Subgroups — C₃₁×#19124 #21475883

Order 7936.

672 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3167	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3076	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3035	C₂	1	1	yes	maximal, characteristic
4	C₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1680	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1679	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1678	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1677	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1672	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1671	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1666	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1665	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1542	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1540	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#135	—	2	1	no	—
17	C₃₁×#533	C₂²	1	1	yes	characteristic
18	C₃₁×#532	C₂²	1	1	yes	characteristic
19	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#498	C₂²	1	1	yes	characteristic
21	C₃₁×#492	C₂²	1	1	yes	characteristic
22	C₃₁×#487	C₂²	1	1	yes	characteristic
23	C₃₁×#486	C₂²	1	1	yes	characteristic
24	C₃₁×#479	—	2	1	no	—
25	C₃₁×#478	—	2	1	no	—
26	C₃₁×#477	C₂²	1	1	yes	characteristic
27	C₃₁×#476	C₂²	1	1	yes	characteristic
28	C₃₁×#475	C₂²	1	1	yes	characteristic
29	C₃₁×#474	C₂²	1	1	yes	characteristic
30	C₃₁×#465	—	2	1	no	—
31	C₃₁×#465	—	2	1	no	—
32	C₃₁×#464	—	2	1	no	—
33	C₃₁×#463	—	2	1	no	—
34	C₃₁×#462	C₂²	1	1	yes	characteristic
35	C₃₁×#461	C₂²	1	1	yes	characteristic
36	C₃₁×#460	C₂²	1	1	yes	characteristic
37	C₃₁×#459	C₂²	1	1	yes	characteristic
38	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
39	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×SD₁₆×C₃₁	—	2	1	no	—
41	C₄×SD₁₆×C₃₁	—	2	1	no	—
42	C₈×D₄×C₃₁	—	2	1	no	—
43	C₈×D₄×C₃₁	—	2	1	no	—
44	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
45	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
46	C₃₁×#431	—	2	1	no	—
47	C₃₁×#431	C₂²	1	1	yes	characteristic
48	C₃₁×#427	—	2	1	no	—
49	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
50	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
51	C₃₁×#416	—	2	1	no	—
52	C₃₁×#416	—	2	1	no	—
53	C₂×C₄×C₈×C₃₁	—	2	1	no	—
54	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
55	C₃₁×#334	C₂²	1	1	yes	characteristic
56	C₃₁×#334	C₂²	1	1	yes	characteristic
57	C₃₁×#334	C₂²	1	1	yes	characteristic
58	C₃₁×#334	C₂²	1	1	yes	characteristic
59	C₃₁×#333	C₂²	1	1	yes	characteristic
60	C₃₁×#333	C₂²	1	1	yes	characteristic
61	C₃₁×#332	C₂²	1	1	yes	characteristic
62	C₃₁×#332	C₂²	1	1	yes	characteristic
63	C₃₁×#332	C₂²	1	1	yes	characteristic
64	C₃₁×#332	C₂²	1	1	yes	characteristic
65	C₃₁×#331	C₂²	1	1	yes	characteristic
66	C₃₁×#331	C₂²	1	1	yes	characteristic
67	C₂×#41×C₃₁	—	2	1	no	—
68	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
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×D₈×C₃₁	—	2	1	no	—
