Subgroups — C₃₁×#60972 #21517731

Order 7936.

1824 conjugacy classes of subgroups.

The subgroup lattice has 570 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×#3256	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3231	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3228	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3210	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3207	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3197	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3185	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3184	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3182	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3111	C₂	1	8	yes	maximal, complemented
12	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#533	C₂²	1	2	yes	—
14	C₃₁×#533	C₂²	1	2	yes	—
15	C₃₁×#533	C₂²	1	2	yes	—
16	C₃₁×#532	C₂²	1	2	yes	—
17	C₃₁×#532	C₂²	1	2	yes	—
18	C₃₁×#532	C₂²	1	2	yes	—
19	C₃₁×#529	—	2	2	no	—
20	C₃₁×#529	—	2	2	no	—
21	C₃₁×#529	—	2	2	no	—
22	C₃₁×#529	—	2	2	no	—
23	C₃₁×#529	C₂²	1	4	yes	—
24	C₃₁×#529	C₂²	1	4	yes	—
25	C₃₁×#525	—	2	2	no	—
26	C₃₁×#525	—	2	2	no	—
27	C₃₁×#525	C₂²	1	4	yes	—
28	C₄×#41×C₃₁	—	2	2	no	—
29	C₄×#41×C₃₁	—	2	2	no	—
30	C₄×#41×C₃₁	C₂²	1	4	yes	—
31	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#500	C₂²	1	2	yes	complemented
33	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
36	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
37	C₃₁×#492	C₂²	1	2	yes	complemented
38	C₃₁×#487	C₂²	1	2	yes	complemented
39	C₃₁×#487	C₂²	1	2	yes	complemented
40	C₃₁×#486	C₂²	1	2	yes	complemented
41	C₃₁×#486	C₂²	1	2	yes	complemented
42	C₃₁×#485	C₂²	1	8	yes	—
43	C₃₁×#482	C₂²	1	8	yes	—
44	C₃₁×#479	C₂²	1	8	yes	complemented
45	C₃₁×#478	C₂²	1	8	yes	complemented
46	C₃₁×#470	C₂²	1	4	yes	—
47	C₃₁×#469	C₂²	1	4	yes	—
48	C₃₁×#468	C₂²	1	8	yes	—
49	C₃₁×#464	C₂²	1	8	yes	complemented
50	C₃₁×#463	C₂²	1	8	yes	complemented
51	C₃₁×#452	C₂²	1	8	yes	—
52	C₃₁×#449	C₂²	1	8	yes	—
53	C₃₁×#442	C₂²	1	4	yes	complemented
54	C₃₁×#441	C₂²	1	4	yes	complemented
55	C₃₁×#440	C₂²	1	8	yes	complemented
56	C₃₁×#432	C₂²	1	4	yes	—
57	C₃₁×#431	C₂²	1	2	yes	—
58	C₃₁×#431	C₂²	1	2	yes	—
59	C₃₁×#423	C₂²	1	2	yes	—
60	C₃₁×#407	C₂²	1	2	yes	—
61	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
62	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
63	C₂×#41×C₃₁	—	2	2	no	—
64	C₂×#41×C₃₁	—	2	2	no	—
65	C₂×#41×C₃₁	C₂³	1	4	yes	—
66	C₂³×C₄×C₃₁	D₄	1	2	yes	—
67	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
69	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
70	C₂×D₈×C₃₁	C₂³	1	4	yes	—
71	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
72	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
73	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
74	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
75	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
76	C₃₁×#127	C₂³	1	2	yes	—
77	C₃₁×#126	—	2	4	no	—
78	C₃₁×#126	—	2	4	no	—
79	C₃₁×#125	—	2	2	no	—
80	C₃₁×#125	—	2	2	no	—
81	C₃₁×#125	C₂³	1	4	yes	—
82	C₃₁×#124	—	2	2	no	—
83	C₃₁×#124	—	2	2	no	—
84	C₃₁×#124	C₂³	1	4	yes	—
85	C₃₁×#123	—	2	4	no	—
86	C₃₁×#123	—	2	4	no	—
