Subgroups — C₃₁×#23377 #21480136

Order 7936.

662 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#126	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1924	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1923	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1923	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1917	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1916	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1916	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1851	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#1851	C₂	1	1	yes	maximal, characteristic
13	C₃₁×#1851	C₂	1	1	yes	maximal, characteristic
14	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1839	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#126	C₄	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#388	C₄	1	1	yes	characteristic
34	C₃₁×#388	C₄	1	1	yes	characteristic
35	C₃₁×#388	C₄	1	1	yes	characteristic
36	C₃₁×#387	C₄	1	1	yes	characteristic
37	C₃₁×#387	C₄	1	1	yes	characteristic
38	C₃₁×#387	C₄	1	1	yes	characteristic
39	C₃₁×#383	C₄	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
44	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
45	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
46	C₄³×C₃₁	C₂²	1	1	yes	characteristic
47	C₃₁×#336	C₂²	1	1	yes	characteristic
48	C₃₁×#336	C₂²	1	1	yes	characteristic
49	C₃₁×#336	C₂²	1	1	yes	characteristic
50	C₃₁×#336	C₂²	1	1	yes	characteristic
51	C₃₁×#336	C₂²	1	1	yes	characteristic
52	C₃₁×#336	C₂²	1	1	yes	characteristic
53	C₃₁×#336	C₂²	1	1	yes	characteristic
54	C₃₁×#336	C₂²	1	1	yes	characteristic
55	C₃₁×#336	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	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₃₁	—	4	1	no	—
61	C₂×M₁₆×C₃₁	—	4	1	no	—
62	C₂×M₁₆×C₃₁	—	4	1	no	—
63	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
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	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#126	—	2	4	no	—
72	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
82	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
91	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
93	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
94	C₃₁×C₄⋊C₈	—	4	1	no	—
95	C₃₁×C₄⋊C₈	—	4	1	no	—
96	C₃₁×C₄⋊C₈	—	4	1	no	—
97	C₃₁×C₄⋊C₈	—	4	1	no	—
98	C₃₁×C₄⋊C₈	—	4	1	no	—
99	C₃₁×C₄⋊C₈	—	4	1	no	—
100	C₃₁×C₄⋊C₈	—	4	1	no	—
101	C₃₁×C₄⋊C₈	—	4	1	no	—
102	C₃₁×C₄⋊C₈	—	4	1	no	—
103	C₃₁×C₂²⋊C₈	—	4	1	no	—
104	C₃₁×C₂²⋊C₈	—	4	1	no	—
105	C₃₁×C₂²⋊C₈	—	4	1	no	—
106	C₃₁×#97	—	4	1	no	—
107	C₄×C₈×C₃₁	—	4	1	no	—
108	C₄×C₈×C₃₁	—	4	1	no	—
109	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
110	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
111	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
112	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
113	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
114	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
115	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
116	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
117	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
118	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
119	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
120	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
121	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	2	1	no	—
128	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
142	M₁₆×C₃₁	—	4	4	no	—
143	C₂×C₈×C₃₁	—	2	1	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	—	2	1	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	2	1	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	—	2	1	no	—
151	C₂×C₈×C₃₁	—	2	1	no	—
152	C₂×C₈×C₃₁	—	2	1	no	—
153	C₂×C₈×C₃₁	—	2	1	no	—
154	C₂×C₈×C₃₁	—	2	1	no	—
155	C₂×C₈×C₃₁	—	2	1	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	—	2	1	no	—
167	C₄⋊C₄×C₃₁	—	2	2	no	—
168	C₄⋊C₄×C₃₁	—	2	2	no	—
169	C₄⋊C₄×C₃₁	—	2	2	no	—
170	C₄⋊C₄×C₃₁	—	2	2	no	—
171	C₄⋊C₄×C₃₁	—	4	1	no	—
172	C₄⋊C₄×C₃₁	M₁₆	1	4	yes	complemented
173	C₂²⋊C₄×C₃₁	—	2	2	no	—
174	C₂²⋊C₄×C₃₁	—	2	2	no	—
175	C₂²⋊C₄×C₃₁	—	2	2	no	—
176	C₂²⋊C₄×C₃₁	—	2	2	no	—
177	C₂²⋊C₄×C₃₁	—	2	2	no	—
178	C₂²⋊C₄×C₃₁	—	4	1	no	—
179	C₄²×C₃₁	—	2	2	no	—
180	C₄²×C₃₁	—	2	2	no	—
181	C₄²×C₃₁	—	2	2	no	—
182	C₄²×C₃₁	—	2	1	no	—
183	C₄²×C₃₁	—	2	1	no	—
184	C₄²×C₃₁	—	2	1	no	—
185	C₄²×C₃₁	—	2	1	no	—
186	C₄²×C₃₁	—	2	1	no	—
187	C₄²×C₃₁	—	2	1	no	—
188	C₄²×C₃₁	—	2	1	no	—
189	C₄²×C₃₁	—	2	1	no	—
