Subgroups — C₃₁×#20000 #21476759

Order 7936.

592 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×#3365	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3361	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3359	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3299	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1711	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1684	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1681	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1680	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1678	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1643	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1640	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1636	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1627	C₂	1	1	yes	maximal, characteristic
14	C₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#559	C₂²	1	1	yes	characteristic
17	C₃₁×#557	C₂²	1	1	yes	characteristic
18	C₃₁×#554	C₂²	1	1	yes	characteristic
19	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#488	—	2	1	no	—
21	C₃₁×#488	—	2	1	no	—
22	C₃₁×#487	—	2	1	no	—
23	C₃₁×#487	C₂²	1	1	yes	characteristic
24	C₃₁×#487	C₂²	1	1	yes	characteristic
25	C₃₁×#487	C₂²	1	1	yes	characteristic
26	C₃₁×#486	C₂²	1	1	yes	characteristic
27	C₃₁×#486	C₂²	1	1	yes	characteristic
28	C₃₁×#486	C₂²	1	1	yes	characteristic
29	C₃₁×#484	—	2	1	no	—
30	C₃₁×#478	—	2	1	no	—
31	C₃₁×#478	—	2	1	no	—
32	C₃₁×#477	C₂²	1	1	yes	characteristic
33	C₃₁×#477	C₂²	1	1	yes	characteristic
34	C₃₁×#475	—	2	1	no	—
35	C₃₁×#475	C₂²	1	1	yes	characteristic
36	C₃₁×#475	C₂²	1	1	yes	characteristic
37	C₃₁×#467	—	2	1	no	—
38	C₃₁×#464	—	2	1	no	—
39	C₃₁×#463	—	2	1	no	—
40	C₃₁×#462	C₂²	1	1	yes	characteristic
41	C₃₁×#461	—	2	1	no	—
42	C₃₁×#461	C₂²	1	1	yes	characteristic
43	C₃₁×#460	C₂²	1	1	yes	characteristic
44	C₃₁×#459	C₂²	1	1	yes	characteristic
45	C₃₁×#451	—	2	1	no	—
46	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
47	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
48	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
49	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
50	C₄×SD₁₆×C₃₁	—	2	1	no	—
51	C₄×SD₁₆×C₃₁	—	2	1	no	—
52	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
53	C₄×D₈×C₃₁	—	2	1	no	—
54	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
55	C₂×C₃₁×#103	—	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×#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	C₈²×C₃₁	C₂²	1	1	yes	characteristic
67	C₂²×Q₈×C₃₁	—	2	1	no	—
68	C₂×Q₁₆×C₃₁	—	2	1	no	—
69	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
70	C₂×SD₁₆×C₃₁	—	2	1	no	—
71	C₂×SD₁₆×C₃₁	—	2	1	no	—
72	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
73	C₂×D₈×C₃₁	—	2	1	no	—
74	C₂²×C₈×C₃₁	—	2	1	no	—
75	C₃₁×C₄⋊Q₈	—	2	1	no	—
76	C₃₁×C₄⋊Q₈	—	2	1	no	—
77	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
78	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₃₁×#127	C₂³	1	1	yes	characteristic
81	C₃₁×#125	—	2	1	no	—
82	C₃₁×#125	—	2	1	no	—
83	C₃₁×#125	—	4	1	no	—
84	C₃₁×#124	—	2	1	no	—
85	C₃₁×#124	—	2	1	no	—
