Subgroups — C₃₁×#21128 #21477887

Order 7936.

862 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#440	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2319	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2276	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1926	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1925	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1921	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1918	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1880	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1880	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1865	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1863	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1776	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₃₁×#103	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#440	C₄	1	8	yes	complemented
22	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#390	C₂²	1	1	yes	characteristic
34	C₃₁×#390	C₂²	1	1	yes	characteristic
35	C₃₁×#389	C₂²	1	1	yes	characteristic
36	C₃₁×#389	C₂²	1	1	yes	characteristic
37	C₃₁×#388	—	2	1	no	—
38	C₃₁×#388	—	2	1	no	—
39	C₃₁×#381	—	2	1	no	—
40	C₃₁×#380	—	2	1	no	—
41	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
42	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
43	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
44	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
45	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
46	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
48	C₃₁×#376	—	2	1	no	—
49	C₃₁×#376	C₂²	1	1	yes	characteristic
50	C₃₁×#340	C₂²	1	1	yes	characteristic
51	C₃₁×#340	C₂²	1	1	yes	characteristic
52	C₃₁×#340	C₂²	1	1	yes	characteristic
53	C₃₁×#340	C₂²	1	1	yes	characteristic
54	C₃₁×#340	C₂²	1	1	yes	characteristic
55	C₃₁×#340	C₂²	1	1	yes	characteristic
56	C₃₁×#336	C₂²	1	1	yes	characteristic
57	C₃₁×#336	C₂²	1	1	yes	characteristic
58	C₃₁×#336	C₂²	1	1	yes	characteristic
59	C₃₁×#336	C₂²	1	1	yes	characteristic
60	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₃₁	—	2	1	no	—
63	C₂×SD₁₆×C₃₁	—	2	2	no	—
64	C₂×SD₁₆×C₃₁	—	2	2	no	—
65	C₂×SD₁₆×C₃₁	C₂×C₄	1	4	yes	—
66	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₄×Q₈×C₃₁	—	2	2	no	—
71	C₄×Q₈×C₃₁	—	2	2	no	—
72	C₄×Q₈×C₃₁	—	2	2	no	—
73	C₄×Q₈×C₃₁	—	2	2	no	—
74	C₄×Q₈×C₃₁	—	2	2	no	—
75	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
76	C₄×D₄×C₃₁	—	2	2	no	—
77	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
78	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
94	C₂×C₄²×C₃₁	—	2	1	no	—
95	C₂×C₄²×C₃₁	—	2	1	no	—
96	C₂×C₄²×C₃₁	—	2	1	no	—
97	C₂×C₄²×C₃₁	—	2	1	no	—
98	C₂×C₄²×C₃₁	—	2	1	no	—
99	C₂×C₄²×C₃₁	—	2	1	no	—
100	C₂×C₄²×C₃₁	—	2	1	no	—
101	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₃₁×#107	—	2	2	no	—
105	C₃₁×#107	C₂×C₄	1	4	yes	—
106	C₃₁×#106	—	2	2	no	—
107	C₃₁×#103	—	2	2	no	—
108	C₃₁×#103	—	2	2	no	—
109	C₃₁×#103	C₂×C₄	1	4	yes	—
110	C₃₁×#102	—	2	2	no	—
