Subgroups — C₃₁×#22503 #21479262

Order 7936.

1246 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#555	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1987	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1895	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1874	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1825	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1736	C₂	1	2	yes	maximal, complemented
12	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#555	—	2	4	no	—
14	C₃₁×#555	C₄	1	8	yes	complemented
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#124	—	2	2	no	—
17	C₂×C₃₁×#121	—	2	1	no	—
18	C₂×C₃₁×#121	C₂²	1	2	yes	—
19	C₂×C₄×D₄×C₃₁	—	2	1	no	—
20	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#117	—	2	1	no	—
23	C₂×C₃₁×#117	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#436	—	2	4	no	—
26	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#389	—	2	1	no	—
33	C₃₁×#385	—	2	2	no	—
34	C₃₁×#384	C₂²	1	1	yes	characteristic
35	C₃₁×#380	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	1	yes	characteristic
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#340	C₂²	1	1	yes	characteristic
44	C₂²×Q₈×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	—	2	2	no	—
46	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
47	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	—	2	1	no	—
49	C₂³×C₄×C₃₁	—	2	2	no	—
50	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
51	C₂×M₁₆×C₃₁	—	2	4	no	—
52	C₂×M₁₆×C₃₁	—	2	4	no	—
53	C₂×M₁₆×C₃₁	—	4	2	no	—
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₃₁×C₄⋊Q₈	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
59	C₃₁×#124	—	2	16	no	—
60	C₃₁×#121	—	2	8	no	—
61	C₃₁×#121	—	2	4	no	—
62	C₃₁×#121	C₂×C₄	1	8	yes	complemented
63	C₄×D₄×C₃₁	—	2	2	no	—
64	C₄×D₄×C₃₁	—	4	4	no	—
65	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
66	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	complemented
67	C₄×D₄×C₃₁	D₄	1	4	yes	—
68	C₃₁×#117	—	2	4	no	—
69	C₃₁×#117	—	2	8	no	—
70	C₃₁×#117	C₂×C₄	1	8	yes	complemented
71	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
77	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
88	C₂×C₄²×C₃₁	—	2	2	no	—
89	C₂×C₄²×C₃₁	—	2	1	no	—
90	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
91	C₃₁×C₄⋊C₈	—	2	2	no	—
92	C₃₁×#102	—	2	2	no	—
93	C₃₁×#102	—	2	2	no	—
94	C₃₁×#102	—	4	2	no	—
95	C₃₁×C₂²⋊C₈	—	2	4	no	—
96	C₄×C₈×C₃₁	—	2	2	no	—
97	C₃₁×#95	—	2	1	no	—
98	C₃₁×#95	—	2	2	no	—
99	C₂⁴×C₃₁	—	2	1	no	—
100	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
101	C₂×Q₈×C₃₁	—	2	4	no	—
102	C₂×Q₈×C₃₁	—	8	1	no	—
103	C₂×Q₈×C₃₁	—	8	1	no	—
104	C₂×D₄×C₃₁	—	2	8	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	1	no	—
108	C₂×D₄×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	—	2	1	no	—
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	8	2	no	—
113	C₂×D₄×C₃₁	—	8	2	no	—
114	C₂×D₄×C₃₁	C₂²⋊C₄	1	4	yes	—
115	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
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	1	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	4	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	4	4	no	—
127	C₂²×C₄×C₃₁	—	4	2	no	—
128	C₂²×C₄×C₃₁	—	4	2	no	—
129	C₂²×C₄×C₃₁	—	4	4	no	—
130	C₂²×C₄×C₃₁	—	8	2	no	—
131	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
132	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
