Subgroups — C₃₁×#60859 #21517618

Order 7936.

1654 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3458	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3455	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3244	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3243	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3242	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3241	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3135	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3091	C₂	1	4	yes	maximal, complemented
12	C₃₁×#3076	C₂	1	4	yes	maximal, complemented
13	C₂×C₃₁×#433	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#544	—	2	2	no	—
15	C₃₁×#544	C₂²	1	4	yes	—
16	C₃₁×#544	C₂²	1	4	yes	—
17	C₃₁×#543	—	2	2	no	—
18	C₃₁×#542	C₂²	1	4	yes	—
19	C₃₁×#541	C₂²	1	2	yes	—
20	C₃₁×#540	C₂²	1	2	yes	—
21	C₃₁×#539	—	2	2	no	—
22	C₃₁×#538	—	2	2	no	—
23	C₃₁×#533	—	2	4	no	—
24	C₃₁×#533	C₂²	1	2	yes	—
25	C₃₁×#533	C₂²	1	1	yes	characteristic
26	C₃₁×#533	C₂²	1	1	yes	characteristic
27	C₃₁×#533	C₂²	1	1	yes	characteristic
28	C₃₁×#532	C₂²	1	1	yes	characteristic
29	C₃₁×#529	—	2	4	no	—
30	C₃₁×#525	C₂²	1	2	yes	—
31	C₄×#41×C₃₁	C₂²	1	2	yes	—
32	C₂×C₃₁×#117	C₂²	1	2	yes	—
33	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
34	C₃₁×#501	C₂²	1	4	yes	complemented
35	C₃₁×#501	C₂²	1	4	yes	complemented
36	C₃₁×#500	C₂²	1	2	yes	complemented
37	C₃₁×#499	C₂²	1	4	yes	complemented
38	C₃₁×#498	C₂²	1	2	yes	complemented
39	C₃₁×#491	C₂²	1	2	yes	complemented
40	C₃₁×#490	C₂²	1	2	yes	complemented
41	C₃₁×#489	C₂²	1	4	yes	complemented
42	C₃₁×#488	C₂²	1	2	yes	complemented
43	C₃₁×#488	C₂²	1	2	yes	complemented
44	C₃₁×#487	C₂²	1	2	yes	complemented
45	C₃₁×#486	C₂²	1	2	yes	complemented
46	C₃₁×#485	C₂²	1	4	yes	—
47	C₃₁×#485	C₂²	1	4	yes	—
48	C₃₁×#484	C₂²	1	4	yes	—
49	C₃₁×#483	C₂²	1	4	yes	—
50	C₃₁×#482	C₂²	1	4	yes	—
51	C₃₁×#482	C₂²	1	4	yes	—
52	C₃₁×#481	C₂²	1	4	yes	—
53	C₃₁×#480	C₂²	1	4	yes	—
54	C₃₁×#479	C₂²	1	4	yes	complemented
55	C₃₁×#479	C₂²	1	4	yes	complemented
56	C₃₁×#478	C₂²	1	4	yes	complemented
57	C₃₁×#478	C₂²	1	4	yes	complemented
58	C₃₁×#477	C₂²	1	4	yes	complemented
59	C₃₁×#476	C₂²	1	4	yes	complemented
60	C₃₁×#475	C₂²	1	4	yes	complemented
61	C₃₁×#474	C₂²	1	4	yes	complemented
62	C₃₁×#433	C₂²	1	4	yes	—
63	C₃₁×#433	C₂²	1	2	yes	—
64	C₃₁×#433	C₂²	1	2	yes	—
65	C₃₁×#427	C₂²	1	4	yes	—
66	C₃₁×#427	C₂²	1	4	yes	—
67	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
68	C₃₁×#416	C₂²	1	4	yes	—
69	C₃₁×#416	C₂²	1	4	yes	—
70	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
71	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
72	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
73	C₂×#41×C₃₁	C₂³	1	2	yes	—
74	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
76	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
77	C₃₁×C₄⋊Q₈	—	2	4	no	—
78	C₃₁×C₄⋊Q₈	—	2	4	no	—
79	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
81	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
