Subgroups — C₃₁×#22562 #21479321

Order 7936.

1206 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2344	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1936	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1926	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1864	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1823	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1807	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1786	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1762	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1752	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×#117	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₂²⋊C₄×C₃₁	—	2	1	no	—
18	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
19	C₂×C₃₁×#102	C₄	1	8	yes	complemented
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
28	C₃₁×#390	C₂²	1	2	yes	—
29	C₃₁×#389	C₂²	1	2	yes	complemented
30	C₃₁×#387	—	2	2	no	—
31	C₃₁×#386	—	2	2	no	—
32	C₃₁×#381	—	2	1	no	—
33	C₃₁×#380	C₂²	1	1	yes	characteristic
34	C₃₁×#380	C₂²	1	1	yes	characteristic
35	C₃₁×#379	—	2	1	no	—
36	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
37	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	complemented
39	C₃₁×#340	C₂²	1	2	yes	complemented
40	C₃₁×#340	C₂²	1	2	yes	complemented
41	C₃₁×#336	C₂²	1	2	yes	complemented
42	C₂⁵×C₃₁	—	2	1	no	—
43	C₂²×D₄×C₃₁	—	2	4	no	—
44	C₂²×D₄×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	—	2	2	no	—
46	C₂²×D₄×C₃₁	—	4	2	no	—
47	C₂²×D₄×C₃₁	C₂×C₄	1	4	yes	—
48	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₃₁	—	2	1	no	—
51	C₂³×C₄×C₃₁	—	2	2	no	—
52	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	—	4	2	no	—
54	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₄×D₄×C₃₁	—	2	4	no	—
59	C₄×D₄×C₃₁	—	4	2	no	—
60	C₃₁×#117	—	4	4	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	complemented
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
75	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	—	2	2	no	—
78	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
79	C₃₁×#106	—	4	2	no	—
80	C₃₁×#102	—	2	4	no	—
81	C₃₁×#102	—	4	2	no	—
82	C₃₁×#102	C₂×C₄	1	4	yes	—
83	C₃₁×#102	C₂×C₄	1	4	yes	—
84	C₃₁×#102	C₂×C₄	1	4	yes	complemented
85	C₃₁×#102	C₂×C₄	1	4	yes	complemented
86	C₃₁×C₂²⋊C₈	—	2	2	no	—
87	C₃₁×#97	—	4	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₃₁×#95	—	2	1	no	—
90	C₃₁×#95	—	2	1	no	—
91	C₃₁×#95	—	2	2	no	—
92	C₃₁×#95	D₄	1	2	yes	—
93	C₂⁴×C₃₁	—	2	1	no	—
94	C₂⁴×C₃₁	—	2	1	no	—
95	C₂⁴×C₃₁	—	2	4	no	—
96	C₂⁴×C₃₁	—	4	4	no	—
97	C₂⁴×C₃₁	—	4	2	no	—
98	C₂⁴×C₃₁	—	4	2	no	—
99	C₂⁴×C₃₁	—	8	2	no	—
100	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×D₄×C₃₁	—	2	8	no	—
102	C₂×D₄×C₃₁	—	2	4	no	—
103	C₂×D₄×C₃₁	—	2	1	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	1	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	4	4	no	—
109	C₂×D₄×C₃₁	—	4	8	no	—
110	C₂×D₄×C₃₁	—	4	4	no	—
111	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
112	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
113	C₂×D₄×C₃₁	#41	1	4	yes	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	2	4	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	4	4	no	—
122	C₂²×C₄×C₃₁	—	4	4	no	—
123	C₂²×C₄×C₃₁	—	4	2	no	—
124	C₂²×C₄×C₃₁	—	4	2	no	—
125	C₂²×C₄×C₃₁	—	8	2	no	—
126	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
127	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	#41	1	2	yes	—
131	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	4	4	no	—
138	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
139	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
140	C₄⋊C₄×C₃₁	—	2	4	no	—
141	C₄⋊C₄×C₃₁	—	2	1	no	—
142	C₄⋊C₄×C₃₁	—	2	1	no	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	4	4	no	—
148	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
149	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
