Subgroups — C₃₁×#60945 #21517704

Order 7936.

1866 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#529	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3250	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3232	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3211	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3189	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3187	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3186	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#3168	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3108	C₂	1	4	yes	maximal, complemented
10	C₃₁×#3105	C₂	1	4	yes	maximal, complemented
11	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#529	—	2	2	no	—
13	C₃₁×#529	—	2	2	no	—
14	C₃₁×#529	—	2	2	no	—
15	C₃₁×#529	—	2	2	no	—
16	C₃₁×#529	C₂²	1	4	yes	—
17	C₃₁×#529	C₂²	1	4	yes	—
18	C₃₁×#529	C₂²	1	4	yes	complemented
19	C₃₁×#529	C₂²	1	4	yes	complemented
20	C₂×C₃₁×#126	—	2	2	no	—
21	C₂×C₃₁×#125	C₂²	1	2	yes	complemented
22	C₂×C₃₁×#124	C₂²	1	2	yes	complemented
23	C₂×C₃₁×#123	—	2	2	no	—
24	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₂×C₃₁×#121	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
27	C₂×C₄×D₄×C₃₁	—	2	2	no	—
28	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
29	C₂×C₃₁×#117	—	2	2	no	—
30	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#499	C₂²	1	4	yes	complemented
32	C₃₁×#495	C₂²	1	4	yes	complemented
33	C₃₁×#488	C₂²	1	8	yes	complemented
34	C₃₁×#484	C₂²	1	4	yes	complemented
35	C₃₁×#483	C₂²	1	4	yes	complemented
36	C₃₁×#481	C₂²	1	4	yes	—
37	C₃₁×#480	C₂²	1	4	yes	—
38	C₃₁×#479	C₂²	1	8	yes	complemented
39	C₃₁×#478	C₂²	1	8	yes	complemented
40	C₃₁×#467	C₂²	1	4	yes	complemented
41	C₃₁×#466	C₂²	1	4	yes	—
42	C₃₁×#465	C₂²	1	8	yes	complemented
43	C₃₁×#464	C₂²	1	8	yes	complemented
44	C₃₁×#463	C₂²	1	8	yes	complemented
45	C₃₁×#451	C₂²	1	4	yes	complemented
46	C₃₁×#450	C₂²	1	4	yes	complemented
47	C₃₁×#448	C₂²	1	4	yes	—
48	C₃₁×#447	C₂²	1	4	yes	—
49	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	complemented
50	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	complemented
51	C₄×D₈×C₃₁	C₂²	1	4	yes	complemented
52	C₃₁×#431	C₂²	1	4	yes	complemented
53	C₃₁×#431	C₂²	1	4	yes	complemented
54	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
55	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
56	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
57	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
58	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
59	C₂²×D₄×C₃₁	—	2	2	no	—
60	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
61	C₂³×C₄×C₃₁	—	2	2	no	—
62	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
66	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
67	C₂×D₈×C₃₁	C₂³	1	4	yes	—
68	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
69	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
70	C₃₁×#126	—	2	4	no	—
