Subgroups — C₃₁×#61284 #21518043

Order 7936.

1882 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3506	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3355	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3354	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3341	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3339	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3334	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3096	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3077	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3058	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2913	C₂	1	2	yes	maximal, complemented
14	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#131	C₂²	1	2	yes	—
16	C₃₁×#559	—	2	4	no	—
17	C₃₁×#556	—	2	2	no	—
18	C₃₁×#555	C₂²	1	4	yes	complemented
19	C₃₁×#553	C₂²	1	8	yes	complemented
20	C₃₁×#550	—	2	2	no	—
21	C₃₁×#547	C₂²	1	4	yes	complemented
22	C₃₁×#538	—	2	4	no	—
23	C₃₁×#535	C₂²	1	4	yes	—
24	C₃₁×#533	C₂²	1	2	yes	—
25	C₃₁×#532	C₂²	1	1	yes	characteristic
26	C₃₁×#525	—	2	4	no	—
27	C₃₁×#520	C₂²	1	4	yes	—
28	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#499	C₂²	1	2	yes	complemented
32	C₃₁×#499	C₂²	1	2	yes	complemented
33	C₃₁×#498	C₂²	1	2	yes	complemented
34	C₃₁×#482	C₂²	1	8	yes	complemented
35	C₃₁×#482	C₂²	1	8	yes	complemented
36	C₃₁×#481	C₂²	1	8	yes	complemented
37	C₃₁×#480	C₂²	1	8	yes	complemented
38	C₃₁×#478	C₂²	1	8	yes	complemented
39	C₃₁×#476	C₂²	1	4	yes	complemented
40	C₃₁×#474	C₂²	1	4	yes	complemented
41	C₃₁×#469	C₂²	1	8	yes	—
42	C₃₁×#466	C₂²	1	4	yes	—
43	C₃₁×#466	C₂²	1	4	yes	—
44	C₄×D₈×C₃₁	C₂²	1	2	yes	—
45	C₄×D₈×C₃₁	C₂²	1	2	yes	—
46	C₄×D₈×C₃₁	C₂²	1	2	yes	—
47	C₄×D₈×C₃₁	C₂²	1	2	yes	—
48	C₃₁×#436	C₂²	1	2	yes	complemented
49	C₃₁×#436	C₂²	1	2	yes	complemented
50	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
51	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
52	C₃₁×#428	C₂²	1	4	yes	—
53	C₃₁×#428	C₂²	1	8	yes	complemented
54	C₃₁×#427	C₂²	1	2	yes	—
55	C₃₁×#427	C₂²	1	2	yes	—
56	C₃₁×#427	C₂²	1	4	yes	complemented
57	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
58	C₂×C₃₁×#107	C₂²	1	4	yes	complemented
59	C₃₁×#423	C₂²	1	1	yes	characteristic
60	C₃₁×#418	C₂²	1	4	yes	—
61	C₂×C₃₁×#102	C₂²	1	2	yes	—
62	C₃₁×#408	C₂²	1	2	yes	—
63	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
64	C₂×#41×C₃₁	—	2	4	no	—
65	C₂×#41×C₃₁	C₂³	1	2	yes	—
66	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
67	C₂³×C₄×C₃₁	—	2	2	no	—
68	C₂×D₈×C₃₁	—	2	4	no	—
69	C₂×D₈×C₃₁	C₂³	1	2	yes	—
70	C₂×D₈×C₃₁	C₂³	1	2	yes	—
71	C₃₁×#131	—	2	8	no	—
72	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
73	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
74	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
76	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
78	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
79	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
80	C₃₁×#127	—	2	4	no	—
81	C₃₁×#127	D₄	1	1	yes	characteristic
82	C₃₁×#126	—	2	8	no	—
83	C₃₁×#125	C₂³	1	4	yes	—
84	C₃₁×#124	—	2	4	no	—
85	C₃₁×#124	—	2	8	no	—
86	C₃₁×#124	D₄	1	2	yes	—
87	C₃₁×#123	—	2	8	no	—
88	C₃₁×#123	—	2	8	no	—
89	C₃₁×C₂²⋊Q₈	—	2	2	no	—
90	C₃₁×C₂²⋊Q₈	—	2	2	no	—
91	C₃₁×C₂²⋊Q₈	—	2	4	no	—
92	C₃₁×#121	—	2	4	no	—
93	C₃₁×#121	—	2	8	no	—
94	C₃₁×#121	D₄	1	2	yes	—
95	C₃₁×#121	D₄	1	2	yes	—
96	C₃₁×#121	C₂³	1	8	yes	—
97	C₃₁×#121	C₂³	1	8	yes	—
