Subgroups — C₃₁×#60967 #21517726

Order 7936.

1824 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3444	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3255	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3254	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3253	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3233	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3212	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3197	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3183	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3116	C₂	1	4	yes	maximal, complemented
10	C₃₁×#3115	C₂	1	4	yes	maximal, complemented
11	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#533	C₂²	1	2	yes	—
13	C₃₁×#533	C₂²	1	2	yes	—
14	C₃₁×#533	C₂²	1	2	yes	—
15	C₃₁×#532	C₂²	1	2	yes	—
16	C₃₁×#532	C₂²	1	2	yes	—
17	C₃₁×#532	C₂²	1	2	yes	—
18	C₃₁×#529	—	2	2	no	—
19	C₃₁×#529	—	2	2	no	—
20	C₃₁×#529	—	2	2	no	—
21	C₃₁×#529	—	2	2	no	—
22	C₃₁×#529	C₂²	1	4	yes	—
23	C₃₁×#529	C₂²	1	4	yes	—
24	C₃₁×#525	—	2	2	no	—
25	C₃₁×#525	—	2	2	no	—
26	C₃₁×#525	C₂²	1	4	yes	—
27	C₄×#41×C₃₁	—	2	2	no	—
28	C₄×#41×C₃₁	—	2	2	no	—
29	C₄×#41×C₃₁	C₂²	1	4	yes	—
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×#485	C₂²	1	8	yes	—
35	C₃₁×#482	C₂²	1	8	yes	—
36	C₃₁×#477	C₂²	1	4	yes	complemented
37	C₃₁×#476	C₂²	1	4	yes	complemented
38	C₃₁×#475	C₂²	1	4	yes	complemented
39	C₃₁×#474	C₂²	1	4	yes	complemented
40	C₃₁×#470	C₂²	1	4	yes	—
41	C₃₁×#469	C₂²	1	4	yes	—
42	C₃₁×#468	C₂²	1	8	yes	—
43	C₃₁×#462	C₂²	1	4	yes	complemented
44	C₃₁×#461	C₂²	1	4	yes	complemented
45	C₃₁×#460	C₂²	1	4	yes	complemented
46	C₃₁×#459	C₂²	1	4	yes	complemented
47	C₃₁×#452	C₂²	1	8	yes	—
48	C₃₁×#449	C₂²	1	8	yes	—
49	C₃₁×#442	C₂²	1	4	yes	complemented
50	C₃₁×#441	C₂²	1	4	yes	complemented
51	C₃₁×#440	C₂²	1	8	yes	complemented
52	C₃₁×#432	C₂²	1	4	yes	—
53	C₃₁×#431	C₂²	1	4	yes	—
54	C₃₁×#423	C₂²	1	1	yes	characteristic
55	C₃₁×#423	C₂²	1	1	yes	characteristic
56	C₃₁×#407	C₂²	1	2	yes	—
57	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
58	C₂×#41×C₃₁	—	2	2	no	—
59	C₂×#41×C₃₁	—	2	2	no	—
60	C₂×#41×C₃₁	C₂³	1	4	yes	—
61	C₂³×C₄×C₃₁	D₄	1	2	yes	—
62	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
64	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
65	C₂×D₈×C₃₁	C₂³	1	4	yes	—
66	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
67	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
68	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
69	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
70	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
71	C₃₁×#127	C₂³	1	2	yes	—
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×#124	—	2	2	no	—
78	C₃₁×#124	—	2	2	no	—
79	C₃₁×#124	C₂³	1	4	yes	—
80	C₃₁×#123	—	2	4	no	—
81	C₃₁×#123	—	2	4	no	—
82	C₃₁×#123	—	2	4	no	—
83	C₃₁×C₂²⋊Q₈	—	2	2	no	—
84	C₃₁×C₂²⋊Q₈	—	2	2	no	—
85	C₃₁×C₂²⋊Q₈	—	2	2	no	—
86	C₃₁×C₂²⋊Q₈	—	2	2	no	—
87	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
88	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
89	C₃₁×#121	—	2	2	no	—
90	C₃₁×#121	—	2	2	no	—