82	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
83	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
84	C₃₁×#127	C₂³	1	1	yes	characteristic
85	C₃₁×#126	—	2	4	no	—
86	C₃₁×#126	—	2	4	no	—
87	C₃₁×#125	—	2	4	no	—
88	C₃₁×#125	—	2	4	no	—
89	C₃₁×#125	—	2	4	no	—
90	C₃₁×#125	C₂³	1	4	yes	—
91	C₃₁×#125	C₂³	1	2	yes	—
92	C₃₁×#125	C₂³	1	2	yes	—
93	C₃₁×#125	C₂³	1	2	yes	—
94	C₃₁×#124	—	2	4	no	—
95	C₃₁×#124	C₂³	1	2	yes	—
96	C₃₁×#123	—	2	2	no	—
97	C₃₁×#123	—	2	4	no	—
98	C₃₁×#123	—	2	4	no	—
99	C₃₁×C₂²⋊Q₈	—	2	2	no	—
100	C₃₁×C₂²⋊Q₈	—	2	4	no	—
101	C₃₁×C₂²⋊Q₈	—	2	2	no	—
102	C₃₁×C₂²⋊Q₈	—	2	2	no	—
103	C₃₁×C₂²⋊Q₈	—	2	4	no	—
104	C₃₁×C₂²⋊Q₈	—	2	4	no	—
105	C₃₁×C₂²⋊Q₈	—	2	4	no	—
106	C₃₁×C₂²⋊Q₈	—	2	4	no	—
107	C₃₁×C₂²⋊Q₈	—	2	4	no	—
108	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
109	C₃₁×#121	—	2	2	no	—
110	C₃₁×#121	—	2	4	no	—
111	C₃₁×#121	C₂³	1	4	yes	—
112	C₃₁×#120	—	2	2	no	—
113	C₄×Q₈×C₃₁	—	2	2	no	—
114	C₄×Q₈×C₃₁	—	2	4	no	—
115	C₄×Q₈×C₃₁	—	2	2	no	—
116	C₄×Q₈×C₃₁	—	2	2	no	—
117	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
118	C₄×D₄×C₃₁	—	2	4	no	—
119	C₄×D₄×C₃₁	—	2	4	no	—
120	C₄×D₄×C₃₁	—	2	2	no	—
121	C₄×D₄×C₃₁	C₂³	1	4	yes	—
122	C₃₁×#117	—	2	2	no	—
123	C₃₁×#117	—	2	2	no	—
124	C₃₁×#117	—	2	2	no	—
125	C₃₁×#117	D₄	1	4	yes	—
126	C₃₁×#117	C₂³	1	2	yes	—
127	C₃₁×#117	C₂³	1	2	yes	—
128	C₃₁×#117	C₂³	1	2	yes	—
129	C₃₁×#117	C₂³	1	4	yes	—
130	C₃₁×#117	C₂³	1	2	yes	—
131	C₃₁×#117	C₂³	1	2	yes	—
132	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
133	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
134	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
135	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
136	C₂×C₄²×C₃₁	—	2	2	no	—
137	C₂×C₄²×C₃₁	—	2	2	no	—
138	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
140	C₃₁×#107	C₂³	1	4	yes	—
141	C₃₁×#107	C₂³	1	4	yes	—
142	C₃₁×#107	C₂³	1	4	yes	—
143	C₃₁×#107	C₂³	1	4	yes	—
144	C₃₁×#106	C₂³	1	4	yes	—
145	C₃₁×#106	C₂³	1	4	yes	—
146	C₃₁×#106	C₂³	1	4	yes	—
147	C₃₁×#106	C₂³	1	4	yes	—
148	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
149	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
150	C₃₁×#103	C₂³	1	4	yes	—
151	C₃₁×#103	C₂³	1	4	yes	—
152	C₃₁×#103	C₂³	1	4	yes	—
153	C₃₁×#103	C₂³	1	4	yes	—
154	C₃₁×#102	C₂³	1	4	yes	—
155	C₃₁×#102	C₂³	1	4	yes	—
156	C₃₁×#102	C₂³	1	4	yes	—
157	C₃₁×#102	C₂³	1	4	yes	—
158	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
159	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
160	C₃₁×#97	C₂³	1	2	yes	—
161	C₃₁×#97	C₂³	1	2	yes	—
162	C₄×C₈×C₃₁	C₂³	1	2	yes	—
163	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
164	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
165	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
166	#41×C₃₁	—	2	8	no	—
167	C₂×Q₈×C₃₁	—	2	2	no	—
168	C₂×Q₈×C₃₁	—	2	4	no	—
169	C₂×Q₈×C₃₁	—	2	2	no	—
170	C₂×Q₈×C₃₁	—	2	2	no	—
171	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
172	C₂×D₄×C₃₁	—	2	2	no	—
173	C₂×D₄×C₃₁	—	2	4	no	—
174	C₂×D₄×C₃₁	—	2	2	no	—
