Subgroups — C₃₁×#33113 #21489872

Order 7936.

1016 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#482	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2856	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1982	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1982	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1947	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1926	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1925	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1875	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#121	—	2	1	no	—
14	C₂×C₃₁×#121	—	2	1	no	—
15	C₂×C₃₁×#121	C₂²	1	2	yes	—
16	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#482	—	2	8	no	—
20	C₂×C₃₁×#107	C₂²	1	2	yes	—
21	C₂×C₃₁×#106	C₂²	1	2	yes	—
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₃₁×#102	C₂²	1	2	yes	—
27	C₂×C₃₁×#102	C₂²	1	2	yes	—
28	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₃₁×#390	—	2	1	no	—
30	C₃₁×#390	—	2	1	no	—
31	C₃₁×#390	C₂²	1	2	yes	—
32	C₃₁×#384	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
38	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
39	C₃₁×#376	—	2	1	no	—
40	C₄³×C₃₁	—	2	1	no	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₂²×D₄×C₃₁	—	2	2	no	—
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₂²×D₄×C₃₁	—	2	1	no	—
47	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
48	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
50	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₃₁×#127	—	2	2	no	—
54	C₃₁×#127	—	2	2	no	—
55	C₃₁×#121	—	2	4	no	—
56	C₃₁×#121	—	2	4	no	—
57	C₃₁×#121	—	4	4	no	—
58	C₃₁×#121	—	4	4	no	—
59	C₃₁×#117	—	2	2	no	—
60	C₃₁×#117	—	2	2	no	—
61	C₃₁×#117	—	2	2	no	—
62	C₃₁×#117	D₄	1	4	yes	complemented
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₄²×C₃₁	—	2	1	no	—
76	C₂×C₄²×C₃₁	—	2	1	no	—
77	C₂×C₄²×C₃₁	—	2	2	no	—
78	C₂×C₄²×C₃₁	—	2	2	no	—
79	C₂×C₄²×C₃₁	—	2	1	no	—
80	C₂×C₄²×C₃₁	—	2	1	no	—
81	C₂×C₄²×C₃₁	—	2	1	no	—
82	C₂×C₄²×C₃₁	—	2	1	no	—
83	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
84	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
87	C₃₁×#107	—	2	4	no	—
88	C₃₁×#106	—	2	4	no	—
89	C₃₁×C₄⋊C₈	—	2	2	no	—
90	C₃₁×C₄⋊C₈	—	2	2	no	—
91	C₃₁×#102	—	2	4	no	—
92	C₃₁×#102	—	2	4	no	—
93	C₃₁×#102	—	2	4	no	—
94	C₃₁×#102	—	2	4	no	—
95	C₃₁×C₂²⋊C₈	—	2	4	no	—
96	C₃₁×#95	—	2	1	no	—
97	C₃₁×#95	—	2	1	no	—
98	C₃₁×#95	—	2	1	no	—
99	C₃₁×#95	—	2	1	no	—
100	C₂⁴×C₃₁	—	2	2	no	—
101	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	4	2	no	—
109	C₂×D₄×C₃₁	—	4	2	no	—
110	C₂×D₄×C₃₁	—	4	4	no	—
111	C₂×D₄×C₃₁	—	4	4	no	—
112	C₂×D₄×C₃₁	—	4	4	no	—
113	C₂×D₄×C₃₁	—	4	2	no	—
114	C₂×D₄×C₃₁	—	4	2	no	—
115	C₂×D₄×C₃₁	—	4	2	no	—
116	C₂×D₄×C₃₁	—	4	2	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
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	2	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	4	1	no	—
132	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	#41	1	2	yes	—
139	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
140	C₂×C₈×C₃₁	—	2	2	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	4	2	no	—
153	C₄⋊C₄×C₃₁	—	4	2	no	—
154	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
