Subgroups — C₃₁×#19762 #21476521

Order 7936.

728 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3391	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3358	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3287	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3278	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1721	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1687	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1682	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1668	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1665	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1641	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1627	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1621	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1621	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1581	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1579	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#137	—	2	1	no	—
17	C₂×C₃₁×#135	—	2	1	no	—
18	C₃₁×#550	C₂²	1	1	yes	characteristic
19	C₃₁×#550	C₂²	1	1	yes	characteristic
20	C₃₁×#550	C₂²	1	1	yes	characteristic
21	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#493	—	2	1	no	—
23	C₃₁×#493	C₂²	1	1	yes	characteristic
24	C₃₁×#493	C₂²	1	1	yes	characteristic
25	C₃₁×#493	C₂²	1	1	yes	characteristic
26	C₃₁×#492	C₂²	1	1	yes	characteristic
27	C₃₁×#486	—	2	1	no	—
28	C₃₁×#486	C₂²	1	1	yes	characteristic
29	C₃₁×#486	C₂²	1	1	yes	characteristic
30	C₃₁×#474	—	2	1	no	—
31	C₃₁×#474	C₂²	1	1	yes	characteristic
32	C₃₁×#468	—	2	1	no	—
33	C₃₁×#466	—	2	1	no	—
34	C₃₁×#462	C₂²	1	1	yes	characteristic
35	C₃₁×#460	C₂²	1	1	yes	characteristic
36	C₃₁×#460	C₂²	1	1	yes	characteristic
37	C₃₁×#460	C₂²	1	1	yes	characteristic
38	C₃₁×#459	—	2	1	no	—
39	C₃₁×#459	—	2	1	no	—
40	C₃₁×#459	—	2	1	no	—
41	C₃₁×#459	C₂²	1	1	yes	characteristic
42	C₃₁×#459	C₂²	1	1	yes	characteristic
43	C₃₁×#459	C₂²	1	1	yes	characteristic
44	C₃₁×#452	—	2	1	no	—
45	C₃₁×#449	—	2	1	no	—
46	C₃₁×#449	—	2	1	no	—
47	C₃₁×#448	—	2	1	no	—
48	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
49	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
50	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
51	C₄×D₈×C₃₁	—	2	1	no	—
52	C₄×D₈×C₃₁	—	2	1	no	—
53	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
54	C₃₁×#436	C₂²	1	1	yes	characteristic
55	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
56	C₃₁×#335	C₂²	1	1	yes	characteristic
57	C₃₁×#334	C₂²	1	1	yes	characteristic
58	C₃₁×#332	C₂²	1	1	yes	characteristic
59	C₃₁×#331	C₂²	1	1	yes	characteristic
60	C₃₁×#331	C₂²	1	1	yes	characteristic