91	C₃₁×#121	—	2	2	no	—
92	C₃₁×#121	—	2	2	no	—
93	C₃₁×#121	C₂³	1	4	yes	—
94	C₃₁×#121	C₂³	1	4	yes	—
95	C₃₁×#120	—	2	4	no	—
96	C₄×Q₈×C₃₁	—	2	2	no	—
97	C₄×Q₈×C₃₁	—	2	2	no	—
98	C₄×Q₈×C₃₁	—	2	2	no	—
99	C₄×Q₈×C₃₁	—	2	2	no	—
100	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
101	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
102	C₄×D₄×C₃₁	—	2	4	no	—
103	C₄×D₄×C₃₁	—	2	4	no	—
104	C₄×D₄×C₃₁	—	2	4	no	—
105	C₄×D₄×C₃₁	—	2	4	no	—
106	C₄×D₄×C₃₁	—	2	4	no	—
107	C₄×D₄×C₃₁	—	2	4	no	—
108	C₄×D₄×C₃₁	—	2	2	no	—
109	C₄×D₄×C₃₁	—	2	2	no	—
110	C₄×D₄×C₃₁	—	2	2	no	—
111	C₄×D₄×C₃₁	—	2	2	no	—
112	C₄×D₄×C₃₁	C₂³	1	4	yes	—
113	C₄×D₄×C₃₁	C₂³	1	4	yes	—
114	C₃₁×#117	—	2	2	no	—
115	C₃₁×#117	—	2	2	no	—
116	C₃₁×#117	—	2	4	no	—
117	C₃₁×#117	—	2	4	no	—
118	C₃₁×#117	—	2	4	no	—
119	C₃₁×#117	C₂³	1	4	yes	—
120	C₂×C₄²×C₃₁	—	2	4	no	—
121	C₂×C₄²×C₃₁	D₄	1	2	yes	—
122	C₂×C₄²×C₃₁	D₄	1	2	yes	—
123	C₂×C₄²×C₃₁	D₄	1	2	yes	—
124	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
127	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
128	C₃₁×#107	C₂³	1	8	yes	—
129	C₃₁×#106	C₂³	1	4	yes	—
130	C₃₁×#106	C₂³	1	4	yes	—
131	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
132	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
133	C₃₁×#103	C₂³	1	8	yes	—
134	C₃₁×#103	C₂³	1	8	yes	—
135	C₃₁×#102	C₂³	1	8	yes	—
136	C₃₁×#102	C₂³	1	8	yes	—
137	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
138	C₃₁×#97	C₂³	1	4	yes	—
139	C₄×C₈×C₃₁	C₂³	1	4	yes	—
140	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
141	#41×C₃₁	—	4	4	no	—
142	#41×C₃₁	—	4	4	no	—
143	#41×C₃₁	—	4	8	no	—
144	#41×C₃₁	—	4	8	no	—
145	C₂×Q₈×C₃₁	—	2	2	no	—
146	C₂×Q₈×C₃₁	—	2	2	no	—
147	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
148	C₂×D₄×C₃₁	—	2	4	no	—
149	C₂×D₄×C₃₁	—	2	4	no	—
150	C₂×D₄×C₃₁	—	2	4	no	—
151	C₂×D₄×C₃₁	—	2	2	no	—
152	C₂×D₄×C₃₁	—	2	2	no	—
153	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₃₁	—	2	4	no	—
155	C₂²×C₄×C₃₁	—	2	4	no	—
156	C₂²×C₄×C₃₁	—	2	4	no	—
157	C₂²×C₄×C₃₁	—	4	4	no	—
158	C₂²×C₄×C₃₁	—	4	1	no	—
159	C₂²×C₄×C₃₁	—	4	1	no	—
160	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
170	Q₁₆×C₃₁	—	4	4	no	—
171	SD₁₆×C₃₁	—	4	8	no	—
172	D₈×C₃₁	—	4	4	no	—
173	M₁₆×C₃₁	—	4	2	no	—
174	M₁₆×C₃₁	—	4	2	no	—
175	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
176	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
177	C₄⋊C₄×C₃₁	—	2	2	no	—
178	C₄⋊C₄×C₃₁	—	2	2	no	—
179	C₄⋊C₄×C₃₁	—	2	2	no	—
180	C₄⋊C₄×C₃₁	—	2	2	no	—
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	C₂⁴	1	4	yes	—
188	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
189	C₂²⋊C₄×C₃₁	—	2	4	no	—
190	C₂²⋊C₄×C₃₁	—	2	4	no	—
191	C₂²⋊C₄×C₃₁	—	2	4	no	—
192	C₂²⋊C₄×C₃₁	—	2	4	no	—
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	C₄²×C₃₁	—	2	4	no	—
199	C₄²×C₃₁	—	2	4	no	—
200	C₄²×C₃₁	—	2	4	no	—
201	C₄²×C₃₁	C₂×D₄	1	2	yes	—
202	C₄²×C₃₁	C₂×D₄	1	2	yes	—
203	C₄²×C₃₁	C₂×D₄	1	2	yes	—
204	C₄²×C₃₁	C₂×D₄	1	2	yes	—
205	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
206	C₄²×C₃₁	C₂×D₄	1	2	yes	—
207	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