76	C₂²×C₈×C₃₁	—	2	1	no	—
77	C₂²×C₈×C₃₁	—	2	1	no	—
78	C₂²×C₈×C₃₁	—	2	1	no	—
79	C₂²×C₈×C₃₁	—	2	1	no	—
80	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
82	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₃₁×#127	C₂³	1	1	yes	characteristic
86	C₃₁×#125	—	2	1	no	—
87	C₃₁×#124	—	2	1	no	—
88	C₃₁×C₂²⋊Q₈	—	2	1	no	—
89	C₃₁×C₂²⋊Q₈	—	2	1	no	—
90	C₃₁×#121	—	2	1	no	—
91	C₃₁×#121	—	2	1	no	—
92	C₄×Q₈×C₃₁	—	2	1	no	—
93	C₄×Q₈×C₃₁	—	2	1	no	—
94	C₄×Q₈×C₃₁	D₄	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	D₄	1	1	yes	characteristic
101	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
102	C₃₁×#117	—	2	1	no	—
103	C₃₁×#117	—	2	1	no	—
104	C₃₁×#117	D₄	1	1	yes	characteristic
105	C₂×C₄²×C₃₁	—	2	1	no	—
106	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
107	C₃₁×#107	—	2	1	no	—
108	C₃₁×#107	—	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×C₄⋊C₈	—	2	1	no	—
114	C₃₁×C₄⋊C₈	—	2	1	no	—
115	C₃₁×C₄⋊C₈	—	2	1	no	—
116	C₃₁×C₄⋊C₈	—	2	1	no	—
117	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
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×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×#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×C₂²⋊C₈	—	2	1	no	—
136	C₃₁×C₂²⋊C₈	—	2	1	no	—
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	D₄	1	1	yes	characteristic
142	C₄×C₈×C₃₁	D₄	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	—	2	2	no	—
146	#41×C₃₁	—	4	2	no	—
147	#41×C₃₁	—	4	2	no	—
148	#41×C₃₁	SD₁₆	1	4	yes	complemented
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×D₄×C₃₁	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	—	2	1	no	—
160	C₂²×C₄×C₃₁	—	2	1	no	—
161	C₂²×C₄×C₃₁	—	2	1	no	—
162	C₂²×C₄×C₃₁	—	4	1	no	—
163	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
166	Q₁₆×C₃₁	—	4	2	no	—
167	SD₁₆×C₃₁	—	4	2	no	—
168	SD₁₆×C₃₁	—	4	2	no	—
169	SD₁₆×C₃₁	—	4	2	no	—
170	SD₁₆×C₃₁	—	4	2	no	—
171	D₈×C₃₁	—	4	2	no	—
172	C₂×C₈×C₃₁	—	2	2	no	—
173	C₂×C₈×C₃₁	—	2	1	no	—
174	C₂×C₈×C₃₁	—	2	1	no	—
175	C₂×C₈×C₃₁	—	2	1	no	—
176	C₂×C₈×C₃₁	—	2	1	no	—
177	C₂×C₈×C₃₁	—	2	1	no	—
178	C₂×C₈×C₃₁	—	2	1	no	—
179	C₂×C₈×C₃₁	—	4	1	no	—
180	C₂×C₈×C₃₁	—	4	2	no	—
181	C₂×C₈×C₃₁	—	4	2	no	—
182	C₂×C₈×C₃₁	—	4	2	no	—
183	C₂×C₈×C₃₁	—	4	2	no	—
184	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
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	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	—	2	1	no	—
196	C₄⋊C₄×C₃₁	—	4	1	no	—
197	C₄⋊C₄×C₃₁	—	4	1	no	—
198	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
199	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
200	C₂²⋊C₄×C₃₁	—	2	1	no	—
201	C₂²⋊C₄×C₃₁	—	2	1	no	—
202	C₂²⋊C₄×C₃₁	—	2	1	no	—
203	C₂²⋊C₄×C₃₁	—	4	1	no	—
204	C₂²⋊C₄×C₃₁	—	4	1	no	—
205	C₂²⋊C₄×C₃₁	—	4	1	no	—
206	C₄²×C₃₁	—	2	1	no	—
207	C₄²×C₃₁	—	2	1	no	—
208	C₄²×C₃₁	—	2	1	no	—
209	C₄²×C₃₁	—	4	1	no	—
210	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
211	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₃₁	#41	1	1	yes	characteristic