175	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₃₁	—	2	4	no	—
177	C₂²×C₄×C₃₁	—	2	4	no	—
178	C₂²×C₄×C₃₁	—	2	2	no	—
179	C₂²×C₄×C₃₁	—	2	4	no	—
180	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
182	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
184	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
186	C₂×C₈×C₃₁	—	2	4	no	—
187	C₂×C₈×C₃₁	—	2	4	no	—
188	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
189	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
190	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
191	C₄⋊C₄×C₃₁	—	2	4	no	—
192	C₄⋊C₄×C₃₁	—	2	4	no	—
193	C₄⋊C₄×C₃₁	—	2	2	no	—
194	C₄⋊C₄×C₃₁	—	2	4	no	—
195	C₄⋊C₄×C₃₁	—	2	4	no	—
196	C₄⋊C₄×C₃₁	—	2	4	no	—
197	C₄⋊C₄×C₃₁	—	2	2	no	—
198	C₄⋊C₄×C₃₁	—	2	2	no	—
199	C₄⋊C₄×C₃₁	—	2	4	no	—
200	C₄⋊C₄×C₃₁	—	2	4	no	—
201	C₄⋊C₄×C₃₁	—	2	4	no	—
202	C₄⋊C₄×C₃₁	—	2	4	no	—
203	C₄⋊C₄×C₃₁	—	2	2	no	—
204	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
205	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
206	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
207	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
208	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
209	C₂²⋊C₄×C₃₁	—	2	4	no	—
210	C₂²⋊C₄×C₃₁	—	2	2	no	—
211	C₂²⋊C₄×C₃₁	—	2	2	no	—
212	C₂²⋊C₄×C₃₁	—	2	2	no	—
213	C₂²⋊C₄×C₃₁	—	2	4	no	—
214	C₂²⋊C₄×C₃₁	—	2	4	no	—
215	C₂²⋊C₄×C₃₁	—	2	4	no	—
216	C₂²⋊C₄×C₃₁	—	2	4	no	—
217	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
218	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
219	C₄²×C₃₁	—	2	4	no	—
220	C₄²×C₃₁	—	2	4	no	—
221	C₄²×C₃₁	—	2	2	no	—
222	C₄²×C₃₁	—	2	2	no	—
223	C₄²×C₃₁	—	2	2	no	—
224	C₄²×C₃₁	C₂×D₄	1	2	yes	—
225	C₄²×C₃₁	C₂×D₄	1	2	yes	—
226	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
227	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
228	C₄²×C₃₁	C₂⁴	1	2	yes	—
229	#60859	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
230	C₂³×C₃₁	—	2	2	no	—
231	C₂³×C₃₁	—	2	2	no	—
232	C₂³×C₃₁	—	4	2	no	—
233	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
234	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
235	Q₈×C₃₁	—	2	4	no	—
236	Q₈×C₃₁	—	8	4	no	—
237	Q₈×C₃₁	—	8	2	no	—
238	Q₈×C₃₁	—	8	2	no	—
239	Q₈×C₃₁	—	8	2	no	—
240	D₄×C₃₁	—	2	4	no	—
241	D₄×C₃₁	—	4	4	no	—
242	D₄×C₃₁	—	8	2	no	—
243	D₄×C₃₁	—	8	4	no	—
244	C₂×C₄×C₃₁	—	2	2	no	—
245	C₂×C₄×C₃₁	—	2	4	no	—
246	C₂×C₄×C₃₁	—	2	4	no	—
247	C₂×C₄×C₃₁	—	2	2	no	—
248	C₂×C₄×C₃₁	—	2	4	no	—
249	C₂×C₄×C₃₁	—	2	4	no	—
250	C₂×C₄×C₃₁	—	2	4	no	—
251	C₂×C₄×C₃₁	—	4	2	no	—
252	C₂×C₄×C₃₁	—	4	2	no	—
253	C₂×C₄×C₃₁	—	4	4	no	—
254	C₂×C₄×C₃₁	—	8	4	no	—
255	C₂×C₄×C₃₁	—	8	4	no	—
256	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
258	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
262	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
263	C₈×C₃₁	—	2	2	no	—
264	C₈×C₃₁	—	2	2	no	—
265	C₈×C₃₁	—	2	4	no	—
266	#3458	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#3455	C₂×C₃₁	1	1	yes	characteristic
268	#3244	C₂×C₃₁	1	4	yes	complemented