61	C₃₁×#331	C₂²	1	1	yes	characteristic
62	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
65	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₃₁×#322	C₂²	1	1	yes	characteristic
67	C₂×#41×C₃₁	—	2	1	no	—
68	C₂×#41×C₃₁	—	2	1	no	—
69	C₂×#41×C₃₁	—	2	1	no	—
70	C₂²×Q₈×C₃₁	—	2	1	no	—
71	C₃₁×#137	—	4	4	no	—
72	C₃₁×#135	—	4	4	no	—
73	C₂×Q₁₆×C₃₁	—	2	1	no	—
74	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
75	C₂×SD₁₆×C₃₁	—	2	1	no	—
76	C₂×SD₁₆×C₃₁	—	2	1	no	—
77	C₂×SD₁₆×C₃₁	—	2	1	no	—
78	C₂×SD₁₆×C₃₁	—	2	1	no	—
79	C₂×SD₁₆×C₃₁	—	2	1	no	—
80	C₂×D₈×C₃₁	—	2	1	no	—
81	C₂×D₈×C₃₁	—	2	1	no	—
82	C₂×D₈×C₃₁	—	2	1	no	—
83	C₂×D₈×C₃₁	—	2	1	no	—
84	C₂×D₈×C₃₁	—	2	1	no	—
85	C₂×D₈×C₃₁	—	2	1	no	—
86	C₂×D₈×C₃₁	—	2	1	no	—
87	C₂×D₈×C₃₁	—	4	1	no	—
88	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
89	C₂×M₁₆×C₃₁	—	2	1	no	—
90	C₂²×C₈×C₃₁	—	2	1	no	—
91	C₃₁×C₄⋊Q₈	—	2	1	no	—
92	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
93	C₃₁×#127	—	2	1	no	—
94	C₃₁×#127	—	2	1	no	—
95	C₃₁×#127	—	2	1	no	—
96	C₃₁×#127	C₂³	1	1	yes	characteristic
97	C₃₁×#127	C₂³	1	1	yes	characteristic
98	C₃₁×#127	C₂³	1	1	yes	characteristic
99	C₃₁×C₂²⋊Q₈	—	2	1	no	—
100	C₃₁×C₂²⋊Q₈	—	4	1	no	—
101	C₃₁×#121	—	2	1	no	—
102	C₃₁×#121	—	2	1	no	—
103	C₃₁×#121	—	2	1	no	—
104	C₃₁×#121	—	4	1	no	—
105	C₃₁×#121	—	4	1	no	—
106	C₃₁×#121	—	4	1	no	—
107	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
108	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
109	C₄×D₄×C₃₁	—	2	1	no	—
110	C₄×D₄×C₃₁	—	2	1	no	—
111	C₄×D₄×C₃₁	—	2	1	no	—
112	C₄×D₄×C₃₁	—	2	1	no	—
113	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
114	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
115	C₃₁×#107	—	2	1	no	—
116	C₃₁×#107	D₄	1	1	yes	characteristic
117	C₃₁×#107	D₄	1	1	yes	characteristic
118	C₃₁×C₄⋊C₈	—	2	1	no	—
119	C₃₁×C₄⋊C₈	—	2	1	no	—
120	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
122	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
123	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
124	C₃₁×#103	—	2	1	no	—
125	C₃₁×#103	D₄	1	1	yes	characteristic
126	C₃₁×#103	D₄	1	1	yes	characteristic
127	C₃₁×#102	—	2	1	no	—
128	C₃₁×#102	—	2	1	no	—
129	C₃₁×#102	—	2	1	no	—
130	C₃₁×#102	—	2	1	no	—
131	C₃₁×#102	—	2	1	no	—
132	C₃₁×#102	—	2	1	no	—
133	C₃₁×#102	—	2	1	no	—
134	C₃₁×#102	—	2	1	no	—
135	C₃₁×#102	—	2	1	no	—
136	C₃₁×#102	—	4	1	no	—
137	C₃₁×#102	D₄	1	1	yes	characteristic
138	C₃₁×#102	D₄	1	1	yes	characteristic
139	C₃₁×C₂²⋊C₈	—	2	1	no	—
140	C₃₁×C₂²⋊C₈	—	2	1	no	—
141	C₄×C₈×C₃₁	—	2	1	no	—
142	C₄×C₈×C₃₁	—	2	1	no	—
143	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
145	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
