Subgroups — C₃₁×#19810 #21476569

Order 7936.

632 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3392	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3383	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3359	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3278	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1714	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1693	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1680	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1666	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1666	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1653	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1651	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1642	C₂	1	1	yes	maximal, characteristic
13	C₃₁×#1626	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1583	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	C₃₁×#557	C₂²	1	1	yes	characteristic
18	C₃₁×#557	C₂²	1	1	yes	characteristic
19	C₃₁×#550	C₂²	1	1	yes	characteristic
20	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#500	C₂²	1	1	yes	characteristic
22	C₃₁×#499	—	2	1	no	—
23	C₃₁×#498	C₂²	1	1	yes	characteristic
24	C₃₁×#492	—	2	1	no	—
25	C₃₁×#492	C₂²	1	1	yes	characteristic
26	C₃₁×#492	C₂²	1	1	yes	characteristic
27	C₃₁×#487	—	2	1	no	—
28	C₃₁×#487	C₂²	1	1	yes	characteristic
29	C₃₁×#486	C₂²	1	1	yes	characteristic
30	C₃₁×#479	—	2	1	no	—
31	C₃₁×#479	—	2	1	no	—
32	C₃₁×#475	—	2	1	no	—
33	C₃₁×#475	C₂²	1	1	yes	characteristic
34	C₃₁×#475	C₂²	1	1	yes	characteristic
35	C₃₁×#475	C₂²	1	1	yes	characteristic
36	C₃₁×#468	—	2	1	no	—
37	C₃₁×#462	C₂²	1	1	yes	characteristic
38	C₃₁×#461	—	2	1	no	—
39	C₃₁×#461	—	2	1	no	—
40	C₃₁×#461	C₂²	1	1	yes	characteristic
41	C₃₁×#461	C₂²	1	1	yes	characteristic
42	C₃₁×#460	—	2	1	no	—
43	C₃₁×#460	C₂²	1	1	yes	characteristic
44	C₃₁×#460	C₂²	1	1	yes	characteristic
45	C₃₁×#452	—	2	1	no	—
46	C₃₁×#448	—	2	1	no	—
47	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
48	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
49	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
50	C₄×Q₁₆×C₃₁	—	2	1	no	—
51	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
52	C₄×SD₁₆×C₃₁	—	2	1	no	—
53	C₄×SD₁₆×C₃₁	—	2	1	no	—
54	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
55	C₃₁×#436	C₂²	1	1	yes	characteristic
56	C₃₁×#334	C₂²	1	1	yes	characteristic
57	C₃₁×#334	C₂²	1	1	yes	characteristic
58	C₃₁×#332	C₂²	1	1	yes	characteristic
59	C₃₁×#332	C₂²	1	1	yes	characteristic
60	C₃₁×#332	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×Q₈⋊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	C₂²×Q₈×C₃₁	—	2	1	no	—
