Subgroups — C₃₁×#22147 #21478906

Order 7936.

988 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#460	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2295	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1982	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1968	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1934	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1877	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1864	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	—	2	1	no	—
16	C₂×C₄×D₄×C₃₁	—	2	1	no	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#460	C₄	1	8	yes	complemented
19	C₂×C₃₁×#106	C₂²	1	2	yes	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#103	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
27	C₃₁×#391	C₂²	1	1	yes	characteristic
28	C₃₁×#391	C₂²	1	1	yes	characteristic
29	C₃₁×#390	—	2	1	no	—
30	C₃₁×#390	—	2	1	no	—
31	C₃₁×#390	C₂²	1	2	yes	—
32	C₃₁×#389	C₂²	1	1	yes	characteristic
33	C₃₁×#389	C₂²	1	1	yes	characteristic
34	C₃₁×#387	—	2	1	no	—
35	C₃₁×#387	—	2	1	no	—
36	C₃₁×#384	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#376	—	2	1	no	—
40	C₄³×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
49	C₂×SD₁₆×C₃₁	—	4	2	no	—
50	C₂×SD₁₆×C₃₁	—	4	2	no	—
51	C₂×SD₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₃₁×#127	—	2	2	no	—
55	C₃₁×#127	C₂×C₄	1	4	yes	—
56	C₄×Q₈×C₃₁	—	2	2	no	—
57	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	complemented
58	C₄×D₄×C₃₁	—	4	2	no	—
59	C₄×D₄×C₃₁	—	4	2	no	—
60	C₄×D₄×C₃₁	—	4	4	no	—
61	C₄×D₄×C₃₁	—	4	4	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
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	2	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
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×C₄²×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	—	2	1	no	—
76	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₃₁	C₂³	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	—	4	2	no	—
84	C₃₁×C₄⋊C₈	—	2	2	no	—
85	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
86	C₃₁×#103	—	4	2	no	—
87	C₃₁×#102	—	4	2	no	—
88	C₃₁×#102	—	4	2	no	—
89	C₃₁×#102	C₂×C₄	1	8	yes	complemented
90	C₄×C₈×C₃₁	—	4	2	no	—
91	C₃₁×#95	—	2	1	no	—
92	C₃₁×#95	—	2	1	no	—
93	C₃₁×#95	—	2	1	no	—
94	C₃₁×#95	—	2	1	no	—
95	C₃₁×#95	—	2	1	no	—
96	C₃₁×#95	—	2	1	no	—
97	C₃₁×#95	—	2	1	no	—
98	C₂⁴×C₃₁	—	2	2	no	—
99	C₂×Q₈×C₃₁	—	2	1	no	—
100	C₂×Q₈×C₃₁	—	2	1	no	—
101	C₂×Q₈×C₃₁	—	2	1	no	—
102	C₂×Q₈×C₃₁	—	2	1	no	—
103	C₂×Q₈×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
105	C₂×D₄×C₃₁	—	2	4	no	—
106	C₂×D₄×C₃₁	—	2	4	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	4	2	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	C₂²×C₄	1	4	yes	—
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	2	no	—
122	C₂²×C₄×C₃₁	—	4	4	no	—
123	C₂²×C₄×C₃₁	—	4	4	no	—
124	C₂²×C₄×C₃₁	—	4	4	no	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
132	SD₁₆×C₃₁	—	4	8	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	—	2	2	no	—
138	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
139	C₄⋊C₄×C₃₁	—	2	4	no	—
140	C₄⋊C₄×C₃₁	—	2	1	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	1	no	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	4	2	no	—
148	C₄⋊C₄×C₃₁	—	4	2	no	—
149	C₄⋊C₄×C₃₁	—	4	2	no	—
150	C₄⋊C₄×C₃₁	—	4	2	no	—
151	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
152	C₂²⋊C₄×C₃₁	—	4	4	no	—
153	C₂²⋊C₄×C₃₁	—	4	4	no	—
154	C₂²⋊C₄×C₃₁	—	4	4	no	—
155	C₄²×C₃₁	—	2	4	no	—
156	C₄²×C₃₁	—	2	2	no	—
157	C₄²×C₃₁	—	2	2	no	—
158	C₄²×C₃₁	—	2	1	no	—
159	C₄²×C₃₁	—	2	1	no	—
160	C₄²×C₃₁	—	2	1	no	—
161	C₄²×C₃₁	—	2	1	no	—
162	C₄²×C₃₁	—	4	1	no	—
163	C₄²×C₃₁	—	4	2	no	—
164	C₄²×C₃₁	SD₁₆	1	4	yes	complemented
