Subgroups — C₃₁×#23339 #21480098

Order 7936.

700 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1924	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1924	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1923	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1923	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1917	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1917	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1916	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1916	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1817	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic
16	C₂×C₃₁×C₂²⋊Q₈	C₄	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₃₁	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×C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×C₂²⋊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×#97	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×#391	C₄	1	1	yes	characteristic
34	C₃₁×#389	C₄	1	1	yes	characteristic
35	C₃₁×#387	C₄	1	1	yes	characteristic
36	C₃₁×#387	C₄	1	1	yes	characteristic
37	C₃₁×#385	C₄	1	1	yes	characteristic
38	C₃₁×#385	C₄	1	1	yes	characteristic
39	C₃₁×#380	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	C₃₁×#336	C₂²	1	1	yes	characteristic
46	C₃₁×#336	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	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
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×C₂²⋊Q₈	—	2	4	no	—
72	C₄×Q₈×C₃₁	—	4	1	no	—
73	C₄×Q₈×C₃₁	—	4	1	no	—
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×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
85	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
86	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄²×C₃₁	—	4	1	no	—
88	C₂×C₄²×C₃₁	—	4	1	no	—
89	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄²×C₃₁	C₂×C₄	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	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
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×C₂²⋊C₈	—	4	1	no	—
107	C₃₁×#97	—	4	1	no	—
108	C₃₁×#97	—	4	1	no	—
109	C₄×C₈×C₃₁	—	4	1	no	—
110	C₄×C₈×C₃₁	—	4	1	no	—
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	C₂×Q₈×C₃₁	—	2	2	no	—
123	C₂×Q₈×C₃₁	—	2	2	no	—
124	C₂×Q₈×C₃₁	—	4	1	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	2	1	no	—
129	C₂²×C₄×C₃₁	—	2	1	no	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	2	1	no	—
134	C₂²×C₄×C₃₁	—	4	1	no	—
135	C₂²×C₄×C₃₁	—	4	1	no	—
136	C₂²×C₄×C₃₁	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
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	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	—	2	1	no	—
174	C₂×C₈×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	—	2	2	no	—
176	C₄⋊C₄×C₃₁	—	2	2	no	—
177	C₄⋊C₄×C₃₁	—	2	2	no	—
178	C₄⋊C₄×C₃₁	—	4	1	no	—
179	C₄⋊C₄×C₃₁	—	4	1	no	—
180	C₄⋊C₄×C₃₁	—	4	1	no	—
181	C₄⋊C₄×C₃₁	—	4	1	no	—
182	C₄⋊C₄×C₃₁	—	4	1	no	—
183	C₄⋊C₄×C₃₁	M₁₆	1	4	yes	—
184	C₂²⋊C₄×C₃₁	—	2	2	no	—
185	C₂²⋊C₄×C₃₁	—	2	2	no	—
186	C₂²⋊C₄×C₃₁	—	2	2	no	—
187	C₂²⋊C₄×C₃₁	—	2	2	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	—	2	1	no	—
195	C₄²×C₃₁	—	2	1	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	#23339	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂³×C₃₁	—	2	1	no	—
204	C₂³×C₃₁	—	2	1	no	—
205	C₂³×C₃₁	—	2	1	no	—
206	C₂³×C₃₁	—	2	1	no	—
207	C₂³×C₃₁	—	2	1	no	—
208	C₂³×C₃₁	—	2	1	no	—
209	C₂³×C₃₁	#131	1	2	yes	—
210	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
211	Q₈×C₃₁	—	4	4	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	—	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	—	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	4	no	—
248	C₂×C₄×C₃₁	—	4	4	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	C₂×M₁₆	1	2	yes	—
254	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
255	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
256	C₂×C₄×C₃₁	#131	1	2	yes	—
257	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
258	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
259	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
260	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
261	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
262	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
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	—	4	1	no	—
268	C₈×C₃₁	—	4	1	no	—
269	C₈×C₃₁	—	4	1	no	—
270	C₈×C₃₁	—	4	1	no	—
271	C₄×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
272	#1985	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
274	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#1924	C₂×C₃₁	1	1	yes	characteristic
277	#1924	C₂×C₃₁	1	1	yes	characteristic
278	#1923	C₂×C₃₁	1	1	yes	characteristic
279	#1923	C₂×C₃₁	1	1	yes	characteristic
280	#1917	C₂×C₃₁	1	1	yes	characteristic, complemented
281	#1917	C₂×C₃₁	1	1	yes	characteristic, complemented
282	#1916	C₂×C₃₁	1	1	yes	characteristic, complemented
283	#1916	C₂×C₃₁	1	1	yes	characteristic, complemented
284	#1817	C₂×C₃₁	1	1	yes	characteristic
285	#1797	C₂×C₃₁	1	1	yes	characteristic
286	C₂²×C₃₁	—	2	2	no	—
287	C₂²×C₃₁	—	2	2	no	—
288	C₂²×C₃₁	—	2	2	no	—
289	C₂²×C₃₁	—	4	1	no	—
290	C₂²×C₃₁	—	4	1	no	—
291	C₂²×C₃₁	—	4	1	no	—
292	C₂²×C₃₁	—	4	1	no	—
293	C₂²×C₃₁	#518	1	1	yes	characteristic