71	C₃₁×#126	—	2	4	no	—
72	C₃₁×#126	—	2	4	no	—
73	C₃₁×#126	—	2	4	no	—
74	C₃₁×#125	—	2	2	no	—
75	C₃₁×#125	—	2	2	no	—
76	C₃₁×#125	C₂³	1	4	yes	—
77	C₃₁×#125	C₂³	1	4	yes	—
78	C₃₁×#124	—	2	2	no	—
79	C₃₁×#124	—	2	2	no	—
80	C₃₁×#124	C₂³	1	4	yes	—
81	C₃₁×#124	C₂³	1	4	yes	—
82	C₃₁×#123	—	2	4	no	—
83	C₃₁×#123	—	2	4	no	—
84	C₃₁×#123	—	2	4	no	—
85	C₃₁×#123	—	2	4	no	—
86	C₃₁×C₂²⋊Q₈	—	2	2	no	—
87	C₃₁×C₂²⋊Q₈	—	2	2	no	—
88	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
89	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
90	C₃₁×#121	—	2	2	no	—
91	C₃₁×#121	—	2	2	no	—
92	C₃₁×#121	C₂³	1	4	yes	—
93	C₃₁×#121	C₂³	1	4	yes	—
94	C₄×Q₈×C₃₁	—	2	2	no	—
95	C₄×Q₈×C₃₁	—	2	2	no	—
96	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
97	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
98	C₄×D₄×C₃₁	—	2	4	no	—
99	C₄×D₄×C₃₁	—	2	4	no	—
100	C₄×D₄×C₃₁	—	2	4	no	—
101	C₄×D₄×C₃₁	—	2	4	no	—
102	C₄×D₄×C₃₁	—	2	2	no	—
103	C₄×D₄×C₃₁	—	2	2	no	—
104	C₄×D₄×C₃₁	C₂³	1	4	yes	—
105	C₄×D₄×C₃₁	C₂³	1	4	yes	—
106	C₃₁×#117	—	2	4	no	—
107	C₃₁×#117	—	2	4	no	—
108	C₃₁×#117	—	2	4	no	—
109	C₃₁×#117	—	2	4	no	—
110	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
113	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
119	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
120	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
121	C₂×C₄²×C₃₁	—	2	2	no	—
122	C₂×C₄²×C₃₁	D₄	1	2	yes	—
123	C₂×C₄²×C₃₁	D₄	1	2	yes	—
124	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
125	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
126	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
129	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
130	C₃₁×#107	C₂³	1	4	yes	—
131	C₃₁×#107	C₂³	1	4	yes	—
132	C₃₁×#106	C₂³	1	8	yes	—
133	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
134	C₃₁×#103	C₂³	1	8	yes	—
135	C₃₁×#103	C₂³	1	8	yes	—
136	C₃₁×#102	C₂³	1	8	yes	—
137	C₃₁×#102	C₂³	1	8	yes	—
138	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
139	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
140	C₄×C₈×C₃₁	C₂³	1	4	yes	—
141	C₄×C₈×C₃₁	C₂³	1	4	yes	—
142	C₂⁴×C₃₁	—	2	2	no	—
143	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂×Q₈×C₃₁	—	2	2	no	—
145	C₂×Q₈×C₃₁	—	4	4	no	—
146	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
147	C₂×D₄×C₃₁	—	2	4	no	—
148	C₂×D₄×C₃₁	—	2	4	no	—
149	C₂×D₄×C₃₁	—	2	2	no	—
150	C₂×D₄×C₃₁	—	4	4	no	—
151	C₂×D₄×C₃₁	—	4	8	no	—
152	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₃₁	—	2	4	no	—
154	C₂²×C₄×C₃₁	—	2	2	no	—
155	C₂²×C₄×C₃₁	—	2	2	no	—
156	C₂²×C₄×C₃₁	—	2	2	no	—
157	C₂²×C₄×C₃₁	—	2	4	no	—
158	C₂²×C₄×C₃₁	—	2	4	no	—
159	C₂²×C₄×C₃₁	—	4	2	no	—
160	C₂²×C₄×C₃₁	—	4	2	no	—
161	C₂²×C₄×C₃₁	—	4	8	no	—
162	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
174	Q₁₆×C₃₁	—	4	4	no	—
175	SD₁₆×C₃₁	—	4	8	no	—
176	D₈×C₃₁	—	4	4	no	—
177	C₂×C₈×C₃₁	—	4	2	no	—
178	C₂×C₈×C₃₁	—	4	2	no	—
179	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