213	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
214	#19124	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂³×C₃₁	—	2	1	no	—
216	C₂³×C₃₁	—	4	1	no	—
217	C₂³×C₃₁	#121	1	1	yes	characteristic
218	Q₈×C₃₁	—	2	1	no	—
219	Q₈×C₃₁	—	4	1	no	—
220	Q₈×C₃₁	—	4	1	no	—
221	Q₈×C₃₁	—	4	1	no	—
222	Q₈×C₃₁	—	4	1	no	—
223	Q₈×C₃₁	—	8	1	no	—
224	Q₈×C₃₁	C₂×SD₁₆	1	2	yes	—
225	D₄×C₃₁	—	2	4	no	—
226	D₄×C₃₁	—	2	1	no	—
227	D₄×C₃₁	—	4	1	no	—
228	D₄×C₃₁	—	4	1	no	—
229	D₄×C₃₁	—	4	1	no	—
230	D₄×C₃₁	—	4	1	no	—
231	D₄×C₃₁	—	8	1	no	—
232	D₄×C₃₁	—	8	2	no	—
233	D₄×C₃₁	C₂×SD₁₆	1	2	yes	—
234	C₂×C₄×C₃₁	—	2	1	no	—
235	C₂×C₄×C₃₁	—	2	1	no	—
236	C₂×C₄×C₃₁	—	2	1	no	—
237	C₂×C₄×C₃₁	—	2	4	no	—
238	C₂×C₄×C₃₁	—	2	1	no	—
239	C₂×C₄×C₃₁	—	2	1	no	—
240	C₂×C₄×C₃₁	—	4	1	no	—
241	C₂×C₄×C₃₁	—	4	2	no	—
242	C₂×C₄×C₃₁	—	4	1	no	—
243	C₂×C₄×C₃₁	—	4	1	no	—
244	C₂×C₄×C₃₁	—	4	1	no	—
245	C₂×C₄×C₃₁	—	4	1	no	—
246	C₂×C₄×C₃₁	—	4	1	no	—
247	C₂×C₄×C₃₁	—	8	2	no	—
248	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
249	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
250	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
251	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
252	C₂×C₄×C₃₁	#135	1	2	yes	—
253	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
256	C₈×C₃₁	—	2	1	no	—
257	C₈×C₃₁	—	2	1	no	—
258	C₈×C₃₁	—	2	1	no	—
259	C₈×C₃₁	—	4	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	#135	1	2	yes	complemented
266	#3167	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#3076	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#3035	C₂×C₃₁	1	1	yes	characteristic
269	C₈×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#1680	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#1679	C₂×C₃₁	1	1	yes	characteristic, complemented
272	#1678	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#1677	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	C₂×#334	C₂×C₃₁	1	1	yes	characteristic, complemented
279	#1542	C₂×C₃₁	1	1	yes	characteristic, complemented
280	#1540	C₂×C₃₁	1	1	yes	characteristic, complemented
281	C₂²×C₃₁	—	2	1	no	—
282	C₂²×C₃₁	—	2	1	no	—
283	C₂²×C₃₁	—	2	2	no	—
284	C₂²×C₃₁	—	4	1	no	—
285	C₂²×C₃₁	—	4	1	no	—
286	C₂²×C₃₁	—	8	1	no	—
287	C₂²×C₃₁	—	8	1	no	—
288	C₂²×C₃₁	—	8	1	no	—
289	C₂²×C₃₁	#465	1	2	yes	—
290	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
291	C₄×C₃₁	—	2	1	no	—
292	C₄×C₃₁	—	2	2	no	—
293	C₄×C₃₁	—	2	1	no	—
294	C₄×C₃₁	—	2	1	no	—
295	C₄×C₃₁	—	4	1	no	—
296	C₄×C₃₁	—	4	1	no	—
297	C₄×C₃₁	—	8	1	no	—
298	C₄×C₃₁	—	8	1	no	—
299	C₄×C₃₁	—	8	1	no	—
300	C₄×C₃₁	#465	1	2	yes	—
301	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
302	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
303	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
304	C₄×C₃₁	#575	1	1	yes	characteristic
305	C₂×#135	—	2	1	no	—
306	#533	C₂²×C₃₁	1	1	yes	characteristic
307	#532	C₂²×C₃₁	1	1	yes	characteristic
308	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
309	#498	C₂²×C₃₁	1	1	yes	characteristic
310	#492	C₂²×C₃₁	1	1	yes	characteristic