87	C₃₁×#123	—	2	4	no	—
88	C₃₁×C₂²⋊Q₈	—	2	2	no	—
89	C₃₁×C₂²⋊Q₈	—	2	2	no	—
90	C₃₁×C₂²⋊Q₈	—	2	2	no	—
91	C₃₁×C₂²⋊Q₈	—	2	2	no	—
92	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
93	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
94	C₃₁×#121	—	2	2	no	—
95	C₃₁×#121	—	2	2	no	—
96	C₃₁×#121	—	2	2	no	—
97	C₃₁×#121	—	2	2	no	—
98	C₃₁×#121	C₂³	1	4	yes	—
99	C₃₁×#121	C₂³	1	4	yes	—
100	C₃₁×#120	—	2	4	no	—
101	C₄×Q₈×C₃₁	—	2	2	no	—
102	C₄×Q₈×C₃₁	—	2	2	no	—
103	C₄×Q₈×C₃₁	—	2	2	no	—
104	C₄×Q₈×C₃₁	—	2	2	no	—
105	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
106	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
107	C₄×D₄×C₃₁	—	2	4	no	—
108	C₄×D₄×C₃₁	—	2	4	no	—
109	C₄×D₄×C₃₁	—	2	4	no	—
110	C₄×D₄×C₃₁	—	2	4	no	—
111	C₄×D₄×C₃₁	—	2	4	no	—
112	C₄×D₄×C₃₁	—	2	4	no	—
113	C₄×D₄×C₃₁	—	2	2	no	—
114	C₄×D₄×C₃₁	—	2	2	no	—
115	C₄×D₄×C₃₁	—	2	2	no	—
116	C₄×D₄×C₃₁	—	2	2	no	—
117	C₄×D₄×C₃₁	C₂³	1	4	yes	—
118	C₄×D₄×C₃₁	C₂³	1	4	yes	—
119	C₃₁×#117	—	2	2	no	—
120	C₃₁×#117	—	2	2	no	—
121	C₃₁×#117	—	2	4	no	—
122	C₃₁×#117	—	2	4	no	—
123	C₃₁×#117	—	2	4	no	—
124	C₃₁×#117	C₂³	1	4	yes	—
125	C₂×C₄²×C₃₁	—	2	4	no	—
126	C₂×C₄²×C₃₁	D₄	1	2	yes	—
127	C₂×C₄²×C₃₁	D₄	1	2	yes	—
128	C₂×C₄²×C₃₁	D₄	1	2	yes	—
129	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
132	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
133	C₃₁×#107	C₂³	1	8	yes	—
134	C₃₁×#106	C₂³	1	4	yes	—
135	C₃₁×#106	C₂³	1	4	yes	—
136	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
137	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
138	C₃₁×#103	C₂³	1	8	yes	—
139	C₃₁×#103	C₂³	1	8	yes	—
140	C₃₁×#102	C₂³	1	8	yes	—
141	C₃₁×#102	C₂³	1	8	yes	—
142	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
143	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
144	C₃₁×#97	C₂³	1	2	yes	—
145	C₃₁×#97	C₂³	1	2	yes	—
146	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
147	C₄×C₈×C₃₁	C₂³	1	2	yes	—
148	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
149	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
150	#41×C₃₁	—	4	8	no	—
151	#41×C₃₁	—	4	8	no	—
152	#41×C₃₁	—	4	8	no	—
153	C₂×Q₈×C₃₁	—	2	2	no	—
154	C₂×Q₈×C₃₁	—	2	2	no	—
155	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
156	C₂×D₄×C₃₁	—	2	4	no	—
157	C₂×D₄×C₃₁	—	2	4	no	—
158	C₂×D₄×C₃₁	—	2	4	no	—
159	C₂×D₄×C₃₁	—	2	2	no	—
160	C₂×D₄×C₃₁	—	2	2	no	—
161	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₃₁	—	2	4	no	—
163	C₂²×C₄×C₃₁	—	2	4	no	—
164	C₂²×C₄×C₃₁	—	2	4	no	—
165	C₂²×C₄×C₃₁	—	4	4	no	—
166	C₂²×C₄×C₃₁	—	4	2	no	—
167	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
177	Q₁₆×C₃₁	—	4	4	no	—
178	SD₁₆×C₃₁	—	4	8	no	—
179	D₈×C₃₁	—	4	4	no	—
180	M₁₆×C₃₁	—	4	2	no	—
181	M₁₆×C₃₁	—	4	2	no	—
182	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
183	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
184	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
185	C₄⋊C₄×C₃₁	—	2	2	no	—
186	C₄⋊C₄×C₃₁	—	2	2	no	—
187	C₄⋊C₄×C₃₁	—	2	2	no	—
188	C₄⋊C₄×C₃₁	—	2	2	no	—
189	C₄⋊C₄×C₃₁	—	2	4	no	—
190	C₄⋊C₄×C₃₁	—	2	4	no	—
191	C₄⋊C₄×C₃₁	—	2	4	no	—
192	C₄⋊C₄×C₃₁	—	2	4	no	—
193	C₄⋊C₄×C₃₁	—	2	4	no	—
194	C₄⋊C₄×C₃₁	—	2	4	no	—
195	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