86	C₃₁×#123	—	4	1	no	—
87	C₃₁×C₂²⋊Q₈	—	2	1	no	—
88	C₃₁×C₂²⋊Q₈	—	2	1	no	—
89	C₃₁×C₂²⋊Q₈	—	4	1	no	—
90	C₃₁×C₂²⋊Q₈	—	4	1	no	—
91	C₄×Q₈×C₃₁	—	2	1	no	—
92	C₄×Q₈×C₃₁	—	2	1	no	—
93	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
96	C₄×D₄×C₃₁	—	2	1	no	—
97	C₄×D₄×C₃₁	—	2	1	no	—
98	C₄×D₄×C₃₁	—	2	1	no	—
99	C₄×D₄×C₃₁	—	2	1	no	—
100	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
102	C₃₁×#107	—	2	1	no	—
103	C₃₁×#107	—	2	1	no	—
104	C₃₁×#107	D₄	1	1	yes	characteristic
105	C₃₁×#106	—	2	1	no	—
106	C₃₁×#106	—	2	1	no	—
107	C₃₁×#106	D₄	1	1	yes	characteristic
108	C₃₁×C₄⋊C₈	—	2	1	no	—
109	C₃₁×C₄⋊C₈	—	2	1	no	—
110	C₃₁×C₄⋊C₈	—	2	1	no	—
111	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
114	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
115	C₃₁×#103	—	2	1	no	—
116	C₃₁×#103	—	2	1	no	—
117	C₃₁×#103	—	2	1	no	—
118	C₃₁×#103	—	2	1	no	—
119	C₃₁×#103	—	2	1	no	—
120	C₃₁×#103	—	2	1	no	—
121	C₃₁×#103	—	2	1	no	—
122	C₃₁×#103	—	2	1	no	—
123	C₃₁×#103	—	2	1	no	—
124	C₃₁×#103	—	2	1	no	—
125	C₃₁×#103	—	4	1	no	—
126	C₃₁×#103	D₄	1	1	yes	characteristic
127	C₃₁×#103	D₄	1	1	yes	characteristic
128	C₃₁×#103	D₄	1	1	yes	characteristic
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	—	4	1	no	—
136	C₃₁×#102	D₄	1	1	yes	characteristic
137	C₃₁×C₂²⋊C₈	—	2	1	no	—
138	C₄×C₈×C₃₁	—	2	1	no	—
139	C₄×C₈×C₃₁	—	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	C₂×Q₈×C₃₁	—	2	1	no	—
145	C₂×Q₈×C₃₁	—	2	1	no	—
146	C₂×Q₈×C₃₁	—	2	1	no	—
147	C₂×Q₈×C₃₁	—	4	1	no	—
148	C₂×Q₈×C₃₁	—	4	2	no	—
149	C₂×Q₈×C₃₁	—	8	1	no	—
150	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₂×D₄×C₃₁	—	2	1	no	—
154	C₂×D₄×C₃₁	—	2	1	no	—
155	C₂×D₄×C₃₁	—	2	1	no	—
156	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	—	2	1	no	—
158	C₂²×C₄×C₃₁	—	4	1	no	—
159	C₂²×C₄×C₃₁	—	4	1	no	—
160	C₂²×C₄×C₃₁	—	4	1	no	—
161	Q₁₆×C₃₁	—	2	2	no	—
162	Q₁₆×C₃₁	—	8	1	no	—
163	SD₁₆×C₃₁	—	2	2	no	—
164	SD₁₆×C₃₁	—	4	2	no	—
165	SD₁₆×C₃₁	—	4	2	no	—
166	D₈×C₃₁	—	4	2	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	—	2	1	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	—	4	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	C₂×D₄	1	1	yes	characteristic
193	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
196	C₂²⋊C₄×C₃₁	—	2	1	no	—
197	C₂²⋊C₄×C₃₁	—	4	1	no	—
198	C₂²⋊C₄×C₃₁	—	4	1	no	—
199	C₂²⋊C₄×C₃₁	—	4	1	no	—
200	C₂²⋊C₄×C₃₁	—	4	1	no	—
201	C₂²⋊C₄×C₃₁	—	4	1	no	—
202	C₄²×C₃₁	—	2	1	no	—
203	C₄²×C₃₁	—	2	1	no	—
204	C₄²×C₃₁	—	2	1	no	—
205	C₄²×C₃₁	—	4	1	no	—
206	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
207	#20000	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂³×C₃₁	—	2	1	no	—
209	C₂³×C₃₁	—	4	1	no	—
210	Q₈×C₃₁	—	2	1	no	—
211	Q₈×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	—	4	1	no	—
216	Q₈×C₃₁	—	4	1	no	—
217	Q₈×C₃₁	—	4	1	no	—
218	Q₈×C₃₁	—	4	1	no	—
219	Q₈×C₃₁	—	8	2	no	—
220	Q₈×C₃₁	—	8	1	no	—
221	Q₈×C₃₁	#135	1	2	yes	—
222	D₄×C₃₁	—	2	1	no	—
223	D₄×C₃₁	—	4	1	no	—
224	D₄×C₃₁	—	4	1	no	—
225	D₄×C₃₁	—	4	1	no	—
226	D₄×C₃₁	—	4	1	no	—
227	D₄×C₃₁	—	8	1	no	—
228	D₄×C₃₁	#135	1	2	yes	—
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	—	2	1	no	—