137	M₁₆×C₃₁	—	4	8	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	#41	1	2	yes	—
147	C₄⋊C₄×C₃₁	—	2	4	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	2	1	no	—
151	C₄⋊C₄×C₃₁	—	2	1	no	—
152	C₄⋊C₄×C₃₁	—	2	2	no	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	2	1	no	—
155	C₄⋊C₄×C₃₁	—	4	4	no	—
156	C₄⋊C₄×C₃₁	—	4	2	no	—
157	C₄⋊C₄×C₃₁	—	8	1	no	—
158	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
159	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
160	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
161	C₄⋊C₄×C₃₁	#41	1	2	yes	—
162	C₂²⋊C₄×C₃₁	—	2	8	no	—
163	C₂²⋊C₄×C₃₁	—	2	8	no	—
164	C₂²⋊C₄×C₃₁	—	2	4	no	—
165	C₂²⋊C₄×C₃₁	—	4	4	no	—
166	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
167	C₄²×C₃₁	—	2	8	no	—
168	C₄²×C₃₁	—	4	2	no	—
169	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
170	C₄²×C₃₁	C₂×D₄	1	2	yes	—
171	#22503	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂³×C₃₁	—	2	2	no	—
173	C₂³×C₃₁	—	2	2	no	—
174	C₂³×C₃₁	—	2	2	no	—
175	C₂³×C₃₁	—	2	2	no	—
176	C₂³×C₃₁	—	2	2	no	—
177	C₂³×C₃₁	—	2	2	no	—
178	C₂³×C₃₁	—	2	2	no	—
179	C₂³×C₃₁	—	4	1	no	—
180	C₂³×C₃₁	—	4	1	no	—
181	C₂³×C₃₁	—	4	1	no	—
182	C₂³×C₃₁	—	4	1	no	—
183	C₂³×C₃₁	—	4	1	no	—
184	C₂³×C₃₁	—	4	1	no	—
185	C₂³×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
186	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
187	Q₈×C₃₁	—	8	4	no	—
188	D₄×C₃₁	—	2	2	no	—
189	D₄×C₃₁	—	2	2	no	—
190	D₄×C₃₁	—	2	8	no	—
191	D₄×C₃₁	—	4	1	no	—
192	D₄×C₃₁	—	4	1	no	—
193	D₄×C₃₁	—	8	8	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	1	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	2	4	no	—
199	C₂×C₄×C₃₁	—	2	4	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	2	4	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	4	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	4	1	no	—
206	C₂×C₄×C₃₁	—	4	1	no	—
207	C₂×C₄×C₃₁	—	4	1	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	8	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	1	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	—	4	1	no	—
218	C₂×C₄×C₃₁	—	8	8	no	—
219	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
220	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
221	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
222	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
223	C₂×C₄×C₃₁	#137	1	2	yes	—
224	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
227	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
228	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
229	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
230	C₈×C₃₁	—	2	2	no	—
231	C₈×C₃₁	—	4	2	no	—
232	C₈×C₃₁	—	4	1	no	—
233	C₂×#555	C₂×C₃₁	1	1	yes	characteristic
234	C₂×#436	C₂×C₃₁	1	1	yes	characteristic, complemented
235	D₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#1990	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#1987	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#1895	C₂×C₃₁	1	2	yes	complemented
239	#1874	C₂×C₃₁	1	2	yes	complemented
240	#1825	C₂×C₃₁	1	2	yes	complemented
241	#1802	C₂×C₃₁	1	1	yes	characteristic, complemented
242	C₄×#102	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#1736	C₂×C₃₁	1	2	yes	complemented
244	C₂²×C₃₁	—	2	4	no	—
245	C₂²×C₃₁	—	2	4	no	—
246	C₂²×C₃₁	—	2	4	no	—
247	C₂²×C₃₁	—	4	2	no	—
248	C₂²×C₃₁	—	4	2	no	—
249	C₂²×C₃₁	—	4	2	no	—
250	C₂²×C₃₁	—	4	2	no	—
251	C₂²×C₃₁	—	4	2	no	—