69	C₃₁×#137	—	4	4	no	—
70	C₂×Q₁₆×C₃₁	—	2	1	no	—
71	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
72	C₂×SD₁₆×C₃₁	—	2	1	no	—
73	C₂×SD₁₆×C₃₁	—	2	1	no	—
74	C₂×SD₁₆×C₃₁	—	2	1	no	—
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×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
81	C₂×M₁₆×C₃₁	—	2	1	no	—
82	C₃₁×C₄⋊Q₈	—	2	1	no	—
83	C₃₁×C₄⋊Q₈	—	2	1	no	—
84	C₃₁×C₄⋊Q₈	—	2	1	no	—
85	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₃₁×#127	—	2	1	no	—
89	C₃₁×#127	C₂³	1	1	yes	characteristic
90	C₃₁×#125	—	2	1	no	—
91	C₃₁×#125	—	2	1	no	—
92	C₃₁×#125	—	4	1	no	—
93	C₃₁×#125	—	4	1	no	—
94	C₃₁×C₂²⋊Q₈	—	2	1	no	—
95	C₃₁×C₂²⋊Q₈	—	4	1	no	—
96	C₃₁×#121	—	2	1	no	—
97	C₃₁×#121	—	4	1	no	—
98	C₄×Q₈×C₃₁	—	2	1	no	—
99	C₄×Q₈×C₃₁	—	2	1	no	—
100	C₄×Q₈×C₃₁	—	2	1	no	—
101	C₄×Q₈×C₃₁	—	2	1	no	—
102	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
103	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
104	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
105	C₄×D₄×C₃₁	—	2	1	no	—
106	C₄×D₄×C₃₁	—	2	1	no	—
107	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
108	C₃₁×#107	—	2	1	no	—
109	C₃₁×#107	—	2	1	no	—
110	C₃₁×#107	D₄	1	1	yes	characteristic
111	C₃₁×#106	—	2	1	no	—
112	C₃₁×#106	—	2	1	no	—
113	C₃₁×#106	—	2	1	no	—
114	C₃₁×#106	—	4	1	no	—
115	C₃₁×#106	D₄	1	1	yes	characteristic
116	C₃₁×C₄⋊C₈	—	2	1	no	—
117	C₃₁×C₄⋊C₈	—	2	1	no	—
118	C₃₁×C₄⋊C₈	—	2	1	no	—
119	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
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×#103	—	2	1	no	—
124	C₃₁×#103	—	2	1	no	—
125	C₃₁×#103	—	2	1	no	—
126	C₃₁×#103	—	2	1	no	—
127	C₃₁×#103	—	2	1	no	—
128	C₃₁×#103	—	2	1	no	—
129	C₃₁×#103	—	2	1	no	—
130	C₃₁×#103	—	2	1	no	—
131	C₃₁×#103	D₄	1	1	yes	characteristic
132	C₃₁×#103	D₄	1	1	yes	characteristic
133	C₃₁×#103	D₄	1	1	yes	characteristic
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×C₂²⋊C₈	—	2	1	no	—
139	C₄×C₈×C₃₁	—	2	1	no	—
140	C₄×C₈×C₃₁	—	2	1	no	—
141	C₄×C₈×C₃₁	—	2	1	no	—
142	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
145	#41×C₃₁	—	4	2	no	—
146	#41×C₃₁	—	8	1	no	—
147	C₂×Q₈×C₃₁	—	2	1	no	—
148	C₂×Q₈×C₃₁	—	2	1	no	—
149	C₂×Q₈×C₃₁	—	2	1	no	—
150	C₂×Q₈×C₃₁	—	4	1	no	—
151	C₂×Q₈×C₃₁	—	4	2	no	—
152	C₂×Q₈×C₃₁	—	8	1	no	—
153	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂×D₄×C₃₁	—	2	1	no	—
157	C₂×D₄×C₃₁	—	2	1	no	—
158	C₂×D₄×C₃₁	—	2	1	no	—
159	C₂×D₄×C₃₁	—	4	1	no	—
160	C₂×D₄×C₃₁	—	4	1	no	—
161	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	—	2	1	no	—
163	C₂²×C₄×C₃₁	—	4	1	no	—
164	C₂²×C₄×C₃₁	—	4	1	no	—
165	Q₁₆×C₃₁	—	4	2	no	—
