Subgroups — C₃₁×#22990 #21479749

Order 7936.

1064 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#436	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₃₁×#121	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1954	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1923	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1870	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1851	C₂	1	2	yes	maximal, complemented
10	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×#121	C₄	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#436	—	2	4	no	—
19	C₃₁×#436	—	2	8	no	—
20	C₃₁×#423	—	2	4	no	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
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	2	yes	—
27	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#390	C₄	1	1	yes	characteristic
34	C₃₁×#389	C₄	1	1	yes	characteristic
35	C₃₁×#388	C₄	1	2	yes	—
36	C₃₁×#386	C₄	1	2	yes	—
37	C₃₁×#380	C₄	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
49	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₃₁	—	2	4	no	—
51	C₂×M₁₆×C₃₁	—	2	4	no	—
52	C₂×M₁₆×C₃₁	—	2	4	no	—
53	C₂×M₁₆×C₃₁	—	2	4	no	—
54	C₂×M₁₆×C₃₁	—	4	2	no	—
55	C₂×M₁₆×C₃₁	—	4	2	no	—
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
59	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₃₁×#121	—	2	4	no	—
63	C₄×D₄×C₃₁	—	2	4	no	—
64	C₄×D₄×C₃₁	—	2	4	no	—
65	C₄×D₄×C₃₁	—	2	2	no	—
66	C₄×D₄×C₃₁	—	2	2	no	—
67	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
72	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
73	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
74	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
75	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
76	C₂×C₄²×C₃₁	—	2	2	no	—
77	C₂×C₄²×C₃₁	—	2	2	no	—
78	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×C₄⋊C₈	—	2	4	no	—
84	C₃₁×C₄⋊C₈	—	2	4	no	—
85	C₃₁×C₄⋊C₈	—	2	2	no	—
86	C₃₁×C₄⋊C₈	D₄	1	4	yes	complemented
87	C₃₁×C₂²⋊C₈	—	2	2	no	—
88	C₃₁×C₂²⋊C₈	—	2	4	no	—
89	C₃₁×C₂²⋊C₈	—	2	4	no	—
90	C₃₁×C₂²⋊C₈	—	4	2	no	—
91	C₃₁×C₂²⋊C₈	D₄	1	4	yes	complemented
92	C₄×C₈×C₃₁	—	2	4	no	—
93	C₄×C₈×C₃₁	—	2	2	no	—
94	C₄×C₈×C₃₁	—	2	2	no	—
95	C₃₁×#95	C₂×C₄	1	2	yes	—
96	C₃₁×#95	C₂×C₄	1	2	yes	—
97	C₃₁×#95	C₂×C₄	1	2	yes	—
98	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
99	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×D₄×C₃₁	—	2	4	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	2	4	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	4	2	no	—
106	C₂²×C₄×C₃₁	—	2	1	no	—
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	2	2	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	4	1	no	—
122	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
128	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₃₁	#41	1	2	yes	—
132	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
133	M₁₆×C₃₁	—	4	8	no	—
134	M₁₆×C₃₁	—	4	8	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	2	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	2	2	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	2	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
151	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
152	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	M₁₆	1	4	yes	complemented
157	C₂²⋊C₄×C₃₁	—	2	4	no	—
158	C₂²⋊C₄×C₃₁	—	2	4	no	—
159	C₂²⋊C₄×C₃₁	—	2	4	no	—
160	C₂²⋊C₄×C₃₁	—	2	4	no	—
161	C₂²⋊C₄×C₃₁	—	2	4	no	—
162	C₄²×C₃₁	—	2	2	no	—
163	C₄²×C₃₁	—	2	2	no	—
164	C₄²×C₃₁	—	2	2	no	—
165	C₄²×C₃₁	—	2	1	no	—
166	C₄²×C₃₁	—	2	1	no	—
167	C₄²×C₃₁	—	2	2	no	—
168	C₄²×C₃₁	C₂×D₄	1	2	yes	—
169	C₄²×C₃₁	#41	1	2	yes	—
170	#22990	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂³×C₃₁	—	2	1	no	—
172	C₂³×C₃₁	—	2	1	no	—
173	C₂³×C₃₁	—	2	1	no	—
174	C₂³×C₃₁	—	2	1	no	—
175	C₂³×C₃₁	—	2	1	no	—
176	C₂³×C₃₁	—	2	1	no	—
177	C₂³×C₃₁	—	2	2	no	—
178	C₂³×C₃₁	—	2	2	no	—
179	C₂³×C₃₁	—	2	2	no	—
180	C₂³×C₃₁	—	2	2	no	—
181	C₂³×C₃₁	—	2	2	no	—
182	C₂³×C₃₁	—	2	2	no	—
183	C₂³×C₃₁	—	2	2	no	—
184	C₂³×C₃₁	C₄×D₄	1	2	yes	—
