Subgroups — C₃₁×#21121 #21477880

Order 7936.

966 conjugacy classes of subgroups.

The subgroup lattice has 632 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×#2295	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1928	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1927	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1921	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1918	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1877	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1877	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1864	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1862	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#1776	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1763	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1762	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#1745	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×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₃₁	—	2	1	no	—
22	C₃₁×#440	C₄	1	8	yes	complemented
23	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#106	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×#103	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×#102	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
34	C₃₁×#391	C₂²	1	1	yes	characteristic
35	C₃₁×#391	C₂²	1	1	yes	characteristic
36	C₃₁×#389	C₂²	1	1	yes	characteristic
37	C₃₁×#389	C₂²	1	1	yes	characteristic
38	C₃₁×#387	—	2	1	no	—
39	C₃₁×#387	—	2	1	no	—
40	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
44	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
45	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
46	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
47	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
48	C₃₁×#376	—	2	1	no	—
49	C₄³×C₃₁	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	C₂³×C₄×C₃₁	—	2	1	no	—
64	C₂³×C₄×C₃₁	—	2	1	no	—
65	C₂×SD₁₆×C₃₁	—	2	2	no	—
66	C₂×SD₁₆×C₃₁	—	2	2	no	—
67	C₂×SD₁₆×C₃₁	C₂×C₄	1	4	yes	—
68	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₃₁	—	2	2	no	—
73	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	complemented
74	C₄×D₄×C₃₁	—	2	2	no	—
75	C₄×D₄×C₃₁	—	2	2	no	—
76	C₄×D₄×C₃₁	—	2	2	no	—
77	C₄×D₄×C₃₁	—	2	2	no	—
78	C₄×D₄×C₃₁	—	2	2	no	—
79	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
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×C₄²×C₃₁	—	2	1	no	—
93	C₂×C₄²×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	—	4	4	no	—
98	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
105	C₃₁×#107	—	2	2	no	—
106	C₃₁×#107	C₂×C₄	1	4	yes	complemented
107	C₃₁×#106	—	2	2	no	—
108	C₃₁×#103	—	2	2	no	—
109	C₃₁×#103	—	2	2	no	—
110	C₃₁×#103	C₂×C₄	1	4	yes	complemented
111	C₃₁×#102	—	2	2	no	—
112	C₃₁×#102	—	2	2	no	—
113	C₃₁×#102	C₂×C₄	1	4	yes	—
114	C₃₁×#97	—	2	2	no	—
115	C₃₁×#97	C₂×C₄	1	4	yes	complemented
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	—	2	1	no	—
122	C₃₁×#95	—	2	1	no	—
123	C₃₁×#95	—	2	1	no	—
124	C₃₁×#95	—	2	1	no	—
125	C₃₁×#95	—	2	1	no	—
126	C₂⁴×C₃₁	—	2	1	no	—
127	C₂×Q₈×C₃₁	—	2	1	no	—
128	C₂×Q₈×C₃₁	—	2	1	no	—
129	C₂×Q₈×C₃₁	—	2	1	no	—
130	C₂×Q₈×C₃₁	—	2	1	no	—
131	C₂×Q₈×C₃₁	—	2	1	no	—
132	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
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	C₂²×C₄	1	2	yes	—
137	C₂×D₄×C₃₁	#41	1	2	yes	—
138	C₂×D₄×C₃₁	#41	1	2	yes	—
139	C₂²×C₄×C₃₁	—	2	4	no	—
140	C₂²×C₄×C₃₁	—	2	4	no	—
141	C₂²×C₄×C₃₁	—	2	1	no	—
142	C₂²×C₄×C₃₁	—	2	1	no	—
143	C₂²×C₄×C₃₁	—	2	1	no	—
144	C₂²×C₄×C₃₁	—	2	1	no	—
145	C₂²×C₄×C₃₁	—	2	1	no	—
146	C₂²×C₄×C₃₁	—	2	4	no	—
147	C₂²×C₄×C₃₁	—	2	1	no	—
148	C₂²×C₄×C₃₁	—	2	1	no	—