311	#487	C₂²×C₃₁	1	1	yes	characteristic
312	#486	C₂²×C₃₁	1	1	yes	characteristic
313	#479	—	2	1	no	—
314	#478	—	2	1	no	—
315	#477	C₂²×C₃₁	1	1	yes	characteristic
316	#476	C₂²×C₃₁	1	1	yes	characteristic
317	#475	C₂²×C₃₁	1	1	yes	characteristic
318	#474	C₂²×C₃₁	1	1	yes	characteristic
319	#465	—	2	1	no	—
320	#465	—	2	1	no	—
321	#464	—	2	1	no	—
322	#463	—	2	1	no	—
323	#462	C₂²×C₃₁	1	1	yes	characteristic
324	#461	C₂²×C₃₁	1	1	yes	characteristic
325	#460	C₂²×C₃₁	1	1	yes	characteristic
326	#459	C₂²×C₃₁	1	1	yes	characteristic
327	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
328	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
329	C₄×SD₁₆	—	2	1	no	—
330	C₄×SD₁₆	—	2	1	no	—
331	C₈×D₄	—	2	1	no	—
332	C₈×D₄	—	2	1	no	—
333	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
334	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
335	#431	—	2	1	no	—
336	#431	C₂²×C₃₁	1	1	yes	characteristic
337	#427	—	2	1	no	—
338	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
339	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
340	#416	—	2	1	no	—
341	#416	—	2	1	no	—
342	C₂×C₄×C₈	—	2	1	no	—
343	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
344	#334	C₂²×C₃₁	1	1	yes	characteristic
345	#334	C₂²×C₃₁	1	1	yes	characteristic
346	#334	C₂²×C₃₁	1	1	yes	characteristic
347	#334	C₂²×C₃₁	1	1	yes	characteristic
348	#333	C₂²×C₃₁	1	1	yes	characteristic
349	#333	C₂²×C₃₁	1	1	yes	characteristic
350	#332	C₂²×C₃₁	1	1	yes	characteristic
351	#332	C₂²×C₃₁	1	1	yes	characteristic
352	#332	C₂²×C₃₁	1	1	yes	characteristic
353	#332	C₂²×C₃₁	1	1	yes	characteristic
354	#331	C₂²×C₃₁	1	1	yes	characteristic
355	#331	C₂²×C₃₁	1	1	yes	characteristic
356	C₂×C₃₁	—	2	2	no	—
357	C₂×C₃₁	—	4	2	no	—
358	C₂×C₃₁	—	16	1	no	—
359	C₂×C₃₁	#3035	1	1	yes	characteristic
360	C₂×C₃₁	C₂×#465	1	1	yes	characteristic
361	C₂×C₃₁	#3067	1	1	yes	characteristic
362	C₂×#41	—	2	1	no	—
363	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
364	#135	—	4	4	no	—
365	C₂×Q₁₆	—	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×D₈	—	2	1	no	—
371	C₂²×C₈	—	2	1	no	—
372	C₂²×C₈	—	2	1	no	—
373	C₂²×C₈	—	2	1	no	—
374	C₂²×C₈	—	2	1	no	—
375	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
376	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
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	C₂³×C₃₁	1	1	yes	characteristic
381	#125	—	2	1	no	—
382	#124	—	2	1	no	—
383	C₂²⋊Q₈	—	2	1	no	—
384	C₂²⋊Q₈	—	2	1	no	—
385	#121	—	2	1	no	—
386	#121	—	2	1	no	—
387	C₄×Q₈	—	2	1	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
390	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
391	C₄×D₄	—	2	1	no	—
392	C₄×D₄	—	2	1	no	—
393	C₄×D₄	—	2	1	no	—
394	C₄×D₄	—	2	1	no	—
395	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
396	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
397	#117	—	2	1	no	—
398	#117	—	2	1	no	—
399	#117	D₄×C₃₁	1	1	yes	characteristic
400	C₂×C₄²	—	2	1	no	—
401	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