190	C₄²×C₃₁	—	2	1	no	—
191	C₄²×C₃₁	—	2	1	no	—
192	C₄²×C₃₁	—	2	1	no	—
193	C₄²×C₃₁	—	2	1	no	—
194	C₄²×C₃₁	—	4	1	no	—
195	#23377	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂³×C₃₁	—	2	1	no	—
197	C₂³×C₃₁	—	2	1	no	—
198	C₂³×C₃₁	—	2	1	no	—
199	C₂³×C₃₁	—	2	1	no	—
200	C₂³×C₃₁	—	2	1	no	—
201	C₂³×C₃₁	—	2	1	no	—
202	C₂³×C₃₁	—	2	1	no	—
203	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	1	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	1	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	1	no	—
213	C₂×C₄×C₃₁	—	2	1	no	—
214	C₂×C₄×C₃₁	—	2	1	no	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	1	no	—
221	C₂×C₄×C₃₁	—	2	1	no	—
222	C₂×C₄×C₃₁	—	2	1	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	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	1	no	—
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	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	—	4	1	no	—
242	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
243	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
244	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
245	C₂×C₄×C₃₁	#131	1	2	yes	—
246	C₂×C₄×C₃₁	#131	1	2	yes	—
247	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
248	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
249	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
250	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
251	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
252	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
253	C₈×C₃₁	—	4	1	no	—
254	C₈×C₃₁	—	4	1	no	—
255	C₈×C₃₁	—	4	1	no	—
256	C₈×C₃₁	—	4	1	no	—
257	C₈×C₃₁	—	4	1	no	—
258	C₈×C₃₁	—	4	1	no	—
259	C₈×C₃₁	—	4	1	no	—
260	C₈×C₃₁	—	4	1	no	—
261	C₄×#126	C₂×C₃₁	1	1	yes	characteristic
262	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
263	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
264	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#1924	C₂×C₃₁	1	1	yes	characteristic
266	#1923	C₂×C₃₁	1	1	yes	characteristic
267	#1923	C₂×C₃₁	1	1	yes	characteristic
268	#1917	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#1916	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#1916	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#1851	C₂×C₃₁	1	1	yes	characteristic
272	#1851	C₂×C₃₁	1	1	yes	characteristic
273	#1851	C₂×C₃₁	1	1	yes	characteristic
274	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#1839	C₂×C₃₁	1	1	yes	characteristic, complemented
276	C₂²×C₃₁	—	4	1	no	—
277	C₂²×C₃₁	—	4	1	no	—
278	C₂²×C₃₁	—	4	1	no	—
279	C₂²×C₃₁	—	4	1	no	—
280	C₂²×C₃₁	—	4	1	no	—
281	C₂²×C₃₁	—	4	1	no	—
282	C₂²×C₃₁	—	4	1	no	—
283	C₂²×C₃₁	#520	1	1	yes	characteristic
284	C₂²×C₃₁	#520	1	1	yes	characteristic
285	C₂²×C₃₁	#520	1	1	yes	characteristic
286	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
287	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
288	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
289	C₂²×C₃₁	#568	1	1	yes	characteristic
290	C₄×C₃₁	—	2	2	no	—
291	C₄×C₃₁	—	2	2	no	—
292	C₄×C₃₁	—	2	2	no	—
293	C₄×C₃₁	—	2	2	no	—
294	C₄×C₃₁	—	2	2	no	—
295	C₄×C₃₁	—	4	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	—	4	1	no	—
302	C₄×C₃₁	—	4	1	no	—
303	C₄×C₃₁	#560	1	1	yes	characteristic
304	C₄×C₃₁	#563	1	1	yes	characteristic
305	C₄×C₃₁	#563	1	1	yes	characteristic
306	C₄×C₃₁	#563	1	1	yes	characteristic
307	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
308	C₂×#126	C₄×C₃₁	1	1	yes	characteristic
309	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
310	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
311	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
312	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
313	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
314	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
315	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
316	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
317	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
318	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
320	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
321	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
322	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
323	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
324	#388	C₄×C₃₁	1	1	yes	characteristic
325	#388	C₄×C₃₁	1	1	yes	characteristic
326	#388	C₄×C₃₁	1	1	yes	characteristic
327	#387	C₄×C₃₁	1	1	yes	characteristic
328	#387	C₄×C₃₁	1	1	yes	characteristic