146	#41×C₃₁	—	4	2	no	—
147	#41×C₃₁	—	4	2	no	—
148	#41×C₃₁	—	4	2	no	—
149	#41×C₃₁	—	8	1	no	—
150	#41×C₃₁	—	8	1	no	—
151	#41×C₃₁	—	8	1	no	—
152	C₂×Q₈×C₃₁	—	2	1	no	—
153	C₂×Q₈×C₃₁	—	4	1	no	—
154	C₂×Q₈×C₃₁	—	4	2	no	—
155	C₂×Q₈×C₃₁	—	8	1	no	—
156	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂×D₄×C₃₁	—	2	1	no	—
159	C₂×D₄×C₃₁	—	2	1	no	—
160	C₂×D₄×C₃₁	—	2	1	no	—
161	C₂×D₄×C₃₁	—	2	1	no	—
162	C₂×D₄×C₃₁	—	2	1	no	—
163	C₂×D₄×C₃₁	—	2	1	no	—
164	C₂×D₄×C₃₁	—	2	1	no	—
165	C₂×D₄×C₃₁	—	2	1	no	—
166	C₂×D₄×C₃₁	—	2	1	no	—
167	C₂×D₄×C₃₁	—	4	1	no	—
168	C₂×D₄×C₃₁	—	4	1	no	—
169	C₂×D₄×C₃₁	—	4	1	no	—
170	C₂×D₄×C₃₁	—	4	1	no	—
171	C₂×D₄×C₃₁	—	4	1	no	—
172	C₂×D₄×C₃₁	—	4	1	no	—
173	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₃₁	—	2	1	no	—
176	C₂²×C₄×C₃₁	—	2	1	no	—
177	C₂²×C₄×C₃₁	—	4	1	no	—
178	C₂²×C₄×C₃₁	—	4	1	no	—
179	C₂²×C₄×C₃₁	—	4	1	no	—
180	C₂²×C₄×C₃₁	—	4	1	no	—
181	Q₁₆×C₃₁	—	4	2	no	—
182	Q₁₆×C₃₁	D₈	1	4	yes	complemented
183	SD₁₆×C₃₁	—	4	2	no	—
184	SD₁₆×C₃₁	—	4	2	no	—
185	SD₁₆×C₃₁	—	4	2	no	—
186	SD₁₆×C₃₁	—	4	2	no	—
187	SD₁₆×C₃₁	—	8	1	no	—
188	D₈×C₃₁	—	4	1	no	—
189	D₈×C₃₁	—	4	2	no	—
190	D₈×C₃₁	—	4	2	no	—
191	D₈×C₃₁	—	4	2	no	—
192	D₈×C₃₁	—	8	1	no	—
193	D₈×C₃₁	—	8	1	no	—
194	D₈×C₃₁	—	8	1	no	—
195	D₈×C₃₁	—	8	1	no	—
196	D₈×C₃₁	—	8	2	no	—
197	M₁₆×C₃₁	—	4	2	no	—
198	C₂×C₈×C₃₁	—	2	1	no	—
199	C₂×C₈×C₃₁	—	2	1	no	—
200	C₂×C₈×C₃₁	—	2	1	no	—
201	C₂×C₈×C₃₁	—	2	1	no	—
202	C₂×C₈×C₃₁	—	2	1	no	—
203	C₂×C₈×C₃₁	—	4	2	no	—
204	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
205	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
206	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
207	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
208	C₄⋊C₄×C₃₁	—	2	1	no	—
209	C₄⋊C₄×C₃₁	—	2	1	no	—
210	C₄⋊C₄×C₃₁	—	2	1	no	—
211	C₄⋊C₄×C₃₁	—	2	1	no	—
212	C₄⋊C₄×C₃₁	—	4	1	no	—
213	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
214	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
215	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
216	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
217	C₂²⋊C₄×C₃₁	—	2	1	no	—
218	C₂²⋊C₄×C₃₁	—	2	1	no	—
219	C₂²⋊C₄×C₃₁	—	4	1	no	—
220	C₂²⋊C₄×C₃₁	—	4	1	no	—
221	C₂²⋊C₄×C₃₁	—	4	1	no	—
222	C₂²⋊C₄×C₃₁	—	4	1	no	—
223	C₄²×C₃₁	—	2	1	no	—
224	C₄²×C₃₁	—	2	1	no	—
225	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
226	#19762	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
227	C₂³×C₃₁	—	2	1	no	—
228	C₂³×C₃₁	—	2	1	no	—
229	C₂³×C₃₁	—	4	1	no	—
230	C₂³×C₃₁	—	4	1	no	—
231	C₂³×C₃₁	—	4	1	no	—
232	Q₈×C₃₁	—	2	1	no	—
233	Q₈×C₃₁	—	2	1	no	—
234	Q₈×C₃₁	—	4	1	no	—
235	Q₈×C₃₁	—	4	1	no	—
236	Q₈×C₃₁	—	8	2	no	—