402	#107	—	2	1	no	—
403	#107	—	2	1	no	—
404	#106	—	2	1	no	—
405	#106	—	2	1	no	—
406	#106	—	2	1	no	—
407	#106	—	2	1	no	—
408	C₄⋊C₈	—	2	1	no	—
409	C₄⋊C₈	—	2	1	no	—
410	C₄⋊C₈	—	2	1	no	—
411	C₄⋊C₈	—	2	1	no	—
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	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
416	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
417	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
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	—	2	1	no	—
424	#102	—	2	1	no	—
425	#102	—	2	1	no	—
426	#102	—	2	1	no	—
427	#102	—	2	1	no	—
428	#102	—	2	1	no	—
429	#102	—	2	1	no	—
430	C₂²⋊C₈	—	2	1	no	—
431	C₂²⋊C₈	—	2	1	no	—
432	C₂²⋊C₈	—	2	1	no	—
433	C₄×C₈	—	2	1	no	—
434	C₄×C₈	—	2	1	no	—
435	C₄×C₈	—	2	1	no	—
436	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
437	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
438	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
439	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
440	C₃₁	#19124	1	1	yes	characteristic, direct product factor, Sylow
441	#41	—	2	2	no	—
442	#41	—	4	2	no	—
443	#41	—	4	2	no	—
444	#41	SD₁₆×C₃₁	1	4	yes	complemented
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	1	no	—
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	—	2	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	—	2	1	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	4	1	no	—
459	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
462	Q₁₆	—	4	2	no	—
463	SD₁₆	—	4	2	no	—
464	SD₁₆	—	4	2	no	—
465	SD₁₆	—	4	2	no	—
466	SD₁₆	—	4	2	no	—
467	D₈	—	4	2	no	—
468	C₂×C₈	—	2	2	no	—
469	C₂×C₈	—	2	1	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	—	4	1	no	—
476	C₂×C₈	—	4	2	no	—
477	C₂×C₈	—	4	2	no	—
478	C₂×C₈	—	4	2	no	—
479	C₂×C₈	—	4	2	no	—
480	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
481	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
482	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
483	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
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	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	4	1	no	—
493	C₄⋊C₄	—	4	1	no	—
494	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
495	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
496	C₂²⋊C₄	—	2	1	no	—
497	C₂²⋊C₄	—	2	1	no	—
498	C₂²⋊C₄	—	2	1	no	—
499	C₂²⋊C₄	—	4	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	C₂×D₄×C₃₁	1	1	yes	characteristic
507	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
508	C₄²	#41×C₃₁	1	1	yes	characteristic
509	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
510	C₂³	—	2	1	no	—
511	C₂³	—	4	1	no	—
512	C₂³	C₃₁×#121	1	1	yes	characteristic
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	—	8	1	no	—
519	Q₈	C₂×SD₁₆×C₃₁	1	2	yes	—
520	D₄	—	2	4	no	—
521	D₄	—	2	1	no	—
522	D₄	—	4	1	no	—
523	D₄	—	4	1	no	—
524	D₄	—	4	1	no	—
525	D₄	—	4	1	no	—
526	D₄	—	8	1	no	—
527	D₄	—	8	2	no	—