185	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
186	D₄×C₃₁	—	4	8	no	—
187	D₄×C₃₁	—	4	4	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	1	no	—
194	C₂×C₄×C₃₁	—	2	1	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	1	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	4	4	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	4	4	no	—
215	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
216	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
217	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
218	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
219	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
220	C₂×C₄×C₃₁	#131	1	2	yes	—
221	C₂×C₄×C₃₁	#131	1	2	yes	—
222	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
226	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
227	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
228	C₈×C₃₁	—	2	4	no	—
229	C₈×C₃₁	—	2	2	no	—
230	C₈×C₃₁	—	2	2	no	—
231	C₈×C₃₁	—	4	2	no	—
232	C₈×C₃₁	—	4	1	no	—
233	C₈×C₃₁	—	4	1	no	—
234	C₂×#436	C₂×C₃₁	1	2	yes	complemented
235	C₂×#436	C₂×C₃₁	1	1	yes	characteristic, complemented
236	C₂×#423	C₂×C₃₁	1	1	yes	characteristic, complemented
237	C₄×#121	C₂×C₃₁	1	1	yes	characteristic
238	#1954	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#1923	C₂×C₃₁	1	2	yes	complemented
241	#1870	C₂×C₃₁	1	2	yes	complemented
242	#1851	C₂×C₃₁	1	2	yes	complemented
243	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
244	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
245	C₂²×C₃₁	—	2	2	no	—
246	C₂²×C₃₁	—	2	2	no	—
247	C₂²×C₃₁	—	2	2	no	—
248	C₂²×C₃₁	—	4	1	no	—
249	C₂²×C₃₁	—	4	1	no	—
250	C₂²×C₃₁	—	4	1	no	—
251	C₂²×C₃₁	—	4	1	no	—
252	C₂²×C₃₁	—	4	2	no	—
253	C₂²×C₃₁	—	4	2	no	—
254	C₂²×C₃₁	—	4	2	no	—
255	C₂²×C₃₁	—	4	2	no	—
256	C₂²×C₃₁	—	4	2	no	—
257	C₂²×C₃₁	—	4	2	no	—
258	C₂²×C₃₁	—	4	2	no	—
259	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
260	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
261	C₂²×C₃₁	#520	1	1	yes	characteristic
262	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
263	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
264	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
265	C₂²×C₃₁	#568	1	1	yes	characteristic
266	C₄×C₃₁	—	2	2	no	—
267	C₄×C₃₁	—	2	2	no	—
268	C₄×C₃₁	—	2	2	no	—
269	C₄×C₃₁	—	2	4	no	—
270	C₄×C₃₁	—	4	2	no	—
271	C₄×C₃₁	—	4	1	no	—
272	C₄×C₃₁	—	4	2	no	—
273	C₄×C₃₁	#436	1	4	yes	complemented
274	C₄×C₃₁	D₄²	1	1	yes	characteristic
275	C₄×C₃₁	#547	1	1	yes	characteristic
276	C₄×C₃₁	#548	1	1	yes	characteristic
277	C₄×C₃₁	#553	1	1	yes	characteristic
278	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
279	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
280	C₂×#121	C₄×C₃₁	1	1	yes	characteristic
281	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
282	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
283	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
284	#436	—	2	4	no	—
285	#436	—	2	8	no	—
286	#423	—	2	4	no	—
287	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
288	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
289	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
293	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
294	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
295	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
297	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
299	#390	C₄×C₃₁	1	1	yes	characteristic
300	#389	C₄×C₃₁	1	1	yes	characteristic
301	#388	C₄×C₃₁	1	2	yes	—
302	#386	C₄×C₃₁	1	2	yes	—
303	#380	C₄×C₃₁	1	1	yes	characteristic
304	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
305	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
306	#336	C₂²×C₃₁	1	1	yes	characteristic
307	#336	C₂²×C₃₁	1	2	yes	—
308	#336	C₂²×C₃₁	1	2	yes	—
309	#336	C₂²×C₃₁	1	1	yes	characteristic
310	C₂×C₃₁	—	4	2	no	—
311	C₂×C₃₁	—	8	2	no	—
312	C₂×C₃₁	#2361	1	1	yes	characteristic