196	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
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	C₂²⋊C₄×C₃₁	—	2	4	no	—
204	C₂²⋊C₄×C₃₁	—	2	4	no	—
205	C₂²⋊C₄×C₃₁	—	2	4	no	—
206	C₄²×C₃₁	—	2	4	no	—
207	C₄²×C₃₁	—	2	4	no	—
208	C₄²×C₃₁	—	2	4	no	—
209	C₄²×C₃₁	C₂×D₄	1	2	yes	—
210	C₄²×C₃₁	C₂×D₄	1	2	yes	—
211	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₃₁	C₂×D₄	1	2	yes	—
213	C₄²×C₃₁	C₂×D₄	1	2	yes	—
214	C₄²×C₃₁	C₂×D₄	1	2	yes	—
215	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
217	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
218	C₄²×C₃₁	C₂⁴	1	2	yes	—
219	#60972	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂³×C₃₁	—	2	4	no	—
221	C₂³×C₃₁	—	4	1	no	—
222	C₂³×C₃₁	—	4	2	no	—
223	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
224	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	Q₈×C₃₁	—	2	8	no	—
226	Q₈×C₃₁	—	4	4	no	—
227	Q₈×C₃₁	—	4	4	no	—
228	D₄×C₃₁	—	2	8	no	—
229	D₄×C₃₁	—	4	8	no	—
230	D₄×C₃₁	—	4	8	no	—
231	D₄×C₃₁	—	4	8	no	—
232	D₄×C₃₁	—	4	4	no	—
233	D₄×C₃₁	—	4	4	no	—
234	C₂×C₄×C₃₁	—	2	4	no	—
235	C₂×C₄×C₃₁	—	2	4	no	—
236	C₂×C₄×C₃₁	—	2	4	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	1	no	—
243	C₂×C₄×C₃₁	—	4	2	no	—
244	C₂×C₄×C₃₁	—	4	2	no	—
245	C₂×C₄×C₃₁	—	4	2	no	—
246	C₂×C₄×C₃₁	—	4	2	no	—
247	C₂×C₄×C₃₁	—	4	8	no	—
248	C₂×C₄×C₃₁	—	4	8	no	—
249	C₂×C₄×C₃₁	—	4	8	no	—
250	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
251	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
253	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
257	C₈×C₃₁	—	2	2	no	—
258	C₈×C₃₁	—	2	4	no	—
259	C₈×C₃₁	—	2	2	no	—
260	#3444	C₂×C₃₁	1	2	yes	complemented
261	#3256	C₂×C₃₁	1	4	yes	complemented
262	#3231	C₂×C₃₁	1	2	yes	complemented
263	#3228	C₂×C₃₁	1	2	yes	complemented
264	#3210	C₂×C₃₁	1	2	yes	complemented
265	#3207	C₂×C₃₁	1	2	yes	complemented
266	#3197	C₂×C₃₁	1	4	yes	complemented
267	#3185	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#3184	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#3182	C₂×C₃₁	1	2	yes	complemented
270	#3111	C₂×C₃₁	1	8	yes	complemented
271	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
272	C₂²×C₃₁	—	2	2	no	—
273	C₂²×C₃₁	—	2	1	no	—
274	C₂²×C₃₁	—	2	2	no	—
275	C₂²×C₃₁	—	2	2	no	—
276	C₂²×C₃₁	—	2	2	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	C₂³×D₄	1	1	yes	characteristic
282	C₄×C₃₁	—	2	2	no	—
283	C₄×C₃₁	—	2	1	no	—
284	C₄×C₃₁	—	2	2	no	—
285	C₄×C₃₁	—	2	2	no	—
286	C₄×C₃₁	—	2	2	no	—
287	C₄×C₃₁	—	4	4	no	—
288	C₄×C₃₁	—	4	4	no	—
289	C₄×C₃₁	—	4	4	no	—
290	C₄×C₃₁	#575	1	4	yes	—
291	C₄×C₃₁	#585	1	2	yes	—
292	#533	C₂²×C₃₁	1	2	yes	—
293	#533	C₂²×C₃₁	1	2	yes	—
294	#533	C₂²×C₃₁	1	2	yes	—
295	#532	C₂²×C₃₁	1	2	yes	—
296	#532	C₂²×C₃₁	1	2	yes	—
297	#532	C₂²×C₃₁	1	2	yes	—
298	#529	—	2	2	no	—
299	#529	—	2	2	no	—
300	#529	—	2	2	no	—
301	#529	—	2	2	no	—
302	#529	C₂²×C₃₁	1	4	yes	—
303	#529	C₂²×C₃₁	1	4	yes	—
304	#525	—	2	2	no	—
305	#525	—	2	2	no	—
306	#525	C₂²×C₃₁	1	4	yes	—
307	C₄×#41	—	2	2	no	—
308	C₄×#41	—	2	2	no	—
309	C₄×#41	C₂²×C₃₁	1	4	yes	—
310	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
311	#500	C₂²×C₃₁	1	2	yes	complemented
312	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