155	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
156	C₄⋊C₄×C₃₁	#41	1	2	yes	—
157	C₄⋊C₄×C₃₁	#41	1	2	yes	—
158	C₂²⋊C₄×C₃₁	—	2	4	no	—
159	C₂²⋊C₄×C₃₁	—	2	4	no	—
160	C₂²⋊C₄×C₃₁	—	4	4	no	—
161	C₂²⋊C₄×C₃₁	—	4	4	no	—
162	C₂²⋊C₄×C₃₁	—	4	4	no	—
163	C₄²×C₃₁	—	2	4	no	—
164	C₄²×C₃₁	—	2	4	no	—
165	C₄²×C₃₁	—	2	2	no	—
166	C₄²×C₃₁	—	2	2	no	—
167	C₄²×C₃₁	—	2	1	no	—
168	C₄²×C₃₁	—	2	1	no	—
169	C₄²×C₃₁	—	4	2	no	—
170	C₄²×C₃₁	—	4	2	no	—
171	C₄²×C₃₁	—	4	2	no	—
172	C₄²×C₃₁	—	4	2	no	—
173	C₄²×C₃₁	#41	1	2	yes	—
174	C₄²×C₃₁	#41	1	2	yes	—
175	#33113	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂³×C₃₁	—	2	1	no	—
177	C₂³×C₃₁	—	2	1	no	—
178	C₂³×C₃₁	—	2	1	no	—
179	C₂³×C₃₁	—	2	1	no	—
180	C₂³×C₃₁	—	2	1	no	—
181	C₂³×C₃₁	—	2	1	no	—
182	C₂³×C₃₁	—	4	2	no	—
183	C₂³×C₃₁	—	4	2	no	—
184	C₂³×C₃₁	—	4	2	no	—
185	C₂³×C₃₁	—	4	2	no	—
186	C₂³×C₃₁	—	4	2	no	—
187	C₂³×C₃₁	—	4	2	no	—
188	C₂³×C₃₁	—	4	2	no	—
189	C₂³×C₃₁	#121	1	2	yes	—
190	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
191	D₄×C₃₁	—	4	2	no	—
192	D₄×C₃₁	—	4	2	no	—
193	D₄×C₃₁	—	4	2	no	—
194	D₄×C₃₁	—	4	2	no	—
195	D₄×C₃₁	—	8	4	no	—
196	D₄×C₃₁	—	8	8	no	—
197	D₄×C₃₁	—	8	4	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	1	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	2	1	no	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	1	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	1	no	—
214	C₂×C₄×C₃₁	—	2	1	no	—
215	C₂×C₄×C₃₁	—	4	4	no	—
216	C₂×C₄×C₃₁	—	4	1	no	—
217	C₂×C₄×C₃₁	—	4	1	no	—
218	C₂×C₄×C₃₁	—	4	1	no	—
219	C₂×C₄×C₃₁	—	4	1	no	—
220	C₂×C₄×C₃₁	—	4	1	no	—
221	C₂×C₄×C₃₁	—	4	1	no	—
222	C₂×C₄×C₃₁	—	4	1	no	—
223	C₂×C₄×C₃₁	—	4	1	no	—
224	C₂×C₄×C₃₁	—	4	4	no	—
225	C₂×C₄×C₃₁	—	4	2	no	—
226	C₂×C₄×C₃₁	—	4	2	no	—
227	C₂×C₄×C₃₁	—	4	2	no	—
228	C₂×C₄×C₃₁	#121	1	2	yes	—
229	C₂×C₄×C₃₁	#123	1	2	yes	—
230	C₂×C₄×C₃₁	#123	1	2	yes	—
231	C₂×C₄×C₃₁	#135	1	2	yes	—
232	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
233	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
234	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
239	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
240	C₈×C₃₁	—	4	2	no	—
241	C₈×C₃₁	—	4	2	no	—
242	C₂×#482	C₂×C₃₁	1	2	yes	complemented
243	#2856	C₂×C₃₁	1	1	yes	characteristic
244	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#1982	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#1982	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
248	#1947	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#1926	C₂×C₃₁	1	2	yes	complemented
250	#1925	C₂×C₃₁	1	2	yes	complemented
251	#1875	C₂×C₃₁	1	2	yes	complemented
252	#1797	C₂×C₃₁	1	1	yes	characteristic, complemented
253	C₂²×C₃₁	—	2	2	no	—
254	C₂²×C₃₁	—	2	2	no	—
255	C₂²×C₃₁	—	2	2	no	—
256	C₂²×C₃₁	—	4	1	no	—
257	C₂²×C₃₁	—	4	1	no	—
258	C₂²×C₃₁	—	4	1	no	—
259	C₂²×C₃₁	—	4	1	no	—
260	C₂²×C₃₁	—	8	2	no	—
261	C₂²×C₃₁	—	8	2	no	—
262	C₂²×C₃₁	—	8	2	no	—
263	C₂²×C₃₁	—	8	2	no	—
264	C₂²×C₃₁	—	8	2	no	—
265	C₂²×C₃₁	—	8	2	no	—
266	C₂²×C₃₁	—	8	2	no	—
267	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
268	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
269	C₂²×C₃₁	#525	1	1	yes	characteristic
270	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