180	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
181	C₄⋊C₄×C₃₁	—	2	4	no	—
182	C₄⋊C₄×C₃₁	—	2	4	no	—
183	C₄⋊C₄×C₃₁	—	2	4	no	—
184	C₄⋊C₄×C₃₁	—	2	4	no	—
185	C₄⋊C₄×C₃₁	—	2	4	no	—
186	C₄⋊C₄×C₃₁	—	2	4	no	—
187	C₄⋊C₄×C₃₁	—	2	4	no	—
188	C₄⋊C₄×C₃₁	—	2	4	no	—
189	C₄⋊C₄×C₃₁	—	2	2	no	—
190	C₄⋊C₄×C₃₁	—	2	2	no	—
191	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
192	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
193	C₂²⋊C₄×C₃₁	—	2	4	no	—
194	C₂²⋊C₄×C₃₁	—	2	4	no	—
195	C₂²⋊C₄×C₃₁	—	2	4	no	—
196	C₂²⋊C₄×C₃₁	—	2	4	no	—
197	C₂²⋊C₄×C₃₁	—	2	4	no	—
198	C₂²⋊C₄×C₃₁	—	2	4	no	—
199	C₂²⋊C₄×C₃₁	—	2	4	no	—
200	C₂²⋊C₄×C₃₁	—	2	4	no	—
201	C₂²⋊C₄×C₃₁	—	2	4	no	—
202	C₂²⋊C₄×C₃₁	—	2	4	no	—
203	C₄²×C₃₁	—	2	4	no	—
204	C₄²×C₃₁	—	2	4	no	—
205	C₄²×C₃₁	C₂×D₄	1	2	yes	—
206	C₄²×C₃₁	C₂×D₄	1	2	yes	—
207	C₄²×C₃₁	C₂×D₄	1	2	yes	—
208	C₄²×C₃₁	C₂×D₄	1	2	yes	—
209	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
211	C₄²×C₃₁	C₂×D₄	1	2	yes	—
212	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
213	C₄²×C₃₁	C₂⁴	1	2	yes	—
214	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
215	#60945	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂³×C₃₁	—	2	4	no	—
217	C₂³×C₃₁	—	4	1	no	—
218	C₂³×C₃₁	—	4	1	no	—
219	C₂³×C₃₁	—	4	4	no	—
220	C₂³×C₃₁	—	4	4	no	—
221	C₂³×C₃₁	—	4	4	no	—
222	C₂³×C₃₁	#135	1	4	yes	—
223	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
225	Q₈×C₃₁	—	2	8	no	—
226	Q₈×C₃₁	—	4	4	no	—
227	D₄×C₃₁	—	2	8	no	—
228	D₄×C₃₁	—	4	4	no	—
229	D₄×C₃₁	—	4	8	no	—
230	D₄×C₃₁	—	4	8	no	—
231	C₂×C₄×C₃₁	—	2	4	no	—
232	C₂×C₄×C₃₁	—	2	4	no	—
233	C₂×C₄×C₃₁	—	2	2	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	4	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	4	no	—
239	C₂×C₄×C₃₁	—	2	4	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	2	4	no	—
242	C₂×C₄×C₃₁	—	4	2	no	—
243	C₂×C₄×C₃₁	—	4	1	no	—
244	C₂×C₄×C₃₁	—	4	1	no	—
245	C₂×C₄×C₃₁	—	4	2	no	—
246	C₂×C₄×C₃₁	—	4	8	no	—
247	C₂×C₄×C₃₁	—	4	4	no	—
248	C₂×C₄×C₃₁	—	4	4	no	—
249	C₂×C₄×C₃₁	—	4	4	no	—
250	C₂×C₄×C₃₁	—	4	8	no	—
251	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
260	C₈×C₃₁	—	2	4	no	—
261	C₈×C₃₁	—	2	4	no	—
262	C₂×#529	C₂×C₃₁	1	2	yes	complemented
263	#3250	C₂×C₃₁	1	4	yes	complemented
264	#3232	C₂×C₃₁	1	4	yes	complemented
265	#3211	C₂×C₃₁	1	4	yes	complemented
266	#3189	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#3187	C₂×C₃₁	1	2	yes	complemented
268	#3186	C₂×C₃₁	1	1	yes	characteristic
269	#3168	C₂×C₃₁	1	4	yes	complemented
270	#3108	C₂×C₃₁	1	4	yes	complemented
271	#3105	C₂×C₃₁	1	4	yes	complemented
272	C₂×#431	C₂×C₃₁	1	1	yes	characteristic, complemented
273	C₂²×C₃₁	—	2	2	no	—
274	C₂²×C₃₁	—	2	1	no	—
275	C₂²×C₃₁	—	2	2	no	—
276	C₂²×C₃₁	—	2	1	no	—
277	C₂²×C₃₁	—	4	4	no	—
278	C₂²×C₃₁	—	4	4	no	—
279	C₂²×C₃₁	—	4	4	no	—
280	C₂²×C₃₁	—	4	4	no	—
281	C₂²×C₃₁	—	8	8	no	—
282	C₂²×C₃₁	C₂×#135	1	4	yes	—
283	C₂²×C₃₁	C₂×#135	1	2	yes	—
284	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
285	C₄×C₃₁	—	2	1	no	—