166	Q₁₆×C₃₁	SD₁₆	1	4	yes	complemented
167	SD₁₆×C₃₁	—	4	1	no	—
168	SD₁₆×C₃₁	—	4	2	no	—
169	SD₁₆×C₃₁	—	4	2	no	—
170	SD₁₆×C₃₁	—	4	2	no	—
171	SD₁₆×C₃₁	—	4	2	no	—
172	SD₁₆×C₃₁	—	8	1	no	—
173	SD₁₆×C₃₁	—	8	1	no	—
174	SD₁₆×C₃₁	—	8	1	no	—
175	SD₁₆×C₃₁	—	8	1	no	—
176	M₁₆×C₃₁	—	4	2	no	—
177	C₂×C₈×C₃₁	—	2	1	no	—
178	C₂×C₈×C₃₁	—	2	1	no	—
179	C₂×C₈×C₃₁	—	2	1	no	—
180	C₂×C₈×C₃₁	—	2	1	no	—
181	C₂×C₈×C₃₁	—	2	1	no	—
182	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
185	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
186	C₄⋊C₄×C₃₁	—	2	1	no	—
187	C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₄⋊C₄×C₃₁	—	2	1	no	—
189	C₄⋊C₄×C₃₁	—	2	1	no	—
190	C₄⋊C₄×C₃₁	—	2	1	no	—
191	C₄⋊C₄×C₃₁	—	2	1	no	—
192	C₄⋊C₄×C₃₁	—	2	1	no	—
193	C₄⋊C₄×C₃₁	—	2	1	no	—
194	C₄⋊C₄×C₃₁	—	2	1	no	—
195	C₄⋊C₄×C₃₁	—	4	1	no	—
196	C₄⋊C₄×C₃₁	—	4	1	no	—
197	C₄⋊C₄×C₃₁	—	4	1	no	—
198	C₄⋊C₄×C₃₁	—	4	1	no	—
199	C₄⋊C₄×C₃₁	—	4	1	no	—
200	C₄⋊C₄×C₃₁	—	4	1	no	—
201	C₄⋊C₄×C₃₁	—	4	1	no	—
202	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
203	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
204	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
205	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
206	C₂²⋊C₄×C₃₁	—	2	1	no	—
207	C₂²⋊C₄×C₃₁	—	4	1	no	—
208	C₂²⋊C₄×C₃₁	—	4	1	no	—
209	C₄²×C₃₁	—	2	1	no	—
210	C₄²×C₃₁	—	2	1	no	—
211	C₄²×C₃₁	—	2	1	no	—
212	C₄²×C₃₁	—	4	1	no	—
213	C₄²×C₃₁	—	4	1	no	—
214	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
215	#19810	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂³×C₃₁	—	2	1	no	—
217	C₂³×C₃₁	—	4	1	no	—
218	Q₈×C₃₁	—	2	1	no	—
219	Q₈×C₃₁	—	2	1	no	—
220	Q₈×C₃₁	—	2	1	no	—
221	Q₈×C₃₁	—	2	1	no	—
222	Q₈×C₃₁	—	4	1	no	—
223	Q₈×C₃₁	—	4	1	no	—
224	Q₈×C₃₁	—	4	1	no	—
225	Q₈×C₃₁	—	4	1	no	—
226	Q₈×C₃₁	—	4	1	no	—
227	Q₈×C₃₁	—	4	1	no	—
228	Q₈×C₃₁	—	8	2	no	—
229	Q₈×C₃₁	C₂×SD₁₆	1	2	yes	—
230	Q₈×C₃₁	C₂×SD₁₆	1	2	yes	—
231	D₄×C₃₁	—	2	1	no	—
232	D₄×C₃₁	—	2	1	no	—
233	D₄×C₃₁	—	4	1	no	—
234	D₄×C₃₁	—	4	1	no	—
235	D₄×C₃₁	—	4	1	no	—
236	D₄×C₃₁	—	4	1	no	—
237	D₄×C₃₁	—	8	2	no	—
238	D₄×C₃₁	—	8	1	no	—
239	D₄×C₃₁	—	16	1	no	—
240	C₂×C₄×C₃₁	—	2	1	no	—
241	C₂×C₄×C₃₁	—	2	1	no	—
242	C₂×C₄×C₃₁	—	2	1	no	—
243	C₂×C₄×C₃₁	—	2	1	no	—
244	C₂×C₄×C₃₁	—	2	1	no	—
245	C₂×C₄×C₃₁	—	2	1	no	—
246	C₂×C₄×C₃₁	—	2	1	no	—
247	C₂×C₄×C₃₁	—	4	1	no	—
248	C₂×C₄×C₃₁	—	4	1	no	—
249	C₂×C₄×C₃₁	—	4	1	no	—
250	C₂×C₄×C₃₁	—	4	1	no	—
251	C₂×C₄×C₃₁	—	4	1	no	—
252	C₂×C₄×C₃₁	—	8	2	no	—
253	C₂×C₄×C₃₁	—	8	2	no	—
254	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
257	C₈×C₃₁	—	2	1	no	—
258	C₈×C₃₁	—	2	1	no	—
259	C₈×C₃₁	—	2	1	no	—
260	C₈×C₃₁	—	4	1	no	—
261	C₈×C₃₁	—	4	1	no	—
