Subgroups — C₃₁×#19982 #21476741

Order 7936.

654 conjugacy classes of subgroups.

The subgroup lattice has 598 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, complemented
2	C₃₁×#3381	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3284	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1705	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1672	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1671	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1666	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1665	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1654	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1653	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1631	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1622	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#135	—	2	1	no	—
17	C₃₁×#557	C₂²	1	1	yes	characteristic
18	C₃₁×#554	C₂²	1	1	yes	characteristic
19	C₃₁×#550	C₂²	1	1	yes	characteristic
20	C₃₁×#547	C₂²	1	1	yes	characteristic
21	C₃₁×#499	—	2	1	no	—
22	C₃₁×#498	C₂²	1	1	yes	characteristic
23	C₃₁×#498	C₂²	1	1	yes	characteristic
24	C₃₁×#498	C₂²	1	1	yes	characteristic
25	C₃₁×#492	—	2	1	no	—
26	C₃₁×#492	C₂²	1	1	yes	characteristic
27	C₃₁×#492	C₂²	1	1	yes	characteristic
28	C₃₁×#492	C₂²	1	1	yes	characteristic
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×#465	—	2	1	no	—
38	C₃₁×#465	—	2	1	no	—
39	C₃₁×#462	C₂²	1	1	yes	characteristic
40	C₃₁×#461	—	2	1	no	—
41	C₃₁×#461	C₂²	1	1	yes	characteristic
42	C₃₁×#460	—	2	1	no	—
43	C₃₁×#460	C₂²	1	1	yes	characteristic
44	C₃₁×#459	C₂²	1	1	yes	characteristic
45	C₃₁×#452	—	2	1	no	—
46	C₃₁×#449	—	2	1	no	—
47	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
48	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
49	C₄×SD₁₆×C₃₁	—	2	1	no	—
50	C₄×SD₁₆×C₃₁	—	2	1	no	—
51	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
52	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
53	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
54	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
55	C₂×C₃₁×#106	—	2	1	no	—
56	C₃₁×#334	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×D₄⋊C₈	C₂²	1	1	yes	characteristic
65	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₈²×C₃₁	C₂²	1	1	yes	characteristic
67	C₂×#41×C₃₁	—	2	1	no	—
68	C₂×#41×C₃₁	—	2	1	no	—
69	C₃₁×#135	—	4	4	no	—
70	C₂×Q₁₆×C₃₁	—	2	1	no	—
71	C₂×SD₁₆×C₃₁	—	2	1	no	—
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	—	4	1	no	—
78	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
79	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
80	C₂×D₈×C₃₁	—	2	1	no	—
81	C₂²×C₈×C₃₁	—	2	1	no	—
82	C₂²×C₈×C₃₁	—	2	1	no	—
83	C₃₁×C₄⋊Q₈	—	2	1	no	—
84	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₃₁×#127	—	2	1	no	—
88	C₃₁×#127	C₂³	1	1	yes	characteristic
89	C₃₁×#125	—	2	1	no	—
90	C₃₁×#125	—	2	1	no	—
91	C₃₁×#125	—	4	1	no	—
92	C₃₁×#123	—	4	1	no	—
93	C₃₁×C₂²⋊Q₈	—	2	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	—	2	1	no	—