269	#3243	C₂×C₃₁	1	4	yes	complemented
270	#3242	C₂×C₃₁	1	4	yes	complemented
271	#3241	C₂×C₃₁	1	4	yes	complemented
272	#3162	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
274	#3135	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#3131	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#3091	C₂×C₃₁	1	4	yes	complemented
277	#3076	C₂×C₃₁	1	4	yes	complemented
278	C₂×#433	C₂×C₃₁	1	1	yes	characteristic, complemented
279	C₂²×C₃₁	—	2	2	no	—
280	C₂²×C₃₁	—	2	4	no	—
281	C₂²×C₃₁	—	2	2	no	—
282	C₂²×C₃₁	—	4	4	no	—
283	C₂²×C₃₁	—	4	2	no	—
284	C₂²×C₃₁	—	8	2	no	—
285	C₂²×C₃₁	#536	1	2	yes	—
286	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
287	C₄×C₃₁	—	2	4	no	—
288	C₄×C₃₁	—	2	2	no	—
289	C₄×C₃₁	—	2	2	no	—
290	C₄×C₃₁	—	2	2	no	—
291	C₄×C₃₁	—	4	4	no	—
292	C₄×C₃₁	—	4	4	no	—
293	C₄×C₃₁	—	4	4	no	—
294	C₄×C₃₁	—	4	2	no	—
295	C₄×C₃₁	#536	1	2	yes	—
296	C₄×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
297	#544	—	2	2	no	—
298	#544	C₂²×C₃₁	1	4	yes	—
299	#544	C₂²×C₃₁	1	4	yes	—
300	#543	—	2	2	no	—
301	#542	C₂²×C₃₁	1	4	yes	—
302	#541	C₂²×C₃₁	1	2	yes	—
303	#540	C₂²×C₃₁	1	2	yes	—
304	#539	—	2	2	no	—
305	#538	—	2	2	no	—
306	#533	—	2	4	no	—
307	#533	C₂²×C₃₁	1	2	yes	—
308	#533	C₂²×C₃₁	1	1	yes	characteristic
309	#533	C₂²×C₃₁	1	1	yes	characteristic
310	#533	C₂²×C₃₁	1	1	yes	characteristic
311	#532	C₂²×C₃₁	1	1	yes	characteristic
312	#529	—	2	4	no	—
313	#525	C₂²×C₃₁	1	2	yes	—
314	C₄×#41	C₂²×C₃₁	1	2	yes	—
315	C₂×#117	C₂²×C₃₁	1	2	yes	—
316	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
317	#501	C₂²×C₃₁	1	4	yes	complemented
318	#501	C₂²×C₃₁	1	4	yes	complemented
319	#500	C₂²×C₃₁	1	2	yes	complemented
320	#499	C₂²×C₃₁	1	4	yes	complemented
321	#498	C₂²×C₃₁	1	2	yes	complemented
322	#491	C₂²×C₃₁	1	2	yes	complemented
323	#490	C₂²×C₃₁	1	2	yes	complemented
324	#489	C₂²×C₃₁	1	4	yes	complemented
325	#488	C₂²×C₃₁	1	2	yes	complemented
326	#488	C₂²×C₃₁	1	2	yes	complemented
327	#487	C₂²×C₃₁	1	2	yes	complemented
328	#486	C₂²×C₃₁	1	2	yes	complemented
329	#485	C₂²×C₃₁	1	4	yes	—
330	#485	C₂²×C₃₁	1	4	yes	—
331	#484	C₂²×C₃₁	1	4	yes	—
332	#483	C₂²×C₃₁	1	4	yes	—
333	#482	C₂²×C₃₁	1	4	yes	—
334	#482	C₂²×C₃₁	1	4	yes	—
335	#481	C₂²×C₃₁	1	4	yes	—
336	#480	C₂²×C₃₁	1	4	yes	—
337	#479	C₂²×C₃₁	1	4	yes	complemented
338	#479	C₂²×C₃₁	1	4	yes	complemented
339	#478	C₂²×C₃₁	1	4	yes	complemented
340	#478	C₂²×C₃₁	1	4	yes	complemented
341	#477	C₂²×C₃₁	1	4	yes	complemented
342	#476	C₂²×C₃₁	1	4	yes	complemented
343	#475	C₂²×C₃₁	1	4	yes	complemented
344	#474	C₂²×C₃₁	1	4	yes	complemented
345	#433	C₂²×C₃₁	1	4	yes	—
346	#433	C₂²×C₃₁	1	2	yes	—
347	#433	C₂²×C₃₁	1	2	yes	—
348	#427	C₂²×C₃₁	1	4	yes	—
349	#427	C₂²×C₃₁	1	4	yes	—
350	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
351	#416	C₂²×C₃₁	1	4	yes	—
352	#416	C₂²×C₃₁	1	4	yes	—
353	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
354	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
355	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
356	C₂×C₃₁	—	2	2	no	—