235	C₂×C₄×C₃₁	—	2	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	—	8	2	no	—
242	C₂×C₄×C₃₁	—	16	1	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	—	2	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	—	4	1	no	—
256	#3365	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#3361	C₂×C₃₁	1	1	yes	characteristic
258	#3359	C₂×C₃₁	1	1	yes	characteristic, complemented
259	#3299	C₂×C₃₁	1	1	yes	characteristic, complemented
260	#1711	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#1684	C₂×C₃₁	1	1	yes	characteristic, complemented
262	#1681	C₂×C₃₁	1	1	yes	characteristic
263	#1680	C₂×C₃₁	1	1	yes	characteristic, complemented
264	#1678	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#1643	C₂×C₃₁	1	1	yes	characteristic, complemented
266	#1640	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#1636	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#1627	C₂×C₃₁	1	1	yes	characteristic
269	C₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
270	C₈×SD₁₆	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	—	8	1	no	—
275	C₂²×C₃₁	—	8	1	no	—
276	C₂²×C₃₁	—	8	1	no	—
277	C₂²×C₃₁	D₄²	1	1	yes	characteristic
278	C₄×C₃₁	—	2	2	no	—
279	C₄×C₃₁	—	2	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	—	4	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×#135	1	1	yes	characteristic
291	C₄×C₃₁	#577	1	1	yes	characteristic
292	C₄×C₃₁	#578	1	1	yes	characteristic
293	#559	C₂²×C₃₁	1	1	yes	characteristic
294	#557	C₂²×C₃₁	1	1	yes	characteristic
295	#554	C₂²×C₃₁	1	1	yes	characteristic
296	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
297	#488	—	2	1	no	—
298	#488	—	2	1	no	—
299	#487	—	2	1	no	—
300	#487	C₂²×C₃₁	1	1	yes	characteristic
301	#487	C₂²×C₃₁	1	1	yes	characteristic
302	#487	C₂²×C₃₁	1	1	yes	characteristic
303	#486	C₂²×C₃₁	1	1	yes	characteristic
304	#486	C₂²×C₃₁	1	1	yes	characteristic
305	#486	C₂²×C₃₁	1	1	yes	characteristic
306	#484	—	2	1	no	—
307	#478	—	2	1	no	—
308	#478	—	2	1	no	—
309	#477	C₂²×C₃₁	1	1	yes	characteristic
310	#477	C₂²×C₃₁	1	1	yes	characteristic
311	#475	—	2	1	no	—
312	#475	C₂²×C₃₁	1	1	yes	characteristic
313	#475	C₂²×C₃₁	1	1	yes	characteristic
314	#467	—	2	1	no	—
315	#464	—	2	1	no	—
316	#463	—	2	1	no	—
317	#462	C₂²×C₃₁	1	1	yes	characteristic
318	#461	—	2	1	no	—
319	#461	C₂²×C₃₁	1	1	yes	characteristic
320	#460	C₂²×C₃₁	1	1	yes	characteristic
321	#459	C₂²×C₃₁	1	1	yes	characteristic
322	#451	—	2	1	no	—
323	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
324	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
325	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
326	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
327	C₄×SD₁₆	—	2	1	no	—
328	C₄×SD₁₆	—	2	1	no	—
329	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
330	C₄×D₈	—	2	1	no	—
331	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
332	C₂×#103	—	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	#331	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	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
342	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