150	C₂²⋊C₄×C₃₁	—	2	2	no	—
151	C₂²⋊C₄×C₃₁	—	2	2	no	—
152	C₂²⋊C₄×C₃₁	—	4	4	no	—
153	C₂²⋊C₄×C₃₁	—	4	4	no	—
154	C₂²⋊C₄×C₃₁	—	4	2	no	—
155	C₂²⋊C₄×C₃₁	—	4	2	no	—
156	C₂²⋊C₄×C₃₁	—	8	2	no	—
157	C₄²×C₃₁	—	2	2	no	—
158	C₄²×C₃₁	—	2	2	no	—
159	C₄²×C₃₁	—	4	4	no	—
160	#22562	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₃₁	—	2	2	no	—
162	C₂³×C₃₁	—	2	2	no	—
163	C₂³×C₃₁	—	2	2	no	—
164	C₂³×C₃₁	—	2	4	no	—
165	C₂³×C₃₁	—	2	2	no	—
166	C₂³×C₃₁	—	4	8	no	—
167	C₂³×C₃₁	—	4	1	no	—
168	C₂³×C₃₁	—	4	1	no	—
169	C₂³×C₃₁	—	4	2	no	—
170	C₂³×C₃₁	—	4	2	no	—
171	C₂³×C₃₁	—	4	1	no	—
172	C₂³×C₃₁	—	4	1	no	—
173	C₂³×C₃₁	—	4	2	no	—
174	C₂³×C₃₁	—	4	8	no	—
175	C₂³×C₃₁	—	8	4	no	—
176	C₂³×C₃₁	—	8	8	no	—
177	C₂³×C₃₁	—	8	8	no	—
178	C₂³×C₃₁	C₄×D₄	1	2	yes	—
179	C₂³×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
180	C₂³×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
181	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
182	D₄×C₃₁	—	2	2	no	—
183	D₄×C₃₁	—	2	2	no	—
184	D₄×C₃₁	—	2	2	no	—
185	D₄×C₃₁	—	2	4	no	—
186	D₄×C₃₁	—	4	8	no	—
187	D₄×C₃₁	—	4	2	no	—
188	D₄×C₃₁	#117	1	4	yes	complemented
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	4	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	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	2	no	—
201	C₂×C₄×C₃₁	—	4	1	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	1	no	—
205	C₂×C₄×C₃₁	—	4	8	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	8	8	no	—
213	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
214	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
216	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
218	C₈×C₃₁	—	2	2	no	—
219	C₈×C₃₁	—	2	2	no	—
220	C₈×C₃₁	—	4	2	no	—
221	C₂²×#102	C₂×C₃₁	1	1	yes	characteristic
222	#2344	C₂×C₃₁	1	1	yes	characteristic, complemented
223	D₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic
224	#1936	C₂×C₃₁	1	2	yes	complemented
225	#1926	C₂×C₃₁	1	2	yes	complemented
226	#1864	C₂×C₃₁	1	2	yes	complemented
227	#1823	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#1807	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#1786	C₂×C₃₁	1	1	yes	characteristic
230	#1762	C₂×C₃₁	1	2	yes	complemented
231	#1752	C₂×C₃₁	1	1	yes	characteristic, complemented
232	C₂²×C₃₁	—	2	2	no	—
233	C₂²×C₃₁	—	2	2	no	—
234	C₂²×C₃₁	—	2	4	no	—
235	C₂²×C₃₁	—	2	4	no	—
236	C₂²×C₃₁	—	2	4	no	—
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	4	4	no	—
240	C₂²×C₃₁	—	4	4	no	—
241	C₂²×C₃₁	—	4	2	no	—
242	C₂²×C₃₁	—	4	2	no	—
243	C₂²×C₃₁	—	4	2	no	—
244	C₂²×C₃₁	—	8	16	no	—
245	C₂²×C₃₁	—	8	2	no	—
246	C₂²×C₃₁	—	8	2	no	—
247	C₂²×C₃₁	#442	1	4	yes	—
248	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
249	C₂²×C₃₁	#546	1	2	yes	—
250	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
251	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
252	C₂²×C₃₁	#577	1	2	yes	—
253	C₄×C₃₁	—	2	2	no	—
254	C₄×C₃₁	—	4	2	no	—
255	C₄×C₃₁	—	4	2	no	—
256	C₄×C₃₁	—	4	2	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	8	2	no	—
259	C₄×C₃₁	—	8	2	no	—
260	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
261	C₄×C₃₁	#518	1	1	yes	characteristic
262	C₄×C₃₁	#551	1	2	yes	—
263	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
264	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
266	C₂×#117	—	2	1	no	—
267	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
268	C₂²×C₂²⋊C₄	—	2	1	no	—
269	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
270	C₂×#102	C₄×C₃₁	1	8	yes	complemented
271	C₂×#102	C₂²×C₃₁	1	2	yes	—
272	C₂×#102	C₂²×C₃₁	1	2	yes	—
273	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
279	#390	C₂²×C₃₁	1	2	yes	—
280	#389	C₂²×C₃₁	1	2	yes	complemented
281	#387	—	2	2	no	—
282	#386	—	2	2	no	—
283	#381	—	2	1	no	—
284	#380	C₂²×C₃₁	1	1	yes	characteristic
285	#380	C₂²×C₃₁	1	1	yes	characteristic
286	#379	—	2	1	no	—
287	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