271	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
272	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
273	C₂²×C₃₁	#575	1	1	yes	characteristic
274	C₄×C₃₁	—	2	2	no	—
275	C₄×C₃₁	—	2	2	no	—
276	C₄×C₃₁	—	2	2	no	—
277	C₄×C₃₁	—	4	1	no	—
278	C₄×C₃₁	—	4	2	no	—
279	C₄×C₃₁	—	4	2	no	—
280	C₄×C₃₁	—	4	2	no	—
281	C₄×C₃₁	—	4	2	no	—
282	C₄×C₃₁	—	8	2	no	—
283	C₄×C₃₁	#482	1	4	yes	complemented
284	C₄×C₃₁	#529	1	1	yes	characteristic
285	C₄×C₃₁	#529	1	1	yes	characteristic
286	C₄×C₃₁	#532	1	1	yes	characteristic
287	C₄×C₃₁	#532	1	1	yes	characteristic
288	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×#121	—	2	1	no	—
290	C₂×#121	—	2	1	no	—
291	C₂×#121	C₂²×C₃₁	1	2	yes	—
292	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
293	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
294	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
295	#482	—	2	8	no	—
296	C₂×#107	C₂²×C₃₁	1	2	yes	—
297	C₂×#106	C₂²×C₃₁	1	2	yes	—
298	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
299	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
300	C₂×#102	C₂²×C₃₁	1	2	yes	—
301	C₂×#102	C₂²×C₃₁	1	2	yes	—
302	C₂×#102	C₂²×C₃₁	1	2	yes	—
303	C₂×#102	C₂²×C₃₁	1	2	yes	—
304	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
305	#390	—	2	1	no	—
306	#390	—	2	1	no	—
307	#390	C₂²×C₃₁	1	2	yes	—
308	#384	C₂²×C₃₁	1	1	yes	characteristic
309	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
310	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
311	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
312	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
313	C₄×C₂²⋊C₄	—	2	1	no	—
314	C₄×C₂²⋊C₄	—	2	1	no	—
315	#376	—	2	1	no	—
316	C₄³	—	2	1	no	—
317	#340	C₂²×C₃₁	1	2	yes	—
318	#340	C₂²×C₃₁	1	2	yes	—
319	#336	C₂²×C₃₁	1	2	yes	—
320	C₂×C₃₁	—	4	2	no	—
321	C₂×C₃₁	—	16	2	no	—
322	C₂×C₃₁	#2854	1	1	yes	characteristic
323	C₂×C₃₁	#3035	1	1	yes	characteristic
324	C₂×C₃₁	#3038	1	1	yes	characteristic
325	C₂×C₃₁	C₂×#482	1	1	yes	characteristic
326	C₂×C₃₁	#3094	1	1	yes	characteristic
327	C₂×C₃₁	#3106	1	1	yes	characteristic
328	C₂×C₃₁	#3115	1	1	yes	characteristic
329	C₂²×D₄	—	2	2	no	—
330	C₂²×D₄	—	2	1	no	—
331	C₂²×D₄	—	2	1	no	—
332	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
333	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
334	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
335	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
336	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
337	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
338	#127	—	2	2	no	—
339	#127	—	2	2	no	—
340	#121	—	2	4	no	—
341	#121	—	2	4	no	—
342	#121	—	4	4	no	—
343	#121	—	4	4	no	—
344	#117	—	2	2	no	—
345	#117	—	2	2	no	—
346	#117	—	2	2	no	—
347	#117	D₄×C₃₁	1	4	yes	complemented
348	C₂×C₄⋊C₄	—	2	1	no	—
349	C₂×C₄⋊C₄	—	2	1	no	—
350	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
351	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
352	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
353	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
354	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
355	C₂×C₂²⋊C₄	—	2	1	no	—
356	C₂×C₂²⋊C₄	—	2	1	no	—
357	C₂×C₂²⋊C₄	—	2	2	no	—
358	C₂×C₂²⋊C₄	—	2	2	no	—