262	C₈×C₃₁	—	4	1	no	—
263	C₈×C₃₁	—	4	1	no	—
264	C₈×C₃₁	—	4	1	no	—
265	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
266	#3392	C₂×C₃₁	1	1	yes	characteristic
267	#3383	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#3359	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#3278	C₂×C₃₁	1	1	yes	characteristic
270	#1714	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#1693	C₂×C₃₁	1	1	yes	characteristic, complemented
272	#1680	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#1666	C₂×C₃₁	1	1	yes	characteristic, complemented
274	#1666	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#1653	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#1651	C₂×C₃₁	1	1	yes	characteristic, complemented
277	#1642	C₂×C₃₁	1	1	yes	characteristic
278	#1626	C₂×C₃₁	1	1	yes	characteristic, complemented
279	#1583	C₂×C₃₁	1	1	yes	characteristic, complemented
280	#1579	C₂×C₃₁	1	1	yes	characteristic, complemented
281	C₂²×C₃₁	—	4	1	no	—
282	C₂²×C₃₁	—	4	1	no	—
283	C₂²×C₃₁	—	4	1	no	—
284	C₂²×C₃₁	—	8	1	no	—
285	C₂²×C₃₁	—	8	1	no	—
286	C₂²×C₃₁	—	8	1	no	—
287	C₂²×C₃₁	D₄²	1	1	yes	characteristic
288	C₄×C₃₁	—	2	2	no	—
289	C₄×C₃₁	—	2	2	no	—
290	C₄×C₃₁	—	2	1	no	—
291	C₄×C₃₁	—	4	1	no	—
292	C₄×C₃₁	—	4	1	no	—
293	C₄×C₃₁	—	4	1	no	—
294	C₄×C₃₁	—	4	1	no	—
295	C₄×C₃₁	—	4	1	no	—
296	C₄×C₃₁	—	8	1	no	—
297	C₄×C₃₁	—	8	1	no	—
298	C₄×C₃₁	—	8	1	no	—
299	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
300	C₄×C₃₁	#575	1	1	yes	characteristic
301	C₄×C₃₁	#577	1	1	yes	characteristic
302	C₄×C₃₁	#577	1	1	yes	characteristic
303	C₂×#137	—	2	1	no	—
304	#557	C₂²×C₃₁	1	1	yes	characteristic
305	#557	C₂²×C₃₁	1	1	yes	characteristic
306	#550	C₂²×C₃₁	1	1	yes	characteristic
307	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
308	#500	C₂²×C₃₁	1	1	yes	characteristic
309	#499	—	2	1	no	—
310	#498	C₂²×C₃₁	1	1	yes	characteristic
311	#492	—	2	1	no	—
312	#492	C₂²×C₃₁	1	1	yes	characteristic
313	#492	C₂²×C₃₁	1	1	yes	characteristic
314	#487	—	2	1	no	—
315	#487	C₂²×C₃₁	1	1	yes	characteristic
316	#486	C₂²×C₃₁	1	1	yes	characteristic
317	#479	—	2	1	no	—
318	#479	—	2	1	no	—
319	#475	—	2	1	no	—
320	#475	C₂²×C₃₁	1	1	yes	characteristic
321	#475	C₂²×C₃₁	1	1	yes	characteristic
322	#475	C₂²×C₃₁	1	1	yes	characteristic
323	#468	—	2	1	no	—
324	#462	C₂²×C₃₁	1	1	yes	characteristic
325	#461	—	2	1	no	—
326	#461	—	2	1	no	—
327	#461	C₂²×C₃₁	1	1	yes	characteristic
328	#461	C₂²×C₃₁	1	1	yes	characteristic
329	#460	—	2	1	no	—
330	#460	C₂²×C₃₁	1	1	yes	characteristic
331	#460	C₂²×C₃₁	1	1	yes	characteristic
332	#452	—	2	1	no	—
333	#448	—	2	1	no	—
334	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
