Subgroups — C₃₁×#29156 #21485915

Order 7936.

912 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2857	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₃₁×#121	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2100	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2094	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2058	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2023	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1920	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1893	C₂	1	2	yes	maximal
9	C₃₁×#1864	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1807	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
15	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₃₁	—	2	1	no	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#106	C₂²	1	2	yes	—
20	C₂×C₃₁×#106	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₃₁×#391	—	2	1	no	—
24	C₃₁×#390	C₂²	1	1	yes	characteristic
25	C₃₁×#389	—	2	1	no	—
26	C₃₁×#389	C₂²	1	1	yes	characteristic
27	C₃₁×#389	C₂²	1	2	yes	—
28	C₃₁×#388	—	2	1	no	—
29	C₃₁×#386	—	2	1	no	—
30	C₃₁×#384	C₂²	1	1	yes	characteristic
31	C₃₁×#380	—	2	1	no	—
32	C₃₁×#380	C₂²	1	1	yes	characteristic
33	C₃₁×#380	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₃₁×#336	C₂²	1	1	yes	characteristic
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	—	2	4	no	—
51	C₂²×C₈×C₃₁	—	2	4	no	—
52	C₂²×C₈×C₃₁	—	4	2	no	—
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₃₁×C₄⋊Q₈	—	4	2	no	—
56	C₃₁×C₂²⋊Q₈	—	4	4	no	—
57	C₃₁×#121	—	2	2	no	—
58	C₃₁×#121	Q₈	1	4	yes	complemented
59	C₄×D₄×C₃₁	—	4	2	no	—
60	C₄×D₄×C₃₁	—	4	4	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄²×C₃₁	—	2	1	no	—
78	C₂×C₄²×C₃₁	—	4	2	no	—
79	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×#106	—	2	4	no	—
84	C₃₁×#106	—	4	2	no	—
85	C₃₁×#102	—	2	4	no	—
86	C₃₁×#102	—	4	2	no	—
87	C₃₁×#95	—	2	1	no	—
88	C₃₁×#95	—	2	1	no	—
89	C₃₁×#95	—	2	1	no	—
90	C₃₁×#95	—	2	2	no	—
91	C₃₁×#95	—	2	2	no	—
92	C₂⁴×C₃₁	—	2	1	no	—
93	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂×Q₈×C₃₁	—	4	2	no	—
95	C₂×Q₈×C₃₁	—	4	2	no	—
96	C₂×Q₈×C₃₁	—	4	2	no	—
97	C₂×D₄×C₃₁	—	2	4	no	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	1	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	4	2	no	—
102	C₂×D₄×C₃₁	—	4	2	no	—
103	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	4	1	no	—
114	C₂²×C₄×C₃₁	—	4	1	no	—
115	C₂²×C₄×C₃₁	—	4	1	no	—
116	C₂²×C₄×C₃₁	—	4	2	no	—
117	C₂²×C₄×C₃₁	—	4	2	no	—
118	C₂²×C₄×C₃₁	—	4	2	no	—
119	C₂²×C₄×C₃₁	SD₁₆	1	4	yes	—
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
126	C₂²×C₄×C₃₁	#41	1	2	yes	—
127	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	—	4	8	no	—
134	C₂×C₈×C₃₁	#41	1	4	yes	—
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	2	2	no	—
141	C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₄⋊C₄×C₃₁	—	4	2	no	—
143	C₄⋊C₄×C₃₁	—	4	2	no	—
144	C₄⋊C₄×C₃₁	—	4	4	no	—
145	C₄⋊C₄×C₃₁	—	4	2	no	—
146	C₄⋊C₄×C₃₁	—	4	2	no	—
147	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
148	C₂²⋊C₄×C₃₁	—	2	4	no	—
149	C₂²⋊C₄×C₃₁	—	4	2	no	—
150	C₂²⋊C₄×C₃₁	—	4	2	no	—
151	C₂²⋊C₄×C₃₁	—	4	4	no	—
152	C₂²⋊C₄×C₃₁	—	4	2	no	—
153	C₂²⋊C₄×C₃₁	—	4	2	no	—
154	C₄²×C₃₁	—	2	2	no	—
155	C₄²×C₃₁	—	2	2	no	—
156	C₄²×C₃₁	—	4	1	no	—
157	C₄²×C₃₁	—	4	1	no	—
158	C₄²×C₃₁	—	4	2	no	—
159	#29156	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₃₁	—	2	1	no	—
161	C₂³×C₃₁	—	2	1	no	—
162	C₂³×C₃₁	—	2	1	no	—
163	C₂³×C₃₁	—	2	1	no	—
164	C₂³×C₃₁	—	2	2	no	—
165	C₂³×C₃₁	—	4	1	no	—
166	C₂³×C₃₁	—	4	1	no	—