286	C₄×C₃₁	—	2	2	no	—
287	C₄×C₃₁	—	2	1	no	—
288	C₄×C₃₁	—	2	2	no	—
289	C₄×C₃₁	—	2	1	no	—
290	C₄×C₃₁	—	2	2	no	—
291	C₄×C₃₁	—	2	1	no	—
292	C₄×C₃₁	—	4	4	no	—
293	C₄×C₃₁	—	4	4	no	—
294	C₄×C₃₁	—	4	4	no	—
295	C₄×C₃₁	#575	1	2	yes	—
296	C₄×C₃₁	#585	1	2	yes	—
297	#529	—	2	2	no	—
298	#529	—	2	2	no	—
299	#529	—	2	2	no	—
300	#529	—	2	2	no	—
301	#529	C₂²×C₃₁	1	4	yes	—
302	#529	C₂²×C₃₁	1	4	yes	—
303	#529	C₂²×C₃₁	1	4	yes	complemented
304	#529	C₂²×C₃₁	1	4	yes	complemented
305	C₂×#126	—	2	2	no	—
306	C₂×#125	C₂²×C₃₁	1	2	yes	complemented
307	C₂×#124	C₂²×C₃₁	1	2	yes	complemented
308	C₂×#123	—	2	2	no	—
309	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
310	C₂×#121	C₂²×C₃₁	1	2	yes	—
311	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
312	C₂×C₄×D₄	—	2	2	no	—
313	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
314	C₂×#117	—	2	2	no	—
315	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
316	#499	C₂²×C₃₁	1	4	yes	complemented
317	#495	C₂²×C₃₁	1	4	yes	complemented
318	#488	C₂²×C₃₁	1	8	yes	complemented
319	#484	C₂²×C₃₁	1	4	yes	complemented
320	#483	C₂²×C₃₁	1	4	yes	complemented
321	#481	C₂²×C₃₁	1	4	yes	—
322	#480	C₂²×C₃₁	1	4	yes	—
323	#479	C₂²×C₃₁	1	8	yes	complemented
324	#478	C₂²×C₃₁	1	8	yes	complemented
325	#467	C₂²×C₃₁	1	4	yes	complemented
326	#466	C₂²×C₃₁	1	4	yes	—
327	#465	C₂²×C₃₁	1	8	yes	complemented
328	#464	C₂²×C₃₁	1	8	yes	complemented
329	#463	C₂²×C₃₁	1	8	yes	complemented
330	#451	C₂²×C₃₁	1	4	yes	complemented
331	#450	C₂²×C₃₁	1	4	yes	complemented
332	#448	C₂²×C₃₁	1	4	yes	—
333	#447	C₂²×C₃₁	1	4	yes	—
334	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	complemented
335	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	complemented
336	C₄×D₈	C₂²×C₃₁	1	4	yes	complemented
337	#431	C₂²×C₃₁	1	4	yes	complemented
338	#431	C₂²×C₃₁	1	4	yes	complemented
339	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
340	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
341	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
342	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
343	C₂×C₃₁	—	2	4	no	—
344	C₂×C₃₁	—	2	2	no	—
345	C₂×C₃₁	—	8	4	no	—
346	C₂×C₃₁	#3449	1	1	yes	characteristic
347	C₂×C₃₁	C₂²×#135	1	1	yes	characteristic
348	C₂×C₃₁	C₂×#575	1	1	yes	characteristic
349	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
350	C₂²×D₄	—	2	2	no	—
351	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
352	C₂³×C₄	—	2	2	no	—
353	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
354	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
355	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
356	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
357	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
358	C₂×D₈	C₂³×C₃₁	1	4	yes	—
359	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
360	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
361	#126	—	2	4	no	—
362	#126	—	2	4	no	—
363	#126	—	2	4	no	—