335	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
336	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
337	C₄×Q₁₆	—	2	1	no	—
338	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
339	C₄×SD₁₆	—	2	1	no	—
340	C₄×SD₁₆	—	2	1	no	—
341	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
342	#436	C₂²×C₃₁	1	1	yes	characteristic
343	#334	C₂²×C₃₁	1	1	yes	characteristic
344	#334	C₂²×C₃₁	1	1	yes	characteristic
345	#332	C₂²×C₃₁	1	1	yes	characteristic
346	#332	C₂²×C₃₁	1	1	yes	characteristic
347	#332	C₂²×C₃₁	1	1	yes	characteristic
348	#331	C₂²×C₃₁	1	1	yes	characteristic
349	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
350	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
351	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
352	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
353	#322	C₂²×C₃₁	1	1	yes	characteristic
354	C₂×C₃₁	—	8	1	no	—
355	C₂×C₃₁	—	16	1	no	—
356	C₂×C₃₁	#3271	1	1	yes	characteristic
357	C₂×C₃₁	D₄×SD₁₆	1	1	yes	characteristic
358	C₂×C₃₁	#3289	1	1	yes	characteristic
359	C₂×#41	—	2	1	no	—
360	C₂²×Q₈	—	2	1	no	—
361	#137	—	4	4	no	—
362	C₂×Q₁₆	—	2	1	no	—
363	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
364	C₂×SD₁₆	—	2	1	no	—
365	C₂×SD₁₆	—	2	1	no	—
366	C₂×SD₁₆	—	2	1	no	—
367	C₂×SD₁₆	—	2	1	no	—
368	C₂×SD₁₆	—	2	1	no	—
369	C₂×SD₁₆	—	2	1	no	—
370	C₂×SD₁₆	—	2	1	no	—
371	C₂×SD₁₆	—	2	1	no	—
372	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
373	C₂×M₁₆	—	2	1	no	—
374	C₄⋊Q₈	—	2	1	no	—
375	C₄⋊Q₈	—	2	1	no	—
376	C₄⋊Q₈	—	2	1	no	—
377	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
378	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
379	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
380	#127	—	2	1	no	—
381	#127	C₂³×C₃₁	1	1	yes	characteristic
382	#125	—	2	1	no	—
383	#125	—	2	1	no	—
384	#125	—	4	1	no	—
385	#125	—	4	1	no	—
386	C₂²⋊Q₈	—	2	1	no	—
387	C₂²⋊Q₈	—	4	1	no	—
388	#121	—	2	1	no	—
389	#121	—	4	1	no	—
390	C₄×Q₈	—	2	1	no	—
391	C₄×Q₈	—	2	1	no	—
392	C₄×Q₈	—	2	1	no	—
393	C₄×Q₈	—	2	1	no	—
394	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
395	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
396	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
397	C₄×D₄	—	2	1	no	—
398	C₄×D₄	—	2	1	no	—
399	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
400	#107	—	2	1	no	—
401	#107	—	2	1	no	—
402	#107	D₄×C₃₁	1	1	yes	characteristic
403	#106	—	2	1	no	—
404	#106	—	2	1	no	—
405	#106	—	2	1	no	—
406	#106	—	4	1	no	—
407	#106	D₄×C₃₁	1	1	yes	characteristic
408	C₄⋊C₈	—	2	1	no	—
409	C₄⋊C₈	—	2	1	no	—
410	C₄⋊C₈	—	2	1	no	—