167	C₂³×C₃₁	—	4	1	no	—
168	C₂³×C₃₁	—	4	1	no	—
169	C₂³×C₃₁	—	4	1	no	—
170	C₂³×C₃₁	—	4	1	no	—
171	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
172	C₂³×C₃₁	C₂×SD₁₆	1	2	yes	—
173	C₂³×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
174	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
175	Q₈×C₃₁	—	8	4	no	—
176	D₄×C₃₁	—	2	4	no	—
177	D₄×C₃₁	—	4	8	no	—
178	D₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	4	no	—
188	C₂×C₄×C₃₁	—	2	4	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	4	4	no	—
194	C₂×C₄×C₃₁	—	4	4	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	1	no	—
200	C₂×C₄×C₃₁	—	4	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	2	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	8	4	no	—
210	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
211	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
212	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
213	C₂×C₄×C₃₁	#135	1	2	yes	—
214	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
217	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
218	C₈×C₃₁	—	2	4	no	—
219	C₈×C₃₁	—	4	2	no	—
220	#2857	C₂×C₃₁	1	1	yes	characteristic, complemented
221	C₄×#121	C₂×C₃₁	1	1	yes	characteristic
222	#2100	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#2094	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#2058	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#2023	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#1920	C₂×C₃₁	1	2	yes	complemented
227	#1893	C₂×C₃₁	1	2	yes	—
228	#1864	C₂×C₃₁	1	2	yes	complemented
229	#1807	C₂×C₃₁	1	2	yes	complemented
230	C₂×#336	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂²×C₃₁	—	2	2	no	—
232	C₂²×C₃₁	—	2	2	no	—
233	C₂²×C₃₁	—	2	2	no	—
234	C₂²×C₃₁	—	2	2	no	—
235	C₂²×C₃₁	—	2	2	no	—
236	C₂²×C₃₁	—	2	2	no	—
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	4	2	no	—
240	C₂²×C₃₁	—	8	1	no	—
241	C₂²×C₃₁	—	8	1	no	—
242	C₂²×C₃₁	—	8	1	no	—
243	C₂²×C₃₁	—	8	1	no	—
244	C₂²×C₃₁	#476	1	4	yes	—
245	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
246	C₂²×C₃₁	#532	1	1	yes	characteristic
247	C₂²×C₃₁	#534	1	1	yes	characteristic
248	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
249	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
250	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
251	C₂²×C₃₁	#575	1	1	yes	characteristic
252	C₄×C₃₁	—	2	2	no	—
253	C₄×C₃₁	—	2	2	no	—
254	C₄×C₃₁	—	2	2	no	—
255	C₄×C₃₁	—	2	2	no	—
256	C₄×C₃₁	—	4	1	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	8	1	no	—
259	C₄×C₃₁	—	8	2	no	—
260	C₄×C₃₁	—	8	1	no	—
261	C₄×C₃₁	—	8	2	no	—
262	C₄×C₃₁	#551	1	2	yes	—
263	C₄×C₃₁	#554	1	2	yes	—
264	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
265	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×C₂²⋊Q₈	—	2	1	no	—
267	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×C₄×D₄	—	2	1	no	—
269	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
270	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×#106	C₂²×C₃₁	1	2	yes	—
272	C₂×#106	C₂²×C₃₁	1	2	yes	—
273	C₂×#102	C₂²×C₃₁	1	2	yes	—
274	C₂×#102	C₂²×C₃₁	1	2	yes	—
275	#391	—	2	1	no	—
276	#390	C₂²×C₃₁	1	1	yes	characteristic
277	#389	—	2	1	no	—
278	#389	C₂²×C₃₁	1	1	yes	characteristic
279	#389	C₂²×C₃₁	1	2	yes	—
280	#388	—	2	1	no	—
281	#386	—	2	1	no	—
282	#384	C₂²×C₃₁	1	1	yes	characteristic
283	#380	—	2	1	no	—
284	#380	C₂²×C₃₁	1	1	yes	characteristic
285	#380	C₂²×C₃₁	1	2	yes	—
286	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
287	C₄×C₂²⋊C₄	—	2	1	no	—
288	#340	C₂²×C₃₁	1	2	yes	—