237	Q₈×C₃₁	C₂×D₈	1	2	yes	—
238	Q₈×C₃₁	C₂×D₈	1	2	yes	—
239	D₄×C₃₁	—	2	1	no	—
240	D₄×C₃₁	—	2	1	no	—
241	D₄×C₃₁	—	2	1	no	—
242	D₄×C₃₁	—	2	1	no	—
243	D₄×C₃₁	—	4	1	no	—
244	D₄×C₃₁	—	4	1	no	—
245	D₄×C₃₁	—	4	1	no	—
246	D₄×C₃₁	—	4	1	no	—
247	D₄×C₃₁	—	4	1	no	—
248	D₄×C₃₁	—	4	1	no	—
249	D₄×C₃₁	—	4	1	no	—
250	D₄×C₃₁	—	4	1	no	—
251	D₄×C₃₁	—	4	1	no	—
252	D₄×C₃₁	—	4	1	no	—
253	D₄×C₃₁	—	4	1	no	—
254	D₄×C₃₁	—	4	1	no	—
255	D₄×C₃₁	—	8	2	no	—
256	D₄×C₃₁	—	8	2	no	—
257	D₄×C₃₁	—	8	2	no	—
258	D₄×C₃₁	—	8	1	no	—
259	D₄×C₃₁	—	8	1	no	—
260	D₄×C₃₁	—	8	1	no	—
261	D₄×C₃₁	—	16	1	no	—
262	D₄×C₃₁	—	16	1	no	—
263	D₄×C₃₁	—	16	1	no	—
264	C₂×C₄×C₃₁	—	2	1	no	—
265	C₂×C₄×C₃₁	—	2	1	no	—
266	C₂×C₄×C₃₁	—	2	1	no	—
267	C₂×C₄×C₃₁	—	2	1	no	—
268	C₂×C₄×C₃₁	—	2	1	no	—
269	C₂×C₄×C₃₁	—	2	1	no	—
270	C₂×C₄×C₃₁	—	4	1	no	—
271	C₂×C₄×C₃₁	—	4	1	no	—
272	C₂×C₄×C₃₁	—	4	1	no	—
273	C₂×C₄×C₃₁	—	4	1	no	—
274	C₂×C₄×C₃₁	—	4	1	no	—
275	C₂×C₄×C₃₁	—	8	2	no	—
276	C₂×C₄×C₃₁	—	8	2	no	—
277	C₂×C₄×C₃₁	—	8	2	no	—
278	C₂×C₄×C₃₁	—	8	2	no	—
279	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
280	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
281	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
282	C₈×C₃₁	—	2	1	no	—
283	C₈×C₃₁	—	2	1	no	—
284	C₈×C₃₁	—	2	1	no	—
285	C₈×C₃₁	—	4	1	no	—
286	C₈×C₃₁	—	4	1	no	—
287	C₈×C₃₁	—	4	1	no	—
288	C₈×C₃₁	—	4	1	no	—
289	C₈×C₃₁	—	4	1	no	—
290	C₈×C₃₁	C₂×D₈	1	2	yes	—
291	#3391	C₂×C₃₁	1	1	yes	characteristic, complemented
292	#3358	C₂×C₃₁	1	1	yes	characteristic, complemented
293	#3287	C₂×C₃₁	1	1	yes	characteristic, complemented
294	#3278	C₂×C₃₁	1	1	yes	characteristic, complemented
295	#1721	C₂×C₃₁	1	1	yes	characteristic, complemented
296	#1687	C₂×C₃₁	1	1	yes	characteristic, complemented
297	#1682	C₂×C₃₁	1	1	yes	characteristic, complemented
298	#1668	C₂×C₃₁	1	1	yes	characteristic, complemented
299	#1665	C₂×C₃₁	1	1	yes	characteristic, complemented
300	#1641	C₂×C₃₁	1	1	yes	characteristic, complemented
301	#1627	C₂×C₃₁	1	1	yes	characteristic, complemented
302	#1621	C₂×C₃₁	1	1	yes	characteristic, complemented
303	#1621	C₂×C₃₁	1	1	yes	characteristic, complemented
304	#1581	C₂×C₃₁	1	1	yes	characteristic, complemented
305	#1579	C₂×C₃₁	1	1	yes	characteristic, complemented
306	C₂²×C₃₁	—	4	1	no	—
307	C₂²×C₃₁	—	4	1	no	—
308	C₂²×C₃₁	—	4	1	no	—
309	C₂²×C₃₁	—	4	1	no	—
310	C₂²×C₃₁	—	4	1	no	—
311	C₂²×C₃₁	—	4	1	no	—
312	C₂²×C₃₁	—	8	1	no	—
313	C₂²×C₃₁	—	8	1	no	—
314	C₂²×C₃₁	—	8	1	no	—
315	C₂²×C₃₁	—	8	1	no	—
316	C₂²×C₃₁	—	8	1	no	—
317	C₂²×C₃₁	—	8	1	no	—
318	C₂²×C₃₁	—	8	1	no	—
319	C₂²×C₃₁	—	8	1	no	—
320	C₂²×C₃₁	—	8	1	no	—
321	C₂²×C₃₁	D₄²	1	1	yes	characteristic
322	C₄×C₃₁	—	2	2	no	—
323	C₄×C₃₁	—	2	2	no	—