209	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
210	C₄²×C₃₁	C₂⁴	1	2	yes	—
211	#60967	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂³×C₃₁	—	2	4	no	—
213	C₂³×C₃₁	—	4	1	no	—
214	C₂³×C₃₁	—	4	2	no	—
215	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
216	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
217	Q₈×C₃₁	—	2	4	no	—
218	Q₈×C₃₁	—	2	4	no	—
219	Q₈×C₃₁	—	4	4	no	—
220	Q₈×C₃₁	—	4	4	no	—
221	D₄×C₃₁	—	2	4	no	—
222	D₄×C₃₁	—	2	4	no	—
223	D₄×C₃₁	—	4	4	no	—
224	D₄×C₃₁	—	4	8	no	—
225	D₄×C₃₁	—	4	8	no	—
226	D₄×C₃₁	—	4	8	no	—
227	D₄×C₃₁	—	4	4	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	4	no	—
230	C₂×C₄×C₃₁	—	2	4	no	—
231	C₂×C₄×C₃₁	—	2	1	no	—
232	C₂×C₄×C₃₁	—	2	1	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	4	no	—
237	C₂×C₄×C₃₁	—	2	4	no	—
238	C₂×C₄×C₃₁	—	4	1	no	—
239	C₂×C₄×C₃₁	—	4	2	no	—
240	C₂×C₄×C₃₁	—	4	2	no	—
241	C₂×C₄×C₃₁	—	4	2	no	—
242	C₂×C₄×C₃₁	—	4	2	no	—
243	C₂×C₄×C₃₁	—	4	8	no	—
244	C₂×C₄×C₃₁	—	4	8	no	—
245	C₂×C₄×C₃₁	—	4	8	no	—
246	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
247	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
248	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
250	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
251	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
253	C₈×C₃₁	—	2	4	no	—
254	C₈×C₃₁	—	2	4	no	—
255	#3444	C₂×C₃₁	1	2	yes	complemented
256	#3255	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#3254	C₂×C₃₁	1	2	yes	complemented
258	#3253	C₂×C₃₁	1	1	yes	characteristic, complemented
259	#3233	C₂×C₃₁	1	4	yes	complemented
260	#3212	C₂×C₃₁	1	4	yes	complemented
261	#3197	C₂×C₃₁	1	4	yes	complemented
262	#3183	C₂×C₃₁	1	4	yes	complemented
263	#3116	C₂×C₃₁	1	4	yes	complemented
264	#3115	C₂×C₃₁	1	4	yes	complemented
265	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
266	C₂²×C₃₁	—	2	2	no	—
267	C₂²×C₃₁	—	2	1	no	—
268	C₂²×C₃₁	—	2	2	no	—
269	C₂²×C₃₁	—	2	2	no	—
270	C₂²×C₃₁	—	2	2	no	—
271	C₂²×C₃₁	—	4	4	no	—
272	C₂²×C₃₁	—	4	4	no	—
273	C₂²×C₃₁	—	4	4	no	—
274	C₂²×C₃₁	—	4	4	no	—
275	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
276	C₄×C₃₁	—	2	2	no	—
277	C₄×C₃₁	—	2	1	no	—
278	C₄×C₃₁	—	2	2	no	—
279	C₄×C₃₁	—	2	2	no	—
280	C₄×C₃₁	—	2	2	no	—
281	C₄×C₃₁	—	4	4	no	—
282	C₄×C₃₁	—	4	4	no	—
283	C₄×C₃₁	—	4	4	no	—
284	C₄×C₃₁	#575	1	4	yes	—
285	C₄×C₃₁	#585	1	1	yes	characteristic
286	C₄×C₃₁	#585	1	1	yes	characteristic
287	#533	C₂²×C₃₁	1	2	yes	—
288	#533	C₂²×C₃₁	1	2	yes	—
289	#533	C₂²×C₃₁	1	2	yes	—
290	#532	C₂²×C₃₁	1	2	yes	—
291	#532	C₂²×C₃₁	1	2	yes	—
292	#532	C₂²×C₃₁	1	2	yes	—
293	#529	—	2	2	no	—
294	#529	—	2	2	no	—
295	#529	—	2	2	no	—
296	#529	—	2	2	no	—
297	#529	C₂²×C₃₁	1	4	yes	—
298	#529	C₂²×C₃₁	1	4	yes	—
299	#525	—	2	2	no	—
300	#525	—	2	2	no	—
301	#525	C₂²×C₃₁	1	4	yes	—
302	C₄×#41	—	2	2	no	—
303	C₄×#41	—	2	2	no	—
304	C₄×#41	C₂²×C₃₁	1	4	yes	—
305	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
306	#499	C₂²×C₃₁	1	4	yes	complemented
307	#495	C₂²×C₃₁	1	4	yes	complemented
308	#488	C₂²×C₃₁	1	8	yes	complemented
309	#485	C₂²×C₃₁	1	8	yes	—
310	#482	C₂²×C₃₁	1	8	yes	—
311	#477	C₂²×C₃₁	1	4	yes	complemented