294	C₂²×C₃₁	#519	1	1	yes	characteristic
295	C₂²×C₃₁	#520	1	1	yes	characteristic
296	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
297	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
298	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
299	C₂²×C₃₁	#568	1	1	yes	characteristic
300	C₄×C₃₁	—	2	2	no	—
301	C₄×C₃₁	—	2	2	no	—
302	C₄×C₃₁	—	2	2	no	—
303	C₄×C₃₁	—	2	2	no	—
304	C₄×C₃₁	—	4	1	no	—
305	C₄×C₃₁	—	4	1	no	—
306	C₄×C₃₁	—	4	1	no	—
307	C₄×C₃₁	—	4	1	no	—
308	C₄×C₃₁	—	4	1	no	—
309	C₄×C₃₁	—	4	1	no	—
310	C₄×C₃₁	—	4	1	no	—
311	C₄×C₃₁	—	4	1	no	—
312	C₄×C₃₁	—	4	1	no	—
313	C₄×C₃₁	#562	1	1	yes	characteristic
314	C₄×C₃₁	#562	1	1	yes	characteristic
315	C₄×C₃₁	#563	1	1	yes	characteristic
316	C₄×C₃₁	#563	1	1	yes	characteristic
317	C₂×C₂²⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
318	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
319	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
320	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
321	C₂×C₄⋊C₈	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	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
325	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
326	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
327	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
328	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
329	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
330	C₂×#97	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	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
334	#391	C₄×C₃₁	1	1	yes	characteristic
335	#389	C₄×C₃₁	1	1	yes	characteristic
336	#387	C₄×C₃₁	1	1	yes	characteristic
337	#387	C₄×C₃₁	1	1	yes	characteristic
338	#385	C₄×C₃₁	1	1	yes	characteristic
339	#385	C₄×C₃₁	1	1	yes	characteristic
340	#380	C₄×C₃₁	1	1	yes	characteristic
341	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
342	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
343	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
344	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
345	C₄×C₂²⋊C₄	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	#336	C₂²×C₃₁	1	1	yes	characteristic
351	#336	C₂²×C₃₁	1	1	yes	characteristic
352	#336	C₂²×C₃₁	1	1	yes	characteristic
353	#336	C₂²×C₃₁	1	1	yes	characteristic
354	#336	C₂²×C₃₁	1	1	yes	characteristic
355	#336	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	—	4	2	no	—
361	C₂×C₃₁	#2378	1	1	yes	characteristic
362	C₂×C₃₁	#2974	1	1	yes	characteristic
363	C₂×C₃₁	#2976	1	1	yes	characteristic
364	C₂×C₃₁	#2984	1	1	yes	characteristic
365	C₂×C₃₁	#2985	1	1	yes	characteristic
366	C₂×C₃₁	#2989	1	1	yes	characteristic
367	C₂×C₃₁	#2994	1	1	yes	characteristic
368	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
369	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
370	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
371	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
372	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
373	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
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	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
379	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
380	C₂²⋊Q₈	—	2	4	no	—
381	C₄×Q₈	—	4	1	no	—
382	C₄×Q₈	—	4	1	no	—
383	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
384	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
385	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
386	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
388	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
390	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
392	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
393	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
394	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
395	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
396	C₂×C₄²	—	4	1	no	—
397	C₂×C₄²	—	4	1	no	—
398	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
399	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
400	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
401	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
402	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
403	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
404	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
405	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
406	C₄⋊C₈	—	4	1	no	—
407	C₄⋊C₈	—	4	1	no	—
408	C₄⋊C₈	—	4	1	no	—
409	C₄⋊C₈	—	4	1	no	—
410	C₄⋊C₈	—	4	1	no	—
411	C₄⋊C₈	—	4	1	no	—
412	C₂²⋊C₈	—	4	1	no	—
413	C₂²⋊C₈	—	4	1	no	—
414	C₂²⋊C₈	—	4	1	no	—
415	C₂²⋊C₈	—	4	1	no	—
416	#97	—	4	1	no	—
417	#97	—	4	1	no	—
418	C₄×C₈	—	4	1	no	—
419	C₄×C₈	—	4	1	no	—
420	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
421	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
422	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
423	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
424	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
425	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
426	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
427	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
428	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