324	C₄×C₃₁	—	2	1	no	—
325	C₄×C₃₁	—	4	1	no	—
326	C₄×C₃₁	—	4	1	no	—
327	C₄×C₃₁	—	4	1	no	—
328	C₄×C₃₁	—	4	1	no	—
329	C₄×C₃₁	—	8	1	no	—
330	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
331	C₄×C₃₁	#575	1	1	yes	characteristic
332	C₄×C₃₁	#576	1	1	yes	characteristic
333	C₄×C₃₁	#577	1	1	yes	characteristic
334	C₂×#137	—	2	1	no	—
335	C₂×#135	—	2	1	no	—
336	#550	C₂²×C₃₁	1	1	yes	characteristic
337	#550	C₂²×C₃₁	1	1	yes	characteristic
338	#550	C₂²×C₃₁	1	1	yes	characteristic
339	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
340	#493	—	2	1	no	—
341	#493	C₂²×C₃₁	1	1	yes	characteristic
342	#493	C₂²×C₃₁	1	1	yes	characteristic
343	#493	C₂²×C₃₁	1	1	yes	characteristic
344	#492	C₂²×C₃₁	1	1	yes	characteristic
345	#486	—	2	1	no	—
346	#486	C₂²×C₃₁	1	1	yes	characteristic
347	#486	C₂²×C₃₁	1	1	yes	characteristic
348	#474	—	2	1	no	—
349	#474	C₂²×C₃₁	1	1	yes	characteristic
350	#468	—	2	1	no	—
351	#466	—	2	1	no	—
352	#462	C₂²×C₃₁	1	1	yes	characteristic
353	#460	C₂²×C₃₁	1	1	yes	characteristic
354	#460	C₂²×C₃₁	1	1	yes	characteristic
355	#460	C₂²×C₃₁	1	1	yes	characteristic
356	#459	—	2	1	no	—
357	#459	—	2	1	no	—
358	#459	—	2	1	no	—
359	#459	C₂²×C₃₁	1	1	yes	characteristic
360	#459	C₂²×C₃₁	1	1	yes	characteristic
361	#459	C₂²×C₃₁	1	1	yes	characteristic
362	#452	—	2	1	no	—
363	#449	—	2	1	no	—
364	#449	—	2	1	no	—
365	#448	—	2	1	no	—
366	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
367	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
368	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
369	C₄×D₈	—	2	1	no	—
370	C₄×D₈	—	2	1	no	—
371	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
372	#436	C₂²×C₃₁	1	1	yes	characteristic
373	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
374	#335	C₂²×C₃₁	1	1	yes	characteristic
375	#334	C₂²×C₃₁	1	1	yes	characteristic
376	#332	C₂²×C₃₁	1	1	yes	characteristic
377	#331	C₂²×C₃₁	1	1	yes	characteristic
378	#331	C₂²×C₃₁	1	1	yes	characteristic
379	#331	C₂²×C₃₁	1	1	yes	characteristic
380	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
381	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
382	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
383	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
384	#322	C₂²×C₃₁	1	1	yes	characteristic
385	C₂×C₃₁	—	8	1	no	—
386	C₂×C₃₁	—	8	1	no	—
387	C₂×C₃₁	—	16	1	no	—
388	C₂×C₃₁	—	16	1	no	—
389	C₂×C₃₁	—	16	1	no	—
390	C₂×C₃₁	#3271	1	1	yes	characteristic
391	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
392	C₂×C₃₁	#3288	1	1	yes	characteristic
393	C₂×#41	—	2	1	no	—
394	C₂×#41	—	2	1	no	—
395	C₂×#41	—	2	1	no	—
396	C₂²×Q₈	—	2	1	no	—
397	#137	—	4	4	no	—