357	C₂×C₃₁	—	4	2	no	—
358	C₂×C₃₁	—	8	2	no	—
359	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
360	C₂×C₃₁	#3584	1	2	yes	—
361	C₂×#41	C₂³×C₃₁	1	2	yes	—
362	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
363	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
364	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
365	C₄⋊Q₈	—	2	4	no	—
366	C₄⋊Q₈	—	2	4	no	—
367	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
368	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
369	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
370	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
371	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
372	#127	C₂³×C₃₁	1	1	yes	characteristic
373	#126	—	2	4	no	—
374	#126	—	2	4	no	—
375	#125	—	2	4	no	—
376	#125	—	2	4	no	—
377	#125	—	2	4	no	—
378	#125	C₂³×C₃₁	1	2	yes	—
379	#125	C₂³×C₃₁	1	4	yes	—
380	#125	C₂³×C₃₁	1	2	yes	—
381	#125	C₂³×C₃₁	1	2	yes	—
382	#124	—	2	4	no	—
383	#124	C₂³×C₃₁	1	2	yes	—
384	#123	—	2	2	no	—
385	#123	—	2	4	no	—
386	#123	—	2	4	no	—
387	C₂²⋊Q₈	—	2	2	no	—
388	C₂²⋊Q₈	—	2	4	no	—
389	C₂²⋊Q₈	—	2	2	no	—
390	C₂²⋊Q₈	—	2	2	no	—
391	C₂²⋊Q₈	—	2	4	no	—
392	C₂²⋊Q₈	—	2	4	no	—
393	C₂²⋊Q₈	—	2	4	no	—
394	C₂²⋊Q₈	—	2	4	no	—
395	C₂²⋊Q₈	—	2	4	no	—
396	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
397	#121	—	2	2	no	—
398	#121	—	2	4	no	—
399	#121	C₂³×C₃₁	1	4	yes	—
400	#120	—	2	2	no	—
401	C₄×Q₈	—	2	2	no	—
402	C₄×Q₈	—	2	4	no	—
403	C₄×Q₈	—	2	2	no	—
404	C₄×Q₈	—	2	2	no	—
405	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
406	C₄×D₄	—	2	4	no	—
407	C₄×D₄	—	2	4	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	C₂³×C₃₁	1	4	yes	—
410	#117	—	2	2	no	—
411	#117	—	2	2	no	—
412	#117	—	2	2	no	—
413	#117	D₄×C₃₁	1	4	yes	—
414	#117	C₂³×C₃₁	1	2	yes	—
415	#117	C₂³×C₃₁	1	2	yes	—
416	#117	C₂³×C₃₁	1	4	yes	—
417	#117	C₂³×C₃₁	1	2	yes	—
418	#117	C₂³×C₃₁	1	2	yes	—
419	#117	C₂³×C₃₁	1	2	yes	—
420	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
421	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
422	C₂×C₂²⋊C₄	—	2	2	no	—
423	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
424	C₂×C₄²	—	2	2	no	—
425	C₂×C₄²	—	2	2	no	—
426	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
427	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
428	#107	C₂³×C₃₁	1	4	yes	—
429	#107	C₂³×C₃₁	1	4	yes	—
430	#107	C₂³×C₃₁	1	4	yes	—
431	#107	C₂³×C₃₁	1	4	yes	—
432	#106	C₂³×C₃₁	1	4	yes	—
433	#106	C₂³×C₃₁	1	4	yes	—
434	#106	C₂³×C₃₁	1	4	yes	—
435	#106	C₂³×C₃₁	1	4	yes	—
436	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
437	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
438	#103	C₂³×C₃₁	1	4	yes	—
439	#103	C₂³×C₃₁	1	4	yes	—
440	#103	C₂³×C₃₁	1	4	yes	—
441	#103	C₂³×C₃₁	1	4	yes	—
442	#102	C₂³×C₃₁	1	4	yes	—
443	#102	C₂³×C₃₁	1	4	yes	—
444	#102	C₂³×C₃₁	1	4	yes	—
445	#102	C₂³×C₃₁	1	4	yes	—
446	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
447	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
448	#97	C₂³×C₃₁	1	2	yes	—
449	#97	C₂³×C₃₁	1	2	yes	—
