Subgroups — C₃₁×#19853 #21476612

Order 7936.

588 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3398	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3381	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3335	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3302	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1716	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1692	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1680	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1671	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1669	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1654	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1653	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1644	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1638	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1582	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#557	C₂²	1	1	yes	characteristic
17	C₃₁×#556	C₂²	1	1	yes	characteristic
18	C₃₁×#554	C₂²	1	1	yes	characteristic
19	C₃₁×#547	C₂²	1	1	yes	characteristic
20	C₃₁×#499	—	2	1	no	—
21	C₃₁×#498	C₂²	1	1	yes	characteristic
22	C₃₁×#498	C₂²	1	1	yes	characteristic
23	C₃₁×#498	C₂²	1	1	yes	characteristic
24	C₃₁×#497	—	2	1	no	—
25	C₃₁×#494	C₂²	1	1	yes	characteristic
26	C₃₁×#487	C₂²	1	1	yes	characteristic
27	C₃₁×#487	C₂²	1	1	yes	characteristic
28	C₃₁×#485	—	2	1	no	—
29	C₃₁×#483	—	2	1	no	—
30	C₃₁×#481	—	2	1	no	—
31	C₃₁×#479	—	2	1	no	—
32	C₃₁×#478	—	2	1	no	—
33	C₃₁×#477	C₂²	1	1	yes	characteristic
34	C₃₁×#476	C₂²	1	1	yes	characteristic
35	C₃₁×#475	C₂²	1	1	yes	characteristic
36	C₃₁×#474	C₂²	1	1	yes	characteristic
37	C₃₁×#474	C₂²	1	1	yes	characteristic
38	C₃₁×#467	—	2	1	no	—
39	C₃₁×#465	—	2	1	no	—
40	C₃₁×#464	—	2	1	no	—
41	C₃₁×#462	—	2	1	no	—
42	C₃₁×#462	C₂²	1	1	yes	characteristic
43	C₃₁×#462	C₂²	1	1	yes	characteristic
44	C₃₁×#461	C₂²	1	1	yes	characteristic
45	C₃₁×#452	—	2	1	no	—
46	C₃₁×#446	C₂²	1	1	yes	characteristic
47	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
48	C₃₁×#441	—	2	1	no	—
49	C₄×SD₁₆×C₃₁	—	2	1	no	—
50	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
51	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
52	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
53	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
54	C₂×C₃₁×#106	—	2	1	no	—
55	C₃₁×#416	—	2	1	no	—
56	C₃₁×#335	C₂²	1	1	yes	characteristic
57	C₃₁×#334	C₂²	1	1	yes	characteristic
58	C₃₁×#333	C₂²	1	1	yes	characteristic
59	C₃₁×#332	C₂²	1	1	yes	characteristic
60	C₃₁×#332	C₂²	1	1	yes	characteristic
61	C₃₁×#332	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×Q₁₆×C₃₁	—	2	1	no	—
69	C₂×Q₁₆×C₃₁	—	2	1	no	—
70	C₂×Q₁₆×C₃₁	—	2	1	no	—
71	C₂×Q₁₆×C₃₁	—	2	1	no	—
72	C₂×SD₁₆×C₃₁	—	2	1	no	—
73	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
74	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
75	C₂²×C₈×C₃₁	—	2	1	no	—
76	C₂²×C₈×C₃₁	—	2	1	no	—