398	#135	—	4	4	no	—
399	C₂×Q₁₆	—	2	1	no	—
400	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
401	C₂×SD₁₆	—	2	1	no	—
402	C₂×SD₁₆	—	2	1	no	—
403	C₂×SD₁₆	—	2	1	no	—
404	C₂×SD₁₆	—	2	1	no	—
405	C₂×SD₁₆	—	2	1	no	—
406	C₂×D₈	—	2	1	no	—
407	C₂×D₈	—	2	1	no	—
408	C₂×D₈	—	2	1	no	—
409	C₂×D₈	—	2	1	no	—
410	C₂×D₈	—	2	1	no	—
411	C₂×D₈	—	2	1	no	—
412	C₂×D₈	—	2	1	no	—
413	C₂×D₈	—	4	1	no	—
414	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
415	C₂×M₁₆	—	2	1	no	—
416	C₂²×C₈	—	2	1	no	—
417	C₄⋊Q₈	—	2	1	no	—
418	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
419	#127	—	2	1	no	—
420	#127	—	2	1	no	—
421	#127	—	2	1	no	—
422	#127	C₂³×C₃₁	1	1	yes	characteristic
423	#127	C₂³×C₃₁	1	1	yes	characteristic
424	#127	C₂³×C₃₁	1	1	yes	characteristic
425	C₂²⋊Q₈	—	2	1	no	—
426	C₂²⋊Q₈	—	4	1	no	—
427	#121	—	2	1	no	—
428	#121	—	2	1	no	—
429	#121	—	2	1	no	—
430	#121	—	4	1	no	—
431	#121	—	4	1	no	—
432	#121	—	4	1	no	—
433	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
434	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
435	C₄×D₄	—	2	1	no	—
436	C₄×D₄	—	2	1	no	—
437	C₄×D₄	—	2	1	no	—
438	C₄×D₄	—	2	1	no	—
439	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
440	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
441	#107	—	2	1	no	—
442	#107	D₄×C₃₁	1	1	yes	characteristic
443	#107	D₄×C₃₁	1	1	yes	characteristic
444	C₄⋊C₈	—	2	1	no	—
445	C₄⋊C₈	—	2	1	no	—
446	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
447	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
448	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
449	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
450	#103	—	2	1	no	—
451	#103	D₄×C₃₁	1	1	yes	characteristic
452	#103	D₄×C₃₁	1	1	yes	characteristic
453	#102	—	2	1	no	—
454	#102	—	2	1	no	—
455	#102	—	2	1	no	—
456	#102	—	2	1	no	—
457	#102	—	2	1	no	—
458	#102	—	2	1	no	—
459	#102	—	2	1	no	—
460	#102	—	2	1	no	—
461	#102	—	2	1	no	—
462	#102	—	4	1	no	—
463	#102	D₄×C₃₁	1	1	yes	characteristic
464	#102	D₄×C₃₁	1	1	yes	characteristic
465	C₂²⋊C₈	—	2	1	no	—
466	C₂²⋊C₈	—	2	1	no	—
467	C₄×C₈	—	2	1	no	—
468	C₄×C₈	—	2	1	no	—
469	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
470	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
471	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
472	C₃₁	#19762	1	1	yes	characteristic, direct product factor, Sylow
473	#41	—	4	2	no	—
474	#41	—	4	2	no	—
475	#41	—	4	2	no	—
476	#41	—	8	1	no	—
477	#41	—	8	1	no	—
478	#41	—	8	1	no	—
479	C₂×Q₈	—	2	1	no	—
480	C₂×Q₈	—	4	1	no	—
481	C₂×Q₈	—	4	2	no	—
482	C₂×Q₈	—	8	1	no	—
483	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