252	C₂²×C₃₁	—	4	2	no	—
253	C₂²×C₃₁	—	4	2	no	—
254	C₂²×C₃₁	—	8	1	no	—
255	C₂²×C₃₁	—	8	1	no	—
256	C₂²×C₃₁	—	8	1	no	—
257	C₂²×C₃₁	—	8	1	no	—
258	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
259	C₂²×C₃₁	#546	1	1	yes	characteristic
260	C₂²×C₃₁	#547	1	1	yes	characteristic
261	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
262	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
263	C₂²×C₃₁	#576	1	1	yes	characteristic
264	C₂²×C₃₁	#577	1	1	yes	characteristic
265	C₄×C₃₁	—	2	4	no	—
266	C₄×C₃₁	—	2	2	no	—
267	C₄×C₃₁	—	4	2	no	—
268	C₄×C₃₁	—	4	4	no	—
269	C₄×C₃₁	—	4	2	no	—
270	C₄×C₃₁	—	8	1	no	—
271	C₄×C₃₁	—	8	2	no	—
272	C₄×C₃₁	#417	1	4	yes	—
273	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
274	C₄×C₃₁	#520	1	1	yes	characteristic
275	C₄×C₃₁	#546	1	1	yes	characteristic
276	C₄×C₃₁	#547	1	1	yes	characteristic
277	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
278	#555	—	2	4	no	—
279	#555	C₄×C₃₁	1	8	yes	complemented
280	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×#124	—	2	2	no	—
282	C₂×#121	—	2	1	no	—
283	C₂×#121	C₂²×C₃₁	1	2	yes	—
284	C₂×C₄×D₄	—	2	1	no	—
285	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
286	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×#117	—	2	1	no	—
288	C₂×#117	C₂²×C₃₁	1	2	yes	—
289	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
290	#436	—	2	4	no	—
291	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
293	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
294	C₂×#102	C₂²×C₃₁	1	2	yes	—
295	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
296	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
297	#389	—	2	1	no	—
298	#385	—	2	2	no	—
299	#384	C₂²×C₃₁	1	1	yes	characteristic
300	#380	C₂²×C₃₁	1	2	yes	—
301	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
302	#340	C₂²×C₃₁	1	2	yes	—
303	#340	C₂²×C₃₁	1	1	yes	characteristic
304	#340	C₂²×C₃₁	1	2	yes	—
305	#340	C₂²×C₃₁	1	2	yes	—
306	#340	C₂²×C₃₁	1	2	yes	—
307	#340	C₂²×C₃₁	1	2	yes	—
308	#340	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×C₃₁	—	4	4	no	—
310	C₂×C₃₁	—	8	2	no	—
311	C₂×C₃₁	#2339	1	1	yes	characteristic
312	C₂×C₃₁	C₂×#417	1	1	yes	characteristic
313	C₂×C₃₁	#2897	1	1	yes	characteristic
314	C₂×C₃₁	#3325	1	1	yes	characteristic
315	C₂×C₃₁	#3329	1	1	yes	characteristic
316	C₂×C₃₁	#3354	1	1	yes	characteristic
317	C₂×C₃₁	#3357	1	1	yes	characteristic
318	C₂²×Q₈	—	2	1	no	—
319	C₂²×D₄	—	2	2	no	—
320	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
321	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
322	C₂³×C₄	—	2	1	no	—
323	C₂³×C₄	—	2	2	no	—
324	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
325	C₂×M₁₆	—	2	4	no	—
326	C₂×M₁₆	—	2	4	no	—
327	C₂×M₁₆	—	4	2	no	—
328	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
329	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
330	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
331	C₄⋊Q₈	—	2	2	no	—
332	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
333	#124	—	2	16	no	—
334	#121	—	2	8	no	—
335	#121	—	2	4	no	—
336	#121	C₂×C₄×C₃₁	1	8	yes	complemented
337	C₄×D₄	—	2	2	no	—
338	C₄×D₄	—	4	4	no	—
339	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
340	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	complemented
341	C₄×D₄	D₄×C₃₁	1	4	yes	—
342	#117	—	2	4	no	—
343	#117	—	2	8	no	—
344	#117	C₂×C₄×C₃₁	1	8	yes	complemented
345	C₂×C₄⋊C₄	—	2	1	no	—