312	#476	C₂²×C₃₁	1	4	yes	complemented
313	#475	C₂²×C₃₁	1	4	yes	complemented
314	#474	C₂²×C₃₁	1	4	yes	complemented
315	#470	C₂²×C₃₁	1	4	yes	—
316	#469	C₂²×C₃₁	1	4	yes	—
317	#468	C₂²×C₃₁	1	8	yes	—
318	#462	C₂²×C₃₁	1	4	yes	complemented
319	#461	C₂²×C₃₁	1	4	yes	complemented
320	#460	C₂²×C₃₁	1	4	yes	complemented
321	#459	C₂²×C₃₁	1	4	yes	complemented
322	#452	C₂²×C₃₁	1	8	yes	—
323	#449	C₂²×C₃₁	1	8	yes	—
324	#442	C₂²×C₃₁	1	4	yes	complemented
325	#441	C₂²×C₃₁	1	4	yes	complemented
326	#440	C₂²×C₃₁	1	8	yes	complemented
327	#432	C₂²×C₃₁	1	4	yes	—
328	#431	C₂²×C₃₁	1	4	yes	—
329	#423	C₂²×C₃₁	1	1	yes	characteristic
330	#423	C₂²×C₃₁	1	1	yes	characteristic
331	#407	C₂²×C₃₁	1	2	yes	—
332	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
333	C₂×C₃₁	—	4	2	no	—
334	C₂×C₃₁	—	4	1	no	—
335	C₂×C₃₁	—	8	4	no	—
336	C₂×C₃₁	#3449	1	1	yes	characteristic
337	C₂×C₃₁	C₂×#575	1	2	yes	—
338	C₂×#41	—	2	2	no	—
339	C₂×#41	—	2	2	no	—
340	C₂×#41	C₂³×C₃₁	1	4	yes	—
341	C₂³×C₄	D₄×C₃₁	1	2	yes	—
342	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
343	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
344	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
345	C₂×D₈	C₂³×C₃₁	1	4	yes	—
346	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
347	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
348	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
349	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
350	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
351	#127	C₂³×C₃₁	1	2	yes	—
352	#126	—	2	4	no	—
353	#126	—	2	4	no	—
354	#125	—	2	2	no	—
355	#125	—	2	2	no	—
356	#125	C₂³×C₃₁	1	4	yes	—
357	#124	—	2	2	no	—
358	#124	—	2	2	no	—
359	#124	C₂³×C₃₁	1	4	yes	—
360	#123	—	2	4	no	—
361	#123	—	2	4	no	—
362	#123	—	2	4	no	—
363	C₂²⋊Q₈	—	2	2	no	—
364	C₂²⋊Q₈	—	2	2	no	—
365	C₂²⋊Q₈	—	2	2	no	—
366	C₂²⋊Q₈	—	2	2	no	—
367	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
368	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
369	#121	—	2	2	no	—
370	#121	—	2	2	no	—
371	#121	—	2	2	no	—
372	#121	—	2	2	no	—
373	#121	C₂³×C₃₁	1	4	yes	—
374	#121	C₂³×C₃₁	1	4	yes	—
375	#120	—	2	4	no	—
376	C₄×Q₈	—	2	2	no	—
377	C₄×Q₈	—	2	2	no	—
378	C₄×Q₈	—	2	2	no	—
379	C₄×Q₈	—	2	2	no	—
380	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
381	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
382	C₄×D₄	—	2	4	no	—
383	C₄×D₄	—	2	4	no	—
384	C₄×D₄	—	2	4	no	—
385	C₄×D₄	—	2	4	no	—
386	C₄×D₄	—	2	4	no	—
387	C₄×D₄	—	2	4	no	—
388	C₄×D₄	—	2	2	no	—
389	C₄×D₄	—	2	2	no	—
390	C₄×D₄	—	2	2	no	—
391	C₄×D₄	—	2	2	no	—
392	C₄×D₄	C₂³×C₃₁	1	4	yes	—
393	C₄×D₄	C₂³×C₃₁	1	4	yes	—
394	#117	—	2	2	no	—
395	#117	—	2	2	no	—
396	#117	—	2	4	no	—
397	#117	—	2	4	no	—
398	#117	—	2	4	no	—
399	#117	C₂³×C₃₁	1	4	yes	—
400	C₂×C₄²	—	2	4	no	—
401	C₂×C₄²	D₄×C₃₁	1	2	yes	—
402	C₂×C₄²	D₄×C₃₁	1	2	yes	—
403	C₂×C₄²	D₄×C₃₁	1	2	yes	—
404	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
405	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
407	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
408	#107	C₂³×C₃₁	1	8	yes	—
409	#106	C₂³×C₃₁	1	4	yes	—