484	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
485	C₂×D₄	—	2	1	no	—
486	C₂×D₄	—	2	1	no	—
487	C₂×D₄	—	2	1	no	—
488	C₂×D₄	—	2	1	no	—
489	C₂×D₄	—	2	1	no	—
490	C₂×D₄	—	2	1	no	—
491	C₂×D₄	—	2	1	no	—
492	C₂×D₄	—	2	1	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	—	4	1	no	—
495	C₂×D₄	—	4	1	no	—
496	C₂×D₄	—	4	1	no	—
497	C₂×D₄	—	4	1	no	—
498	C₂×D₄	—	4	1	no	—
499	C₂×D₄	—	4	1	no	—
500	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
501	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
502	C₂²×C₄	—	2	1	no	—
503	C₂²×C₄	—	2	1	no	—
504	C₂²×C₄	—	4	1	no	—
505	C₂²×C₄	—	4	1	no	—
506	C₂²×C₄	—	4	1	no	—
507	C₂²×C₄	—	4	1	no	—
508	Q₁₆	—	4	2	no	—
509	Q₁₆	D₈×C₃₁	1	4	yes	complemented
510	SD₁₆	—	4	2	no	—
511	SD₁₆	—	4	2	no	—
512	SD₁₆	—	4	2	no	—
513	SD₁₆	—	4	2	no	—
514	SD₁₆	—	8	1	no	—
515	D₈	—	4	1	no	—
516	D₈	—	4	2	no	—
517	D₈	—	4	2	no	—
518	D₈	—	4	2	no	—
519	D₈	—	8	1	no	—
520	D₈	—	8	1	no	—
521	D₈	—	8	1	no	—
522	D₈	—	8	1	no	—
523	D₈	—	8	2	no	—
524	M₁₆	—	4	2	no	—
525	C₂×C₈	—	2	1	no	—
526	C₂×C₈	—	2	1	no	—
527	C₂×C₈	—	2	1	no	—
528	C₂×C₈	—	2	1	no	—
529	C₂×C₈	—	2	1	no	—
530	C₂×C₈	—	4	2	no	—
531	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
532	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
533	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
534	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	—	2	1	no	—
537	C₄⋊C₄	—	2	1	no	—
538	C₄⋊C₄	—	2	1	no	—
539	C₄⋊C₄	—	4	1	no	—
540	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
541	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
542	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
543	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
544	C₂²⋊C₄	—	2	1	no	—
545	C₂²⋊C₄	—	2	1	no	—
546	C₂²⋊C₄	—	4	1	no	—
547	C₂²⋊C₄	—	4	1	no	—
548	C₂²⋊C₄	—	4	1	no	—
549	C₂²⋊C₄	—	4	1	no	—
550	C₄²	—	2	1	no	—
551	C₄²	—	2	1	no	—
552	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
553	C₂³	—	2	1	no	—
554	C₂³	—	2	1	no	—
555	C₂³	—	4	1	no	—
556	C₂³	—	4	1	no	—
557	C₂³	—	4	1	no	—
558	Q₈	—	2	1	no	—
559	Q₈	—	2	1	no	—
560	Q₈	—	4	1	no	—
561	Q₈	—	4	1	no	—
562	Q₈	—	8	2	no	—
563	Q₈	C₂×D₈×C₃₁	1	2	yes	—
564	Q₈	C₂×D₈×C₃₁	1	2	yes	—
565	D₄	—	2	1	no	—
566	D₄	—	2	1	no	—
567	D₄	—	2	1	no	—
568	D₄	—	2	1	no	—
569	D₄	—	4	1	no	—
570	D₄	—	4	1	no	—
571	D₄	—	4	1	no	—
572	D₄	—	4	1	no	—
573	D₄	—	4	1	no	—
574	D₄	—	4	1	no	—
575	D₄	—	4	1	no	—
576	D₄	—	4	1	no	—
577	D₄	—	4	1	no	—
578	D₄	—	4	1	no	—
579	D₄	—	4	1	no	—
580	D₄	—	4	1	no	—
581	D₄	—	8	2	no	—
582	D₄	—	8	2	no	—
583	D₄	—	8	2	no	—
584	D₄	—	8	1	no	—
585	D₄	—	8	1	no	—
586	D₄	—	8	1	no	—
587	D₄	—	16	1	no	—
588	D₄	—	16	1	no	—
589	D₄	—	16	1	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	8	2	no	—
