	#97	C₂³×C₃₁	1	2	yes	—
394	C₄×C₈	C₂³×C₃₁	1	2	yes	—
395	C₄×C₈	C₂³×C₃₁	1	2	yes	—
396	C₄×C₈	C₂³×C₃₁	1	2	yes	—
397	C₃₁	#60820	1	1	yes	characteristic, direct product factor, Sylow
398	C₂⁴	—	2	2	no	—
399	#41	—	2	4	no	—
400	#41	—	2	8	no	—
401	#41	—	4	8	no	—
402	#41	—	4	16	no	—
403	#41	—	4	8	no	—
404	#41	—	8	8	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	2	4	no	—
407	C₂×Q₈	—	2	2	no	—
408	C₂×Q₈	—	2	8	no	—
409	C₂×Q₈	—	4	8	no	—
410	C₂×Q₈	—	4	8	no	—
411	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
412	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
413	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	2	2	no	—
416	C₂×D₄	—	2	8	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	8	no	—
422	C₂×D₄	—	8	4	no	—
423	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
424	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
425	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
426	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	8	no	—
432	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
433	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
434	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
435	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
436	SD₁₆	—	4	16	no	—
437	SD₁₆	—	4	8	no	—
438	D₈	—	4	8	no	—
439	M₁₆	—	2	4	no	—
440	C₂×C₈	—	2	4	no	—
441	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
442	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	8	no	—
446	C₄⋊C₄	—	2	4	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
450	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
451	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
452	C₂²⋊C₄	—	2	2	no	—
453	C₂²⋊C₄	—	2	8	no	—
454	C₂²⋊C₄	—	2	8	no	—
455	C₂²⋊C₄	—	2	8	no	—
456	C₂²⋊C₄	—	2	4	no	—
457	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
458	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
459	C₄²	—	2	2	no	—
460	C₄²	C₂×D₄×C₃₁	1	4	yes	—
461	C₄²	C₂×D₄×C₃₁	1	2	yes	—
462	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
463	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
464	C₄²	C₂⁴×C₃₁	1	2	yes	—
465	C₂³	—	2	2	no	—
466	C₂³	—	2	4	no	—
467	C₂³	—	4	4	no	—
468	C₂³	—	8	4	no	—
469	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
470	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
471	Q₈	—	2	2	no	—
472	Q₈	—	2	8	no	—
473	Q₈	—	2	4	no	—
474	Q₈	—	4	8	no	—
475	Q₈	—	4	4	no	—
476	Q₈	—	4	4	no	—
477	D₄	—	2	4	no	—
478	D₄	—	2	2	no	—
479	D₄	—	2	4	no	—
480	D₄	—	2	8	no	—
481	D₄	—	4	8	no	—
482	D₄	—	4	4	no	—
483	D₄	—	4	4	no	—
484	D₄	—	4	4	no	—
485	D₄	—	4	8	no	—
486	D₄	—	8	4	no	—
487	D₄	—	8	8	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	8	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	4	8	no	—
495	C₂×C₄	—	4	8	no	—
496	C₂×C₄	—	4	4	no	—
497	C₂×C₄	—	4	4	no	—
498	C₂×C₄	—	8	8	no	—
499	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
500	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
501	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
502	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
503	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
504	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
505	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
506	C₈	—	2	4	no	—
507	C₈	—	2	4	no	—
508	C₂²	—	2	2	no	—
509	C₂²	—	2	2	no	—
510	C₂²	—	2	4	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	4	no	—
513	C₂²	—	4	8	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	16	4	no	—
516	C₂²	C₃₁×#535	1	2	yes	—
517	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
518	C₄	—	2	2	no	—
519	C₄	—	2	4	no	—
520	C₄	—	2	2	no	—
521	C₄	—	4	4	no	—
522	C₄	—	4	2	no	—
523	C₄	—	4	4	no	—
524	C₄	C₃₁×#535	1	2	yes	—
525	C₄	C₃₁×#577	1	4	yes	—
526	C₄	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
527	C₂	—	2	2	no	—
528	C₂	—	4	2	no	—
529	C₂	—	8	4	no	—
530	C₂	—	8	2	no	—
531	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
532	C₂	C₂×C₃₁×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3508	C₂	semidirect
2. C₃₁×#3499	C₂	semidirect
3. C₃₁×#3221	C₂	semidirect
4. C₃₁×#3207	C₂	semidirect
5. C₃₁×#3206	C₂	semidirect
6. C₃₁×#3151	C₂	semidirect
7. C₂×C₃₁×#492	C₂	semidirect
8. C₃₁×#3139	C₂	semidirect
9. C₃₁×#3131	C₂	semidirect
10. C₃₁×#3042	C₂	semidirect
11. C₃₁×#3010	C₂	semidirect
12. C₃₁×#2922	C₂	semidirect
24. C₃₁×#536	C₂²	semidirect
29. C₃₁×#532	C₂²	semidirect
31. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
40. C₃₁×#491	C₂²	semidirect
41. C₃₁×#488	C₂²	semidirect
42. C₃₁×#487	C₂²	semidirect
43. C₃₁×#487	C₂²	semidirect
44. C₃₁×#486	C₂²	semidirect
45. C₃₁×#463	C₂²	semidirect
46. C₃₁×#461	C₂²	semidirect
47. C₃₁×#449	C₂²	semidirect
48. C₃₁×#448	C₂²	semidirect
49. C₃₁×#433	C₂²	semidirect
50. C₃₁×#431	C₂²	semidirect
198. #60820	C₃₁	direct (N × H)
242. #3508	C₂×C₃₁	semidirect
243. #3499	C₂×C₃₁	semidirect
244. #3221	C₂×C₃₁	semidirect
245. #3207	C₂×C₃₁	semidirect
246. #3206	C₂×C₃₁	semidirect
247. #3151	C₂×C₃₁	semidirect
248. C₂×#492	C₂×C₃₁	semidirect
249. #3139	C₂×C₃₁	semidirect
250. #3131	C₂×C₃₁	semidirect
251. #3042	C₂×C₃₁	semidirect
252. #3010	C₂×C₃₁	semidirect
253. #2922	C₂×C₃₁	semidirect
284. #536	C₂²×C₃₁	semidirect
289. #532	C₂²×C₃₁	semidirect
291. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
300. #491	C₂²×C₃₁	semidirect
301. #488	C₂²×C₃₁	semidirect
302. #487	C₂²×C₃₁	semidirect
303. #487	C₂²×C₃₁	semidirect
304. #486	C₂²×C₃₁	semidirect
305. #463	C₂²×C₃₁	semidirect
306. #461	C₂²×C₃₁	semidirect
307. #449	C₂²×C₃₁	semidirect
308. #448	C₂²×C₃₁	semidirect
309. #433	C₂²×C₃₁	semidirect
310. #431	C₂²×C₃₁	semidirect
397. C₃₁	#60820	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×#3508	2
C₃₁×#3499	2
C₃₁×#3221	2
C₃₁×#3207	2
C₃₁×#3206	2
C₃₁×#3151	2
C₂×C₃₁×#492	2
C₃₁×#3139	2
C₃₁×#3131	2
C₃₁×#3042	2
C₃₁×#3010	2
C₃₁×#2922	2
#60820	31

Subgroups — C31×#60820 #21517579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60820 #21517579