410	#106	C₂³×C₃₁	1	4	yes	—
411	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
412	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
413	#103	C₂³×C₃₁	1	8	yes	—
414	#103	C₂³×C₃₁	1	8	yes	—
415	#102	C₂³×C₃₁	1	8	yes	—
416	#102	C₂³×C₃₁	1	8	yes	—
417	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
418	#97	C₂³×C₃₁	1	4	yes	—
419	C₄×C₈	C₂³×C₃₁	1	4	yes	—
420	C₃₁	#60967	1	1	yes	characteristic, direct product factor, Sylow
421	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
422	#41	—	4	4	no	—
423	#41	—	4	4	no	—
424	#41	—	4	8	no	—
425	#41	—	4	8	no	—
426	C₂×Q₈	—	2	2	no	—
427	C₂×Q₈	—	2	2	no	—
428	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
429	C₂×D₄	—	2	4	no	—
430	C₂×D₄	—	2	4	no	—
431	C₂×D₄	—	2	4	no	—
432	C₂×D₄	—	2	2	no	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	4	4	no	—
439	C₂²×C₄	—	4	1	no	—
440	C₂²×C₄	—	4	1	no	—
441	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
442	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
443	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
444	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
445	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
446	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
447	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
448	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
449	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
450	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
451	Q₁₆	—	4	4	no	—
452	SD₁₆	—	4	8	no	—
453	D₈	—	4	4	no	—
454	M₁₆	—	4	2	no	—
455	M₁₆	—	4	2	no	—
456	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
457	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
458	C₄⋊C₄	—	2	2	no	—
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₄⋊C₄	—	2	2	no	—
462	C₄⋊C₄	—	2	4	no	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	4	no	—
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	4	no	—
468	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
469	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
470	C₂²⋊C₄	—	2	4	no	—
471	C₂²⋊C₄	—	2	4	no	—
472	C₂²⋊C₄	—	2	4	no	—
473	C₂²⋊C₄	—	2	4	no	—
474	C₂²⋊C₄	—	2	4	no	—
475	C₂²⋊C₄	—	2	4	no	—
476	C₂²⋊C₄	—	2	4	no	—
477	C₂²⋊C₄	—	2	4	no	—
478	C₂²⋊C₄	—	2	4	no	—
479	C₄²	—	2	4	no	—
480	C₄²	—	2	4	no	—
481	C₄²	—	2	4	no	—
482	C₄²	C₂×D₄×C₃₁	1	2	yes	—
483	C₄²	C₂×D₄×C₃₁	1	2	yes	—
484	C₄²	C₂×D₄×C₃₁	1	2	yes	—
485	C₄²	C₂×D₄×C₃₁	1	2	yes	—
486	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
487	C₄²	C₂×D₄×C₃₁	1	2	yes	—
488	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
489	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
490	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
491	C₄²	C₂⁴×C₃₁	1	2	yes	—
492	C₂³	—	2	4	no	—
493	C₂³	—	4	1	no	—
494	C₂³	—	4	2	no	—
495	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
496	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
497	Q₈	—	2	4	no	—
498	Q₈	—	2	4	no	—
499	Q₈	—	4	4	no	—
500	Q₈	—	4	4	no	—
501	D₄	—	2	4	no	—
502	D₄	—	2	4	no	—
503	D₄	—	4	4	no	—
504	D₄	—	4	8	no	—
505	D₄	—	4	8	no	—
506	D₄	—	4	8	no	—