359	C₂×C₂²⋊C₄	—	2	2	no	—
360	C₂×C₄²	—	2	1	no	—
361	C₂×C₄²	—	2	1	no	—
362	C₂×C₄²	—	2	2	no	—
363	C₂×C₄²	—	2	2	no	—
364	C₂×C₄²	—	2	1	no	—
365	C₂×C₄²	—	2	1	no	—
366	C₂×C₄²	—	2	1	no	—
367	C₂×C₄²	—	2	1	no	—
368	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
369	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
370	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
371	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
372	#107	—	2	4	no	—
373	#106	—	2	4	no	—
374	C₄⋊C₈	—	2	2	no	—
375	C₄⋊C₈	—	2	2	no	—
376	#102	—	2	4	no	—
377	#102	—	2	4	no	—
378	#102	—	2	4	no	—
379	#102	—	2	4	no	—
380	C₂²⋊C₈	—	2	4	no	—
381	#95	—	2	1	no	—
382	#95	—	2	1	no	—
383	#95	—	2	1	no	—
384	#95	—	2	1	no	—
385	C₃₁	#33113	1	1	yes	characteristic, direct product factor, Sylow
386	C₂⁴	—	2	2	no	—
387	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	—	2	2	no	—
394	C₂×D₄	—	4	2	no	—
395	C₂×D₄	—	4	2	no	—
396	C₂×D₄	—	4	4	no	—
397	C₂×D₄	—	4	4	no	—
398	C₂×D₄	—	4	4	no	—
399	C₂×D₄	—	4	2	no	—
400	C₂×D₄	—	4	2	no	—
401	C₂×D₄	—	4	2	no	—
402	C₂×D₄	—	4	2	no	—
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	1	no	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	1	no	—
408	C₂²×C₄	—	2	1	no	—
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	4	1	no	—
418	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
419	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
420	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
421	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
422	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
423	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	#41×C₃₁	1	2	yes	—
425	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	2	no	—
429	C₂×C₈	—	2	2	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	—	2	2	no	—
432	C₄⋊C₄	—	2	1	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	2	no	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	—	4	2	no	—
439	C₄⋊C₄	—	4	2	no	—
440	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
441	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
442	C₄⋊C₄	#41×C₃₁	1	2	yes	—
443	C₄⋊C₄	#41×C₃₁	1	2	yes	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	2	4	no	—
446	C₂²⋊C₄	—	4	4	no	—
447	C₂²⋊C₄	—	4	4	no	—
448	C₂²⋊C₄	—	4	4	no	—
449	C₄²	—	2	4	no	—
450	C₄²	—	2	4	no	—
451	C₄²	—	2	2	no	—
452	C₄²	—	2	2	no	—
453	C₄²	—	2	1	no	—
454	C₄²	—	2	1	no	—
455	C₄²	—	4	2	no	—
456	C₄²	—	4	2	no	—
457	C₄²	—	4	2	no	—
458	C₄²	—	4	2	no	—
459	C₄²	#41×C₃₁	1	2	yes	—
460	C₄²	#41×C₃₁	1	2	yes	—
461	C₂³	—	2	1	no	—
462	C₂³	—	2	1	no	—
463	C₂³	—	2	1	no	—
464	C₂³	—	2	1	no	—
465	C₂³	—	2	1	no	—
466	C₂³	—	2	1	no	—
467	C₂³	—	4	2	no	—
468	C₂³	—	4	2	no	—
469	C₂³	—	4	2	no	—
470	C₂³	—	4	2	no	—
471	C₂³	—	4	2	no	—
472	C₂³	—	4	2	no	—
473	C₂³	—	4	2	no	—
474	C₂³	C₃₁×#121	1	2	yes	—
475	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
476	D₄	—	4	2	no	—
477	D₄	—	4	2	no	—
478	D₄	—	4	2	no	—
479	D₄	—	4	2	no	—
480	D₄	—	8	4	no	—
481	D₄	—	8	8	no	—
482	D₄	—	8	4	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	4	4	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	1	no	—
505	C₂×C₄	—	4	1	no	—
506	C₂×C₄	—	4	1	no	—
507	C₂×C₄	—	4	1	no	—
508	C₂×C₄	—	4	1	no	—
509	C₂×C₄	—	4	4	no	—