165	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
166	C₄²×C₃₁	#41	1	2	yes	—
167	#22147	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂³×C₃₁	—	2	4	no	—
169	C₂³×C₃₁	—	4	2	no	—
170	C₂³×C₃₁	—	4	2	no	—
171	C₂³×C₃₁	—	4	2	no	—
172	C₂³×C₃₁	—	4	2	no	—
173	C₂³×C₃₁	—	4	2	no	—
174	C₂³×C₃₁	—	4	2	no	—
175	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
176	Q₈×C₃₁	—	2	2	no	—
177	Q₈×C₃₁	—	2	2	no	—
178	Q₈×C₃₁	—	4	1	no	—
179	Q₈×C₃₁	—	4	1	no	—
180	D₄×C₃₁	—	2	4	no	—
181	D₄×C₃₁	—	2	4	no	—
182	D₄×C₃₁	—	4	8	no	—
183	D₄×C₃₁	—	4	2	no	—
184	D₄×C₃₁	—	4	2	no	—
185	D₄×C₃₁	—	4	8	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	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	—	2	1	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	1	no	—
199	C₂×C₄×C₃₁	—	4	1	no	—
200	C₂×C₄×C₃₁	—	4	1	no	—
201	C₂×C₄×C₃₁	—	4	1	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	1	no	—
205	C₂×C₄×C₃₁	—	4	1	no	—
206	C₂×C₄×C₃₁	—	4	2	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	1	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	8	8	no	—
213	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
214	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
215	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
216	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
217	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
218	C₂×C₄×C₃₁	#135	1	2	yes	—
219	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
220	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
222	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
223	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
224	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
225	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
226	C₈×C₃₁	—	2	4	no	—
227	C₈×C₃₁	—	4	2	no	—
228	C₂×#460	C₂×C₃₁	1	1	yes	characteristic
229	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
230	#2295	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#1982	C₂×C₃₁	1	1	yes	characteristic
232	#1968	C₂×C₃₁	1	2	yes	complemented
233	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#1934	C₂×C₃₁	1	2	yes	complemented
235	#1877	C₂×C₃₁	1	2	yes	complemented
236	#1864	C₂×C₃₁	1	2	yes	complemented
237	#1803	C₂×C₃₁	1	1	yes	characteristic, complemented
238	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
239	C₂²×C₃₁	—	4	4	no	—
240	C₂²×C₃₁	—	4	4	no	—
241	C₂²×C₃₁	—	4	4	no	—
242	C₂²×C₃₁	—	8	2	no	—
243	C₂²×C₃₁	—	8	2	no	—
244	C₂²×C₃₁	—	8	2	no	—
245	C₂²×C₃₁	—	8	2	no	—
246	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
247	C₂²×C₃₁	#535	1	1	yes	characteristic
248	C₂²×C₃₁	#537	1	1	yes	characteristic
249	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
250	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
251	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
252	C₂²×C₃₁	#575	1	1	yes	characteristic
253	C₄×C₃₁	—	2	2	no	—
254	C₄×C₃₁	—	2	2	no	—
255	C₄×C₃₁	—	2	2	no	—
256	C₄×C₃₁	—	2	2	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	—	4	1	no	—
260	C₄×C₃₁	—	8	2	no	—
261	C₄×C₃₁	—	8	1	no	—
262	C₄×C₃₁	—	8	1	no	—
263	C₄×C₃₁	C₄×SD₁₆	1	4	yes	—
264	C₄×C₃₁	#518	1	1	yes	characteristic
265	C₄×C₃₁	#520	1	1	yes	characteristic
266	C₄×C₃₁	#540	1	1	yes	characteristic
267	C₄×C₃₁	#542	1	1	yes	characteristic
268	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
269	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×C₄×D₄	—	2	1	no	—
272	C₂×C₄×D₄	—	2	1	no	—
273	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
274	#460	C₄×C₃₁	1	8	yes	complemented
275	C₂×#106	C₂²×C₃₁	1	2	yes	—
276	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×#103	C₂²×C₃₁	1	2	yes	—
279	C₂×#102	C₂²×C₃₁	1	2	yes	—
280	C₂×#102	C₂²×C₃₁	1	2	yes	—