77	C₃₁×C₄⋊Q₈	—	2	1	no	—
78	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₃₁×#126	—	4	1	no	—
83	C₃₁×#125	—	2	1	no	—
84	C₃₁×#125	—	2	1	no	—
85	C₃₁×#125	—	4	1	no	—
86	C₃₁×#124	—	2	1	no	—
87	C₃₁×#123	—	4	1	no	—
88	C₃₁×C₂²⋊Q₈	—	2	1	no	—
89	C₃₁×C₂²⋊Q₈	—	2	1	no	—
90	C₃₁×C₂²⋊Q₈	—	2	1	no	—
91	C₃₁×C₂²⋊Q₈	—	4	1	no	—
92	C₃₁×#121	—	2	1	no	—
93	C₄×Q₈×C₃₁	—	2	1	no	—
94	C₄×Q₈×C₃₁	—	2	1	no	—
95	C₄×Q₈×C₃₁	—	2	1	no	—
96	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
97	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
98	C₄×D₄×C₃₁	—	2	1	no	—
99	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
101	C₃₁×#117	—	2	1	no	—
102	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
104	C₃₁×#107	—	2	1	no	—
105	C₃₁×#107	—	2	1	no	—
106	C₃₁×#107	—	2	1	no	—
107	C₃₁×#106	—	2	1	no	—
108	C₃₁×#106	—	2	1	no	—
109	C₃₁×#106	—	2	1	no	—
110	C₃₁×#106	—	2	1	no	—
111	C₃₁×#106	—	2	1	no	—
112	C₃₁×#106	—	2	1	no	—
113	C₃₁×#106	—	4	1	no	—
114	C₃₁×#106	D₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
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×#103	—	2	1	no	—
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	—	4	1	no	—
130	C₃₁×#103	D₄	1	1	yes	characteristic
131	C₃₁×#103	D₄	1	1	yes	characteristic
132	C₃₁×#102	—	2	1	no	—
133	C₃₁×#102	—	2	1	no	—
134	C₃₁×#102	—	2	1	no	—
135	C₃₁×#102	D₄	1	1	yes	characteristic
136	C₃₁×#102	D₄	1	1	yes	characteristic
137	C₃₁×C₂²⋊C₈	—	2	1	no	—
138	C₃₁×C₂²⋊C₈	—	2	1	no	—
139	C₃₁×#97	—	2	1	no	—
140	C₄×C₈×C₃₁	—	2	1	no	—
141	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
144	#41×C₃₁	—	4	2	no	—
145	#41×C₃₁	—	8	1	no	—
146	C₂×Q₈×C₃₁	—	2	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	C₂×D₄	1	1	yes	characteristic
151	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₂×D₄×C₃₁	—	4	1	no	—
153	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	—	2	1	no	—
156	C₂²×C₄×C₃₁	—	2	1	no	—
157	C₂²×C₄×C₃₁	—	4	1	no	—
158	C₂²×C₄×C₃₁	—	4	1	no	—
159	Q₁₆×C₃₁	—	4	2	no	—
160	Q₁₆×C₃₁	—	8	1	no	—
161	Q₁₆×C₃₁	—	8	1	no	—
162	Q₁₆×C₃₁	—	8	1	no	—
163	SD₁₆×C₃₁	—	2	2	no	—
164	SD₁₆×C₃₁	—	2	2	no	—
165	SD₁₆×C₃₁	—	8	1	no	—
166	C₂×C₈×C₃₁	—	2	1	no	—
167	C₂×C₈×C₃₁	—	2	1	no	—
168	C₂×C₈×C₃₁	—	2	1	no	—
169	C₂×C₈×C₃₁	—	2	1	no	—
170	C₂×C₈×C₃₁	—	2	1	no	—
171	C₂×C₈×C₃₁	—	4	2	no	—
172	C₂×C₈×C₃₁	—	8	1	no	—
173	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	2	1	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	—	2	1	no	—
181	C₄⋊C₄×C₃₁	—	2	1	no	—
182	C₄⋊C₄×C₃₁	—	2	1	no	—
183	C₄⋊C₄×C₃₁	—	2	1	no	—
184	C₄⋊C₄×C₃₁	—	2	1	no	—
185	C₄⋊C₄×C₃₁	—	2	1	no	—
186	C₄⋊C₄×C₃₁	—	2	1	no	—
187	C₄⋊C₄×C₃₁	—	4	1	no	—
188	C₄⋊C₄×C₃₁	—	4	1	no	—
189	C₄⋊C₄×C₃₁	—	4	1	no	—
190	C₄⋊C₄×C₃₁	—	4	1	no	—
191	C₄⋊C₄×C₃₁	—	4	1	no	—
192	C₄⋊C₄×C₃₁	—	4	1	no	—
193	C₄⋊C₄×C₃₁	—	4	1	no	—
194	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
196	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
197	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