149	C₂²×C₄×C₃₁	—	4	2	no	—
150	C₂²×C₄×C₃₁	—	4	2	no	—
151	C₂²×C₄×C₃₁	—	4	2	no	—
152	C₂²×C₄×C₃₁	—	4	2	no	—
153	C₂²×C₄×C₃₁	—	4	2	no	—
154	C₂²×C₄×C₃₁	—	4	2	no	—
155	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
162	SD₁₆×C₃₁	—	2	8	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	—	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	—	2	1	no	—
172	C₂×C₈×C₃₁	—	2	1	no	—
173	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
174	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
175	C₄⋊C₄×C₃₁	—	2	4	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	—	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	—	4	2	no	—
185	C₄⋊C₄×C₃₁	—	4	2	no	—
186	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
187	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
188	C₄⋊C₄×C₃₁	#41	1	2	yes	—
189	C₄⋊C₄×C₃₁	#41	1	2	yes	—
190	C₂²⋊C₄×C₃₁	—	2	4	no	—
191	C₂²⋊C₄×C₃₁	—	2	4	no	—
192	C₂²⋊C₄×C₃₁	—	2	4	no	—
193	C₄²×C₃₁	—	2	4	no	—
194	C₄²×C₃₁	—	2	2	no	—
195	C₄²×C₃₁	—	2	2	no	—
196	C₄²×C₃₁	—	2	1	no	—
197	C₄²×C₃₁	—	2	1	no	—
198	C₄²×C₃₁	—	2	1	no	—
199	C₄²×C₃₁	—	2	1	no	—
200	C₄²×C₃₁	—	4	1	no	—
201	C₄²×C₃₁	—	4	1	no	—
202	C₄²×C₃₁	—	4	2	no	—
203	C₄²×C₃₁	—	4	2	no	—
204	C₄²×C₃₁	—	4	2	no	—
205	C₄²×C₃₁	—	4	2	no	—
206	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
207	C₄²×C₃₁	#41	1	2	yes	—
208	#21121	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂³×C₃₁	—	2	2	no	—
210	C₂³×C₃₁	—	2	2	no	—
211	C₂³×C₃₁	—	2	2	no	—
212	C₂³×C₃₁	—	4	1	no	—
213	C₂³×C₃₁	—	4	1	no	—
214	C₂³×C₃₁	—	4	1	no	—
215	C₂³×C₃₁	—	4	1	no	—
216	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
217	Q₈×C₃₁	—	2	2	no	—
218	Q₈×C₃₁	—	2	2	no	—
219	Q₈×C₃₁	—	4	1	no	—
220	Q₈×C₃₁	—	4	1	no	—
221	D₄×C₃₁	—	2	2	no	—
222	D₄×C₃₁	—	2	2	no	—
223	D₄×C₃₁	—	2	2	no	—
224	D₄×C₃₁	#117	1	4	yes	—
225	C₂×C₄×C₃₁	—	2	1	no	—
226	C₂×C₄×C₃₁	—	2	1	no	—
227	C₂×C₄×C₃₁	—	2	1	no	—
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	2	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	1	no	—
237	C₂×C₄×C₃₁	—	2	1	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	2	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	—	2	1	no	—
244	C₂×C₄×C₃₁	—	4	8	no	—
245	C₂×C₄×C₃₁	—	4	8	no	—
246	C₂×C₄×C₃₁	—	4	8	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	—	4	1	no	—
258	C₂×C₄×C₃₁	—	4	1	no	—
259	C₂×C₄×C₃₁	—	4	1	no	—
260	C₂×C₄×C₃₁	—	4	1	no	—
261	C₂×C₄×C₃₁	—	4	1	no	—
262	C₂×C₄×C₃₁	—	4	1	no	—
263	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
264	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
265	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
266	C₂×C₄×C₃₁	#137	1	2	yes	—
267	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
268	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
269	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
270	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
271	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
272	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
273	C₈×C₃₁	—	2	2	no	—
274	C₈×C₃₁	—	2	2	no	—
275	C₈×C₃₁	—	4	1	no	—
276	C₈×C₃₁	—	4	1	no	—
277	C₂×#440	C₂×C₃₁	1	1	yes	characteristic
278	#2295	C₂×C₃₁	1	1	yes	characteristic, complemented
279	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic
280	#1928	C₂×C₃₁	1	1	yes	characteristic, complemented
281	#1927	C₂×C₃₁	1	1	yes	characteristic, complemented
282	#1921	C₂×C₃₁	1	1	yes	characteristic, complemented
283	#1918	C₂×C₃₁	1	1	yes	characteristic, complemented
284	#1877	C₂×C₃₁	1	1	yes	characteristic
285	#1877	C₂×C₃₁	1	1	yes	characteristic
286	#1864	C₂×C₃₁	1	1	yes	characteristic
287	#1862	C₂×C₃₁	1	1	yes	characteristic
288	#1776	C₂×C₃₁	1	1	yes	characteristic, complemented