289	#340	C₂²×C₃₁	1	2	yes	—
290	#340	C₂²×C₃₁	1	2	yes	—
291	#340	C₂²×C₃₁	1	2	yes	—
292	#336	C₂²×C₃₁	1	1	yes	characteristic
293	#336	C₂²×C₃₁	1	1	yes	characteristic
294	#336	C₂²×C₃₁	1	2	yes	—
295	C₂×C₃₁	—	2	4	no	—
296	C₂×C₃₁	—	8	2	no	—
297	C₂×C₃₁	#2564	1	1	yes	characteristic
298	C₂×C₃₁	C₂×#476	1	1	yes	characteristic
299	C₂×C₃₁	#3076	1	1	yes	characteristic
300	C₂×C₃₁	#3167	1	1	yes	characteristic
301	C₂×C₃₁	#3180	1	1	yes	characteristic
302	C₂×C₃₁	#3187	1	1	yes	characteristic
303	C₂×C₃₁	#3188	1	1	yes	characteristic
304	C₂²×Q₈	—	2	1	no	—
305	C₂²×D₄	—	2	1	no	—
306	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
307	C₂³×C₄	—	2	1	no	—
308	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
309	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
310	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
311	C₂²×C₈	—	2	4	no	—
312	C₂²×C₈	—	2	4	no	—
313	C₂²×C₈	—	4	2	no	—
314	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
315	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
316	C₄⋊Q₈	—	4	2	no	—
317	C₂²⋊Q₈	—	4	4	no	—
318	#121	—	2	2	no	—
319	#121	Q₈×C₃₁	1	4	yes	complemented
320	C₄×D₄	—	4	2	no	—
321	C₄×D₄	—	4	4	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	—	2	2	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	—	2	1	no	—
327	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
328	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
329	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
330	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
331	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
332	C₂×C₂²⋊C₄	—	2	1	no	—
333	C₂×C₂²⋊C₄	—	2	1	no	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	—	2	1	no	—
336	C₂×C₂²⋊C₄	—	2	1	no	—
337	C₂×C₂²⋊C₄	—	2	1	no	—
338	C₂×C₄²	—	2	1	no	—
339	C₂×C₄²	—	4	2	no	—
340	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
341	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
342	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
343	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
344	#106	—	2	4	no	—
345	#106	—	4	2	no	—
346	#102	—	2	4	no	—
347	#102	—	4	2	no	—
348	#95	—	2	1	no	—
349	#95	—	2	1	no	—
350	#95	—	2	1	no	—
351	#95	—	2	2	no	—
352	#95	—	2	2	no	—
353	C₃₁	#29156	1	1	yes	characteristic, direct product factor, Sylow
354	C₂⁴	—	2	1	no	—
355	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
356	C₂×Q₈	—	4	2	no	—
357	C₂×Q₈	—	4	2	no	—
358	C₂×Q₈	—	4	2	no	—
359	C₂×D₄	—	2	4	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	1	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	4	2	no	—
364	C₂×D₄	—	4	2	no	—
365	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	4	1	no	—
376	C₂²×C₄	—	4	1	no	—
377	C₂²×C₄	—	4	1	no	—
378	C₂²×C₄	—	4	2	no	—
379	C₂²×C₄	—	4	2	no	—
380	C₂²×C₄	—	4	2	no	—
381	C₂²×C₄	SD₁₆×C₃₁	1	4	yes	—
382	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
383	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
386	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
387	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
388	C₂²×C₄	#41×C₃₁	1	2	yes	—
389	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	4	8	no	—
396	C₂×C₈	#41×C₃₁	1	4	yes	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	4	2	no	—
405	C₄⋊C₄	—	4	2	no	—
406	C₄⋊C₄	—	4	4	no	—
407	C₄⋊C₄	—	4	2	no	—
408	C₄⋊C₄	—	4	2	no	—
409	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
410	C₂²⋊C₄	—	2	4	no	—
411	C₂²⋊C₄	—	4	2	no	—
412	C₂²⋊C₄	—	4	2	no	—
413	C₂²⋊C₄	—	4	4	no	—
414	C₂²⋊C₄	—	4	2	no	—
415	C₂²⋊C₄	—	4	2	no	—