429	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
430	C₃₁	#23339	1	1	yes	characteristic, direct product factor, Sylow
431	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
432	C₂×Q₈	—	2	2	no	—
433	C₂×Q₈	—	2	2	no	—
434	C₂×Q₈	—	4	1	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	4	1	no	—
445	C₂²×C₄	—	4	1	no	—
446	C₂²×C₄	—	4	1	no	—
447	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
448	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
449	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
450	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
451	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
452	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
453	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
454	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
455	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
456	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
457	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
458	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
459	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
460	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
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	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	—	2	1	no	—
476	C₂×C₈	—	2	1	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	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	2	1	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	4	1	no	—
489	C₄⋊C₄	—	4	1	no	—
490	C₄⋊C₄	—	4	1	no	—
491	C₄⋊C₄	—	4	1	no	—
492	C₄⋊C₄	—	4	1	no	—
493	C₄⋊C₄	M₁₆×C₃₁	1	4	yes	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	2	2	no	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₄²	—	2	1	no	—
499	C₄²	—	2	1	no	—
500	C₄²	—	2	1	no	—
501	C₄²	—	2	1	no	—
502	C₄²	—	2	1	no	—
503	C₄²	—	2	1	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	2	1	no	—
506	C₄²	—	2	1	no	—
507	C₄²	—	2	1	no	—
508	C₄²	—	2	1	no	—
509	C₄²	—	2	1	no	—
510	C₄²	—	4	1	no	—
511	C₄²	—	4	1	no	—
512	C₂³	—	2	1	no	—
513	C₂³	—	2	1	no	—
514	C₂³	—	2	1	no	—
515	C₂³	—	2	1	no	—
516	C₂³	—	2	1	no	—
517	C₂³	—	2	1	no	—
518	C₂³	C₃₁×#131	1	2	yes	—
519	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
520	Q₈	—	4	4	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	—	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	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	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	4	4	no	—
557	C₂×C₄	—	4	4	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	C₂×M₁₆×C₃₁	1	2	yes	—
563	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
564	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
565	C₂×C₄	C₃₁×#131	1	2	yes	—
566	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
567	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
568	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
569	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
570	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
571	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
572	C₈	—	4	1	no	—
573	C₈	—	4	1	no	—
574	C₈	—	4	1	no	—
575	C₈	—	4	1	no	—
576	C₈	—	4	1	no	—
577	C₈	—	4	1	no	—
578	C₈	—	4	1	no	—
579	C₈	—	4	1	no	—
580	C₂²	—	2	2	no	—
581	C₂²	—	2	2	no	—
582	C₂²	—	2	2	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	1	no	—
585	C₂²	—	4	1	no	—
586	C₂²	—	4	1	no	—
587	C₂²	C₃₁×#518	1	1	yes	characteristic
588	C₂²	C₃₁×#519	1	1	yes	characteristic
589	C₂²	C₃₁×#520	1	1	yes	characteristic
590	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
591	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
592	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
593	C₂²	C₃₁×#568	1	1	yes	characteristic
594	C₄	—	2	2	no	—
595	C₄	—	2	2	no	—
596	C₄	—	2	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	4	1	no	—
599	C₄	—	4	1	no	—
600	C₄	—	4	1	no	—
601	C₄	—	4	1	no	—
602	C₄	—	4	1	no	—
603	C₄	—	4	1	no	—
604	C₄	—	4	1	no	—
605	C₄	—	4	1	no	—
606	C₄	—	4	1	no	—
607	C₄	C₃₁×#562	1	1	yes	characteristic
608	C₄	C₃₁×#562	1	1	yes	characteristic
609	C₄	C₃₁×#563	1	1	yes	characteristic
610	C₄	C₃₁×#563	1	1	yes	characteristic
611	C₂	—	4	2	no	—
612	C₂	C₃₁×#2378	1	1	yes	characteristic
613	C₂	C₃₁×#2974	1	1	yes	characteristic
614	C₂	C₃₁×#2976	1	1	yes	characteristic
615	C₂	C₃₁×#2984	1	1	yes	characteristic
616	C₂	C₃₁×#2985	1	1	yes	characteristic
617	C₂	C₃₁×#2989	1	1	yes	characteristic
618	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×#1985	C₂	semidirect
3. C₃₁×#1984	C₂	semidirect
4. C₃₁×#1940	C₂	semidirect
5. C₃₁×#1939	C₂	semidirect
10. C₃₁×#1917	C₂	semidirect
11. C₃₁×#1917	C₂	semidirect
12. C₃₁×#1916	C₂	semidirect
13. C₃₁×#1916	C₂	semidirect
202. #23339	C₃₁	direct (N × H)
272. #1985	C₂×C₃₁	semidirect
273. #1984	C₂×C₃₁	semidirect
274. #1940	C₂×C₃₁	semidirect
275. #1939	C₂×C₃₁	semidirect
280. #1917	C₂×C₃₁	semidirect
281. #1917	C₂×C₃₁	semidirect
282. #1916	C₂×C₃₁	semidirect
283. #1916	C₂×C₃₁	semidirect
430. C₃₁	#23339	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×C₂²⋊Q₈	2
C₃₁×#1985	2
C₃₁×#1984	2
C₃₁×#1940	2
C₃₁×#1939	2
C₃₁×#1924	2
C₃₁×#1923	2
C₃₁×#1917	2
C₃₁×#1916	2
C₃₁×#1817	2
C₃₁×#1797	2
#23339	31

Subgroups — C31×#23339 #21480098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23339 #21480098