198	C₂²⋊C₄×C₃₁	—	2	1	no	—
199	C₂²⋊C₄×C₃₁	—	2	1	no	—
200	C₂²⋊C₄×C₃₁	—	4	1	no	—
201	C₂²⋊C₄×C₃₁	—	4	1	no	—
202	C₂²⋊C₄×C₃₁	—	4	1	no	—
203	C₂²⋊C₄×C₃₁	—	4	1	no	—
204	C₄²×C₃₁	—	2	1	no	—
205	C₄²×C₃₁	—	2	1	no	—
206	C₄²×C₃₁	—	4	1	no	—
207	C₄²×C₃₁	—	4	1	no	—
208	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
209	#19853	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂³×C₃₁	—	2	1	no	—
211	C₂³×C₃₁	—	2	1	no	—
212	Q₈×C₃₁	—	2	1	no	—
213	Q₈×C₃₁	—	2	1	no	—
214	Q₈×C₃₁	—	2	1	no	—
215	Q₈×C₃₁	—	2	1	no	—
216	Q₈×C₃₁	—	4	1	no	—
217	Q₈×C₃₁	—	4	1	no	—
218	Q₈×C₃₁	—	4	1	no	—
219	Q₈×C₃₁	—	4	1	no	—
220	Q₈×C₃₁	—	4	1	no	—
221	Q₈×C₃₁	—	4	1	no	—
222	Q₈×C₃₁	—	8	1	no	—
223	D₄×C₃₁	—	2	1	no	—
224	D₄×C₃₁	—	2	1	no	—
225	D₄×C₃₁	—	2	1	no	—
226	D₄×C₃₁	—	8	2	no	—
227	D₄×C₃₁	#137	1	2	yes	—
228	C₂×C₄×C₃₁	—	2	1	no	—
229	C₂×C₄×C₃₁	—	2	1	no	—
230	C₂×C₄×C₃₁	—	2	1	no	—
231	C₂×C₄×C₃₁	—	2	1	no	—
232	C₂×C₄×C₃₁	—	2	1	no	—
233	C₂×C₄×C₃₁	—	2	1	no	—
234	C₂×C₄×C₃₁	—	4	1	no	—
235	C₂×C₄×C₃₁	—	4	1	no	—
236	C₂×C₄×C₃₁	—	4	1	no	—
237	C₂×C₄×C₃₁	—	4	1	no	—
238	C₂×C₄×C₃₁	—	4	1	no	—
239	C₂×C₄×C₃₁	—	4	1	no	—
240	C₂×C₄×C₃₁	—	4	1	no	—
241	C₂×C₄×C₃₁	—	4	1	no	—
242	C₂×C₄×C₃₁	—	8	2	no	—
243	C₂×C₄×C₃₁	—	16	1	no	—
244	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
247	C₈×C₃₁	—	2	1	no	—
248	C₈×C₃₁	—	2	1	no	—
249	C₈×C₃₁	—	2	1	no	—
250	C₈×C₃₁	—	4	1	no	—
251	C₈×C₃₁	—	4	1	no	—
252	C₈×C₃₁	—	4	1	no	—
253	C₈×C₃₁	—	4	1	no	—
254	C₈×C₃₁	—	4	1	no	—
255	C₈×C₃₁	#135	1	2	yes	—
256	#3398	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#3381	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#3335	C₂×C₃₁	1	1	yes	characteristic
259	#3302	C₂×C₃₁	1	1	yes	characteristic
260	#1716	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#1692	C₂×C₃₁	1	1	yes	characteristic, complemented
262	#1680	C₂×C₃₁	1	1	yes	characteristic, complemented
263	#1671	C₂×C₃₁	1	1	yes	characteristic, complemented
264	#1669	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#1654	C₂×C₃₁	1	1	yes	characteristic
266	#1653	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#1644	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#1638	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#1582	C₂×C₃₁	1	1	yes	characteristic, complemented
270	SD₁₆⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
271	C₂²×C₃₁	—	2	2	no	—
272	C₂²×C₃₁	—	4	1	no	—
273	C₂²×C₃₁	—	4	1	no	—
274	C₂²×C₃₁	—	4	1	no	—
275	C₂²×C₃₁	—	4	1	no	—
276	C₂²×C₃₁	—	4	1	no	—
277	C₂²×C₃₁	D₄²	1	1	yes	characteristic
278	C₄×C₃₁	—	2	1	no	—
279	C₄×C₃₁	—	4	1	no	—
280	C₄×C₃₁	—	4	1	no	—
281	C₄×C₃₁	—	4	1	no	—
282	C₄×C₃₁	—	4	1	no	—
283	C₄×C₃₁	—	4	1	no	—
284	C₄×C₃₁	—	4	1	no	—
285	C₄×C₃₁	—	8	1	no	—
286	C₄×C₃₁	—	8	1	no	—
287	C₄×C₃₁	—	8	1	no	—
288	C₄×C₃₁	—	8	1	no	—
289	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
290	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
291	C₄×C₃₁	#577	1	1	yes	characteristic
292	C₄×C₃₁	#577	1	1	yes	characteristic