98	C₃₁×#120	—	2	8	no	—
99	C₄×Q₈×C₃₁	—	2	4	no	—
100	C₄×Q₈×C₃₁	—	2	2	no	—
101	C₄×Q₈×C₃₁	—	2	2	no	—
102	C₄×D₄×C₃₁	—	2	8	no	—
103	C₄×D₄×C₃₁	—	2	4	no	—
104	C₄×D₄×C₃₁	—	2	8	no	—
105	C₄×D₄×C₃₁	—	2	2	no	—
106	C₄×D₄×C₃₁	—	2	2	no	—
107	C₄×D₄×C₃₁	C₂³	1	4	yes	—
108	C₄×D₄×C₃₁	C₂³	1	4	yes	—
109	C₄×D₄×C₃₁	C₂³	1	2	yes	—
110	C₄×D₄×C₃₁	C₂³	1	2	yes	—
111	C₃₁×#117	—	2	4	no	—
112	C₃₁×#117	C₂³	1	4	yes	—
113	C₃₁×#117	C₂³	1	8	yes	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
117	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
118	C₃₁×#107	C₂³	1	2	yes	—
119	C₃₁×#107	C₂³	1	2	yes	—
120	C₃₁×#107	C₂³	1	2	yes	—
121	C₃₁×#107	C₂³	1	8	yes	—
122	C₃₁×#107	C₂³	1	2	yes	—
123	C₃₁×#107	C₂³	1	2	yes	—
124	C₃₁×#107	C₂³	1	2	yes	—
125	C₃₁×#106	C₂³	1	4	yes	—
126	C₃₁×#106	C₂³	1	4	yes	—
127	C₃₁×#106	C₂³	1	8	yes	—
128	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
129	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
130	C₃₁×#102	C₂³	1	4	yes	—
131	C₃₁×#102	C₂³	1	4	yes	—
132	C₃₁×#102	C₂³	1	8	yes	—
133	C₃₁×#102	C₂³	1	8	yes	—
134	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
135	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
136	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
137	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
140	C₂⁴×C₃₁	—	2	2	no	—
141	#41×C₃₁	—	2	4	no	—
142	#41×C₃₁	—	2	4	no	—
143	#41×C₃₁	—	8	8	no	—
144	C₂×Q₈×C₃₁	—	2	2	no	—
145	C₂×Q₈×C₃₁	—	2	2	no	—
146	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
147	C₂×D₄×C₃₁	—	2	8	no	—
148	C₂×D₄×C₃₁	—	2	8	no	—
149	C₂×D₄×C₃₁	—	2	2	no	—
150	C₂×D₄×C₃₁	—	2	4	no	—
151	C₂×D₄×C₃₁	—	2	4	no	—
152	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
153	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
154	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
155	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
156	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₃₁	—	2	2	no	—
158	C₂²×C₄×C₃₁	—	2	8	no	—
159	C₂²×C₄×C₃₁	—	2	4	no	—
160	C₂²×C₄×C₃₁	—	2	4	no	—
161	C₂²×C₄×C₃₁	—	8	4	no	—
162	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
166	D₈×C₃₁	—	2	4	no	—
167	D₈×C₃₁	—	2	4	no	—
168	M₁₆×C₃₁	—	2	4	no	—
169	M₁₆×C₃₁	—	2	8	no	—
170	C₂×C₈×C₃₁	—	2	2	no	—
171	C₂×C₈×C₃₁	—	2	8	no	—
172	C₂×C₈×C₃₁	#41	1	4	yes	complemented
173	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
174	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
175	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
176	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
177	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
178	C₄⋊C₄×C₃₁	—	2	2	no	—
179	C₄⋊C₄×C₃₁	—	2	8	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	2	no	—
184	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
185	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
186	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
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	8	no	—
191	C₂²⋊C₄×C₃₁	—	2	4	no	—
192	C₂²⋊C₄×C₃₁	—	2	4	no	—
193	C₂²⋊C₄×C₃₁	—	2	8	no	—
194	C₂²⋊C₄×C₃₁	—	2	8	no	—
195	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
196	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
197	C₄²×C₃₁	—	2	8	no	—
198	C₄²×C₃₁	—	2	4	no	—
199	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
200	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
201	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
202	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