343	C₈²	C₂²×C₃₁	1	1	yes	characteristic
344	C₂×C₃₁	—	4	2	no	—
345	C₂×C₃₁	—	16	1	no	—
346	C₂×C₃₁	#3282	1	1	yes	characteristic
347	C₂×C₃₁	#3285	1	1	yes	characteristic
348	C₂×C₃₁	#3290	1	1	yes	characteristic
349	C₂²×Q₈	—	2	1	no	—
350	C₂×Q₁₆	—	2	1	no	—
351	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
352	C₂×SD₁₆	—	2	1	no	—
353	C₂×SD₁₆	—	2	1	no	—
354	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
355	C₂×D₈	—	2	1	no	—
356	C₂²×C₈	—	2	1	no	—
357	C₄⋊Q₈	—	2	1	no	—
358	C₄⋊Q₈	—	2	1	no	—
359	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
360	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
361	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
362	#127	C₂³×C₃₁	1	1	yes	characteristic
363	#125	—	2	1	no	—
364	#125	—	2	1	no	—
365	#125	—	4	1	no	—
366	#124	—	2	1	no	—
367	#124	—	2	1	no	—
368	#123	—	4	1	no	—
369	C₂²⋊Q₈	—	2	1	no	—
370	C₂²⋊Q₈	—	2	1	no	—
371	C₂²⋊Q₈	—	4	1	no	—
372	C₂²⋊Q₈	—	4	1	no	—
373	C₄×Q₈	—	2	1	no	—
374	C₄×Q₈	—	2	1	no	—
375	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
376	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
377	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
378	C₄×D₄	—	2	1	no	—
379	C₄×D₄	—	2	1	no	—
380	C₄×D₄	—	2	1	no	—
381	C₄×D₄	—	2	1	no	—
382	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
383	C₂×C₄⋊C₄	—	2	1	no	—
384	#107	—	2	1	no	—
385	#107	—	2	1	no	—
386	#107	D₄×C₃₁	1	1	yes	characteristic
387	#106	—	2	1	no	—
388	#106	—	2	1	no	—
389	#106	D₄×C₃₁	1	1	yes	characteristic
390	C₄⋊C₈	—	2	1	no	—
391	C₄⋊C₈	—	2	1	no	—
392	C₄⋊C₈	—	2	1	no	—
393	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
394	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
395	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
396	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
397	#103	—	2	1	no	—
398	#103	—	2	1	no	—
399	#103	—	2	1	no	—
400	#103	—	2	1	no	—
401	#103	—	2	1	no	—
402	#103	—	2	1	no	—
403	#103	—	2	1	no	—
404	#103	—	2	1	no	—
405	#103	—	2	1	no	—
406	#103	—	2	1	no	—
407	#103	—	4	1	no	—
408	#103	D₄×C₃₁	1	1	yes	characteristic
409	#103	D₄×C₃₁	1	1	yes	characteristic
410	#103	D₄×C₃₁	1	1	yes	characteristic
411	#102	—	2	1	no	—
412	#102	—	2	1	no	—
413	#102	—	2	1	no	—
414	#102	—	2	1	no	—
415	#102	—	2	1	no	—
416	#102	—	2	1	no	—
417	#102	—	4	1	no	—
418	#102	D₄×C₃₁	1	1	yes	characteristic
419	C₂²⋊C₈	—	2	1	no	—
420	C₄×C₈	—	2	1	no	—
421	C₄×C₈	—	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	#20000	1	1	yes	characteristic, direct product factor, Sylow
427	C₂×Q₈	—	2	1	no	—
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	—	4	1	no	—
431	C₂×Q₈	—	4	2	no	—
432	C₂×Q₈	—	8	1	no	—
433	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
434	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
435	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
436	C₂×D₄	—	2	1	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	4	1	no	—
442	C₂²×C₄	—	4	1	no	—
443	C₂²×C₄	—	4	1	no	—
444	Q₁₆	—	2	2	no	—
445	Q₁₆	—	8	1	no	—
446	SD₁₆	—	2	2	no	—
447	SD₁₆	—	4	2	no	—
448	SD₁₆	—	4	2	no	—