450	C₄×C₈	C₂³×C₃₁	1	2	yes	—
451	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
452	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
453	C₃₁	#60859	1	1	yes	characteristic, direct product factor, Sylow
454	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
455	#41	—	2	8	no	—
456	C₂×Q₈	—	2	2	no	—
457	C₂×Q₈	—	2	4	no	—
458	C₂×Q₈	—	2	2	no	—
459	C₂×Q₈	—	2	2	no	—
460	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
461	C₂×D₄	—	2	2	no	—
462	C₂×D₄	—	2	4	no	—
463	C₂×D₄	—	2	2	no	—
464	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
470	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
472	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
473	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
474	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
475	C₂×C₈	—	2	4	no	—
476	C₂×C₈	—	2	4	no	—
477	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
478	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
479	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
480	C₄⋊C₄	—	2	4	no	—
481	C₄⋊C₄	—	2	4	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	4	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	4	no	—
490	C₄⋊C₄	—	2	4	no	—
491	C₄⋊C₄	—	2	4	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
494	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
495	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
496	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
497	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	—	2	2	no	—
500	C₂²⋊C₄	—	2	2	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	—	2	4	no	—
505	C₂²⋊C₄	—	2	4	no	—
506	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
507	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
508	C₄²	—	2	2	no	—
509	C₄²	—	2	2	no	—
510	C₄²	—	2	2	no	—
511	C₄²	—	2	4	no	—
512	C₄²	—	2	4	no	—
513	C₄²	C₂×D₄×C₃₁	1	2	yes	—
514	C₄²	C₂×D₄×C₃₁	1	2	yes	—
515	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
516	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
517	C₄²	C₂⁴×C₃₁	1	2	yes	—
518	C₂³	—	2	2	no	—
519	C₂³	—	2	2	no	—
520	C₂³	—	4	2	no	—
521	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
522	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
523	Q₈	—	2	4	no	—
524	Q₈	—	8	4	no	—
525	Q₈	—	8	2	no	—
526	Q₈	—	8	2	no	—
527	Q₈	—	8	2	no	—
528	D₄	—	2	4	no	—
529	D₄	—	4	4	no	—
530	D₄	—	8	2	no	—
531	D₄	—	8	4	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	4	2	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	4	4	no	—
542	C₂×C₄	—	8	4	no	—
543	C₂×C₄	—	8	4	no	—
544	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
545	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
546	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
547	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
548	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
549	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
550	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
551	C₈	—	2	2	no	—