313	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
314	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
315	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
316	#492	C₂²×C₃₁	1	2	yes	complemented
317	#487	C₂²×C₃₁	1	2	yes	complemented
318	#487	C₂²×C₃₁	1	2	yes	complemented
319	#486	C₂²×C₃₁	1	2	yes	complemented
320	#486	C₂²×C₃₁	1	2	yes	complemented
321	#485	C₂²×C₃₁	1	8	yes	—
322	#482	C₂²×C₃₁	1	8	yes	—
323	#479	C₂²×C₃₁	1	8	yes	complemented
324	#478	C₂²×C₃₁	1	8	yes	complemented
325	#470	C₂²×C₃₁	1	4	yes	—
326	#469	C₂²×C₃₁	1	4	yes	—
327	#468	C₂²×C₃₁	1	8	yes	—
328	#464	C₂²×C₃₁	1	8	yes	complemented
329	#463	C₂²×C₃₁	1	8	yes	complemented
330	#452	C₂²×C₃₁	1	8	yes	—
331	#449	C₂²×C₃₁	1	8	yes	—
332	#442	C₂²×C₃₁	1	4	yes	complemented
333	#441	C₂²×C₃₁	1	4	yes	complemented
334	#440	C₂²×C₃₁	1	8	yes	complemented
335	#432	C₂²×C₃₁	1	4	yes	—
336	#431	C₂²×C₃₁	1	2	yes	—
337	#431	C₂²×C₃₁	1	2	yes	—
338	#423	C₂²×C₃₁	1	2	yes	—
339	#407	C₂²×C₃₁	1	2	yes	—
340	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
341	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
342	C₂×C₃₁	—	4	2	no	—
343	C₂×C₃₁	—	4	1	no	—
344	C₂×C₃₁	—	8	4	no	—
345	C₂×C₃₁	#3449	1	1	yes	characteristic
346	C₂×C₃₁	C₂×#575	1	2	yes	—
347	C₂×#41	—	2	2	no	—
348	C₂×#41	—	2	2	no	—
349	C₂×#41	C₂³×C₃₁	1	4	yes	—
350	C₂³×C₄	D₄×C₃₁	1	2	yes	—
351	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
352	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
353	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
354	C₂×D₈	C₂³×C₃₁	1	4	yes	—
355	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
356	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
357	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
358	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
359	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
360	#127	C₂³×C₃₁	1	2	yes	—
361	#126	—	2	4	no	—
362	#126	—	2	4	no	—
363	#125	—	2	2	no	—
364	#125	—	2	2	no	—
365	#125	C₂³×C₃₁	1	4	yes	—
366	#124	—	2	2	no	—
367	#124	—	2	2	no	—
368	#124	C₂³×C₃₁	1	4	yes	—
369	#123	—	2	4	no	—
370	#123	—	2	4	no	—
371	#123	—	2	4	no	—
372	C₂²⋊Q₈	—	2	2	no	—
373	C₂²⋊Q₈	—	2	2	no	—
374	C₂²⋊Q₈	—	2	2	no	—
375	C₂²⋊Q₈	—	2	2	no	—
376	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
377	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
378	#121	—	2	2	no	—
379	#121	—	2	2	no	—
380	#121	—	2	2	no	—
381	#121	—	2	2	no	—
382	#121	C₂³×C₃₁	1	4	yes	—
383	#121	C₂³×C₃₁	1	4	yes	—
384	#120	—	2	4	no	—
385	C₄×Q₈	—	2	2	no	—
386	C₄×Q₈	—	2	2	no	—
387	C₄×Q₈	—	2	2	no	—
388	C₄×Q₈	—	2	2	no	—
389	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
390	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
391	C₄×D₄	—	2	4	no	—
392	C₄×D₄	—	2	4	no	—
393	C₄×D₄	—	2	4	no	—
394	C₄×D₄	—	2	4	no	—
395	C₄×D₄	—	2	4	no	—
396	C₄×D₄	—	2	4	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	—	2	2	no	—
401	C₄×D₄	C₂³×C₃₁	1	4	yes	—
402	C₄×D₄	C₂³×C₃₁	1	4	yes	—
403	#117	—	2	2	no	—
404	#117	—	2	2	no	—
405	#117	—	2	4	no	—
406	#117	—	2	4	no	—
407	#117	—	2	4	no	—
408	#117	C₂³×C₃₁	1	4	yes	—
409	C₂×C₄²	—	2	4	no	—
410	C₂×C₄²	D₄×C₃₁	1	2	yes	—
411	C₂×C₄²	D₄×C₃₁	1	2	yes	—
412	C₂×C₄²	D₄×C₃₁	1	2	yes	—