602	C₂×C₄	—	8	2	no	—
603	C₂×C₄	—	8	2	no	—
604	C₂×C₄	—	8	2	no	—
605	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
606	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
607	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
608	C₈	—	2	1	no	—
609	C₈	—	2	1	no	—
610	C₈	—	2	1	no	—
611	C₈	—	4	1	no	—
612	C₈	—	4	1	no	—
613	C₈	—	4	1	no	—
614	C₈	—	4	1	no	—
615	C₈	—	4	1	no	—
616	C₈	C₂×D₈×C₃₁	1	2	yes	—
617	C₂²	—	4	1	no	—
618	C₂²	—	4	1	no	—
619	C₂²	—	4	1	no	—
620	C₂²	—	4	1	no	—
621	C₂²	—	4	1	no	—
622	C₂²	—	4	1	no	—
623	C₂²	—	8	1	no	—
624	C₂²	—	8	1	no	—
625	C₂²	—	8	1	no	—
626	C₂²	—	8	1	no	—
627	C₂²	—	8	1	no	—
628	C₂²	—	8	1	no	—
629	C₂²	—	8	1	no	—
630	C₂²	—	8	1	no	—
631	C₂²	—	8	1	no	—
632	C₂²	D₄²×C₃₁	1	1	yes	characteristic
633	C₄	—	2	2	no	—
634	C₄	—	2	2	no	—
635	C₄	—	2	1	no	—
636	C₄	—	4	1	no	—
637	C₄	—	4	1	no	—
638	C₄	—	4	1	no	—
639	C₄	—	4	1	no	—
640	C₄	—	8	1	no	—
641	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
642	C₄	C₃₁×#575	1	1	yes	characteristic
643	C₄	C₃₁×#576	1	1	yes	characteristic
644	C₄	C₃₁×#577	1	1	yes	characteristic
645	C₂	—	8	1	no	—
646	C₂	—	8	1	no	—
647	C₂	—	16	1	no	—
648	C₂	—	16	1	no	—
649	C₂	—	16	1	no	—
650	C₂	C₃₁×#3271	1	1	yes	characteristic
651	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
652	C₂	C₃₁×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3391	C₂	semidirect
2. C₃₁×#3358	C₂	semidirect
3. C₃₁×#3287	C₂	semidirect
4. C₃₁×#3278	C₂	semidirect
5. C₃₁×#1721	C₂	semidirect
6. C₃₁×#1687	C₂	semidirect
7. C₃₁×#1682	C₂	semidirect
8. C₃₁×#1668	C₂	semidirect
9. C₃₁×#1665	C₂	semidirect
10. C₃₁×#1641	C₂	semidirect
11. C₃₁×#1627	C₂	semidirect
12. C₃₁×#1621	C₂	semidirect
13. C₃₁×#1621	C₂	semidirect
14. C₃₁×#1581	C₂	semidirect
15. C₃₁×#1579	C₂	semidirect
182. Q₁₆×C₃₁	D₈	semidirect
226. #19762	C₃₁	direct (N × H)
291. #3391	C₂×C₃₁	semidirect
292. #3358	C₂×C₃₁	semidirect
293. #3287	C₂×C₃₁	semidirect
294. #3278	C₂×C₃₁	semidirect
295. #1721	C₂×C₃₁	semidirect
296. #1687	C₂×C₃₁	semidirect
297. #1682	C₂×C₃₁	semidirect
298. #1668	C₂×C₃₁	semidirect
299. #1665	C₂×C₃₁	semidirect
300. #1641	C₂×C₃₁	semidirect
301. #1627	C₂×C₃₁	semidirect
302. #1621	C₂×C₃₁	semidirect
303. #1621	C₂×C₃₁	semidirect
304. #1581	C₂×C₃₁	semidirect
305. #1579	C₂×C₃₁	semidirect
472. C₃₁	#19762	direct (N × H)
509. Q₁₆	D₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3391	2
C₃₁×#3358	2
C₃₁×#3287	2
C₃₁×#3278	2
C₃₁×#1721	2
C₃₁×#1687	2
C₃₁×#1682	2
C₃₁×#1668	2
C₃₁×#1665	2
C₃₁×#1641	2
C₃₁×#1627	2
C₃₁×#1621	2
C₃₁×#1581	2
C₃₁×#1579	2
#19762	31

Subgroups — C31×#19762 #21476521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19762 #21476521