364	#126	—	2	4	no	—
365	#125	—	2	2	no	—
366	#125	—	2	2	no	—
367	#125	C₂³×C₃₁	1	4	yes	—
368	#125	C₂³×C₃₁	1	4	yes	—
369	#124	—	2	2	no	—
370	#124	—	2	2	no	—
371	#124	C₂³×C₃₁	1	4	yes	—
372	#124	C₂³×C₃₁	1	4	yes	—
373	#123	—	2	4	no	—
374	#123	—	2	4	no	—
375	#123	—	2	4	no	—
376	#123	—	2	4	no	—
377	C₂²⋊Q₈	—	2	2	no	—
378	C₂²⋊Q₈	—	2	2	no	—
379	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
380	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
381	#121	—	2	2	no	—
382	#121	—	2	2	no	—
383	#121	C₂³×C₃₁	1	4	yes	—
384	#121	C₂³×C₃₁	1	4	yes	—
385	C₄×Q₈	—	2	2	no	—
386	C₄×Q₈	—	2	2	no	—
387	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
388	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
389	C₄×D₄	—	2	4	no	—
390	C₄×D₄	—	2	4	no	—
391	C₄×D₄	—	2	4	no	—
392	C₄×D₄	—	2	4	no	—
393	C₄×D₄	—	2	2	no	—
394	C₄×D₄	—	2	2	no	—
395	C₄×D₄	C₂³×C₃₁	1	4	yes	—
396	C₄×D₄	C₂³×C₃₁	1	4	yes	—
397	#117	—	2	4	no	—
398	#117	—	2	4	no	—
399	#117	—	2	4	no	—
400	#117	—	2	4	no	—
401	C₂×C₄⋊C₄	—	2	2	no	—
402	C₂×C₄⋊C₄	—	2	2	no	—
403	C₂×C₄⋊C₄	—	2	2	no	—
404	C₂×C₄⋊C₄	—	2	2	no	—
405	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
407	C₂×C₂²⋊C₄	—	2	2	no	—
408	C₂×C₂²⋊C₄	—	2	2	no	—
409	C₂×C₂²⋊C₄	—	2	2	no	—
410	C₂×C₂²⋊C₄	—	2	2	no	—
411	C₂×C₂²⋊C₄	—	2	2	no	—
412	C₂×C₄²	—	2	2	no	—
413	C₂×C₄²	D₄×C₃₁	1	2	yes	—
414	C₂×C₄²	D₄×C₃₁	1	2	yes	—
415	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
416	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
417	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
418	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
419	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
420	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
421	#107	C₂³×C₃₁	1	4	yes	—
422	#107	C₂³×C₃₁	1	4	yes	—
423	#106	C₂³×C₃₁	1	8	yes	—
424	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
425	#103	C₂³×C₃₁	1	8	yes	—
426	#103	C₂³×C₃₁	1	8	yes	—
427	#102	C₂³×C₃₁	1	8	yes	—
428	#102	C₂³×C₃₁	1	8	yes	—
429	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
430	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
431	C₄×C₈	C₂³×C₃₁	1	4	yes	—
432	C₄×C₈	C₂³×C₃₁	1	4	yes	—
433	C₃₁	#60945	1	1	yes	characteristic, direct product factor, Sylow
434	C₂⁴	—	2	2	no	—
435	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	—	4	4	no	—
438	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	4	4	no	—
443	C₂×D₄	—	4	8	no	—
444	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	2	4	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	—	4	2	no	—
453	C₂²×C₄	—	4	8	no	—
454	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
455	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
456	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
457	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
458	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
459	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
461	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