289	#1763	C₂×C₃₁	1	1	yes	characteristic, complemented
290	#1762	C₂×C₃₁	1	1	yes	characteristic
291	#1745	C₂×C₃₁	1	1	yes	characteristic, complemented
292	C₂²×C₃₁	—	2	4	no	—
293	C₂²×C₃₁	—	4	2	no	—
294	C₂²×C₃₁	—	4	2	no	—
295	C₂²×C₃₁	—	4	2	no	—
296	C₂²×C₃₁	—	4	2	no	—
297	C₂²×C₃₁	—	4	2	no	—
298	C₂²×C₃₁	—	4	2	no	—
299	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
300	C₂²×C₃₁	#525	1	1	yes	characteristic
301	C₂²×C₃₁	#525	1	1	yes	characteristic
302	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
303	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
304	C₂²×C₃₁	#575	1	1	yes	characteristic
305	C₂²×C₃₁	#575	1	1	yes	characteristic
306	C₄×C₃₁	—	2	2	no	—
307	C₄×C₃₁	—	2	2	no	—
308	C₄×C₃₁	—	2	2	no	—
309	C₄×C₃₁	—	2	2	no	—
310	C₄×C₃₁	—	4	2	no	—
311	C₄×C₃₁	—	4	1	no	—
312	C₄×C₃₁	—	4	2	no	—
313	C₄×C₃₁	—	4	1	no	—
314	C₄×C₃₁	—	4	2	no	—
315	C₄×C₃₁	—	4	2	no	—
316	C₄×C₃₁	—	4	2	no	—
317	C₄×C₃₁	—	8	1	no	—
318	C₄×C₃₁	—	8	1	no	—
319	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
320	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
321	C₄×C₃₁	#529	1	1	yes	characteristic
322	C₄×C₃₁	#529	1	1	yes	characteristic
323	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
324	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
325	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
326	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
327	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
328	C₂²×C₄²	—	2	1	no	—
329	#440	C₄×C₃₁	1	8	yes	complemented
330	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
332	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
333	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
334	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
335	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
336	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
337	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
339	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
340	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
341	#391	C₂²×C₃₁	1	1	yes	characteristic
342	#391	C₂²×C₃₁	1	1	yes	characteristic
343	#389	C₂²×C₃₁	1	1	yes	characteristic
344	#389	C₂²×C₃₁	1	1	yes	characteristic
345	#387	—	2	1	no	—
346	#387	—	2	1	no	—
347	C₄×C₄⋊C₄	—	2	1	no	—
348	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
349	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
350	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
351	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
352	C₄×C₂²⋊C₄	—	2	1	no	—
353	C₄×C₂²⋊C₄	—	2	1	no	—
354	C₄×C₂²⋊C₄	—	2	1	no	—
355	#376	—	2	1	no	—
356	C₄³	C₂²×C₃₁	1	1	yes	characteristic
357	#340	C₂²×C₃₁	1	1	yes	characteristic
358	#340	C₂²×C₃₁	1	1	yes	characteristic
359	#340	C₂²×C₃₁	1	1	yes	characteristic
360	#340	C₂²×C₃₁	1	1	yes	characteristic
361	#340	C₂²×C₃₁	1	1	yes	characteristic
362	#340	C₂²×C₃₁	1	1	yes	characteristic
363	#336	C₂²×C₃₁	1	1	yes	characteristic
364	#336	C₂²×C₃₁	1	1	yes	characteristic
365	#336	C₂²×C₃₁	1	1	yes	characteristic
366	#336	C₂²×C₃₁	1	1	yes	characteristic
367	C₂×C₃₁	—	4	4	no	—
368	C₂×C₃₁	#2290	1	1	yes	characteristic
369	C₂×C₃₁	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
370	C₂×C₃₁	C₄×#137	1	1	yes	characteristic
371	C₂×C₃₁	#3110	1	1	yes	characteristic
372	C₂×C₃₁	#3111	1	1	yes	characteristic
373	C₂×C₃₁	#3114	1	1	yes	characteristic
374	C₂×C₃₁	#3116	1	1	yes	characteristic
375	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
376	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
377	C₂³×C₄	—	2	1	no	—
378	C₂³×C₄	—	2	1	no	—
379	C₂³×C₄	—	2	1	no	—
380	C₂×SD₁₆	—	2	2	no	—
381	C₂×SD₁₆	—	2	2	no	—