203	#61284	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂³×C₃₁	—	2	4	no	—
205	C₂³×C₃₁	—	4	4	no	—
206	C₂³×C₃₁	—	8	2	no	—
207	C₂³×C₃₁	—	8	2	no	—
208	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
209	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
210	Q₈×C₃₁	—	2	4	no	—
211	Q₈×C₃₁	—	4	4	no	—
212	Q₈×C₃₁	—	8	2	no	—
213	D₄×C₃₁	—	2	8	no	—
214	D₄×C₃₁	—	2	4	no	—
215	D₄×C₃₁	—	2	4	no	—
216	D₄×C₃₁	—	2	4	no	—
217	D₄×C₃₁	—	2	4	no	—
218	D₄×C₃₁	—	4	2	no	—
219	D₄×C₃₁	—	4	4	no	—
220	D₄×C₃₁	—	8	8	no	—
221	D₄×C₃₁	—	8	8	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	4	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	4	no	—
228	C₂×C₄×C₃₁	—	4	1	no	—
229	C₂×C₄×C₃₁	—	4	8	no	—
230	C₂×C₄×C₃₁	—	8	4	no	—
231	C₂×C₄×C₃₁	—	8	2	no	—
232	C₂×C₄×C₃₁	—	8	8	no	—
233	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
234	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
240	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
241	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
242	C₈×C₃₁	—	2	4	no	—
243	C₈×C₃₁	—	2	1	no	—
244	C₈×C₃₁	—	2	1	no	—
245	C₈×C₃₁	C₂×#41	1	2	yes	—
246	C₈×C₃₁	C₂×#41	1	2	yes	—
247	#3506	C₂×C₃₁	1	2	yes	complemented
248	#3355	C₂×C₃₁	1	4	yes	complemented
249	#3354	C₂×C₃₁	1	4	yes	complemented
250	#3341	C₂×C₃₁	1	2	yes	complemented
251	#3339	C₂×C₃₁	1	4	yes	complemented
252	#3334	C₂×C₃₁	1	2	yes	complemented
253	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#3096	C₂×C₃₁	1	4	yes	complemented
255	#3077	C₂×C₃₁	1	2	yes	complemented
256	#3058	C₂×C₃₁	1	2	yes	complemented
257	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#2931	C₂×C₃₁	1	1	yes	characteristic, complemented
259	#2913	C₂×C₃₁	1	2	yes	complemented
260	C₂²×C₃₁	—	2	1	no	—
261	C₂²×C₃₁	—	2	1	no	—
262	C₂²×C₃₁	—	2	4	no	—
263	C₂²×C₃₁	—	2	4	no	—
264	C₂²×C₃₁	—	2	4	no	—
265	C₂²×C₃₁	—	4	4	no	—
266	C₂²×C₃₁	—	4	4	no	—
267	C₂²×C₃₁	—	4	4	no	—
268	C₂²×C₃₁	—	8	8	no	—
269	C₂²×C₃₁	#547	1	2	yes	—
270	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
271	C₄×C₃₁	—	2	1	no	—
272	C₄×C₃₁	—	2	1	no	—
273	C₄×C₃₁	—	2	4	no	—
274	C₄×C₃₁	—	2	1	no	—
275	C₄×C₃₁	—	4	4	no	—
276	C₄×C₃₁	—	4	4	no	—
277	C₄×C₃₁	—	4	4	no	—
278	C₄×C₃₁	#547	1	2	yes	—
279	C₄×C₃₁	#576	1	2	yes	—
280	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
281	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
282	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×#131	C₂²×C₃₁	1	2	yes	—
284	#559	—	2	4	no	—
285	#556	—	2	2	no	—
286	#555	C₂²×C₃₁	1	4	yes	complemented
287	#553	C₂²×C₃₁	1	8	yes	complemented
288	#550	—	2	2	no	—
289	#547	C₂²×C₃₁	1	4	yes	complemented
290	#538	—	2	4	no	—
291	#535	C₂²×C₃₁	1	4	yes	—
292	#533	C₂²×C₃₁	1	2	yes	—
293	#532	C₂²×C₃₁	1	1	yes	characteristic
294	#525	—	2	4	no	—
295	#520	C₂²×C₃₁	1	4	yes	—
296	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
297	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
298	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
299	#499	C₂²×C₃₁	1	2	yes	complemented
300	#499	C₂²×C₃₁	1	2	yes	complemented
301	#498	C₂²×C₃₁	1	2	yes	complemented
302	#482	C₂²×C₃₁	1	8	yes	complemented
303	#482	C₂²×C₃₁	1	8	yes	complemented
304	#481	C₂²×C₃₁	1	8	yes	complemented
305	#480	C₂²×C₃₁	1	8	yes	complemented
306	#478	C₂²×C₃₁	1	8	yes	complemented
307	#476	C₂²×C₃₁	1	4	yes	complemented
308	#474	C₂²×C₃₁	1	4	yes	complemented
309	#469	C₂²×C₃₁	1	8	yes	—
310	#466	C₂²×C₃₁	1	4	yes	—
311	#466	C₂²×C₃₁	1	4	yes	—