463	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
464	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
465	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
466	Q₁₆	—	4	4	no	—
467	SD₁₆	—	4	8	no	—
468	D₈	—	4	4	no	—
469	C₂×C₈	—	4	2	no	—
470	C₂×C₈	—	4	2	no	—
471	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
472	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
473	C₄⋊C₄	—	2	4	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	—	2	4	no	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	4	no	—
480	C₄⋊C₄	—	2	4	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
484	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	—	2	4	no	—
489	C₂²⋊C₄	—	2	4	no	—
490	C₂²⋊C₄	—	2	4	no	—
491	C₂²⋊C₄	—	2	4	no	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₂²⋊C₄	—	2	4	no	—
494	C₂²⋊C₄	—	2	4	no	—
495	C₄²	—	2	4	no	—
496	C₄²	—	2	4	no	—
497	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
498	C₄²	C₂×D₄×C₃₁	1	2	yes	—
499	C₄²	C₂×D₄×C₃₁	1	2	yes	—
500	C₄²	C₂×D₄×C₃₁	1	2	yes	—
501	C₄²	C₂×D₄×C₃₁	1	2	yes	—
502	C₄²	C₂×D₄×C₃₁	1	2	yes	—
503	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
504	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
505	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
506	C₄²	C₂⁴×C₃₁	1	2	yes	—
507	C₂³	—	2	4	no	—
508	C₂³	—	4	1	no	—
509	C₂³	—	4	1	no	—
510	C₂³	—	4	4	no	—
511	C₂³	—	4	4	no	—
512	C₂³	—	4	4	no	—
513	C₂³	C₃₁×#135	1	4	yes	—
514	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
515	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
516	Q₈	—	2	8	no	—
517	Q₈	—	4	4	no	—
518	D₄	—	2	8	no	—
519	D₄	—	4	4	no	—
520	D₄	—	4	8	no	—
521	D₄	—	4	8	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	2	no	—
537	C₂×C₄	—	4	8	no	—
538	C₂×C₄	—	4	4	no	—
539	C₂×C₄	—	4	4	no	—
540	C₂×C₄	—	4	4	no	—
541	C₂×C₄	—	4	8	no	—
542	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
543	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
544	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
545	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
546	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
547	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
548	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
549	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
550	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
551	C₈	—	2	4	no	—
552	C₈	—	2	4	no	—
553	C₂²	—	2	2	no	—
554	C₂²	—	2	1	no	—
555	C₂²	—	2	2	no	—
556	C₂²	—	2	1	no	—
557	C₂²	—	4	4	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	4	4	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	8	8	no	—
562	C₂²	C₂×C₃₁×#135	1	4	yes	—
563	C₂²	C₂×C₃₁×#135	1	2	yes	—
564	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
565	C₄	—	2	2	no	—
566	C₄	—	2	1	no	—
567	C₄	—	2	1	no	—
568	C₄	—	2	2	no	—
569	C₄	—	2	1	no	—
570	C₄	—	2	2	no	—
571	C₄	—	2	1	no	—
572	C₄	—	4	4	no	—
573	C₄	—	4	4	no	—
574	C₄	—	4	4	no	—