98	C₃₁×#121	—	4	1	no	—
99	C₄×Q₈×C₃₁	—	2	1	no	—
100	C₄×Q₈×C₃₁	—	2	1	no	—
101	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
102	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
103	C₄×D₄×C₃₁	—	2	1	no	—
104	C₄×D₄×C₃₁	—	2	1	no	—
105	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
106	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
108	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
109	C₃₁×#107	—	2	1	no	—
110	C₃₁×#107	—	2	1	no	—
111	C₃₁×#106	—	2	1	no	—
112	C₃₁×#106	—	2	1	no	—
113	C₃₁×#106	—	2	1	no	—
114	C₃₁×#106	—	2	1	no	—
115	C₃₁×#106	—	2	1	no	—
116	C₃₁×#106	—	2	1	no	—
117	C₃₁×#106	—	4	1	no	—
118	C₃₁×#106	D₄	1	1	yes	characteristic
119	C₃₁×#106	D₄	1	1	yes	characteristic
120	C₃₁×C₄⋊C₈	—	2	1	no	—
121	C₃₁×C₄⋊C₈	—	2	1	no	—
122	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
123	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
124	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
125	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
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	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	—	2	1	no	—
136	C₃₁×#102	D₄	1	1	yes	characteristic
137	C₃₁×#102	D₄	1	1	yes	characteristic
138	C₃₁×C₂²⋊C₈	—	2	1	no	—
139	C₃₁×C₂²⋊C₈	—	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	—	4	2	no	—
147	#41×C₃₁	—	8	1	no	—
148	#41×C₃₁	—	8	1	no	—
149	C₂×Q₈×C₃₁	—	2	1	no	—
150	C₂×Q₈×C₃₁	—	2	1	no	—
151	C₂×Q₈×C₃₁	—	2	1	no	—
152	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂×D₄×C₃₁	—	2	1	no	—
155	C₂×D₄×C₃₁	—	2	1	no	—
156	C₂×D₄×C₃₁	—	2	1	no	—
157	C₂×D₄×C₃₁	—	4	1	no	—
158	C₂×D₄×C₃₁	—	4	1	no	—
159	C₂×D₄×C₃₁	—	4	1	no	—
160	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	—	2	1	no	—
163	C₂²×C₄×C₃₁	—	2	1	no	—
164	C₂²×C₄×C₃₁	—	4	1	no	—
165	C₂²×C₄×C₃₁	—	4	1	no	—
166	C₂²×C₄×C₃₁	—	4	1	no	—
167	Q₁₆×C₃₁	—	4	2	no	—
168	SD₁₆×C₃₁	—	2	2	no	—
169	SD₁₆×C₃₁	—	4	2	no	—
170	SD₁₆×C₃₁	—	4	2	no	—
171	SD₁₆×C₃₁	—	8	1	no	—
172	SD₁₆×C₃₁	—	8	1	no	—
173	SD₁₆×C₃₁	—	8	2	no	—
174	SD₁₆×C₃₁	—	8	1	no	—
175	SD₁₆×C₃₁	—	8	1	no	—
176	SD₁₆×C₃₁	SD₁₆	1	4	yes	complemented
177	D₈×C₃₁	—	4	2	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	—	2	1	no	—
183	C₂×C₈×C₃₁	—	4	2	no	—
184	C₂×C₈×C₃₁	—	4	2	no	—
185	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
186	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
187	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
196	C₄⋊C₄×C₃₁	—	2	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	—	4	1	no	—
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	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
207	C₂²⋊C₄×C₃₁	—	2	1	no	—
208	C₂²⋊C₄×C₃₁	—	2	1	no	—
209	C₂²⋊C₄×C₃₁	—	4	1	no	—
210	C₂²⋊C₄×C₃₁	—	4	1	no	—
211	C₂²⋊C₄×C₃₁	—	4	1	no	—
212	C₄²×C₃₁	—	2	1	no	—
213	C₄²×C₃₁	—	2	1	no	—
214	C₄²×C₃₁	—	4	1	no	—
215	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
216	#19982	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂³×C₃₁	—	2	1	no	—