313	C₂×C₃₁	C₂×#436	1	1	yes	characteristic
314	C₂×C₃₁	#2931	1	1	yes	characteristic
315	C₂×C₃₁	#2933	1	1	yes	characteristic
316	C₂×C₃₁	D₄×M₁₆	1	1	yes	characteristic
317	C₂×C₃₁	#2991	1	1	yes	characteristic
318	C₂×C₃₁	#2993	1	1	yes	characteristic
319	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
320	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
321	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
322	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
323	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
324	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
325	C₂×M₁₆	—	2	4	no	—
326	C₂×M₁₆	—	2	4	no	—
327	C₂×M₁₆	—	2	4	no	—
328	C₂×M₁₆	—	2	4	no	—
329	C₂×M₁₆	—	4	2	no	—
330	C₂×M₁₆	—	4	2	no	—
331	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
332	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
333	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
334	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
335	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
336	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
337	#121	—	2	4	no	—
338	C₄×D₄	—	2	4	no	—
339	C₄×D₄	—	2	4	no	—
340	C₄×D₄	—	2	2	no	—
341	C₄×D₄	—	2	2	no	—
342	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
343	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
344	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
345	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
346	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
347	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
348	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
349	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
350	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
351	C₂×C₄²	—	2	2	no	—
352	C₂×C₄²	—	2	2	no	—
353	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
354	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
355	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
356	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
357	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
358	C₄⋊C₈	—	2	4	no	—
359	C₄⋊C₈	—	2	4	no	—
360	C₄⋊C₈	—	2	2	no	—
361	C₄⋊C₈	D₄×C₃₁	1	4	yes	complemented
362	C₂²⋊C₈	—	2	2	no	—
363	C₂²⋊C₈	—	2	4	no	—
364	C₂²⋊C₈	—	2	4	no	—
365	C₂²⋊C₈	—	4	2	no	—
366	C₂²⋊C₈	D₄×C₃₁	1	4	yes	complemented
367	C₄×C₈	—	2	4	no	—
368	C₄×C₈	—	2	2	no	—
369	C₄×C₈	—	2	2	no	—
370	#95	C₂×C₄×C₃₁	1	2	yes	—
371	#95	C₂×C₄×C₃₁	1	2	yes	—
372	#95	C₂×C₄×C₃₁	1	2	yes	—
373	C₃₁	#22990	1	1	yes	characteristic, direct product factor, Sylow
374	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
375	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
376	C₂×D₄	—	2	4	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	4	2	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	4	1	no	—
398	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
399	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
400	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
401	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
402	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
403	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
404	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
405	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
406	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
407	C₂²×C₄	#41×C₃₁	1	2	yes	—
408	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
409	M₁₆	—	4	8	no	—
410	M₁₆	—	4	8	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	1	no	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
427	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
428	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	M₁₆×C₃₁	1	4	yes	complemented
433	C₂²⋊C₄	—	2	4	no	—
434	C₂²⋊C₄	—	2	4	no	—
435	C₂²⋊C₄	—	2	4	no	—
436	C₂²⋊C₄	—	2	4	no	—
437	C₂²⋊C₄	—	2	4	no	—
438	C₄²	—	2	2	no	—
439	C₄²	—	2	2	no	—
440	C₄²	—	2	2	no	—
441	C₄²	—	2	1	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	2	no	—
444	C₄²	C₂×D₄×C₃₁	1	2	yes	—