575	C₄	C₃₁×#575	1	2	yes	—
576	C₄	C₃₁×#585	1	2	yes	—
577	C₂	—	2	4	no	—
578	C₂	—	2	2	no	—
579	C₂	—	8	4	no	—
580	C₂	C₃₁×#3449	1	1	yes	characteristic
581	C₂	C₂²×C₃₁×#135	1	1	yes	characteristic
582	C₂	C₂×C₃₁×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂³×D₄
prime radical	C₃₁×#3186	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#529	C₂	semidirect
2. C₃₁×#3250	C₂	semidirect
3. C₃₁×#3232	C₂	semidirect
4. C₃₁×#3211	C₂	semidirect
5. C₃₁×#3189	C₂	semidirect
6. C₃₁×#3187	C₂	semidirect
8. C₃₁×#3168	C₂	semidirect
9. C₃₁×#3108	C₂	semidirect
10. C₃₁×#3105	C₂	semidirect
11. C₂×C₃₁×#431	C₂	semidirect
18. C₃₁×#529	C₂²	semidirect
19. C₃₁×#529	C₂²	semidirect
21. C₂×C₃₁×#125	C₂²	semidirect
22. C₂×C₃₁×#124	C₂²	semidirect
26. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
28. C₂×C₄×D₄×C₃₁	C₂²	semidirect
31. C₃₁×#499	C₂²	semidirect
32. C₃₁×#495	C₂²	semidirect
33. C₃₁×#488	C₂²	semidirect
34. C₃₁×#484	C₂²	semidirect
35. C₃₁×#483	C₂²	semidirect
38. C₃₁×#479	C₂²	semidirect
39. C₃₁×#478	C₂²	semidirect
40. C₃₁×#467	C₂²	semidirect
42. C₃₁×#465	C₂²	semidirect
43. C₃₁×#464	C₂²	semidirect
44. C₃₁×#463	C₂²	semidirect
45. C₃₁×#451	C₂²	semidirect
46. C₃₁×#450	C₂²	semidirect
49. C₄×Q₁₆×C₃₁	C₂²	semidirect
50. C₄×SD₁₆×C₃₁	C₂²	semidirect
51. C₄×D₈×C₃₁	C₂²	semidirect
52. C₃₁×#431	C₂²	semidirect
53. C₃₁×#431	C₂²	semidirect
54. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
57. C₂×C₄×C₈×C₃₁	C₂²	semidirect
215. #60945	C₃₁	direct (N × H)
262. C₂×#529	C₂×C₃₁	semidirect
263. #3250	C₂×C₃₁	semidirect
264. #3232	C₂×C₃₁	semidirect
265. #3211	C₂×C₃₁	semidirect
266. #3189	C₂×C₃₁	semidirect
267. #3187	C₂×C₃₁	semidirect
269. #3168	C₂×C₃₁	semidirect
270. #3108	C₂×C₃₁	semidirect
271. #3105	C₂×C₃₁	semidirect
272. C₂×#431	C₂×C₃₁	semidirect
303. #529	C₂²×C₃₁	semidirect
304. #529	C₂²×C₃₁	semidirect
306. C₂×#125	C₂²×C₃₁	semidirect
307. C₂×#124	C₂²×C₃₁	semidirect
311. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
313. C₂×C₄×D₄	C₂²×C₃₁	semidirect
316. #499	C₂²×C₃₁	semidirect
317. #495	C₂²×C₃₁	semidirect
318. #488	C₂²×C₃₁	semidirect
319. #484	C₂²×C₃₁	semidirect
320. #483	C₂²×C₃₁	semidirect
323. #479	C₂²×C₃₁	semidirect
324. #478	C₂²×C₃₁	semidirect
325. #467	C₂²×C₃₁	semidirect
327. #465	C₂²×C₃₁	semidirect
328. #464	C₂²×C₃₁	semidirect
329. #463	C₂²×C₃₁	semidirect
330. #451	C₂²×C₃₁	semidirect
331. #450	C₂²×C₃₁	semidirect
334. C₄×Q₁₆	C₂²×C₃₁	semidirect
335. C₄×SD₁₆	C₂²×C₃₁	semidirect
336. C₄×D₈	C₂²×C₃₁	semidirect
337. #431	C₂²×C₃₁	semidirect
338. #431	C₂²×C₃₁	semidirect
339. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
342. C₂×C₄×C₈	C₂²×C₃₁	semidirect
433. C₃₁	#60945	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#529	2
C₃₁×#3250	2
C₃₁×#3232	2
C₃₁×#3211	2
C₃₁×#3189	2
C₃₁×#3187	2
C₃₁×#3186	2
C₃₁×#3168	2
C₃₁×#3108	2
C₃₁×#3105	2
C₂×C₃₁×#431	2
#60945	31

Subgroups — C31×#60945 #21517704

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60945 #21517704