382	C₂×SD₁₆	C₂×C₄×C₃₁	1	4	yes	—
383	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
384	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
385	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
386	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
387	C₄×Q₈	—	2	2	no	—
388	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
389	C₄×D₄	—	2	2	no	—
390	C₄×D₄	—	2	2	no	—
391	C₄×D₄	—	2	2	no	—
392	C₄×D₄	—	2	2	no	—
393	C₄×D₄	—	2	2	no	—
394	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
395	C₂×C₄⋊C₄	—	2	1	no	—
396	C₂×C₄⋊C₄	—	2	1	no	—
397	C₂×C₄⋊C₄	—	2	1	no	—
398	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
399	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
400	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
401	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
402	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
403	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
404	C₂×C₂²⋊C₄	—	2	1	no	—
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₂²⋊C₄	—	2	1	no	—
407	C₂×C₄²	—	2	1	no	—
408	C₂×C₄²	—	2	1	no	—
409	C₂×C₄²	—	2	1	no	—
410	C₂×C₄²	—	2	1	no	—
411	C₂×C₄²	—	2	1	no	—
412	C₂×C₄²	—	4	4	no	—
413	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
414	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
415	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
416	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
417	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
418	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
420	#107	—	2	2	no	—
421	#107	C₂×C₄×C₃₁	1	4	yes	complemented
422	#106	—	2	2	no	—
423	#103	—	2	2	no	—
424	#103	—	2	2	no	—
425	#103	C₂×C₄×C₃₁	1	4	yes	complemented
426	#102	—	2	2	no	—
427	#102	—	2	2	no	—
428	#102	C₂×C₄×C₃₁	1	4	yes	—
429	#97	—	2	2	no	—
430	#97	C₂×C₄×C₃₁	1	4	yes	complemented
431	#95	—	2	1	no	—
432	#95	—	2	1	no	—
433	#95	—	2	1	no	—
434	#95	—	2	1	no	—
435	#95	—	2	1	no	—
436	#95	—	2	1	no	—
437	#95	—	2	1	no	—
438	#95	—	2	1	no	—
439	#95	—	2	1	no	—
440	#95	—	2	1	no	—
441	C₃₁	#21121	1	1	yes	characteristic, direct product factor, Sylow
442	C₂⁴	—	2	1	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	1	no	—
448	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	1	no	—
452	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
453	C₂×D₄	#41×C₃₁	1	2	yes	—
454	C₂×D₄	#41×C₃₁	1	2	yes	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	4	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	4	2	no	—
466	C₂²×C₄	—	4	2	no	—
467	C₂²×C₄	—	4	2	no	—
468	C₂²×C₄	—	4	2	no	—
469	C₂²×C₄	—	4	2	no	—
470	C₂²×C₄	—	4	2	no	—
471	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
474	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
478	SD₁₆	—	2	8	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	2	1	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
490	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
491	C₄⋊C₄	—	2	4	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	—	2	1	no	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	4	2	no	—
501	C₄⋊C₄	—	4	2	no	—
502	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
503	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
504	C₄⋊C₄	#41×C₃₁	1	2	yes	—
505	C₄⋊C₄	#41×C₃₁	1	2	yes	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₄²	—	2	4	no	—
510	C₄²	—	2	2	no	—
511	C₄²	—	2	2	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	2	1	no	—
514	C₄²	—	2	1	no	—
515	C₄²	—	2	1	no	—
516	C₄²	—	4	1	no	—
517	C₄²	—	4	1	no	—
518	C₄²	—	4	2	no	—
519	C₄²	—	4	2	no	—
520	C₄²	—	4	2	no	—
521	C₄²	—	4	2	no	—
522	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