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	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
415	#103	—	2	1	no	—
416	#103	—	2	1	no	—
417	#103	—	2	1	no	—
418	#103	—	2	1	no	—
419	#103	—	2	1	no	—
420	#103	—	2	1	no	—
421	#103	—	2	1	no	—
422	#103	—	2	1	no	—
423	#103	D₄×C₃₁	1	1	yes	characteristic
424	#103	D₄×C₃₁	1	1	yes	characteristic
425	#103	D₄×C₃₁	1	1	yes	characteristic
426	#102	—	2	1	no	—
427	#102	—	2	1	no	—
428	#102	—	4	1	no	—
429	#102	D₄×C₃₁	1	1	yes	characteristic
430	C₂²⋊C₈	—	2	1	no	—
431	C₄×C₈	—	2	1	no	—
432	C₄×C₈	—	2	1	no	—
433	C₄×C₈	—	2	1	no	—
434	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
435	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
436	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
437	C₃₁	#19810	1	1	yes	characteristic, direct product factor, Sylow
438	#41	—	4	2	no	—
439	#41	—	8	1	no	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	—	2	1	no	—
442	C₂×Q₈	—	2	1	no	—
443	C₂×Q₈	—	4	1	no	—
444	C₂×Q₈	—	4	2	no	—
445	C₂×Q₈	—	8	1	no	—
446	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
447	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
448	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	1	no	—
452	C₂×D₄	—	4	1	no	—
453	C₂×D₄	—	4	1	no	—
454	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	4	1	no	—
457	C₂²×C₄	—	4	1	no	—
458	Q₁₆	—	4	2	no	—
459	Q₁₆	SD₁₆×C₃₁	1	4	yes	complemented
460	SD₁₆	—	4	1	no	—
461	SD₁₆	—	4	2	no	—
462	SD₁₆	—	4	2	no	—
463	SD₁₆	—	4	2	no	—
464	SD₁₆	—	4	2	no	—
465	SD₁₆	—	8	1	no	—
466	SD₁₆	—	8	1	no	—
467	SD₁₆	—	8	1	no	—
468	SD₁₆	—	8	1	no	—
469	M₁₆	—	4	2	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
477	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
478	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	4	1	no	—
489	C₄⋊C₄	—	4	1	no	—
490	C₄⋊C₄	—	4	1	no	—
491	C₄⋊C₄	—	4	1	no	—
492	C₄⋊C₄	—	4	1	no	—
493	C₄⋊C₄	—	4	1	no	—
494	C₄⋊C₄	—	4	1	no	—
495	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
496	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
497	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
498	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
499	C₂²⋊C₄	—	2	1	no	—
500	C₂²⋊C₄	—	4	1	no	—
501	C₂²⋊C₄	—	4	1	no	—
502	C₄²	—	2	1	no	—
503	C₄²	—	2	1	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	4	1	no	—
506	C₄²	—	4	1	no	—
507	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
508	C₂³	—	2	1	no	—
509	C₂³	—	4	1	no	—
510	Q₈	—	2	1	no	—
511	Q₈	—	2	1	no	—
512	Q₈	—	2	1	no	—
513	Q₈	—	2	1	no	—
514	Q₈	—	4	1	no	—
515	Q₈	—	4	1	no	—
516	Q₈	—	4	1	no	—
517	Q₈	—	4	1	no	—
518	Q₈	—	4	1	no	—
519	Q₈	—	4	1	no	—
520	Q₈	—	8	2	no	—