329	#387	C₄×C₃₁	1	1	yes	characteristic
330	#383	C₄×C₃₁	1	1	yes	characteristic
331	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
332	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
333	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
334	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
335	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
336	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
337	C₄³	C₂²×C₃₁	1	1	yes	characteristic
338	#336	C₂²×C₃₁	1	1	yes	characteristic
339	#336	C₂²×C₃₁	1	1	yes	characteristic
340	#336	C₂²×C₃₁	1	1	yes	characteristic
341	#336	C₂²×C₃₁	1	1	yes	characteristic
342	#336	C₂²×C₃₁	1	1	yes	characteristic
343	#336	C₂²×C₃₁	1	1	yes	characteristic
344	#336	C₂²×C₃₁	1	1	yes	characteristic
345	#336	C₂²×C₃₁	1	1	yes	characteristic
346	#336	C₂²×C₃₁	1	1	yes	characteristic
347	#336	C₂²×C₃₁	1	1	yes	characteristic
348	#336	C₂²×C₃₁	1	1	yes	characteristic
349	#336	C₂²×C₃₁	1	1	yes	characteristic
350	C₂×C₃₁	—	8	1	no	—
351	C₂×C₃₁	#2383	1	1	yes	characteristic
352	C₂×C₃₁	#2989	1	1	yes	characteristic
353	C₂×C₃₁	#2990	1	1	yes	characteristic
354	C₂×C₃₁	#2991	1	1	yes	characteristic
355	C₂×C₃₁	#2992	1	1	yes	characteristic
356	C₂×C₃₁	#2993	1	1	yes	characteristic
357	C₂×C₃₁	#2994	1	1	yes	characteristic
358	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
359	C₂×M₁₆	—	4	1	no	—
360	C₂×M₁₆	—	4	1	no	—
361	C₂×M₁₆	—	4	1	no	—
362	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
363	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
364	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
365	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
366	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
367	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
368	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
369	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
370	#126	—	2	4	no	—
371	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
372	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
373	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
374	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
375	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
377	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
378	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
379	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
380	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
381	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
382	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
383	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
384	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
385	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
386	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
387	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
388	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
389	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
391	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
392	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
393	C₄⋊C₈	—	4	1	no	—
394	C₄⋊C₈	—	4	1	no	—
395	C₄⋊C₈	—	4	1	no	—
396	C₄⋊C₈	—	4	1	no	—
397	C₄⋊C₈	—	4	1	no	—
398	C₄⋊C₈	—	4	1	no	—
399	C₄⋊C₈	—	4	1	no	—
400	C₄⋊C₈	—	4	1	no	—
401	C₄⋊C₈	—	4	1	no	—
402	C₂²⋊C₈	—	4	1	no	—
403	C₂²⋊C₈	—	4	1	no	—
404	C₂²⋊C₈	—	4	1	no	—
405	#97	—	4	1	no	—
406	C₄×C₈	—	4	1	no	—
407	C₄×C₈	—	4	1	no	—
408	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
409	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
410	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
411	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
412	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
413	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
414	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
415	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
416	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
417	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
418	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
419	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
420	C₃₁	#23377	1	1	yes	characteristic, direct product factor, Sylow
421	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
429	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
430	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
431	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
432	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