111	C₃₁×#102	—	2	2	no	—
112	C₃₁×#102	C₂×C₄	1	4	yes	—
113	C₃₁×#97	C₂×C₄	1	4	yes	—
114	C₄×C₈×C₃₁	—	2	2	no	—
115	C₃₁×#95	—	2	1	no	—
116	C₃₁×#95	—	2	1	no	—
117	C₃₁×#95	—	2	1	no	—
118	C₃₁×#95	—	2	1	no	—
119	C₃₁×#95	—	2	1	no	—
120	C₃₁×#95	—	2	1	no	—
121	C₃₁×#95	D₄	1	1	yes	characteristic
122	C₃₁×#95	D₄	1	1	yes	characteristic
123	C₃₁×#95	D₄	1	1	yes	characteristic
124	C₃₁×#95	D₄	1	1	yes	characteristic
125	C₂⁴×C₃₁	—	2	1	no	—
126	C₂×Q₈×C₃₁	—	2	1	no	—
127	C₂×Q₈×C₃₁	—	2	1	no	—
128	C₂×Q₈×C₃₁	—	2	1	no	—
129	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
130	C₂×Q₈×C₃₁	#41	1	2	yes	—
131	C₂×Q₈×C₃₁	#41	1	2	yes	—
132	C₂×D₄×C₃₁	—	2	1	no	—
133	C₂×D₄×C₃₁	—	2	1	no	—
134	C₂×D₄×C₃₁	—	2	1	no	—
135	C₂×D₄×C₃₁	—	2	1	no	—
136	C₂×D₄×C₃₁	—	2	1	no	—
137	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
138	C₂²×C₄×C₃₁	—	2	1	no	—
139	C₂²×C₄×C₃₁	—	2	1	no	—
140	C₂²×C₄×C₃₁	—	2	1	no	—
141	C₂²×C₄×C₃₁	—	2	1	no	—
142	C₂²×C₄×C₃₁	—	2	1	no	—
143	C₂²×C₄×C₃₁	—	2	4	no	—
144	C₂²×C₄×C₃₁	—	2	1	no	—
145	C₂²×C₄×C₃₁	—	2	1	no	—
146	C₂²×C₄×C₃₁	—	8	1	no	—
147	C₂²×C₄×C₃₁	—	8	1	no	—
148	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
155	SD₁₆×C₃₁	—	2	8	no	—
156	C₂×C₈×C₃₁	—	2	1	no	—
157	C₂×C₈×C₃₁	—	2	1	no	—
158	C₂×C₈×C₃₁	—	2	1	no	—
159	C₂×C₈×C₃₁	—	2	1	no	—
160	C₂×C₈×C₃₁	—	2	1	no	—
161	C₂×C₈×C₃₁	—	2	1	no	—
162	C₂×C₈×C₃₁	—	2	1	no	—
163	C₂×C₈×C₃₁	—	2	1	no	—
164	C₂×C₈×C₃₁	—	2	1	no	—
165	C₂×C₈×C₃₁	—	2	1	no	—
166	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
167	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
168	C₄⋊C₄×C₃₁	—	2	4	no	—
169	C₄⋊C₄×C₃₁	—	2	4	no	—
170	C₄⋊C₄×C₃₁	—	2	4	no	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	2	1	no	—
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	2	1	no	—
179	C₄⋊C₄×C₃₁	—	8	1	no	—
180	C₄⋊C₄×C₃₁	—	8	1	no	—
181	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
182	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
183	C₄⋊C₄×C₃₁	#41	1	2	yes	—
184	C₄⋊C₄×C₃₁	#41	1	2	yes	—
185	C₂²⋊C₄×C₃₁	—	2	4	no	—
186	C₂²⋊C₄×C₃₁	—	8	1	no	—
187	C₂²⋊C₄×C₃₁	—	8	1	no	—
188	C₂²⋊C₄×C₃₁	—	8	1	no	—
189	C₂²⋊C₄×C₃₁	—	8	1	no	—
190	C₄²×C₃₁	—	2	4	no	—
191	C₄²×C₃₁	—	2	4	no	—
192	C₄²×C₃₁	—	2	4	no	—
193	C₄²×C₃₁	—	2	1	no	—
194	C₄²×C₃₁	—	2	1	no	—
195	C₄²×C₃₁	—	2	1	no	—
196	C₄²×C₃₁	—	2	1	no	—
197	C₄²×C₃₁	—	8	1	no	—
198	C₄²×C₃₁	—	8	1	no	—
199	C₄²×C₃₁	—	8	1	no	—
200	C₄²×C₃₁	—	8	1	no	—
201	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
202	C₄²×C₃₁	#41	1	2	yes	—
203	#21128	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂³×C₃₁	—	2	2	no	—
205	C₂³×C₃₁	—	4	1	no	—
206	C₂³×C₃₁	—	4	1	no	—
207	C₂³×C₃₁	—	4	1	no	—
208	C₂³×C₃₁	—	4	1	no	—
209	C₂³×C₃₁	—	4	1	no	—
210	C₂³×C₃₁	—	4	1	no	—
211	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
212	Q₈×C₃₁	—	2	2	no	—
213	Q₈×C₃₁	—	2	2	no	—
214	Q₈×C₃₁	—	2	2	no	—
215	Q₈×C₃₁	#117	1	4	yes	—
216	D₄×C₃₁	—	2	2	no	—
217	D₄×C₃₁	—	2	2	no	—
218	D₄×C₃₁	—	4	1	no	—
219	D₄×C₃₁	—	4	1	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	1	no	—