281	C₂×#102	C₂²×C₃₁	1	2	yes	—
282	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
283	#391	C₂²×C₃₁	1	1	yes	characteristic
284	#391	C₂²×C₃₁	1	1	yes	characteristic
285	#390	—	2	1	no	—
286	#390	—	2	1	no	—
287	#390	C₂²×C₃₁	1	2	yes	—
288	#389	C₂²×C₃₁	1	1	yes	characteristic
289	#389	C₂²×C₃₁	1	1	yes	characteristic
290	#387	—	2	1	no	—
291	#387	—	2	1	no	—
292	#384	C₂²×C₃₁	1	1	yes	characteristic
293	C₄×C₄⋊C₄	—	2	1	no	—
294	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
295	#376	—	2	1	no	—
296	C₄³	C₂²×C₃₁	1	1	yes	characteristic
297	#340	C₂²×C₃₁	1	2	yes	—
298	#340	C₂²×C₃₁	1	2	yes	—
299	#336	C₂²×C₃₁	1	2	yes	—
300	C₂×C₃₁	—	8	4	no	—
301	C₂×C₃₁	#2324	1	1	yes	characteristic
302	C₂×C₃₁	C₂×C₄×SD₁₆	1	1	yes	characteristic
303	C₂×C₃₁	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
304	C₂×C₃₁	#3208	1	1	yes	characteristic
305	C₂×C₃₁	#3212	1	1	yes	characteristic
306	C₂×C₃₁	#3246	1	1	yes	characteristic
307	C₂×C₃₁	#3247	1	1	yes	characteristic
308	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
309	C₂²×D₄	—	2	1	no	—
310	C₂²×D₄	—	2	1	no	—
311	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
312	C₂³×C₄	—	2	2	no	—
313	C₂×SD₁₆	—	4	2	no	—
314	C₂×SD₁₆	—	4	2	no	—
315	C₂×SD₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
316	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
317	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
318	#127	—	2	2	no	—
319	#127	C₂×C₄×C₃₁	1	4	yes	—
320	C₄×Q₈	—	2	2	no	—
321	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
322	C₄×D₄	—	4	2	no	—
323	C₄×D₄	—	4	2	no	—
324	C₄×D₄	—	4	4	no	—
325	C₄×D₄	—	4	4	no	—
326	C₂×C₄⋊C₄	—	2	1	no	—
327	C₂×C₄⋊C₄	—	2	1	no	—
328	C₂×C₄⋊C₄	—	2	1	no	—
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
332	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
333	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
334	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₂²⋊C₄	—	2	2	no	—
337	C₂×C₂²⋊C₄	—	2	2	no	—
338	C₂×C₄²	—	2	1	no	—
339	C₂×C₄²	—	2	1	no	—
340	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
341	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
342	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
343	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
344	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
345	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
346	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
347	#106	—	4	2	no	—
348	C₄⋊C₈	—	2	2	no	—
349	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	complemented
350	#103	—	4	2	no	—
351	#102	—	4	2	no	—
352	#102	—	4	2	no	—
353	#102	C₂×C₄×C₃₁	1	8	yes	complemented
354	C₄×C₈	—	4	2	no	—
355	#95	—	2	1	no	—
356	#95	—	2	1	no	—
357	#95	—	2	1	no	—
358	#95	—	2	1	no	—
359	#95	—	2	1	no	—
360	#95	—	2	1	no	—
361	#95	—	2	1	no	—
362	C₃₁	#22147	1	1	yes	characteristic, direct product factor, Sylow
363	C₂⁴	—	2	2	no	—
364	C₂×Q₈	—	2	1	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	—	2	1	no	—
368	C₂×Q₈	—	2	1	no	—
369	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	2	4	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	4	2	no	—
378	C₂×D₄	—	4	2	no	—
379	C₂×D₄	—	4	2	no	—
380	C₂×D₄	—	4	2	no	—
381	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
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	—	4	4	no	—
388	C₂²×C₄	—	4	4	no	—
389	C₂²×C₄	—	4	4	no	—
390	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
395	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
396	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
397	SD₁₆	—	4	8	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	4	2	no	—
413	C₄⋊C₄	—	4	2	no	—
414	C₄⋊C₄	—	4	2	no	—
415	C₄⋊C₄	—	4	2	no	—