312	C₄×D₈	C₂²×C₃₁	1	2	yes	—
313	C₄×D₈	C₂²×C₃₁	1	2	yes	—
314	C₄×D₈	C₂²×C₃₁	1	2	yes	—
315	C₄×D₈	C₂²×C₃₁	1	2	yes	—
316	#436	C₂²×C₃₁	1	2	yes	complemented
317	#436	C₂²×C₃₁	1	2	yes	complemented
318	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
319	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
320	#428	C₂²×C₃₁	1	4	yes	—
321	#428	C₂²×C₃₁	1	8	yes	complemented
322	#427	C₂²×C₃₁	1	2	yes	—
323	#427	C₂²×C₃₁	1	2	yes	—
324	#427	C₂²×C₃₁	1	4	yes	complemented
325	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
326	C₂×#107	C₂²×C₃₁	1	4	yes	complemented
327	#423	C₂²×C₃₁	1	1	yes	characteristic
328	#418	C₂²×C₃₁	1	4	yes	—
329	C₂×#102	C₂²×C₃₁	1	2	yes	—
330	#408	C₂²×C₃₁	1	2	yes	—
331	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
332	C₂×C₃₁	—	2	2	no	—
333	C₂×C₃₁	—	4	4	no	—
334	C₂×C₃₁	—	8	4	no	—
335	C₂×C₃₁	C₂×#547	1	1	yes	characteristic
336	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
337	C₂×C₃₁	#3584	1	1	yes	characteristic
338	C₂×#41	—	2	4	no	—
339	C₂×#41	C₂³×C₃₁	1	2	yes	—
340	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
341	C₂³×C₄	—	2	2	no	—
342	C₂×D₈	—	2	4	no	—
343	C₂×D₈	C₂³×C₃₁	1	2	yes	—
344	C₂×D₈	C₂³×C₃₁	1	2	yes	—
345	#131	—	2	8	no	—
346	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
347	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
348	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
349	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
350	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
351	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
352	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
353	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
354	#127	—	2	4	no	—
355	#127	D₄×C₃₁	1	1	yes	characteristic
356	#126	—	2	8	no	—
357	#125	C₂³×C₃₁	1	4	yes	—
358	#124	—	2	4	no	—
359	#124	—	2	8	no	—
360	#124	D₄×C₃₁	1	2	yes	—
361	#123	—	2	8	no	—
362	#123	—	2	8	no	—
363	C₂²⋊Q₈	—	2	2	no	—
364	C₂²⋊Q₈	—	2	2	no	—
365	C₂²⋊Q₈	—	2	4	no	—
366	#121	—	2	4	no	—
367	#121	—	2	8	no	—
368	#121	D₄×C₃₁	1	2	yes	—
369	#121	D₄×C₃₁	1	2	yes	—
370	#121	C₂³×C₃₁	1	8	yes	—
371	#121	C₂³×C₃₁	1	8	yes	—
372	#120	—	2	8	no	—
373	C₄×Q₈	—	2	4	no	—
374	C₄×Q₈	—	2	2	no	—
375	C₄×Q₈	—	2	2	no	—
376	C₄×D₄	—	2	8	no	—
377	C₄×D₄	—	2	4	no	—
378	C₄×D₄	—	2	8	no	—
379	C₄×D₄	—	2	2	no	—
380	C₄×D₄	—	2	2	no	—
381	C₄×D₄	C₂³×C₃₁	1	4	yes	—
382	C₄×D₄	C₂³×C₃₁	1	4	yes	—
383	C₄×D₄	C₂³×C₃₁	1	2	yes	—
384	C₄×D₄	C₂³×C₃₁	1	2	yes	—
385	#117	—	2	4	no	—
386	#117	C₂³×C₃₁	1	4	yes	—
387	#117	C₂³×C₃₁	1	8	yes	—
388	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
389	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
390	C₂×C₂²⋊C₄	—	2	2	no	—
391	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
392	#107	C₂³×C₃₁	1	2	yes	—
393	#107	C₂³×C₃₁	1	2	yes	—
394	#107	C₂³×C₃₁	1	2	yes	—
395	#107	C₂³×C₃₁	1	8	yes	—
396	#107	C₂³×C₃₁	1	2	yes	—
397	#107	C₂³×C₃₁	1	2	yes	—
398	#107	C₂³×C₃₁	1	2	yes	—
399	#106	C₂³×C₃₁	1	4	yes	—
400	#106	C₂³×C₃₁	1	4	yes	—
401	#106	C₂³×C₃₁	1	8	yes	—
402	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
403	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
404	#102	C₂³×C₃₁	1	4	yes	—
405	#102	C₂³×C₃₁	1	4	yes	—
406	#102	C₂³×C₃₁	1	8	yes	—
407	#102	C₂³×C₃₁	1	8	yes	—
408	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
409	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
410	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