224	C₂×C₄×C₃₁	—	2	1	no	—
225	C₂×C₄×C₃₁	—	2	1	no	—
226	C₂×C₄×C₃₁	—	2	1	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	2	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	2	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	1	no	—
236	C₂×C₄×C₃₁	—	2	1	no	—
237	C₂×C₄×C₃₁	—	2	1	no	—
238	C₂×C₄×C₃₁	—	2	1	no	—
239	C₂×C₄×C₃₁	—	2	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	—	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	—	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	—	4	1	no	—
255	C₂×C₄×C₃₁	—	4	1	no	—
256	C₂×C₄×C₃₁	—	4	1	no	—
257	C₂×C₄×C₃₁	—	8	4	no	—
258	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
259	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
260	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
261	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
262	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
263	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
264	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
265	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
266	C₈×C₃₁	—	2	2	no	—
267	C₈×C₃₁	—	2	2	no	—
268	C₈×C₃₁	—	4	1	no	—
269	C₈×C₃₁	—	4	1	no	—
270	C₂×#440	C₂×C₃₁	1	1	yes	characteristic
271	#2319	C₂×C₃₁	1	1	yes	characteristic
272	#2276	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#1926	C₂×C₃₁	1	1	yes	characteristic
274	#1925	C₂×C₃₁	1	1	yes	characteristic
275	#1921	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#1918	C₂×C₃₁	1	1	yes	characteristic, complemented
277	#1880	C₂×C₃₁	1	1	yes	characteristic
278	#1880	C₂×C₃₁	1	1	yes	characteristic
279	#1865	C₂×C₃₁	1	1	yes	characteristic, complemented
280	#1863	C₂×C₃₁	1	1	yes	characteristic, complemented
281	#1776	C₂×C₃₁	1	1	yes	characteristic, complemented
282	C₄×#103	C₂×C₃₁	1	1	yes	characteristic, complemented
283	C₄×#102	C₂×C₃₁	1	1	yes	characteristic
284	#1746	C₂×C₃₁	1	1	yes	characteristic, complemented
285	C₂²×C₃₁	—	4	2	no	—
286	C₂²×C₃₁	—	4	2	no	—
287	C₂²×C₃₁	—	4	2	no	—
288	C₂²×C₃₁	—	8	1	no	—
289	C₂²×C₃₁	—	8	1	no	—
290	C₂²×C₃₁	—	8	1	no	—
291	C₂²×C₃₁	—	8	1	no	—
292	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
293	C₂²×C₃₁	#525	1	1	yes	characteristic
294	C₂²×C₃₁	#525	1	1	yes	characteristic
295	C₂²×C₃₁	#576	1	1	yes	characteristic
296	C₂²×C₃₁	#577	1	1	yes	characteristic
297	C₂²×C₃₁	#577	1	1	yes	characteristic
298	C₂²×C₃₁	#578	1	1	yes	characteristic
299	C₄×C₃₁	—	2	2	no	—
300	C₄×C₃₁	—	2	2	no	—
301	C₄×C₃₁	—	2	4	no	—
302	C₄×C₃₁	—	4	1	no	—
303	C₄×C₃₁	—	4	2	no	—
304	C₄×C₃₁	—	4	1	no	—
305	C₄×C₃₁	—	4	1	no	—
306	C₄×C₃₁	—	4	2	no	—
307	C₄×C₃₁	—	4	1	no	—
308	C₄×C₃₁	—	4	2	no	—
309	C₄×C₃₁	—	4	2	no	—
310	C₄×C₃₁	—	8	1	no	—
311	C₄×C₃₁	—	8	1	no	—
312	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
313	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
314	C₄×C₃₁	#529	1	1	yes	characteristic
315	C₄×C₃₁	#529	1	1	yes	characteristic
316	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
317	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