510	C₂×C₄	—	4	2	no	—
511	C₂×C₄	—	4	2	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	C₃₁×#121	1	2	yes	—
514	C₂×C₄	C₃₁×#123	1	2	yes	—
515	C₂×C₄	C₃₁×#123	1	2	yes	—
516	C₂×C₄	C₃₁×#135	1	2	yes	—
517	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
518	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
519	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
520	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
521	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
522	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
523	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
524	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
525	C₈	—	4	2	no	—
526	C₈	—	4	2	no	—
527	C₂²	—	2	2	no	—
528	C₂²	—	2	2	no	—
529	C₂²	—	2	2	no	—
530	C₂²	—	4	1	no	—
531	C₂²	—	4	1	no	—
532	C₂²	—	4	1	no	—
533	C₂²	—	4	1	no	—
534	C₂²	—	8	2	no	—
535	C₂²	—	8	2	no	—
536	C₂²	—	8	2	no	—
537	C₂²	—	8	2	no	—
538	C₂²	—	8	2	no	—
539	C₂²	—	8	2	no	—
540	C₂²	—	8	2	no	—
541	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
542	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
543	C₂²	C₃₁×#525	1	1	yes	characteristic
544	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
545	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
546	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
547	C₂²	C₃₁×#575	1	1	yes	characteristic
548	C₄	—	2	2	no	—
549	C₄	—	2	2	no	—
550	C₄	—	2	2	no	—
551	C₄	—	4	1	no	—
552	C₄	—	4	2	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	2	no	—
555	C₄	—	4	2	no	—
556	C₄	—	8	2	no	—
557	C₄	C₃₁×#482	1	4	yes	complemented
558	C₄	C₃₁×#529	1	1	yes	characteristic
559	C₄	C₃₁×#529	1	1	yes	characteristic
560	C₄	C₃₁×#532	1	1	yes	characteristic
561	C₄	C₃₁×#532	1	1	yes	characteristic
562	C₂	—	4	2	no	—
563	C₂	—	16	2	no	—
564	C₂	C₃₁×#2854	1	1	yes	characteristic
565	C₂	C₃₁×#3035	1	1	yes	characteristic
566	C₂	C₃₁×#3038	1	1	yes	characteristic
567	C₂	C₂×C₃₁×#482	1	1	yes	characteristic
568	C₂	C₃₁×#3094	1	1	yes	characteristic
569	C₂	C₃₁×#3106	1	1	yes	characteristic
570	C₂	C₃₁×#3115	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#2856	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#482	C₂	semidirect
3. C₄×C₃₁×#117	C₂	semidirect
4. C₃₁×#1982	C₂	semidirect
5. C₃₁×#1982	C₂	semidirect
6. C₃₁×#1948	C₂	semidirect
7. C₃₁×#1947	C₂	semidirect
8. C₃₁×#1926	C₂	semidirect
9. C₃₁×#1925	C₂	semidirect
10. C₃₁×#1875	C₂	semidirect
11. C₃₁×#1797	C₂	semidirect
62. C₃₁×#117	D₄	semidirect
175. #33113	C₃₁	direct (N × H)
242. C₂×#482	C₂×C₃₁	semidirect
244. C₄×#117	C₂×C₃₁	semidirect
245. #1982	C₂×C₃₁	semidirect
246. #1982	C₂×C₃₁	semidirect
247. #1948	C₂×C₃₁	semidirect
248. #1947	C₂×C₃₁	semidirect
249. #1926	C₂×C₃₁	semidirect
250. #1925	C₂×C₃₁	semidirect
251. #1875	C₂×C₃₁	semidirect
252. #1797	C₂×C₃₁	semidirect
283. C₄×C₃₁	#482	semidirect
347. #117	D₄×C₃₁	semidirect
385. C₃₁	#33113	direct (N × H)
557. C₄	C₃₁×#482	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#482	2
C₃₁×#2856	2
C₄×C₃₁×#117	2
C₃₁×#1982	2
C₃₁×#1948	2
C₃₁×#1947	2
C₃₁×#1926	2
C₃₁×#1925	2
C₃₁×#1875	2
C₃₁×#1797	2
#33113	31

Subgroups — C31×#33113 #21489872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33113 #21489872