218	C₂³×C₃₁	—	2	1	no	—
219	C₂³×C₃₁	—	4	1	no	—
220	Q₈×C₃₁	—	2	1	no	—
221	Q₈×C₃₁	—	2	1	no	—
222	Q₈×C₃₁	—	2	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	—	8	1	no	—
228	Q₈×C₃₁	C₂×SD₁₆	1	2	yes	—
229	D₄×C₃₁	—	2	1	no	—
230	D₄×C₃₁	—	2	1	no	—
231	D₄×C₃₁	—	2	1	no	—
232	D₄×C₃₁	—	4	1	no	—
233	D₄×C₃₁	—	4	1	no	—
234	D₄×C₃₁	—	4	1	no	—
235	D₄×C₃₁	—	4	1	no	—
236	D₄×C₃₁	—	8	2	no	—
237	D₄×C₃₁	—	8	1	no	—
238	D₄×C₃₁	—	8	2	no	—
239	D₄×C₃₁	—	16	1	no	—
240	D₄×C₃₁	C₂×SD₁₆	1	2	yes	—
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	—	4	1	no	—
253	C₂×C₄×C₃₁	—	4	1	no	—
254	C₂×C₄×C₃₁	—	8	2	no	—
255	C₂×C₄×C₃₁	—	8	2	no	—
256	C₂×C₄×C₃₁	—	16	1	no	—
257	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
260	C₈×C₃₁	—	2	2	no	—
261	C₈×C₃₁	—	2	1	no	—
262	C₈×C₃₁	—	2	1	no	—
263	C₈×C₃₁	—	4	1	no	—
264	C₈×C₃₁	—	4	1	no	—
265	C₈×C₃₁	—	4	1	no	—
266	C₈×C₃₁	—	4	1	no	—
267	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
268	C₈×C₃₁	#135	1	2	yes	complemented
269	#3392	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#3381	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#3335	C₂×C₃₁	1	1	yes	characteristic, complemented
272	#3284	C₂×C₃₁	1	1	yes	characteristic
273	#1705	C₂×C₃₁	1	1	yes	characteristic, complemented
274	#1672	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#1671	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#1666	C₂×C₃₁	1	1	yes	characteristic, complemented
277	#1665	C₂×C₃₁	1	1	yes	characteristic, complemented
278	#1654	C₂×C₃₁	1	1	yes	characteristic, complemented
279	#1653	C₂×C₃₁	1	1	yes	characteristic, complemented
280	#1631	C₂×C₃₁	1	1	yes	characteristic, complemented
281	#1622	C₂×C₃₁	1	1	yes	characteristic, complemented
282	C₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
283	C₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
284	C₂²×C₃₁	—	2	2	no	—
285	C₂²×C₃₁	—	4	1	no	—
286	C₂²×C₃₁	—	4	1	no	—
287	C₂²×C₃₁	—	4	1	no	—
288	C₂²×C₃₁	—	4	1	no	—
289	C₂²×C₃₁	—	4	1	no	—
290	C₂²×C₃₁	—	8	1	no	—
291	C₂²×C₃₁	—	8	1	no	—
292	C₂²×C₃₁	—	8	1	no	—
293	C₂²×C₃₁	D₄²	1	1	yes	characteristic
294	C₄×C₃₁	—	2	2	no	—
295	C₄×C₃₁	—	2	1	no	—
296	C₄×C₃₁	—	4	1	no	—
297	C₄×C₃₁	—	4	1	no	—
298	C₄×C₃₁	—	4	1	no	—
299	C₄×C₃₁	—	4	1	no	—
300	C₄×C₃₁	—	4	1	no	—
301	C₄×C₃₁	—	8	1	no	—
302	C₄×C₃₁	—	8	1	no	—
303	C₄×C₃₁	—	8	1	no	—
304	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
305	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
306	C₄×C₃₁	#577	1	1	yes	characteristic
307	C₄×C₃₁	#577	1	1	yes	characteristic
308	C₂×#135	—	2	1	no	—
309	#557	C₂²×C₃₁	1	1	yes	characteristic
310	#554	C₂²×C₃₁	1	1	yes	characteristic
311	#550	C₂²×C₃₁	1	1	yes	characteristic
312	#547	C₂²×C₃₁	1	1	yes	characteristic
313	#499	—	2	1	no	—
314	#498	C₂²×C₃₁	1	1	yes	characteristic
315	#498	C₂²×C₃₁	1	1	yes	characteristic
316	#498	C₂²×C₃₁	1	1	yes	characteristic
317	#492	—	2	1	no	—
318	#492	C₂²×C₃₁	1	1	yes	characteristic
319	#492	C₂²×C₃₁	1	1	yes	characteristic
320	#492	C₂²×C₃₁	1	1	yes	characteristic
321	#483	—	2	1	no	—