528	D₄	C₂×SD₁₆×C₃₁	1	2	yes	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	2	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	8	2	no	—
543	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
544	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
545	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
546	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
547	C₂×C₄	C₃₁×#135	1	2	yes	—
548	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
549	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
550	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
551	C₈	—	2	1	no	—
552	C₈	—	2	1	no	—
553	C₈	—	2	1	no	—
554	C₈	—	4	1	no	—
555	C₈	—	4	1	no	—
556	C₈	—	4	1	no	—
557	C₈	—	4	1	no	—
558	C₈	—	4	1	no	—
559	C₈	—	4	1	no	—
560	C₈	C₃₁×#135	1	2	yes	complemented
561	C₂²	—	2	1	no	—
562	C₂²	—	2	1	no	—
563	C₂²	—	2	2	no	—
564	C₂²	—	4	1	no	—
565	C₂²	—	4	1	no	—
566	C₂²	—	8	1	no	—
567	C₂²	—	8	1	no	—
568	C₂²	—	8	1	no	—
569	C₂²	C₃₁×#465	1	2	yes	—
570	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
571	C₄	—	2	1	no	—
572	C₄	—	2	2	no	—
573	C₄	—	2	1	no	—
574	C₄	—	2	1	no	—
575	C₄	—	4	1	no	—
576	C₄	—	4	1	no	—
577	C₄	—	8	1	no	—
578	C₄	—	8	1	no	—
579	C₄	—	8	1	no	—
580	C₄	C₃₁×#465	1	2	yes	—
581	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
582	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
583	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
584	C₄	C₃₁×#575	1	1	yes	characteristic
585	C₂	—	2	2	no	—
586	C₂	—	4	2	no	—
587	C₂	—	16	1	no	—
588	C₂	C₃₁×#3035	1	1	yes	characteristic
589	C₂	C₂×C₃₁×#465	1	1	yes	characteristic
590	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×#121
prime radical	C₃₁×#3035	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3167	C₂	semidirect
2. C₃₁×#3076	C₂	semidirect
4. C₈×#41×C₃₁	C₂	semidirect
5. C₃₁×#1680	C₂	semidirect
6. C₃₁×#1679	C₂	semidirect
7. C₃₁×#1678	C₂	semidirect
8. C₃₁×#1677	C₂	semidirect
9. C₃₁×#1672	C₂	semidirect
10. C₃₁×#1671	C₂	semidirect
11. C₃₁×#1666	C₂	semidirect
12. C₃₁×#1665	C₂	semidirect
13. C₂×C₃₁×#334	C₂	semidirect
14. C₃₁×#1542	C₂	semidirect
15. C₃₁×#1540	C₂	semidirect
148. #41×C₃₁	SD₁₆	semidirect
214. #19124	C₃₁	direct (N × H)
265. C₈×C₃₁	#135	semidirect
266. #3167	C₂×C₃₁	semidirect
267. #3076	C₂×C₃₁	semidirect
269. C₈×#41	C₂×C₃₁	semidirect
270. #1680	C₂×C₃₁	semidirect
271. #1679	C₂×C₃₁	semidirect
272. #1678	C₂×C₃₁	semidirect
273. #1677	C₂×C₃₁	semidirect
274. #1672	C₂×C₃₁	semidirect
275. #1671	C₂×C₃₁	semidirect
276. #1666	C₂×C₃₁	semidirect
277. #1665	C₂×C₃₁	semidirect
278. C₂×#334	C₂×C₃₁	semidirect
279. #1542	C₂×C₃₁	semidirect
280. #1540	C₂×C₃₁	semidirect
440. C₃₁	#19124	direct (N × H)
444. #41	SD₁₆×C₃₁	semidirect
560. 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×#3167	2
C₃₁×#3076	2
C₃₁×#3035	2
C₈×#41×C₃₁	2
C₃₁×#1680	2
C₃₁×#1679	2
C₃₁×#1678	2
C₃₁×#1677	2
C₃₁×#1672	2
C₃₁×#1671	2
C₃₁×#1666	2
C₃₁×#1665	2
C₂×C₃₁×#334	2
C₃₁×#1542	2
C₃₁×#1540	2
#19124	31

Subgroups — C31×#19124 #21475883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19124 #21475883