521	Q₈	C₂×SD₁₆×C₃₁	1	2	yes	—
522	Q₈	C₂×SD₁₆×C₃₁	1	2	yes	—
523	D₄	—	2	1	no	—
524	D₄	—	2	1	no	—
525	D₄	—	4	1	no	—
526	D₄	—	4	1	no	—
527	D₄	—	4	1	no	—
528	D₄	—	4	1	no	—
529	D₄	—	8	2	no	—
530	D₄	—	8	1	no	—
531	D₄	—	16	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	8	2	no	—
545	C₂×C₄	—	8	2	no	—
546	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
547	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
548	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
549	C₈	—	2	1	no	—
550	C₈	—	2	1	no	—
551	C₈	—	2	1	no	—
552	C₈	—	4	1	no	—
553	C₈	—	4	1	no	—
554	C₈	—	4	1	no	—
555	C₈	—	4	1	no	—
556	C₈	—	4	1	no	—
557	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
558	C₂²	—	4	1	no	—
559	C₂²	—	4	1	no	—
560	C₂²	—	4	1	no	—
561	C₂²	—	8	1	no	—
562	C₂²	—	8	1	no	—
563	C₂²	—	8	1	no	—
564	C₂²	D₄²×C₃₁	1	1	yes	characteristic
565	C₄	—	2	2	no	—
566	C₄	—	2	2	no	—
567	C₄	—	2	1	no	—
568	C₄	—	4	1	no	—
569	C₄	—	4	1	no	—
570	C₄	—	4	1	no	—
571	C₄	—	4	1	no	—
572	C₄	—	4	1	no	—
573	C₄	—	8	1	no	—
574	C₄	—	8	1	no	—
575	C₄	—	8	1	no	—
576	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
577	C₄	C₃₁×#575	1	1	yes	characteristic
578	C₄	C₃₁×#577	1	1	yes	characteristic
579	C₄	C₃₁×#577	1	1	yes	characteristic
580	C₂	—	8	1	no	—
581	C₂	—	16	1	no	—
582	C₂	C₃₁×#3271	1	1	yes	characteristic
583	C₂	D₄×SD₁₆×C₃₁	1	1	yes	characteristic
584	C₂	C₃₁×#3289	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#550	C₂²
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
2. C₃₁×#3383	C₂	semidirect
3. C₃₁×#3359	C₂	semidirect
5. C₃₁×#1714	C₂	semidirect
6. C₃₁×#1693	C₂	semidirect
7. C₃₁×#1680	C₂	semidirect
8. C₃₁×#1666	C₂	semidirect
9. C₃₁×#1666	C₂	semidirect
10. C₃₁×#1653	C₂	semidirect
11. C₃₁×#1651	C₂	semidirect
13. C₃₁×#1626	C₂	semidirect
14. C₃₁×#1583	C₂	semidirect
15. C₃₁×#1579	C₂	semidirect
166. Q₁₆×C₃₁	SD₁₆	semidirect
215. #19810	C₃₁	direct (N × H)
267. #3383	C₂×C₃₁	semidirect
268. #3359	C₂×C₃₁	semidirect
270. #1714	C₂×C₃₁	semidirect
271. #1693	C₂×C₃₁	semidirect
272. #1680	C₂×C₃₁	semidirect
273. #1666	C₂×C₃₁	semidirect
274. #1666	C₂×C₃₁	semidirect
275. #1653	C₂×C₃₁	semidirect
276. #1651	C₂×C₃₁	semidirect
278. #1626	C₂×C₃₁	semidirect
279. #1583	C₂×C₃₁	semidirect
280. #1579	C₂×C₃₁	semidirect
437. C₃₁	#19810	direct (N × H)
459. Q₁₆	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3392	2
C₃₁×#3383	2
C₃₁×#3359	2
C₃₁×#3278	2
C₃₁×#1714	2
C₃₁×#1693	2
C₃₁×#1680	2
C₃₁×#1666	2
C₃₁×#1653	2
C₃₁×#1651	2
C₃₁×#1642	2
C₃₁×#1626	2
C₃₁×#1583	2
C₃₁×#1579	2
#19810	31

Subgroups — C31×#19810 #21476569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19810 #21476569