318	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
320	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
321	#440	C₄×C₃₁	1	8	yes	complemented
322	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
323	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
324	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
325	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
326	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
327	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
328	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
329	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
330	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
332	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
333	#390	C₂²×C₃₁	1	1	yes	characteristic
334	#390	C₂²×C₃₁	1	1	yes	characteristic
335	#389	C₂²×C₃₁	1	1	yes	characteristic
336	#389	C₂²×C₃₁	1	1	yes	characteristic
337	#388	—	2	1	no	—
338	#388	—	2	1	no	—
339	#381	—	2	1	no	—
340	#380	—	2	1	no	—
341	C₄×C₄⋊C₄	—	2	1	no	—
342	C₄×C₄⋊C₄	—	2	1	no	—
343	C₄×C₄⋊C₄	—	2	1	no	—
344	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
345	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
346	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
347	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
348	#376	—	2	1	no	—
349	#376	C₂²×C₃₁	1	1	yes	characteristic
350	#340	C₂²×C₃₁	1	1	yes	characteristic
351	#340	C₂²×C₃₁	1	1	yes	characteristic
352	#340	C₂²×C₃₁	1	1	yes	characteristic
353	#340	C₂²×C₃₁	1	1	yes	characteristic
354	#340	C₂²×C₃₁	1	1	yes	characteristic
355	#340	C₂²×C₃₁	1	1	yes	characteristic
356	#336	C₂²×C₃₁	1	1	yes	characteristic
357	#336	C₂²×C₃₁	1	1	yes	characteristic
358	#336	C₂²×C₃₁	1	1	yes	characteristic
359	#336	C₂²×C₃₁	1	1	yes	characteristic
360	C₂×C₃₁	—	8	2	no	—
361	C₂×C₃₁	#2290	1	1	yes	characteristic
362	C₂×C₃₁	#2946	1	1	yes	characteristic
363	C₂×C₃₁	#2947	1	1	yes	characteristic
364	C₂×C₃₁	#3119	1	1	yes	characteristic
365	C₂×C₃₁	#3121	1	1	yes	characteristic
366	C₂×C₃₁	#3124	1	1	yes	characteristic
367	C₂×C₃₁	#3126	1	1	yes	characteristic
368	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
369	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
370	C₂³×C₄	—	2	1	no	—
371	C₂×SD₁₆	—	2	2	no	—
372	C₂×SD₁₆	—	2	2	no	—
373	C₂×SD₁₆	C₂×C₄×C₃₁	1	4	yes	—
374	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
375	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
376	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
377	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
378	C₄×Q₈	—	2	2	no	—
379	C₄×Q₈	—	2	2	no	—
380	C₄×Q₈	—	2	2	no	—
381	C₄×Q₈	—	2	2	no	—
382	C₄×Q₈	—	2	2	no	—
383	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
384	C₄×D₄	—	2	2	no	—
385	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
386	C₂×C₄⋊C₄	—	2	1	no	—
387	C₂×C₄⋊C₄	—	2	1	no	—
388	C₂×C₄⋊C₄	—	2	1	no	—
389	C₂×C₄⋊C₄	—	2	1	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
394	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