288	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
289	#340	C₂²×C₃₁	1	2	yes	complemented
290	#340	C₂²×C₃₁	1	2	yes	complemented
291	#340	C₂²×C₃₁	1	2	yes	complemented
292	#336	C₂²×C₃₁	1	2	yes	complemented
293	C₂×C₃₁	—	2	4	no	—
294	C₂×C₃₁	—	4	4	no	—
295	C₂×C₃₁	—	8	2	no	—
296	C₂×C₃₁	#2342	1	1	yes	characteristic
297	C₂×C₃₁	C₂×#442	1	1	yes	characteristic
298	C₂×C₃₁	#2949	1	1	yes	characteristic
299	C₂×C₃₁	#3306	1	2	yes	—
300	C₂×C₃₁	#3310	1	2	yes	—
301	C₂⁵	—	2	1	no	—
302	C₂²×D₄	—	2	4	no	—
303	C₂²×D₄	—	2	1	no	—
304	C₂²×D₄	—	2	2	no	—
305	C₂²×D₄	—	4	2	no	—
306	C₂²×D₄	C₂×C₄×C₃₁	1	4	yes	—
307	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
308	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
309	C₂³×C₄	—	2	1	no	—
310	C₂³×C₄	—	2	2	no	—
311	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂²×C₈	—	4	2	no	—
313	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
314	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
315	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
316	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
317	C₄×D₄	—	2	4	no	—
318	C₄×D₄	—	4	2	no	—
319	#117	—	4	4	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	2	no	—
322	C₂×C₄⋊C₄	—	2	2	no	—
323	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	complemented
324	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
325	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
326	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
327	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
328	C₂×C₂²⋊C₄	—	2	1	no	—
329	C₂×C₂²⋊C₄	—	2	1	no	—
330	C₂×C₂²⋊C₄	—	2	2	no	—
331	C₂×C₂²⋊C₄	—	2	2	no	—
332	C₂×C₂²⋊C₄	—	2	2	no	—
333	C₂×C₂²⋊C₄	—	4	4	no	—
334	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
335	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
336	C₂×C₄²	—	2	2	no	—
337	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
338	#106	—	4	2	no	—
339	#102	—	2	4	no	—
340	#102	—	4	2	no	—
341	#102	C₂×C₄×C₃₁	1	4	yes	—
342	#102	C₂×C₄×C₃₁	1	4	yes	—
343	#102	C₂×C₄×C₃₁	1	4	yes	complemented
344	#102	C₂×C₄×C₃₁	1	4	yes	complemented
345	C₂²⋊C₈	—	2	2	no	—
346	#97	—	4	2	no	—
347	#95	—	2	2	no	—
348	#95	—	2	1	no	—
349	#95	—	2	1	no	—
350	#95	—	2	2	no	—
351	#95	D₄×C₃₁	1	2	yes	—
352	C₃₁	#22562	1	1	yes	characteristic, direct product factor, Sylow
353	C₂⁴	—	2	1	no	—
354	C₂⁴	—	2	1	no	—
355	C₂⁴	—	2	4	no	—
356	C₂⁴	—	4	4	no	—
357	C₂⁴	—	4	2	no	—
358	C₂⁴	—	4	2	no	—
359	C₂⁴	—	8	2	no	—
360	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
361	C₂×D₄	—	2	8	no	—
362	C₂×D₄	—	2	4	no	—
363	C₂×D₄	—	2	1	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	4	4	no	—
369	C₂×D₄	—	4	8	no	—
370	C₂×D₄	—	4	4	no	—
371	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
372	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
373	C₂×D₄	#41×C₃₁	1	4	yes	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	4	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	4	4	no	—
382	C₂²×C₄	—	4	4	no	—
383	C₂²×C₄	—	4	2	no	—
384	C₂²×C₄	—	4	2	no	—
385	C₂²×C₄	—	8	2	no	—
386	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
387	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
388	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
390	C₂²×C₄	#41×C₃₁	1	2	yes	—
391	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	4	4	no	—
398	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
399	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	4	4	no	—
408	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
409	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
410	C₂²⋊C₄	—	2	2	no	—
411	C₂²⋊C₄	—	2	2	no	—
412	C₂²⋊C₄	—	4	4	no	—
413	C₂²⋊C₄	—	4	4	no	—
414	C₂²⋊C₄	—	4	2	no	—
415	C₂²⋊C₄	—	4	2	no	—
416	C₂²⋊C₄	—	8	2	no	—
417	C₄²	—	2	2	no	—
418	C₄²	—	2	2	no	—
419	C₄²	—	4	4	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	4	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	4	8	no	—
426	C₂³	—	4	1	no	—
427	C₂³	—	4	1	no	—