411	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
412	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
413	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
414	C₃₁	#61284	1	1	yes	characteristic, direct product factor, Sylow
415	C₂⁴	—	2	2	no	—
416	#41	—	2	4	no	—
417	#41	—	2	4	no	—
418	#41	—	8	8	no	—
419	C₂×Q₈	—	2	2	no	—
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
422	C₂×D₄	—	2	8	no	—
423	C₂×D₄	—	2	8	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	4	no	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
428	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
429	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
430	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
431	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	2	8	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	8	4	no	—
437	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
438	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
439	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
440	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
441	D₈	—	2	4	no	—
442	D₈	—	2	4	no	—
443	M₁₆	—	2	4	no	—
444	M₁₆	—	2	8	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	—	2	8	no	—
447	C₂×C₈	#41×C₃₁	1	4	yes	complemented
448	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
449	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
450	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
451	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
452	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	—	2	8	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	2	no	—
459	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
460	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
461	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
462	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
463	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	8	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	4	no	—
468	C₂²⋊C₄	—	2	8	no	—
469	C₂²⋊C₄	—	2	8	no	—
470	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
471	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
472	C₄²	—	2	8	no	—
473	C₄²	—	2	4	no	—
474	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
477	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
478	C₂³	—	2	4	no	—
479	C₂³	—	4	4	no	—
480	C₂³	—	8	2	no	—
481	C₂³	—	8	2	no	—
482	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
483	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
484	Q₈	—	2	4	no	—
485	Q₈	—	4	4	no	—
486	Q₈	—	8	2	no	—
487	D₄	—	2	8	no	—
488	D₄	—	2	4	no	—
489	D₄	—	2	4	no	—
490	D₄	—	2	4	no	—
491	D₄	—	2	4	no	—
492	D₄	—	4	2	no	—
493	D₄	—	4	4	no	—
494	D₄	—	8	8	no	—
495	D₄	—	8	8	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	8	no	—
504	C₂×C₄	—	8	4	no	—
505	C₂×C₄	—	8	2	no	—
506	C₂×C₄	—	8	8	no	—
507	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
508	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
509	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
510	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
511	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
512	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
513	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
514	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
515	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
516	C₈	—	2	4	no	—
517	C₈	—	2	1	no	—