416	C₄²	—	2	2	no	—
417	C₄²	—	2	2	no	—
418	C₄²	—	4	1	no	—
419	C₄²	—	4	1	no	—
420	C₄²	—	4	2	no	—
421	C₂³	—	2	1	no	—
422	C₂³	—	2	1	no	—
423	C₂³	—	2	1	no	—
424	C₂³	—	2	1	no	—
425	C₂³	—	2	2	no	—
426	C₂³	—	4	1	no	—
427	C₂³	—	4	1	no	—
428	C₂³	—	4	1	no	—
429	C₂³	—	4	1	no	—
430	C₂³	—	4	1	no	—
431	C₂³	—	4	1	no	—
432	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
433	C₂³	C₂×SD₁₆×C₃₁	1	2	yes	—
434	C₂³	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
435	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
436	Q₈	—	8	4	no	—
437	D₄	—	2	4	no	—
438	D₄	—	4	8	no	—
439	D₄	—	4	2	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	4	4	no	—
455	C₂×C₄	—	4	4	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	2	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	1	no	—
470	C₂×C₄	—	8	4	no	—
471	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
472	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
473	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
474	C₂×C₄	C₃₁×#135	1	2	yes	—
475	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
476	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
477	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
478	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
479	C₈	—	2	4	no	—
480	C₈	—	4	2	no	—
481	C₂²	—	2	2	no	—
482	C₂²	—	2	2	no	—
483	C₂²	—	2	2	no	—
484	C₂²	—	2	2	no	—
485	C₂²	—	2	2	no	—
486	C₂²	—	2	2	no	—
487	C₂²	—	4	2	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	8	1	no	—
491	C₂²	—	8	1	no	—
492	C₂²	—	8	1	no	—
493	C₂²	—	8	1	no	—
494	C₂²	C₃₁×#476	1	4	yes	—
495	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
496	C₂²	C₃₁×#532	1	1	yes	characteristic
497	C₂²	C₃₁×#534	1	1	yes	characteristic
498	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
499	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
500	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
501	C₂²	C₃₁×#575	1	1	yes	characteristic
502	C₄	—	2	2	no	—
503	C₄	—	2	2	no	—
504	C₄	—	2	2	no	—
505	C₄	—	2	2	no	—
506	C₄	—	4	1	no	—
507	C₄	—	4	2	no	—
508	C₄	—	8	1	no	—
509	C₄	—	8	2	no	—
510	C₄	—	8	1	no	—
511	C₄	—	8	2	no	—
512	C₄	C₃₁×#551	1	2	yes	—
513	C₄	C₃₁×#554	1	2	yes	—
514	C₂	—	2	4	no	—
515	C₂	—	8	2	no	—
516	C₂	C₃₁×#2564	1	1	yes	characteristic
517	C₂	C₂×C₃₁×#476	1	1	yes	characteristic
518	C₂	C₃₁×#3076	1	1	yes	characteristic
519	C₂	C₃₁×#3167	1	1	yes	characteristic
520	C₂	C₃₁×#3180	1	1	yes	characteristic
521	C₂	C₃₁×#3187	1	1	yes	characteristic
522	C₂	C₃₁×#3188	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#121	C₂²
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. C₃₁×#2857	C₂	semidirect
3. C₃₁×#2100	C₂	semidirect
4. C₃₁×#2094	C₂	semidirect
5. C₃₁×#2058	C₂	semidirect
6. C₃₁×#2023	C₂	semidirect
7. C₃₁×#1920	C₂	semidirect
9. C₃₁×#1864	C₂	semidirect
10. C₃₁×#1807	C₂	semidirect
11. C₂×C₃₁×#336	C₂	semidirect
58. C₃₁×#121	Q₈	semidirect
159. #29156	C₃₁	direct (N × H)
220. #2857	C₂×C₃₁	semidirect
222. #2100	C₂×C₃₁	semidirect
223. #2094	C₂×C₃₁	semidirect
224. #2058	C₂×C₃₁	semidirect
225. #2023	C₂×C₃₁	semidirect
226. #1920	C₂×C₃₁	semidirect
228. #1864	C₂×C₃₁	semidirect
229. #1807	C₂×C₃₁	semidirect
230. C₂×#336	C₂×C₃₁	semidirect
319. #121	Q₈×C₃₁	semidirect
353. C₃₁	#29156	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2857	2
C₄×C₃₁×#121	2
C₃₁×#2100	2
C₃₁×#2094	2
C₃₁×#2058	2
C₃₁×#2023	2
C₃₁×#1920	2
C₃₁×#1893	2
C₃₁×#1864	2
C₃₁×#1807	2
C₂×C₃₁×#336	2
#29156	31

Subgroups — C31×#29156 #21485915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29156 #21485915