293	#557	C₂²×C₃₁	1	1	yes	characteristic
294	#556	C₂²×C₃₁	1	1	yes	characteristic
295	#554	C₂²×C₃₁	1	1	yes	characteristic
296	#547	C₂²×C₃₁	1	1	yes	characteristic
297	#499	—	2	1	no	—
298	#498	C₂²×C₃₁	1	1	yes	characteristic
299	#498	C₂²×C₃₁	1	1	yes	characteristic
300	#498	C₂²×C₃₁	1	1	yes	characteristic
301	#497	—	2	1	no	—
302	#494	C₂²×C₃₁	1	1	yes	characteristic
303	#487	C₂²×C₃₁	1	1	yes	characteristic
304	#487	C₂²×C₃₁	1	1	yes	characteristic
305	#485	—	2	1	no	—
306	#483	—	2	1	no	—
307	#481	—	2	1	no	—
308	#479	—	2	1	no	—
309	#478	—	2	1	no	—
310	#477	C₂²×C₃₁	1	1	yes	characteristic
311	#476	C₂²×C₃₁	1	1	yes	characteristic
312	#475	C₂²×C₃₁	1	1	yes	characteristic
313	#474	C₂²×C₃₁	1	1	yes	characteristic
314	#474	C₂²×C₃₁	1	1	yes	characteristic
315	#467	—	2	1	no	—
316	#465	—	2	1	no	—
317	#464	—	2	1	no	—
318	#462	—	2	1	no	—
319	#462	C₂²×C₃₁	1	1	yes	characteristic
320	#462	C₂²×C₃₁	1	1	yes	characteristic
321	#461	C₂²×C₃₁	1	1	yes	characteristic
322	#452	—	2	1	no	—
323	#446	C₂²×C₃₁	1	1	yes	characteristic
324	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
325	#441	—	2	1	no	—
326	C₄×SD₁₆	—	2	1	no	—
327	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
328	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
329	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
330	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×#106	—	2	1	no	—
332	#416	—	2	1	no	—
333	#335	C₂²×C₃₁	1	1	yes	characteristic
334	#334	C₂²×C₃₁	1	1	yes	characteristic
335	#333	C₂²×C₃₁	1	1	yes	characteristic
336	#332	C₂²×C₃₁	1	1	yes	characteristic
337	#332	C₂²×C₃₁	1	1	yes	characteristic
338	#332	C₂²×C₃₁	1	1	yes	characteristic
339	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
340	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
341	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
342	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
343	#322	C₂²×C₃₁	1	1	yes	characteristic
344	C₂×C₃₁	—	4	2	no	—
345	C₂×C₃₁	—	8	1	no	—
346	C₂×C₃₁	#3266	1	1	yes	characteristic
347	C₂×C₃₁	#3284	1	1	yes	characteristic
348	C₂×C₃₁	#3289	1	1	yes	characteristic
349	C₂×#41	—	2	1	no	—
350	C₂×Q₁₆	—	2	1	no	—
351	C₂×Q₁₆	—	2	1	no	—
352	C₂×Q₁₆	—	2	1	no	—
353	C₂×Q₁₆	—	2	1	no	—
354	C₂×SD₁₆	—	2	1	no	—
355	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
356	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
357	C₂²×C₈	—	2	1	no	—
358	C₂²×C₈	—	2	1	no	—
359	C₄⋊Q₈	—	2	1	no	—
360	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
361	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
362	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
363	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
364	#126	—	4	1	no	—
365	#125	—	2	1	no	—
366	#125	—	2	1	no	—
367	#125	—	4	1	no	—
368	#124	—	2	1	no	—
369	#123	—	4	1	no	—
370	C₂²⋊Q₈	—	2	1	no	—
371	C₂²⋊Q₈	—	2	1	no	—
372	C₂²⋊Q₈	—	2	1	no	—
373	C₂²⋊Q₈	—	4	1	no	—
374	#121	—	2	1	no	—
375	C₄×Q₈	—	2	1	no	—
376	C₄×Q₈	—	2	1	no	—
377	C₄×Q₈	—	2	1	no	—