507	D₄	—	4	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	4	8	no	—
524	C₂×C₄	—	4	8	no	—
525	C₂×C₄	—	4	8	no	—
526	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
527	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
528	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
529	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
530	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
531	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
532	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
533	C₈	—	2	4	no	—
534	C₈	—	2	4	no	—
535	C₂²	—	2	2	no	—
536	C₂²	—	2	1	no	—
537	C₂²	—	2	2	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	4	4	no	—
541	C₂²	—	4	4	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	4	4	no	—
544	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
545	C₄	—	2	2	no	—
546	C₄	—	2	1	no	—
547	C₄	—	2	2	no	—
548	C₄	—	2	2	no	—
549	C₄	—	2	2	no	—
550	C₄	—	4	4	no	—
551	C₄	—	4	4	no	—
552	C₄	—	4	4	no	—
553	C₄	C₃₁×#575	1	4	yes	—
554	C₄	C₃₁×#585	1	1	yes	characteristic
555	C₄	C₃₁×#585	1	1	yes	characteristic
556	C₂	—	4	2	no	—
557	C₂	—	4	1	no	—
558	C₂	—	8	4	no	—
559	C₂	C₃₁×#3449	1	1	yes	characteristic
560	C₂	C₂×C₃₁×#575	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 ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3444	C₂	semidirect
2. C₃₁×#3255	C₂	semidirect
3. C₃₁×#3254	C₂	semidirect
4. C₃₁×#3253	C₂	semidirect
5. C₃₁×#3233	C₂	semidirect
6. C₃₁×#3212	C₂	semidirect
7. C₃₁×#3197	C₂	semidirect
8. C₃₁×#3183	C₂	semidirect
9. C₃₁×#3116	C₂	semidirect
10. C₃₁×#3115	C₂	semidirect
11. C₃₁×#2920	C₂	semidirect
31. C₃₁×#499	C₂²	semidirect
32. C₃₁×#495	C₂²	semidirect
33. C₃₁×#488	C₂²	semidirect
36. C₃₁×#477	C₂²	semidirect
37. C₃₁×#476	C₂²	semidirect
38. C₃₁×#475	C₂²	semidirect
39. C₃₁×#474	C₂²	semidirect
43. C₃₁×#462	C₂²	semidirect
44. C₃₁×#461	C₂²	semidirect
45. C₃₁×#460	C₂²	semidirect
46. C₃₁×#459	C₂²	semidirect
49. C₃₁×#442	C₂²	semidirect
50. C₃₁×#441	C₂²	semidirect
51. C₃₁×#440	C₂²	semidirect
211. #60967	C₃₁	direct (N × H)
255. #3444	C₂×C₃₁	semidirect
256. #3255	C₂×C₃₁	semidirect
257. #3254	C₂×C₃₁	semidirect
258. #3253	C₂×C₃₁	semidirect
259. #3233	C₂×C₃₁	semidirect
260. #3212	C₂×C₃₁	semidirect
261. #3197	C₂×C₃₁	semidirect
262. #3183	C₂×C₃₁	semidirect
263. #3116	C₂×C₃₁	semidirect
264. #3115	C₂×C₃₁	semidirect
265. #2920	C₂×C₃₁	semidirect
306. #499	C₂²×C₃₁	semidirect
307. #495	C₂²×C₃₁	semidirect
308. #488	C₂²×C₃₁	semidirect
311. #477	C₂²×C₃₁	semidirect
312. #476	C₂²×C₃₁	semidirect
313. #475	C₂²×C₃₁	semidirect
314. #474	C₂²×C₃₁	semidirect
318. #462	C₂²×C₃₁	semidirect
319. #461	C₂²×C₃₁	semidirect
320. #460	C₂²×C₃₁	semidirect
321. #459	C₂²×C₃₁	semidirect
324. #442	C₂²×C₃₁	semidirect
325. #441	C₂²×C₃₁	semidirect
326. #440	C₂²×C₃₁	semidirect
420. C₃₁	#60967	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×#3444	2
C₃₁×#3255	2
C₃₁×#3254	2
C₃₁×#3253	2
C₃₁×#3233	2
C₃₁×#3212	2
C₃₁×#3197	2
C₃₁×#3183	2
C₃₁×#3116	2
C₃₁×#3115	2
C₃₁×#2920	2
#60967	31

Subgroups — C31×#60967 #21517726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60967 #21517726