322	#481	—	2	1	no	—
323	#479	—	2	1	no	—
324	#478	—	2	1	no	—
325	#477	C₂²×C₃₁	1	1	yes	characteristic
326	#476	C₂²×C₃₁	1	1	yes	characteristic
327	#475	C₂²×C₃₁	1	1	yes	characteristic
328	#474	C₂²×C₃₁	1	1	yes	characteristic
329	#465	—	2	1	no	—
330	#465	—	2	1	no	—
331	#462	C₂²×C₃₁	1	1	yes	characteristic
332	#461	—	2	1	no	—
333	#461	C₂²×C₃₁	1	1	yes	characteristic
334	#460	—	2	1	no	—
335	#460	C₂²×C₃₁	1	1	yes	characteristic
336	#459	C₂²×C₃₁	1	1	yes	characteristic
337	#452	—	2	1	no	—
338	#449	—	2	1	no	—
339	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
340	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
341	C₄×SD₁₆	—	2	1	no	—
342	C₄×SD₁₆	—	2	1	no	—
343	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
344	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
345	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
346	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
347	C₂×#106	—	2	1	no	—
348	#334	C₂²×C₃₁	1	1	yes	characteristic
349	#334	C₂²×C₃₁	1	1	yes	characteristic
350	#333	C₂²×C₃₁	1	1	yes	characteristic
351	#332	C₂²×C₃₁	1	1	yes	characteristic
352	#332	C₂²×C₃₁	1	1	yes	characteristic
353	#331	C₂²×C₃₁	1	1	yes	characteristic
354	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
355	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
356	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
357	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
358	C₈²	C₂²×C₃₁	1	1	yes	characteristic
359	C₂×C₃₁	—	4	2	no	—
360	C₂×C₃₁	—	8	1	no	—
361	C₂×C₃₁	—	16	1	no	—
362	C₂×C₃₁	D₄×SD₁₆	1	1	yes	characteristic
363	C₂×C₃₁	#3284	1	1	yes	characteristic
364	C₂×C₃₁	#3291	1	1	yes	characteristic
365	C₂×#41	—	2	1	no	—
366	C₂×#41	—	2	1	no	—
367	#135	—	4	4	no	—
368	C₂×Q₁₆	—	2	1	no	—
369	C₂×SD₁₆	—	2	1	no	—
370	C₂×SD₁₆	—	2	1	no	—
371	C₂×SD₁₆	—	2	1	no	—
372	C₂×SD₁₆	—	2	1	no	—
373	C₂×SD₁₆	—	2	1	no	—
374	C₂×SD₁₆	—	2	1	no	—
375	C₂×SD₁₆	—	4	1	no	—
376	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
377	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
378	C₂×D₈	—	2	1	no	—
379	C₂²×C₈	—	2	1	no	—
380	C₂²×C₈	—	2	1	no	—
381	C₄⋊Q₈	—	2	1	no	—
382	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
383	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
384	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
385	#127	—	2	1	no	—
386	#127	C₂³×C₃₁	1	1	yes	characteristic
387	#125	—	2	1	no	—
388	#125	—	2	1	no	—
389	#125	—	4	1	no	—
390	#123	—	4	1	no	—
391	C₂²⋊Q₈	—	2	1	no	—
392	C₂²⋊Q₈	—	2	1	no	—
393	C₂²⋊Q₈	—	4	1	no	—
394	#121	—	2	1	no	—
395	#121	—	2	1	no	—
396	#121	—	4	1	no	—
397	C₄×Q₈	—	2	1	no	—
398	C₄×Q₈	—	2	1	no	—
399	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
400	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
401	C₄×D₄	—	2	1	no	—
402	C₄×D₄	—	2	1	no	—
403	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
404	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
405	C₂×C₄⋊C₄	—	2	1	no	—
406	C₂×C₄⋊C₄	—	2	1	no	—
407	#107	—	2	1	no	—