395	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
396	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
397	C₂×C₂²⋊C₄	—	2	1	no	—
398	C₂×C₂²⋊C₄	—	2	1	no	—
399	C₂×C₂²⋊C₄	—	2	1	no	—
400	C₂×C₂²⋊C₄	—	2	1	no	—
401	C₂×C₂²⋊C₄	—	2	1	no	—
402	C₂×C₄²	—	2	1	no	—
403	C₂×C₄²	—	2	1	no	—
404	C₂×C₄²	—	2	1	no	—
405	C₂×C₄²	—	2	1	no	—
406	C₂×C₄²	—	2	1	no	—
407	C₂×C₄²	—	2	1	no	—
408	C₂×C₄²	—	2	1	no	—
409	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
410	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
411	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
412	#107	—	2	2	no	—
413	#107	C₂×C₄×C₃₁	1	4	yes	—
414	#106	—	2	2	no	—
415	#103	—	2	2	no	—
416	#103	—	2	2	no	—
417	#103	C₂×C₄×C₃₁	1	4	yes	—
418	#102	—	2	2	no	—
419	#102	—	2	2	no	—
420	#102	C₂×C₄×C₃₁	1	4	yes	—
421	#97	C₂×C₄×C₃₁	1	4	yes	—
422	C₄×C₈	—	2	2	no	—
423	#95	—	2	1	no	—
424	#95	—	2	1	no	—
425	#95	—	2	1	no	—
426	#95	—	2	1	no	—
427	#95	—	2	1	no	—
428	#95	—	2	1	no	—
429	#95	D₄×C₃₁	1	1	yes	characteristic
430	#95	D₄×C₃₁	1	1	yes	characteristic
431	#95	D₄×C₃₁	1	1	yes	characteristic
432	#95	D₄×C₃₁	1	1	yes	characteristic
433	C₃₁	#21128	1	1	yes	characteristic, direct product factor, Sylow
434	C₂⁴	—	2	1	no	—
435	C₂×Q₈	—	2	1	no	—
436	C₂×Q₈	—	2	1	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
439	C₂×Q₈	#41×C₃₁	1	2	yes	—
440	C₂×Q₈	#41×C₃₁	1	2	yes	—
441	C₂×D₄	—	2	1	no	—
442	C₂×D₄	—	2	1	no	—
443	C₂×D₄	—	2	1	no	—
444	C₂×D₄	—	2	1	no	—
445	C₂×D₄	—	2	1	no	—
446	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	8	1	no	—
456	C₂²×C₄	—	8	1	no	—
457	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
458	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
459	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
464	SD₁₆	—	2	8	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	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	C₂²×C₄×C₃₁	1	2	yes	—
476	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
477	C₄⋊C₄	—	2	4	no	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	4	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	—	8	1	no	—
489	C₄⋊C₄	—	8	1	no	—
490	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
491	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
492	C₄⋊C₄	#41×C₃₁	1	2	yes	—
493	C₄⋊C₄	#41×C₃₁	1	2	yes	—
494	C₂²⋊C₄	—	2	4	no	—
495	C₂²⋊C₄	—	8	1	no	—
496	C₂²⋊C₄	—	8	1	no	—
497	C₂²⋊C₄	—	8	1	no	—
498	C₂²⋊C₄	—	8	1	no	—
499	C₄²	—	2	4	no	—
500	C₄²	—	2	4	no	—
501	C₄²	—	2	4	no	—
502	C₄²	—	2	1	no	—
503	C₄²	—	2	1	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	2	1	no	—
506	C₄²	—	8	1	no	—
507	C₄²	—	8	1	no	—
508	C₄²	—	8	1	no	—
509	C₄²	—	8	1	no	—
510	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
511	C₄²	#41×C₃₁	1	2	yes	—
512	C₂³	—	2	2	no	—
513	C₂³	—	4	1	no	—
514	C₂³	—	4	1	no	—
515	C₂³	—	4	1	no	—
516	C₂³	—	4	1	no	—
517	C₂³	—	4	1	no	—
518	C₂³	—	4	1	no	—
519	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
520	Q₈	—	2	2	no	—
521	Q₈	—	2	2	no	—