378	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
379	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
380	C₄×D₄	—	2	1	no	—
381	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
382	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
383	#117	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	1	no	—
385	C₂×C₄⋊C₄	—	2	1	no	—
386	#107	—	2	1	no	—
387	#107	—	2	1	no	—
388	#107	—	2	1	no	—
389	#106	—	2	1	no	—
390	#106	—	2	1	no	—
391	#106	—	2	1	no	—
392	#106	—	2	1	no	—
393	#106	—	2	1	no	—
394	#106	—	2	1	no	—
395	#106	—	4	1	no	—
396	#106	D₄×C₃₁	1	1	yes	characteristic
397	#106	D₄×C₃₁	1	1	yes	characteristic
398	C₄⋊C₈	—	2	1	no	—
399	C₄⋊C₈	—	2	1	no	—
400	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
401	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
402	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
403	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
404	#103	—	2	1	no	—
405	#103	—	2	1	no	—
406	#103	—	2	1	no	—
407	#103	—	2	1	no	—
408	#103	—	2	1	no	—
409	#103	—	2	1	no	—
410	#103	—	2	1	no	—
411	#103	—	4	1	no	—
412	#103	D₄×C₃₁	1	1	yes	characteristic
413	#103	D₄×C₃₁	1	1	yes	characteristic
414	#102	—	2	1	no	—
415	#102	—	2	1	no	—
416	#102	—	2	1	no	—
417	#102	D₄×C₃₁	1	1	yes	characteristic
418	#102	D₄×C₃₁	1	1	yes	characteristic
419	C₂²⋊C₈	—	2	1	no	—
420	C₂²⋊C₈	—	2	1	no	—
421	#97	—	2	1	no	—
422	C₄×C₈	—	2	1	no	—
423	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
424	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
425	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
426	C₃₁	#19853	1	1	yes	characteristic, direct product factor, Sylow
427	#41	—	4	2	no	—
428	#41	—	8	1	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	—	2	1	no	—
431	C₂×Q₈	—	2	1	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
434	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
435	C₂×D₄	—	4	1	no	—
436	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
437	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	4	1	no	—
441	C₂²×C₄	—	4	1	no	—
442	Q₁₆	—	4	2	no	—
443	Q₁₆	—	8	1	no	—
444	Q₁₆	—	8	1	no	—
445	Q₁₆	—	8	1	no	—
446	SD₁₆	—	2	2	no	—
447	SD₁₆	—	2	2	no	—
448	SD₁₆	—	8	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	4	2	no	—
455	C₂×C₈	—	8	1	no	—
456	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
457	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
458	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
459	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	4	1	no	—
471	C₄⋊C₄	—	4	1	no	—
472	C₄⋊C₄	—	4	1	no	—
473	C₄⋊C₄	—	4	1	no	—
474	C₄⋊C₄	—	4	1	no	—
475	C₄⋊C₄	—	4	1	no	—
476	C₄⋊C₄	—	4	1	no	—
477	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
478	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
479	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
480	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
481	C₂²⋊C₄	—	2	1	no	—
482	C₂²⋊C₄	—	2	1	no	—