433	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
434	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
435	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
436	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
437	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
438	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
439	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
440	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
441	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
442	M₁₆	—	4	4	no	—
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	1	no	—
446	C₂×C₈	—	2	1	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	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	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	2	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	4	1	no	—
472	C₄⋊C₄	M₁₆×C₃₁	1	4	yes	complemented
473	C₂²⋊C₄	—	2	2	no	—
474	C₂²⋊C₄	—	2	2	no	—
475	C₂²⋊C₄	—	2	2	no	—
476	C₂²⋊C₄	—	2	2	no	—
477	C₂²⋊C₄	—	2	2	no	—
478	C₂²⋊C₄	—	4	1	no	—
479	C₄²	—	2	2	no	—
480	C₄²	—	2	2	no	—
481	C₄²	—	2	2	no	—
482	C₄²	—	2	1	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	2	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	—	2	1	no	—
489	C₄²	—	2	1	no	—
490	C₄²	—	2	1	no	—
491	C₄²	—	2	1	no	—
492	C₄²	—	2	1	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	4	1	no	—
495	C₂³	—	2	1	no	—
496	C₂³	—	2	1	no	—
497	C₂³	—	2	1	no	—
498	C₂³	—	2	1	no	—
499	C₂³	—	2	1	no	—
500	C₂³	—	2	1	no	—
501	C₂³	—	2	1	no	—
502	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
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	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	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	1	no	—
536	C₂×C₄	—	2	1	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	C₂×M₁₆×C₃₁	1	2	yes	—
542	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
543	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
544	C₂×C₄	C₃₁×#131	1	2	yes	—
545	C₂×C₄	C₃₁×#131	1	2	yes	—
546	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
547	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
548	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
549	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
550	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
551	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
552	C₈	—	4	1	no	—
553	C₈	—	4	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	C₂²	—	4	1	no	—
561	C₂²	—	4	1	no	—
562	C₂²	—	4	1	no	—
563	C₂²	—	4	1	no	—
564	C₂²	—	4	1	no	—
565	C₂²	—	4	1	no	—
566	C₂²	—	4	1	no	—
567	C₂²	C₃₁×#520	1	1	yes	characteristic
568	C₂²	C₃₁×#520	1	1	yes	characteristic
569	C₂²	C₃₁×#520	1	1	yes	characteristic
570	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
571	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
572	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
573	C₂²	C₃₁×#568	1	1	yes	characteristic
574	C₄	—	2	2	no	—
575	C₄	—	2	2	no	—
576	C₄	—	2	2	no	—
577	C₄	—	2	2	no	—
578	C₄	—	2	2	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	—	4	1	no	—
585	C₄	—	4	1	no	—
586	C₄	—	4	1	no	—
587	C₄	C₃₁×#560	1	1	yes	characteristic
588	C₄	C₃₁×#563	1	1	yes	characteristic
589	C₄	C₃₁×#563	1	1	yes	characteristic
590	C₄	C₃₁×#563	1	1	yes	characteristic
591	C₂	—	8	1	no	—
592	C₂	C₃₁×#2383	1	1	yes	characteristic
593	C₂	C₃₁×#2989	1	1	yes	characteristic
594	C₂	C₃₁×#2990	1	1	yes	characteristic
595	C₂	C₃₁×#2991	1	1	yes	characteristic
596	C₂	C₃₁×#2992	1	1	yes	characteristic
597	C₂	C₃₁×#2993	1	1	yes	characteristic
598	C₂	C₃₁×#2994	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1940	C₂	semidirect
3. C₃₁×#1939	C₂	semidirect
4. C₃₁×#1939	C₂	semidirect
8. C₃₁×#1917	C₂	semidirect
9. C₃₁×#1916	C₂	semidirect
10. C₃₁×#1916	C₂	semidirect
14. C₃₁×#1842	C₂	semidirect
15. C₃₁×#1839	C₂	semidirect
172. C₄⋊C₄×C₃₁	M₁₆	semidirect
195. #23377	C₃₁	direct (N × H)
262. #1940	C₂×C₃₁	semidirect
263. #1939	C₂×C₃₁	semidirect
264. #1939	C₂×C₃₁	semidirect
268. #1917	C₂×C₃₁	semidirect
269. #1916	C₂×C₃₁	semidirect
270. #1916	C₂×C₃₁	semidirect
274. #1842	C₂×C₃₁	semidirect
275. #1839	C₂×C₃₁	semidirect
420. C₃₁	#23377	direct (N × H)
472. C₄⋊C₄	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#126	2
C₃₁×#1940	2
C₃₁×#1939	2
C₃₁×#1924	2
C₃₁×#1923	2
C₃₁×#1917	2
C₃₁×#1916	2
C₃₁×#1851	2
C₃₁×#1842	2
C₃₁×#1839	2
#23377	31

Subgroups — C31×#23377 #21480136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23377 #21480136