523	C₄²	#41×C₃₁	1	2	yes	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	4	1	no	—
528	C₂³	—	4	1	no	—
529	C₂³	—	4	1	no	—
530	C₂³	—	4	1	no	—
531	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
532	Q₈	—	2	2	no	—
533	Q₈	—	2	2	no	—
534	Q₈	—	4	1	no	—
535	Q₈	—	4	1	no	—
536	D₄	—	2	2	no	—
537	D₄	—	2	2	no	—
538	D₄	—	2	2	no	—
539	D₄	C₃₁×#117	1	4	yes	—
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	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	4	8	no	—
560	C₂×C₄	—	4	8	no	—
561	C₂×C₄	—	4	8	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
579	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
580	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
581	C₂×C₄	C₃₁×#137	1	2	yes	—
582	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
583	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
584	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
585	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
586	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
587	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
588	C₈	—	2	2	no	—
589	C₈	—	2	2	no	—
590	C₈	—	4	1	no	—
591	C₈	—	4	1	no	—
592	C₂²	—	2	4	no	—
593	C₂²	—	4	2	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	4	2	no	—
596	C₂²	—	4	2	no	—
597	C₂²	—	4	2	no	—
598	C₂²	—	4	2	no	—
599	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
600	C₂²	C₃₁×#525	1	1	yes	characteristic
601	C₂²	C₃₁×#525	1	1	yes	characteristic
602	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
603	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
604	C₂²	C₃₁×#575	1	1	yes	characteristic
605	C₂²	C₃₁×#575	1	1	yes	characteristic
606	C₄	—	2	2	no	—
607	C₄	—	2	2	no	—
608	C₄	—	2	2	no	—
609	C₄	—	2	2	no	—
610	C₄	—	4	2	no	—
611	C₄	—	4	1	no	—
612	C₄	—	4	2	no	—
613	C₄	—	4	1	no	—
614	C₄	—	4	2	no	—
615	C₄	—	4	2	no	—
616	C₄	—	4	2	no	—
617	C₄	—	8	1	no	—
618	C₄	—	8	1	no	—
619	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
620	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
621	C₄	C₃₁×#529	1	1	yes	characteristic
622	C₄	C₃₁×#529	1	1	yes	characteristic
623	C₂	—	4	4	no	—
624	C₂	C₃₁×#2290	1	1	yes	characteristic
625	C₂	C₄×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
626	C₂	C₄×C₃₁×#137	1	1	yes	characteristic
627	C₂	C₃₁×#3110	1	1	yes	characteristic
628	C₂	C₃₁×#3111	1	1	yes	characteristic
629	C₂	C₃₁×#3114	1	1	yes	characteristic
630	C₂	C₃₁×#3116	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 ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2295	C₂	semidirect
4. C₃₁×#1928	C₂	semidirect
5. C₃₁×#1927	C₂	semidirect
6. C₃₁×#1921	C₂	semidirect
7. C₃₁×#1918	C₂	semidirect
12. C₃₁×#1776	C₂	semidirect
13. C₃₁×#1763	C₂	semidirect
15. C₃₁×#1745	C₂	semidirect
22. C₃₁×#440	C₄	semidirect
73. C₄×Q₈×C₃₁	C₂×C₄	semidirect
106. C₃₁×#107	C₂×C₄	semidirect
110. C₃₁×#103	C₂×C₄	semidirect
115. C₃₁×#97	C₂×C₄	semidirect
208. #21121	C₃₁	direct (N × H)
278. #2295	C₂×C₃₁	semidirect
280. #1928	C₂×C₃₁	semidirect
281. #1927	C₂×C₃₁	semidirect
282. #1921	C₂×C₃₁	semidirect
283. #1918	C₂×C₃₁	semidirect
288. #1776	C₂×C₃₁	semidirect
289. #1763	C₂×C₃₁	semidirect
291. #1745	C₂×C₃₁	semidirect
329. #440	C₄×C₃₁	semidirect
388. C₄×Q₈	C₂×C₄×C₃₁	semidirect
421. #107	C₂×C₄×C₃₁	semidirect
425. #103	C₂×C₄×C₃₁	semidirect
430. #97	C₂×C₄×C₃₁	semidirect
441. C₃₁	#21121	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×#2295	2
C₄²×D₄×C₃₁	2
C₃₁×#1928	2
C₃₁×#1927	2
C₃₁×#1921	2
C₃₁×#1918	2
C₃₁×#1877	2
C₃₁×#1864	2
C₃₁×#1862	2
C₃₁×#1776	2
C₃₁×#1763	2
C₃₁×#1762	2
C₃₁×#1745	2
#21121	31

Subgroups — C31×#21121 #21477880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21121 #21477880