483	C₂²⋊C₄	—	4	1	no	—
484	C₂²⋊C₄	—	4	1	no	—
485	C₂²⋊C₄	—	4	1	no	—
486	C₂²⋊C₄	—	4	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	—	2	1	no	—
489	C₄²	—	4	1	no	—
490	C₄²	—	4	1	no	—
491	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
492	C₂³	—	2	1	no	—
493	C₂³	—	2	1	no	—
494	Q₈	—	2	1	no	—
495	Q₈	—	2	1	no	—
496	Q₈	—	2	1	no	—
497	Q₈	—	2	1	no	—
498	Q₈	—	4	1	no	—
499	Q₈	—	4	1	no	—
500	Q₈	—	4	1	no	—
501	Q₈	—	4	1	no	—
502	Q₈	—	4	1	no	—
503	Q₈	—	4	1	no	—
504	Q₈	—	8	1	no	—
505	D₄	—	2	1	no	—
506	D₄	—	2	1	no	—
507	D₄	—	2	1	no	—
508	D₄	—	8	2	no	—
509	D₄	C₃₁×#137	1	2	yes	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	4	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	8	2	no	—
525	C₂×C₄	—	16	1	no	—
526	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
528	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
529	C₈	—	2	1	no	—
530	C₈	—	2	1	no	—
531	C₈	—	2	1	no	—
532	C₈	—	4	1	no	—
533	C₈	—	4	1	no	—
534	C₈	—	4	1	no	—
535	C₈	—	4	1	no	—
536	C₈	—	4	1	no	—
537	C₈	C₃₁×#135	1	2	yes	—
538	C₂²	—	2	2	no	—
539	C₂²	—	4	1	no	—
540	C₂²	—	4	1	no	—
541	C₂²	—	4	1	no	—
542	C₂²	—	4	1	no	—
543	C₂²	—	4	1	no	—
544	C₂²	D₄²×C₃₁	1	1	yes	characteristic
545	C₄	—	2	1	no	—
546	C₄	—	4	1	no	—
547	C₄	—	4	1	no	—
548	C₄	—	4	1	no	—
549	C₄	—	4	1	no	—
550	C₄	—	4	1	no	—
551	C₄	—	4	1	no	—
552	C₄	—	8	1	no	—
553	C₄	—	8	1	no	—
554	C₄	—	8	1	no	—
555	C₄	—	8	1	no	—
556	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
557	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
558	C₄	C₃₁×#577	1	1	yes	characteristic
559	C₄	C₃₁×#577	1	1	yes	characteristic
560	C₂	—	4	2	no	—
561	C₂	—	8	1	no	—
562	C₂	C₃₁×#3266	1	1	yes	characteristic
563	C₂	C₃₁×#3284	1	1	yes	characteristic
564	C₂	C₃₁×#3289	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#547	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
1. C₃₁×#3398	C₂	semidirect
2. C₃₁×#3381	C₂	semidirect
5. C₃₁×#1716	C₂	semidirect
6. C₃₁×#1692	C₂	semidirect
7. C₃₁×#1680	C₂	semidirect
8. C₃₁×#1671	C₂	semidirect
9. C₃₁×#1669	C₂	semidirect
11. C₃₁×#1653	C₂	semidirect
12. C₃₁×#1644	C₂	semidirect
13. C₃₁×#1638	C₂	semidirect
14. C₃₁×#1582	C₂	semidirect
15. C₃₁×SD₁₆⋊C₈	C₂	semidirect
209. #19853	C₃₁	direct (N × H)
256. #3398	C₂×C₃₁	semidirect
257. #3381	C₂×C₃₁	semidirect
260. #1716	C₂×C₃₁	semidirect
261. #1692	C₂×C₃₁	semidirect
262. #1680	C₂×C₃₁	semidirect
263. #1671	C₂×C₃₁	semidirect
264. #1669	C₂×C₃₁	semidirect
266. #1653	C₂×C₃₁	semidirect
267. #1644	C₂×C₃₁	semidirect
268. #1638	C₂×C₃₁	semidirect
269. #1582	C₂×C₃₁	semidirect
270. SD₁₆⋊C₈	C₂×C₃₁	semidirect
426. C₃₁	#19853	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×#3398	2
C₃₁×#3381	2
C₃₁×#3335	2
C₃₁×#3302	2
C₃₁×#1716	2
C₃₁×#1692	2
C₃₁×#1680	2
C₃₁×#1671	2
C₃₁×#1669	2
C₃₁×#1654	2
C₃₁×#1653	2
C₃₁×#1644	2
C₃₁×#1638	2
C₃₁×#1582	2
C₃₁×SD₁₆⋊C₈	2
#19853	31

Subgroups — C31×#19853 #21476612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19853 #21476612