413	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
414	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
415	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
416	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
417	#107	C₂³×C₃₁	1	8	yes	—
418	#106	C₂³×C₃₁	1	4	yes	—
419	#106	C₂³×C₃₁	1	4	yes	—
420	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
421	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
422	#103	C₂³×C₃₁	1	8	yes	—
423	#103	C₂³×C₃₁	1	8	yes	—
424	#102	C₂³×C₃₁	1	8	yes	—
425	#102	C₂³×C₃₁	1	8	yes	—
426	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
427	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
428	#97	C₂³×C₃₁	1	2	yes	—
429	#97	C₂³×C₃₁	1	2	yes	—
430	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
431	C₄×C₈	C₂³×C₃₁	1	2	yes	—
432	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
433	C₃₁	#60972	1	1	yes	characteristic, direct product factor, Sylow
434	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
435	#41	—	4	8	no	—
436	#41	—	4	8	no	—
437	#41	—	4	8	no	—
438	C₂×Q₈	—	2	2	no	—
439	C₂×Q₈	—	2	2	no	—
440	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	2	4	no	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	4	4	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
453	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
454	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
455	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
456	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
457	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
458	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
459	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
460	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
461	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
462	Q₁₆	—	4	4	no	—
463	SD₁₆	—	4	8	no	—
464	D₈	—	4	4	no	—
465	M₁₆	—	4	2	no	—
466	M₁₆	—	4	2	no	—
467	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
468	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
469	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	2	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	C₂⁴×C₃₁	1	4	yes	—
481	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	4	no	—
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	C₄²	—	2	4	no	—
492	C₄²	—	2	4	no	—
493	C₄²	—	2	4	no	—
494	C₄²	C₂×D₄×C₃₁	1	2	yes	—
495	C₄²	C₂×D₄×C₃₁	1	2	yes	—
496	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
497	C₄²	C₂×D₄×C₃₁	1	2	yes	—
498	C₄²	C₂×D₄×C₃₁	1	2	yes	—
499	C₄²	C₂×D₄×C₃₁	1	2	yes	—
500	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
501	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
502	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
503	C₄²	C₂⁴×C₃₁	1	2	yes	—
504	C₂³	—	2	4	no	—
505	C₂³	—	4	1	no	—
506	C₂³	—	4	2	no	—
507	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
508	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
509	Q₈	—	2	8	no	—
510	Q₈	—	4	4	no	—
511	Q₈	—	4	4	no	—
512	D₄	—	2	8	no	—
513	D₄	—	4	8	no	—
514	D₄	—	4	8	no	—
515	D₄	—	4	8	no	—
516	D₄	—	4	4	no	—
517	D₄	—	4	4	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	2	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	8	no	—