518	C₈	—	2	1	no	—
519	C₈	C₂×#41×C₃₁	1	2	yes	—
520	C₈	C₂×#41×C₃₁	1	2	yes	—
521	C₂²	—	2	1	no	—
522	C₂²	—	2	1	no	—
523	C₂²	—	2	4	no	—
524	C₂²	—	2	4	no	—
525	C₂²	—	2	4	no	—
526	C₂²	—	4	4	no	—
527	C₂²	—	4	4	no	—
528	C₂²	—	4	4	no	—
529	C₂²	—	8	8	no	—
530	C₂²	C₃₁×#547	1	2	yes	—
531	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
532	C₄	—	2	4	no	—
533	C₄	—	2	1	no	—
534	C₄	—	2	1	no	—
535	C₄	—	2	1	no	—
536	C₄	—	4	4	no	—
537	C₄	—	4	4	no	—
538	C₄	—	4	4	no	—
539	C₄	C₃₁×#547	1	2	yes	—
540	C₄	C₃₁×#576	1	2	yes	—
541	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
542	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
543	C₂	—	2	2	no	—
544	C₂	—	4	4	no	—
545	C₂	—	8	4	no	—
546	C₂	C₂×C₃₁×#547	1	1	yes	characteristic
547	C₂	C₂×C₃₁×#576	1	1	yes	characteristic
548	C₂	C₃₁×#3584	1	1	yes	characteristic

	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 ◁ C₃₁×#532 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3506	C₂	semidirect
2. C₃₁×#3355	C₂	semidirect
3. C₃₁×#3354	C₂	semidirect
4. C₃₁×#3341	C₂	semidirect
5. C₃₁×#3339	C₂	semidirect
6. C₃₁×#3334	C₂	semidirect
7. C₃₁×#3160	C₂	semidirect
8. C₃₁×#3096	C₂	semidirect
9. C₃₁×#3077	C₂	semidirect
10. C₃₁×#3058	C₂	semidirect
11. C₂×C₄×D₈×C₃₁	C₂	semidirect
12. C₃₁×#2931	C₂	semidirect
13. C₃₁×#2913	C₂	semidirect
18. C₃₁×#555	C₂²	semidirect
19. C₃₁×#553	C₂²	semidirect
21. C₃₁×#547	C₂²	semidirect
29. C₃₁×#500	C₂²	semidirect
30. C₃₁×#500	C₂²	semidirect
31. C₃₁×#499	C₂²	semidirect
32. C₃₁×#499	C₂²	semidirect
33. C₃₁×#498	C₂²	semidirect
34. C₃₁×#482	C₂²	semidirect
35. C₃₁×#482	C₂²	semidirect
36. C₃₁×#481	C₂²	semidirect
37. C₃₁×#480	C₂²	semidirect
38. C₃₁×#478	C₂²	semidirect
39. C₃₁×#476	C₂²	semidirect
40. C₃₁×#474	C₂²	semidirect
48. C₃₁×#436	C₂²	semidirect
49. C₃₁×#436	C₂²	semidirect
50. C₈×D₄×C₃₁	C₂²	semidirect
51. C₈×D₄×C₃₁	C₂²	semidirect
53. C₃₁×#428	C₂²	semidirect
56. C₃₁×#427	C₂²	semidirect
58. C₂×C₃₁×#107	C₂²	semidirect
172. C₂×C₈×C₃₁	#41	semidirect
203. #61284	C₃₁	direct (N × H)
247. #3506	C₂×C₃₁	semidirect
248. #3355	C₂×C₃₁	semidirect
249. #3354	C₂×C₃₁	semidirect
250. #3341	C₂×C₃₁	semidirect
251. #3339	C₂×C₃₁	semidirect
252. #3334	C₂×C₃₁	semidirect
253. #3160	C₂×C₃₁	semidirect
254. #3096	C₂×C₃₁	semidirect
255. #3077	C₂×C₃₁	semidirect
256. #3058	C₂×C₃₁	semidirect
257. C₂×C₄×D₈	C₂×C₃₁	semidirect
258. #2931	C₂×C₃₁	semidirect
259. #2913	C₂×C₃₁	semidirect
286. #555	C₂²×C₃₁	semidirect
287. #553	C₂²×C₃₁	semidirect
289. #547	C₂²×C₃₁	semidirect
297. #500	C₂²×C₃₁	semidirect
298. #500	C₂²×C₃₁	semidirect
299. #499	C₂²×C₃₁	semidirect
300. #499	C₂²×C₃₁	semidirect
301. #498	C₂²×C₃₁	semidirect
302. #482	C₂²×C₃₁	semidirect
303. #482	C₂²×C₃₁	semidirect
304. #481	C₂²×C₃₁	semidirect
305. #480	C₂²×C₃₁	semidirect
306. #478	C₂²×C₃₁	semidirect
307. #476	C₂²×C₃₁	semidirect
308. #474	C₂²×C₃₁	semidirect
316. #436	C₂²×C₃₁	semidirect
317. #436	C₂²×C₃₁	semidirect
318. C₈×D₄	C₂²×C₃₁	semidirect
319. C₈×D₄	C₂²×C₃₁	semidirect
321. #428	C₂²×C₃₁	semidirect
324. #427	C₂²×C₃₁	semidirect
326. C₂×#107	C₂²×C₃₁	semidirect
414. C₃₁	#61284	direct (N × H)
447. C₂×C₈	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3506	2
C₃₁×#3355	2
C₃₁×#3354	2
C₃₁×#3341	2
C₃₁×#3339	2
C₃₁×#3334	2
C₃₁×#3160	2
C₃₁×#3096	2
C₃₁×#3077	2
C₃₁×#3058	2
C₂×C₄×D₈×C₃₁	2
C₃₁×#2931	2
C₃₁×#2913	2
#61284	31

Subgroups — C31×#61284 #21518043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61284 #21518043