522	Q₈	—	2	2	no	—
523	Q₈	C₃₁×#117	1	4	yes	—
524	D₄	—	2	2	no	—
525	D₄	—	2	2	no	—
526	D₄	—	4	1	no	—
527	D₄	—	4	1	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	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	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	8	4	no	—
566	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
567	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
568	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
569	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
570	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
571	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
572	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
573	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
574	C₈	—	2	2	no	—
575	C₈	—	2	2	no	—
576	C₈	—	4	1	no	—
577	C₈	—	4	1	no	—
578	C₂²	—	4	2	no	—
579	C₂²	—	4	2	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	8	1	no	—
582	C₂²	—	8	1	no	—
583	C₂²	—	8	1	no	—
584	C₂²	—	8	1	no	—
585	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
586	C₂²	C₃₁×#525	1	1	yes	characteristic
587	C₂²	C₃₁×#525	1	1	yes	characteristic
588	C₂²	C₃₁×#576	1	1	yes	characteristic
589	C₂²	C₃₁×#577	1	1	yes	characteristic
590	C₂²	C₃₁×#577	1	1	yes	characteristic
591	C₂²	C₃₁×#578	1	1	yes	characteristic
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	2	4	no	—
595	C₄	—	4	1	no	—
596	C₄	—	4	2	no	—
597	C₄	—	4	1	no	—
598	C₄	—	4	1	no	—
599	C₄	—	4	2	no	—
600	C₄	—	4	1	no	—
601	C₄	—	4	2	no	—
602	C₄	—	4	2	no	—
603	C₄	—	8	1	no	—
604	C₄	—	8	1	no	—
605	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
606	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
607	C₄	C₃₁×#529	1	1	yes	characteristic
608	C₄	C₃₁×#529	1	1	yes	characteristic
609	C₂	—	8	2	no	—
610	C₂	C₃₁×#2290	1	1	yes	characteristic
611	C₂	C₃₁×#2946	1	1	yes	characteristic
612	C₂	C₃₁×#2947	1	1	yes	characteristic
613	C₂	C₃₁×#3119	1	1	yes	characteristic
614	C₂	C₃₁×#3121	1	1	yes	characteristic
615	C₂	C₃₁×#3124	1	1	yes	characteristic
616	C₂	C₃₁×#3126	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2276	C₂	semidirect
6. C₃₁×#1921	C₂	semidirect
7. C₃₁×#1918	C₂	semidirect
10. C₃₁×#1865	C₂	semidirect
11. C₃₁×#1863	C₂	semidirect
12. C₃₁×#1776	C₂	semidirect
13. C₄×C₃₁×#103	C₂	semidirect
15. C₃₁×#1746	C₂	semidirect
21. C₃₁×#440	C₄	semidirect
203. #21128	C₃₁	direct (N × H)
272. #2276	C₂×C₃₁	semidirect
275. #1921	C₂×C₃₁	semidirect
276. #1918	C₂×C₃₁	semidirect
279. #1865	C₂×C₃₁	semidirect
280. #1863	C₂×C₃₁	semidirect
281. #1776	C₂×C₃₁	semidirect
282. C₄×#103	C₂×C₃₁	semidirect
284. #1746	C₂×C₃₁	semidirect
321. #440	C₄×C₃₁	semidirect
433. C₃₁	#21128	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#440	2
C₃₁×#2319	2
C₃₁×#2276	2
C₃₁×#1926	2
C₃₁×#1925	2
C₃₁×#1921	2
C₃₁×#1918	2
C₃₁×#1880	2
C₃₁×#1865	2
C₃₁×#1863	2
C₃₁×#1776	2
C₄×C₃₁×#103	2
C₄×C₃₁×#102	2
C₃₁×#1746	2
#21128	31

Subgroups — C31×#21128 #21477887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21128 #21477887