449	D₈	—	4	2	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	—	2	1	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	—	4	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	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
477	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
478	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
479	C₂²⋊C₄	—	2	1	no	—
480	C₂²⋊C₄	—	4	1	no	—
481	C₂²⋊C₄	—	4	1	no	—
482	C₂²⋊C₄	—	4	1	no	—
483	C₂²⋊C₄	—	4	1	no	—
484	C₂²⋊C₄	—	4	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	2	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	—	4	1	no	—
489	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
490	C₂³	—	2	1	no	—
491	C₂³	—	4	1	no	—
492	Q₈	—	2	1	no	—
493	Q₈	—	2	1	no	—
494	Q₈	—	2	1	no	—
495	Q₈	—	2	1	no	—
496	Q₈	—	2	1	no	—
497	Q₈	—	4	1	no	—
498	Q₈	—	4	1	no	—
499	Q₈	—	4	1	no	—
500	Q₈	—	4	1	no	—
501	Q₈	—	8	2	no	—
502	Q₈	—	8	1	no	—
503	Q₈	C₃₁×#135	1	2	yes	—
504	D₄	—	2	1	no	—
505	D₄	—	4	1	no	—
506	D₄	—	4	1	no	—
507	D₄	—	4	1	no	—
508	D₄	—	4	1	no	—
509	D₄	—	8	1	no	—
510	D₄	C₃₁×#135	1	2	yes	—
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	—	2	1	no	—
517	C₂×C₄	—	2	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	—	8	2	no	—
524	C₂×C₄	—	16	1	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	—	2	1	no	—
533	C₈	—	4	1	no	—
534	C₈	—	4	1	no	—
535	C₈	—	4	1	no	—
536	C₈	—	4	1	no	—
537	C₈	—	4	1	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	4	1	no	—
540	C₂²	—	4	1	no	—
541	C₂²	—	8	1	no	—
542	C₂²	—	8	1	no	—
543	C₂²	—	8	1	no	—
544	C₂²	D₄²×C₃₁	1	1	yes	characteristic
545	C₄	—	2	2	no	—
546	C₄	—	2	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	—	4	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×#135	1	1	yes	characteristic
558	C₄	C₃₁×#577	1	1	yes	characteristic
559	C₄	C₃₁×#578	1	1	yes	characteristic
560	C₂	—	4	2	no	—
561	C₂	—	16	1	no	—
562	C₂	C₃₁×#3282	1	1	yes	characteristic
563	C₂	C₃₁×#3285	1	1	yes	characteristic
564	C₂	C₃₁×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#459	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×#3365	C₂	semidirect
3. C₃₁×#3359	C₂	semidirect
4. C₃₁×#3299	C₂	semidirect
5. C₃₁×#1711	C₂	semidirect
6. C₃₁×#1684	C₂	semidirect
8. C₃₁×#1680	C₂	semidirect
9. C₃₁×#1678	C₂	semidirect
10. C₃₁×#1643	C₂	semidirect
11. C₃₁×#1640	C₂	semidirect
12. C₃₁×#1636	C₂	semidirect
14. C₈×Q₁₆×C₃₁	C₂	semidirect
15. C₈×SD₁₆×C₃₁	C₂	semidirect
207. #20000	C₃₁	direct (N × H)
256. #3365	C₂×C₃₁	semidirect
258. #3359	C₂×C₃₁	semidirect
259. #3299	C₂×C₃₁	semidirect
260. #1711	C₂×C₃₁	semidirect
261. #1684	C₂×C₃₁	semidirect
263. #1680	C₂×C₃₁	semidirect
264. #1678	C₂×C₃₁	semidirect
265. #1643	C₂×C₃₁	semidirect
266. #1640	C₂×C₃₁	semidirect
267. #1636	C₂×C₃₁	semidirect
269. C₈×Q₁₆	C₂×C₃₁	semidirect
270. C₈×SD₁₆	C₂×C₃₁	semidirect
426. C₃₁	#20000	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×#3365	2
C₃₁×#3361	2
C₃₁×#3359	2
C₃₁×#3299	2
C₃₁×#1711	2
C₃₁×#1684	2
C₃₁×#1681	2
C₃₁×#1680	2
C₃₁×#1678	2
C₃₁×#1643	2
C₃₁×#1640	2
C₃₁×#1636	2
C₃₁×#1627	2
C₈×Q₁₆×C₃₁	2
C₈×SD₁₆×C₃₁	2
#20000	31

Subgroups — C31×#20000 #21476759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20000 #21476759