445	C₄²	#41×C₃₁	1	2	yes	—
446	C₂³	—	2	1	no	—
447	C₂³	—	2	1	no	—
448	C₂³	—	2	1	no	—
449	C₂³	—	2	1	no	—
450	C₂³	—	2	1	no	—
451	C₂³	—	2	1	no	—
452	C₂³	—	2	2	no	—
453	C₂³	—	2	2	no	—
454	C₂³	—	2	2	no	—
455	C₂³	—	2	2	no	—
456	C₂³	—	2	2	no	—
457	C₂³	—	2	2	no	—
458	C₂³	—	2	2	no	—
459	C₂³	C₄×D₄×C₃₁	1	2	yes	—
460	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
461	D₄	—	4	8	no	—
462	D₄	—	4	4	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	4	4	no	—
485	C₂×C₄	—	4	4	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
491	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
492	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
493	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
494	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
495	C₂×C₄	C₃₁×#131	1	2	yes	—
496	C₂×C₄	C₃₁×#131	1	2	yes	—
497	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
498	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
499	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
500	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
501	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
502	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
503	C₈	—	2	4	no	—
504	C₈	—	2	2	no	—
505	C₈	—	2	2	no	—
506	C₈	—	4	2	no	—
507	C₈	—	4	1	no	—
508	C₈	—	4	1	no	—
509	C₂²	—	2	2	no	—
510	C₂²	—	2	2	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	4	1	no	—
513	C₂²	—	4	1	no	—
514	C₂²	—	4	1	no	—
515	C₂²	—	4	1	no	—
516	C₂²	—	4	2	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	2	no	—
521	C₂²	—	4	2	no	—
522	C₂²	—	4	2	no	—
523	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
524	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
525	C₂²	C₃₁×#520	1	1	yes	characteristic
526	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
527	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
528	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
529	C₂²	C₃₁×#568	1	1	yes	characteristic
530	C₄	—	2	2	no	—
531	C₄	—	2	2	no	—
532	C₄	—	2	2	no	—
533	C₄	—	2	4	no	—
534	C₄	—	4	2	no	—
535	C₄	—	4	1	no	—
536	C₄	—	4	2	no	—
537	C₄	C₃₁×#436	1	4	yes	complemented
538	C₄	D₄²×C₃₁	1	1	yes	characteristic
539	C₄	C₃₁×#547	1	1	yes	characteristic
540	C₄	C₃₁×#548	1	1	yes	characteristic
541	C₄	C₃₁×#553	1	1	yes	characteristic
542	C₂	—	4	2	no	—
543	C₂	—	8	2	no	—
544	C₂	C₃₁×#2361	1	1	yes	characteristic
545	C₂	C₂×C₃₁×#436	1	1	yes	characteristic
546	C₂	C₃₁×#2931	1	1	yes	characteristic
547	C₂	C₃₁×#2933	1	1	yes	characteristic
548	C₂	D₄×M₁₆×C₃₁	1	1	yes	characteristic
549	C₂	C₃₁×#2991	1	1	yes	characteristic
550	C₂	C₃₁×#2993	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	C₂×C₃₁×#121	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
1. C₂×C₃₁×#436	C₂	semidirect
2. C₂×C₃₁×#436	C₂	semidirect
3. C₂×C₃₁×#423	C₂	semidirect
5. C₃₁×#1954	C₂	semidirect
6. C₃₁×#1939	C₂	semidirect
7. C₃₁×#1923	C₂	semidirect
8. C₃₁×#1870	C₂	semidirect
9. C₃₁×#1851	C₂	semidirect
10. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
11. C₄×C₃₁×C₂²⋊C₈	C₂	semidirect
86. C₃₁×C₄⋊C₈	D₄	semidirect
91. C₃₁×C₂²⋊C₈	D₄	semidirect
156. C₄⋊C₄×C₃₁	M₁₆	semidirect
170. #22990	C₃₁	direct (N × H)
234. C₂×#436	C₂×C₃₁	semidirect
235. C₂×#436	C₂×C₃₁	semidirect
236. C₂×#423	C₂×C₃₁	semidirect
238. #1954	C₂×C₃₁	semidirect
239. #1939	C₂×C₃₁	semidirect
240. #1923	C₂×C₃₁	semidirect
241. #1870	C₂×C₃₁	semidirect
242. #1851	C₂×C₃₁	semidirect
243. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
244. C₄×C₂²⋊C₈	C₂×C₃₁	semidirect
273. C₄×C₃₁	#436	semidirect
361. C₄⋊C₈	D₄×C₃₁	semidirect
366. C₂²⋊C₈	D₄×C₃₁	semidirect
373. C₃₁	#22990	direct (N × H)
432. C₄⋊C₄	M₁₆×C₃₁	semidirect
537. C₄	C₃₁×#436	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#436	2
C₂×C₃₁×#423	2
C₄×C₃₁×#121	2
C₃₁×#1954	2
C₃₁×#1939	2
C₃₁×#1923	2
C₃₁×#1870	2
C₃₁×#1851	2
C₈×C₄⋊C₄×C₃₁	2
C₄×C₃₁×C₂²⋊C₈	2
#22990	31

Subgroups — C31×#22990 #21479749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22990 #21479749