408	#107	—	2	1	no	—
409	#106	—	2	1	no	—
410	#106	—	2	1	no	—
411	#106	—	2	1	no	—
412	#106	—	2	1	no	—
413	#106	—	2	1	no	—
414	#106	—	2	1	no	—
415	#106	—	4	1	no	—
416	#106	D₄×C₃₁	1	1	yes	characteristic
417	#106	D₄×C₃₁	1	1	yes	characteristic
418	C₄⋊C₈	—	2	1	no	—
419	C₄⋊C₈	—	2	1	no	—
420	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
421	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
422	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
423	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
424	#103	—	2	1	no	—
425	#103	—	2	1	no	—
426	#103	—	2	1	no	—
427	#103	—	2	1	no	—
428	#103	D₄×C₃₁	1	1	yes	characteristic
429	#103	D₄×C₃₁	1	1	yes	characteristic
430	#102	—	2	1	no	—
431	#102	—	2	1	no	—
432	#102	—	2	1	no	—
433	#102	—	2	1	no	—
434	#102	D₄×C₃₁	1	1	yes	characteristic
435	#102	D₄×C₃₁	1	1	yes	characteristic
436	C₂²⋊C₈	—	2	1	no	—
437	C₂²⋊C₈	—	2	1	no	—
438	C₄×C₈	—	2	1	no	—
439	C₄×C₈	—	2	1	no	—
440	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
441	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
442	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
443	C₃₁	#19982	1	1	yes	characteristic, direct product factor, Sylow
444	#41	—	4	2	no	—
445	#41	—	4	2	no	—
446	#41	—	8	1	no	—
447	#41	—	8	1	no	—
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
452	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
453	C₂×D₄	—	2	1	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	1	no	—
456	C₂×D₄	—	4	1	no	—
457	C₂×D₄	—	4	1	no	—
458	C₂×D₄	—	4	1	no	—
459	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	4	1	no	—
464	C₂²×C₄	—	4	1	no	—
465	C₂²×C₄	—	4	1	no	—
466	Q₁₆	—	4	2	no	—
467	SD₁₆	—	2	2	no	—
468	SD₁₆	—	4	2	no	—
469	SD₁₆	—	4	2	no	—
470	SD₁₆	—	8	1	no	—
471	SD₁₆	—	8	1	no	—
472	SD₁₆	—	8	2	no	—
473	SD₁₆	—	8	1	no	—
474	SD₁₆	—	8	1	no	—
475	SD₁₆	SD₁₆×C₃₁	1	4	yes	complemented
476	D₈	—	4	2	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	4	2	no	—
483	C₂×C₈	—	4	2	no	—
484	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
485	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
486	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
487	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	4	1	no	—
497	C₄⋊C₄	—	4	1	no	—
498	C₄⋊C₄	—	4	1	no	—
499	C₄⋊C₄	—	4	1	no	—
500	C₄⋊C₄	—	4	1	no	—
501	C₄⋊C₄	—	4	1	no	—
502	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
503	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
504	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
505	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
506	C₂²⋊C₄	—	2	1	no	—
507	C₂²⋊C₄	—	2	1	no	—
508	C₂²⋊C₄	—	4	1	no	—
509	C₂²⋊C₄	—	4	1	no	—
510	C₂²⋊C₄	—	4	1	no	—
511	C₄²	—	2	1	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	4	1	no	—
514	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
515	C₂³	—	2	1	no	—
516	C₂³	—	2	1	no	—
517	C₂³	—	4	1	no	—
518	Q₈	—	2	1	no	—
519	Q₈	—	2	1	no	—
520	Q₈	—	2	1	no	—
521	Q₈	—	4	1	no	—
522	Q₈	—	4	1	no	—
523	Q₈	—	4	1	no	—
524	Q₈	—	4	1	no	—