552	C₈	—	2	2	no	—
553	C₈	—	2	4	no	—
554	C₂²	—	2	2	no	—
555	C₂²	—	2	2	no	—
556	C₂²	—	2	4	no	—
557	C₂²	—	4	2	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	8	2	no	—
560	C₂²	C₃₁×#536	1	2	yes	—
561	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
562	C₄	—	2	2	no	—
563	C₄	—	2	4	no	—
564	C₄	—	2	2	no	—
565	C₄	—	2	2	no	—
566	C₄	—	4	2	no	—
567	C₄	—	4	4	no	—
568	C₄	—	4	4	no	—
569	C₄	—	4	4	no	—
570	C₄	C₃₁×#536	1	2	yes	—
571	C₄	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
572	C₂	—	2	2	no	—
573	C₂	—	4	2	no	—
574	C₂	—	8	2	no	—
575	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
576	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂³×D₄
prime radical	C₃₁×#3455	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3458	C₂	semidirect
3. C₃₁×#3244	C₂	semidirect
4. C₃₁×#3243	C₂	semidirect
5. C₃₁×#3242	C₂	semidirect
6. C₃₁×#3241	C₂	semidirect
7. C₃₁×#3162	C₂	semidirect
8. C₃₁×#3160	C₂	semidirect
9. C₃₁×#3135	C₂	semidirect
10. C₃₁×#3131	C₂	semidirect
11. C₃₁×#3091	C₂	semidirect
12. C₃₁×#3076	C₂	semidirect
13. C₂×C₃₁×#433	C₂	semidirect
34. C₃₁×#501	C₂²	semidirect
35. C₃₁×#501	C₂²	semidirect
36. C₃₁×#500	C₂²	semidirect
37. C₃₁×#499	C₂²	semidirect
38. C₃₁×#498	C₂²	semidirect
39. C₃₁×#491	C₂²	semidirect
40. C₃₁×#490	C₂²	semidirect
41. C₃₁×#489	C₂²	semidirect
42. C₃₁×#488	C₂²	semidirect
43. C₃₁×#488	C₂²	semidirect
44. C₃₁×#487	C₂²	semidirect
45. C₃₁×#486	C₂²	semidirect
54. C₃₁×#479	C₂²	semidirect
55. C₃₁×#479	C₂²	semidirect
56. C₃₁×#478	C₂²	semidirect
57. C₃₁×#478	C₂²	semidirect
58. C₃₁×#477	C₂²	semidirect
59. C₃₁×#476	C₂²	semidirect
60. C₃₁×#475	C₂²	semidirect
61. C₃₁×#474	C₂²	semidirect
229. #60859	C₃₁	direct (N × H)
266. #3458	C₂×C₃₁	semidirect
268. #3244	C₂×C₃₁	semidirect
269. #3243	C₂×C₃₁	semidirect
270. #3242	C₂×C₃₁	semidirect
271. #3241	C₂×C₃₁	semidirect
272. #3162	C₂×C₃₁	semidirect
273. #3160	C₂×C₃₁	semidirect
274. #3135	C₂×C₃₁	semidirect
275. #3131	C₂×C₃₁	semidirect
276. #3091	C₂×C₃₁	semidirect
277. #3076	C₂×C₃₁	semidirect
278. C₂×#433	C₂×C₃₁	semidirect
317. #501	C₂²×C₃₁	semidirect
318. #501	C₂²×C₃₁	semidirect
319. #500	C₂²×C₃₁	semidirect
320. #499	C₂²×C₃₁	semidirect
321. #498	C₂²×C₃₁	semidirect
322. #491	C₂²×C₃₁	semidirect
323. #490	C₂²×C₃₁	semidirect
324. #489	C₂²×C₃₁	semidirect
325. #488	C₂²×C₃₁	semidirect
326. #488	C₂²×C₃₁	semidirect
327. #487	C₂²×C₃₁	semidirect
328. #486	C₂²×C₃₁	semidirect
337. #479	C₂²×C₃₁	semidirect
338. #479	C₂²×C₃₁	semidirect
339. #478	C₂²×C₃₁	semidirect
340. #478	C₂²×C₃₁	semidirect
341. #477	C₂²×C₃₁	semidirect
342. #476	C₂²×C₃₁	semidirect
343. #475	C₂²×C₃₁	semidirect
344. #474	C₂²×C₃₁	semidirect
453. C₃₁	#60859	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×#3458	2
C₃₁×#3455	2
C₃₁×#3244	2
C₃₁×#3243	2
C₃₁×#3242	2
C₃₁×#3241	2
C₃₁×#3162	2
C₃₁×#3160	2
C₃₁×#3135	2
C₃₁×#3131	2
C₃₁×#3091	2
C₃₁×#3076	2
C₂×C₃₁×#433	2
#60859	31

Subgroups — C31×#60859 #21517618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60859 #21517618