532	C₂×C₄	—	4	8	no	—
533	C₂×C₄	—	4	8	no	—
534	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
535	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
536	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
537	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
538	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
539	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
540	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
541	C₈	—	2	2	no	—
542	C₈	—	2	4	no	—
543	C₈	—	2	2	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	1	no	—
546	C₂²	—	2	2	no	—
547	C₂²	—	2	2	no	—
548	C₂²	—	2	2	no	—
549	C₂²	—	4	4	no	—
550	C₂²	—	4	4	no	—
551	C₂²	—	4	4	no	—
552	C₂²	—	4	4	no	—
553	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
554	C₄	—	2	2	no	—
555	C₄	—	2	1	no	—
556	C₄	—	2	2	no	—
557	C₄	—	2	2	no	—
558	C₄	—	2	2	no	—
559	C₄	—	4	4	no	—
560	C₄	—	4	4	no	—
561	C₄	—	4	4	no	—
562	C₄	C₃₁×#575	1	4	yes	—
563	C₄	C₃₁×#585	1	2	yes	—
564	C₂	—	4	2	no	—
565	C₂	—	4	1	no	—
566	C₂	—	8	4	no	—
567	C₂	C₃₁×#3449	1	1	yes	characteristic
568	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×#3256	C₂	semidirect
3. C₃₁×#3231	C₂	semidirect
4. C₃₁×#3228	C₂	semidirect
5. C₃₁×#3210	C₂	semidirect
6. C₃₁×#3207	C₂	semidirect
7. C₃₁×#3197	C₂	semidirect
8. C₃₁×#3185	C₂	semidirect
9. C₃₁×#3184	C₂	semidirect
10. C₃₁×#3182	C₂	semidirect
11. C₃₁×#3111	C₂	semidirect
12. C₃₁×#2920	C₂	semidirect
32. C₃₁×#500	C₂²	semidirect
33. C₃₁×#498	C₂²	semidirect
34. C₃₁×#498	C₂²	semidirect
35. C₃₁×#494	C₂²	semidirect
36. C₃₁×#493	C₂²	semidirect
37. C₃₁×#492	C₂²	semidirect
38. C₃₁×#487	C₂²	semidirect
39. C₃₁×#487	C₂²	semidirect
40. C₃₁×#486	C₂²	semidirect
41. C₃₁×#486	C₂²	semidirect
44. C₃₁×#479	C₂²	semidirect
45. C₃₁×#478	C₂²	semidirect
49. C₃₁×#464	C₂²	semidirect
50. C₃₁×#463	C₂²	semidirect
53. C₃₁×#442	C₂²	semidirect
54. C₃₁×#441	C₂²	semidirect
55. C₃₁×#440	C₂²	semidirect
219. #60972	C₃₁	direct (N × H)
260. #3444	C₂×C₃₁	semidirect
261. #3256	C₂×C₃₁	semidirect
262. #3231	C₂×C₃₁	semidirect
263. #3228	C₂×C₃₁	semidirect
264. #3210	C₂×C₃₁	semidirect
265. #3207	C₂×C₃₁	semidirect
266. #3197	C₂×C₃₁	semidirect
267. #3185	C₂×C₃₁	semidirect
268. #3184	C₂×C₃₁	semidirect
269. #3182	C₂×C₃₁	semidirect
270. #3111	C₂×C₃₁	semidirect
271. #2920	C₂×C₃₁	semidirect
311. #500	C₂²×C₃₁	semidirect
312. #498	C₂²×C₃₁	semidirect
313. #498	C₂²×C₃₁	semidirect
314. #494	C₂²×C₃₁	semidirect
315. #493	C₂²×C₃₁	semidirect
316. #492	C₂²×C₃₁	semidirect
317. #487	C₂²×C₃₁	semidirect
318. #487	C₂²×C₃₁	semidirect
319. #486	C₂²×C₃₁	semidirect
320. #486	C₂²×C₃₁	semidirect
323. #479	C₂²×C₃₁	semidirect
324. #478	C₂²×C₃₁	semidirect
328. #464	C₂²×C₃₁	semidirect
329. #463	C₂²×C₃₁	semidirect
332. #442	C₂²×C₃₁	semidirect
333. #441	C₂²×C₃₁	semidirect
334. #440	C₂²×C₃₁	semidirect
433. C₃₁	#60972	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×#3256	2
C₃₁×#3231	2
C₃₁×#3228	2
C₃₁×#3210	2
C₃₁×#3207	2
C₃₁×#3197	2
C₃₁×#3185	2
C₃₁×#3184	2
C₃₁×#3182	2
C₃₁×#3111	2
C₃₁×#2920	2
#60972	31

Subgroups — C31×#60972 #21517731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60972 #21517731