346	C₂×C₄⋊C₄	—	2	4	no	—
347	C₂×C₄⋊C₄	—	2	2	no	—
348	C₂×C₄⋊C₄	—	2	1	no	—
349	C₂×C₄⋊C₄	—	2	1	no	—
350	C₂×C₄⋊C₄	—	4	2	no	—
351	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
352	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
353	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
354	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
355	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
356	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
357	C₂×C₂²⋊C₄	—	2	2	no	—
358	C₂×C₂²⋊C₄	—	2	2	no	—
359	C₂×C₂²⋊C₄	—	2	1	no	—
360	C₂×C₂²⋊C₄	—	2	2	no	—
361	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
362	C₂×C₄²	—	2	2	no	—
363	C₂×C₄²	—	2	1	no	—
364	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
365	C₄⋊C₈	—	2	2	no	—
366	#102	—	2	2	no	—
367	#102	—	2	2	no	—
368	#102	—	4	2	no	—
369	C₂²⋊C₈	—	2	4	no	—
370	C₄×C₈	—	2	2	no	—
371	#95	—	2	1	no	—
372	#95	—	2	2	no	—
373	C₃₁	#22503	1	1	yes	characteristic, direct product factor, Sylow
374	C₂⁴	—	2	1	no	—
375	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
376	C₂×Q₈	—	2	4	no	—
377	C₂×Q₈	—	8	1	no	—
378	C₂×Q₈	—	8	1	no	—
379	C₂×D₄	—	2	8	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	1	no	—
383	C₂×D₄	—	2	1	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	8	2	no	—
388	C₂×D₄	—	8	2	no	—
389	C₂×D₄	C₂²⋊C₄×C₃₁	1	4	yes	—
390	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
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	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	4	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	4	4	no	—
402	C₂²×C₄	—	4	2	no	—
403	C₂²×C₄	—	4	2	no	—
404	C₂²×C₄	—	4	4	no	—
405	C₂²×C₄	—	8	2	no	—
406	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
407	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
408	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
409	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
410	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
411	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
412	M₁₆	—	4	8	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	1	no	—
416	C₂×C₈	—	2	2	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	#41×C₃₁	1	2	yes	—
422	C₄⋊C₄	—	2	4	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	2	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	4	4	no	—
431	C₄⋊C₄	—	4	2	no	—
432	C₄⋊C₄	—	8	1	no	—
433	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
434	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
435	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
436	C₄⋊C₄	#41×C₃₁	1	2	yes	—
437	C₂²⋊C₄	—	2	8	no	—
438	C₂²⋊C₄	—	2	8	no	—
439	C₂²⋊C₄	—	2	4	no	—
440	C₂²⋊C₄	—	4	4	no	—
441	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
442	C₄²	—	2	8	no	—
443	C₄²	—	4	2	no	—
444	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
445	C₄²	C₂×D₄×C₃₁	1	2	yes	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	2	no	—
452	C₂³	—	2	2	no	—
453	C₂³	—	4	1	no	—
454	C₂³	—	4	1	no	—
455	C₂³	—	4	1	no	—
456	C₂³	—	4	1	no	—
457	C₂³	—	4	1	no	—
458	C₂³	—	4	1	no	—
459	C₂³	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
460	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
461	Q₈	—	8	4	no	—
462	D₄	—	2	8	no	—
463	D₄	—	2	2	no	—
464	D₄	—	2	2	no	—
465	D₄	—	4	1	no	—
466	D₄	—	4	1	no	—
467	D₄	—	8	8	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	8	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	8	8	no	—