428	C₂³	—	4	2	no	—
429	C₂³	—	4	8	no	—
430	C₂³	—	4	2	no	—
431	C₂³	—	4	1	no	—
432	C₂³	—	4	1	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	8	4	no	—
435	C₂³	—	8	8	no	—
436	C₂³	—	8	8	no	—
437	C₂³	C₄×D₄×C₃₁	1	2	yes	—
438	C₂³	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
439	C₂³	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
440	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
441	D₄	—	2	2	no	—
442	D₄	—	2	2	no	—
443	D₄	—	2	2	no	—
444	D₄	—	2	4	no	—
445	D₄	—	4	8	no	—
446	D₄	—	4	2	no	—
447	D₄	C₃₁×#117	1	4	yes	complemented
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	4	8	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	8	8	no	—
472	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
473	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
474	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
475	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
476	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
477	C₈	—	2	2	no	—
478	C₈	—	2	2	no	—
479	C₈	—	4	2	no	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	2	no	—
482	C₂²	—	2	4	no	—
483	C₂²	—	2	4	no	—
484	C₂²	—	2	4	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	4	no	—
488	C₂²	—	4	4	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	8	16	no	—
493	C₂²	—	8	2	no	—
494	C₂²	—	8	2	no	—
495	C₂²	C₃₁×#442	1	4	yes	—
496	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
497	C₂²	C₃₁×#546	1	2	yes	—
498	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
499	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
500	C₂²	C₃₁×#577	1	2	yes	—
501	C₄	—	2	2	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	2	no	—
506	C₄	—	8	2	no	—
507	C₄	—	8	2	no	—
508	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
509	C₄	C₃₁×#518	1	1	yes	characteristic
510	C₄	C₃₁×#551	1	2	yes	—
511	C₂	—	2	4	no	—
512	C₂	—	4	4	no	—
513	C₂	—	8	2	no	—
514	C₂	C₃₁×#2342	1	1	yes	characteristic
515	C₂	C₂×C₃₁×#442	1	1	yes	characteristic
516	C₂	C₃₁×#2949	1	1	yes	characteristic
517	C₂	C₃₁×#3306	1	2	yes	—
518	C₂	C₃₁×#3310	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₂²⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2344	C₂	semidirect
4. C₃₁×#1936	C₂	semidirect
5. C₃₁×#1926	C₂	semidirect
6. C₃₁×#1864	C₂	semidirect
7. C₃₁×#1823	C₂	semidirect
8. C₃₁×#1807	C₂	semidirect
10. C₃₁×#1762	C₂	semidirect
11. C₃₁×#1752	C₂	semidirect
18. C₂×C₃₁×#106	C₂²	semidirect
19. C₂×C₃₁×#102	C₄	semidirect
27. C₂×C₃₁×#97	C₂²	semidirect
29. C₃₁×#389	C₂²	semidirect
36. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
38. C₃₁×#340	C₂²	semidirect
39. C₃₁×#340	C₂²	semidirect
40. C₃₁×#340	C₂²	semidirect
41. C₃₁×#336	C₂²	semidirect
54. C₂²×C₈×C₃₁	C₂×C₄	semidirect
64. C₂×C₄⋊C₄×C₃₁	C₂×C₄	semidirect
84. C₃₁×#102	C₂×C₄	semidirect
85. C₃₁×#102	C₂×C₄	semidirect
160. #22562	C₃₁	direct (N × H)
188. D₄×C₃₁	#117	semidirect
222. #2344	C₂×C₃₁	semidirect
224. #1936	C₂×C₃₁	semidirect
225. #1926	C₂×C₃₁	semidirect
226. #1864	C₂×C₃₁	semidirect
227. #1823	C₂×C₃₁	semidirect
228. #1807	C₂×C₃₁	semidirect
230. #1762	C₂×C₃₁	semidirect
231. #1752	C₂×C₃₁	semidirect
269. C₂×#106	C₂²×C₃₁	semidirect
270. C₂×#102	C₄×C₃₁	semidirect
278. C₂×#97	C₂²×C₃₁	semidirect
280. #389	C₂²×C₃₁	semidirect
287. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
289. #340	C₂²×C₃₁	semidirect
290. #340	C₂²×C₃₁	semidirect
291. #340	C₂²×C₃₁	semidirect
292. #336	C₂²×C₃₁	semidirect
313. C₂²×C₈	C₂×C₄×C₃₁	semidirect
323. C₂×C₄⋊C₄	C₂×C₄×C₃₁	semidirect
343. #102	C₂×C₄×C₃₁	semidirect
344. #102	C₂×C₄×C₃₁	semidirect
352. C₃₁	#22562	direct (N × H)
447. D₄	C₃₁×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#102	2
C₃₁×#2344	2
D₄×C₂²⋊C₄×C₃₁	2
C₃₁×#1936	2
C₃₁×#1926	2
C₃₁×#1864	2
C₃₁×#1823	2
C₃₁×#1807	2
C₃₁×#1786	2
C₃₁×#1762	2
C₃₁×#1752	2
#22562	31

Subgroups — C31×#22562 #21479321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22562 #21479321