525	Q₈	—	8	1	no	—
526	Q₈	C₂×SD₁₆×C₃₁	1	2	yes	—
527	D₄	—	2	1	no	—
528	D₄	—	2	1	no	—
529	D₄	—	2	1	no	—
530	D₄	—	4	1	no	—
531	D₄	—	4	1	no	—
532	D₄	—	4	1	no	—
533	D₄	—	4	1	no	—
534	D₄	—	8	2	no	—
535	D₄	—	8	1	no	—
536	D₄	—	8	2	no	—
537	D₄	—	16	1	no	—
538	D₄	C₂×SD₁₆×C₃₁	1	2	yes	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	8	2	no	—
553	C₂×C₄	—	8	2	no	—
554	C₂×C₄	—	16	1	no	—
555	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
556	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
557	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
558	C₈	—	2	2	no	—
559	C₈	—	2	1	no	—
560	C₈	—	2	1	no	—
561	C₈	—	4	1	no	—
562	C₈	—	4	1	no	—
563	C₈	—	4	1	no	—
564	C₈	—	4	1	no	—
565	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
566	C₈	C₃₁×#135	1	2	yes	complemented
567	C₂²	—	2	2	no	—
568	C₂²	—	4	1	no	—
569	C₂²	—	4	1	no	—
570	C₂²	—	4	1	no	—
571	C₂²	—	4	1	no	—
572	C₂²	—	4	1	no	—
573	C₂²	—	8	1	no	—
574	C₂²	—	8	1	no	—
575	C₂²	—	8	1	no	—
576	C₂²	D₄²×C₃₁	1	1	yes	characteristic
577	C₄	—	2	2	no	—
578	C₄	—	2	1	no	—
579	C₄	—	4	1	no	—
580	C₄	—	4	1	no	—
581	C₄	—	4	1	no	—
582	C₄	—	4	1	no	—
583	C₄	—	4	1	no	—
584	C₄	—	8	1	no	—
585	C₄	—	8	1	no	—
586	C₄	—	8	1	no	—
587	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
588	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
589	C₄	C₃₁×#577	1	1	yes	characteristic
590	C₄	C₃₁×#577	1	1	yes	characteristic
591	C₂	—	4	2	no	—
592	C₂	—	8	1	no	—
593	C₂	—	16	1	no	—
594	C₂	D₄×SD₁₆×C₃₁	1	1	yes	characteristic
595	C₂	C₃₁×#3284	1	1	yes	characteristic
596	C₂	C₃₁×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3284	C₂
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×#3392	C₂	semidirect
2. C₃₁×#3381	C₂	semidirect
3. C₃₁×#3335	C₂	semidirect
5. C₃₁×#1705	C₂	semidirect
6. C₃₁×#1672	C₂	semidirect
7. C₃₁×#1671	C₂	semidirect
8. C₃₁×#1666	C₂	semidirect
9. C₃₁×#1665	C₂	semidirect
10. C₃₁×#1654	C₂	semidirect
11. C₃₁×#1653	C₂	semidirect
12. C₃₁×#1631	C₂	semidirect
13. C₃₁×#1622	C₂	semidirect
14. C₈×SD₁₆×C₃₁	C₂	semidirect
15. C₈×SD₁₆×C₃₁	C₂	semidirect
176. SD₁₆×C₃₁	SD₁₆	semidirect
216. #19982	C₃₁	direct (N × H)
268. C₈×C₃₁	#135	semidirect
269. #3392	C₂×C₃₁	semidirect
270. #3381	C₂×C₃₁	semidirect
271. #3335	C₂×C₃₁	semidirect
273. #1705	C₂×C₃₁	semidirect
274. #1672	C₂×C₃₁	semidirect
275. #1671	C₂×C₃₁	semidirect
276. #1666	C₂×C₃₁	semidirect
277. #1665	C₂×C₃₁	semidirect
278. #1654	C₂×C₃₁	semidirect
279. #1653	C₂×C₃₁	semidirect
280. #1631	C₂×C₃₁	semidirect
281. #1622	C₂×C₃₁	semidirect
282. C₈×SD₁₆	C₂×C₃₁	semidirect
283. C₈×SD₁₆	C₂×C₃₁	semidirect
443. C₃₁	#19982	direct (N × H)
475. SD₁₆	SD₁₆×C₃₁	semidirect
566. C₈	C₃₁×#135	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×#3381	2
C₃₁×#3335	2
C₃₁×#3284	2
C₃₁×#1705	2
C₃₁×#1672	2
C₃₁×#1671	2
C₃₁×#1666	2
C₃₁×#1665	2
C₃₁×#1654	2
C₃₁×#1653	2
C₃₁×#1631	2
C₃₁×#1622	2
C₈×SD₁₆×C₃₁	2
#19982	31

Subgroups — C31×#19982 #21476741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19982 #21476741