493	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
494	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
495	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
496	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
497	C₂×C₄	C₃₁×#137	1	2	yes	—
498	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
499	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
500	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
501	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
502	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
503	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
504	C₈	—	2	2	no	—
505	C₈	—	4	2	no	—
506	C₈	—	4	1	no	—
507	C₂²	—	2	4	no	—
508	C₂²	—	2	4	no	—
509	C₂²	—	2	4	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	4	2	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	4	2	no	—
517	C₂²	—	8	1	no	—
518	C₂²	—	8	1	no	—
519	C₂²	—	8	1	no	—
520	C₂²	—	8	1	no	—
521	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
522	C₂²	C₃₁×#546	1	1	yes	characteristic
523	C₂²	C₃₁×#547	1	1	yes	characteristic
524	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
525	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
526	C₂²	C₃₁×#576	1	1	yes	characteristic
527	C₂²	C₃₁×#577	1	1	yes	characteristic
528	C₄	—	2	4	no	—
529	C₄	—	2	2	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	4	no	—
532	C₄	—	4	2	no	—
533	C₄	—	8	1	no	—
534	C₄	—	8	2	no	—
535	C₄	C₃₁×#417	1	4	yes	—
536	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
537	C₄	C₃₁×#520	1	1	yes	characteristic
538	C₄	C₃₁×#546	1	1	yes	characteristic
539	C₄	C₃₁×#547	1	1	yes	characteristic
540	C₂	—	4	4	no	—
541	C₂	—	8	2	no	—
542	C₂	C₃₁×#2339	1	1	yes	characteristic
543	C₂	C₂×C₃₁×#417	1	1	yes	characteristic
544	C₂	C₃₁×#2897	1	1	yes	characteristic
545	C₂	C₃₁×#3325	1	1	yes	characteristic
546	C₂	C₃₁×#3329	1	1	yes	characteristic
547	C₂	C₃₁×#3354	1	1	yes	characteristic
548	C₂	C₃₁×#3357	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	C₂×C₃₁×#555	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#436	C₂	semidirect
3. D₄×C₄⋊C₄×C₃₁	C₂	semidirect
4. C₃₁×#1990	C₂	semidirect
5. C₃₁×#1987	C₂	semidirect
6. C₃₁×#1895	C₂	semidirect
7. C₃₁×#1874	C₂	semidirect
8. C₃₁×#1825	C₂	semidirect
9. C₃₁×#1802	C₂	semidirect
10. C₄×C₃₁×#102	C₂	semidirect
11. C₃₁×#1736	C₂	semidirect
14. C₃₁×#555	C₄	semidirect
58. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
62. C₃₁×#121	C₂×C₄	semidirect
66. C₄×D₄×C₃₁	C₂×C₄	semidirect
70. C₃₁×#117	C₂×C₄	semidirect
171. #22503	C₃₁	direct (N × H)
234. C₂×#436	C₂×C₃₁	semidirect
235. D₄×C₄⋊C₄	C₂×C₃₁	semidirect
236. #1990	C₂×C₃₁	semidirect
237. #1987	C₂×C₃₁	semidirect
238. #1895	C₂×C₃₁	semidirect
239. #1874	C₂×C₃₁	semidirect
240. #1825	C₂×C₃₁	semidirect
241. #1802	C₂×C₃₁	semidirect
242. C₄×#102	C₂×C₃₁	semidirect
243. #1736	C₂×C₃₁	semidirect
279. #555	C₄×C₃₁	semidirect
332. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
336. #121	C₂×C₄×C₃₁	semidirect
340. C₄×D₄	C₂×C₄×C₃₁	semidirect
344. #117	C₂×C₄×C₃₁	semidirect
373. C₃₁	#22503	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×#555	2
C₂×C₃₁×#436	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#1990	2
C₃₁×#1987	2
C₃₁×#1895	2
C₃₁×#1874	2
C₃₁×#1825	2
C₃₁×#1802	2
C₄×C₃₁×#102	2
C₃₁×#1736	2
#22503	31

Subgroups — C31×#22503 #21479262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22503 #21479262