416	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
417	C₂²⋊C₄	—	4	4	no	—
418	C₂²⋊C₄	—	4	4	no	—
419	C₂²⋊C₄	—	4	4	no	—
420	C₄²	—	2	4	no	—
421	C₄²	—	2	2	no	—
422	C₄²	—	2	2	no	—
423	C₄²	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	4	1	no	—
428	C₄²	—	4	2	no	—
429	C₄²	SD₁₆×C₃₁	1	4	yes	complemented
430	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
431	C₄²	#41×C₃₁	1	2	yes	—
432	C₂³	—	2	4	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	4	2	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	4	2	no	—
439	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
440	Q₈	—	2	2	no	—
441	Q₈	—	2	2	no	—
442	Q₈	—	4	1	no	—
443	Q₈	—	4	1	no	—
444	D₄	—	2	4	no	—
445	D₄	—	2	4	no	—
446	D₄	—	4	8	no	—
447	D₄	—	4	8	no	—
448	D₄	—	4	2	no	—
449	D₄	—	4	2	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	8	8	no	—
477	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
478	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
479	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
480	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
481	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
482	C₂×C₄	C₃₁×#135	1	2	yes	—
483	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
484	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
485	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
486	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
487	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
488	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
489	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
490	C₈	—	2	4	no	—
491	C₈	—	4	2	no	—
492	C₂²	—	4	4	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	8	2	no	—
496	C₂²	—	8	2	no	—
497	C₂²	—	8	2	no	—
498	C₂²	—	8	2	no	—
499	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
500	C₂²	C₃₁×#535	1	1	yes	characteristic
501	C₂²	C₃₁×#537	1	1	yes	characteristic
502	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
503	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
504	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
505	C₂²	C₃₁×#575	1	1	yes	characteristic
506	C₄	—	2	2	no	—
507	C₄	—	2	2	no	—
508	C₄	—	2	2	no	—
509	C₄	—	2	2	no	—
510	C₄	—	4	2	no	—
511	C₄	—	4	2	no	—
512	C₄	—	4	1	no	—
513	C₄	—	8	2	no	—
514	C₄	—	8	1	no	—
515	C₄	—	8	1	no	—
516	C₄	C₄×SD₁₆×C₃₁	1	4	yes	—
517	C₄	C₃₁×#518	1	1	yes	characteristic
518	C₄	C₃₁×#520	1	1	yes	characteristic
519	C₄	C₃₁×#540	1	1	yes	characteristic
520	C₄	C₃₁×#542	1	1	yes	characteristic
521	C₂	—	8	4	no	—
522	C₂	C₃₁×#2324	1	1	yes	characteristic
523	C₂	C₂×C₄×SD₁₆×C₃₁	1	1	yes	characteristic
524	C₂	C₄×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
525	C₂	C₃₁×#3208	1	1	yes	characteristic
526	C₂	C₃₁×#3212	1	1	yes	characteristic
527	C₂	C₃₁×#3246	1	1	yes	characteristic
528	C₂	C₃₁×#3247	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×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2295	C₂	semidirect
5. C₃₁×#1968	C₂	semidirect
6. C₃₁×#1948	C₂	semidirect
7. C₃₁×#1934	C₂	semidirect
8. C₃₁×#1877	C₂	semidirect
9. C₃₁×#1864	C₂	semidirect
10. C₃₁×#1803	C₂	semidirect
11. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
18. C₃₁×#460	C₄	semidirect
51. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
57. C₄×Q₈×C₃₁	C₂×C₄	semidirect
85. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
89. C₃₁×#102	C₂×C₄	semidirect
164. C₄²×C₃₁	SD₁₆	semidirect
167. #22147	C₃₁	direct (N × H)
230. #2295	C₂×C₃₁	semidirect
232. #1968	C₂×C₃₁	semidirect
233. #1948	C₂×C₃₁	semidirect
234. #1934	C₂×C₃₁	semidirect
235. #1877	C₂×C₃₁	semidirect
236. #1864	C₂×C₃₁	semidirect
237. #1803	C₂×C₃₁	semidirect
238. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
274. #460	C₄×C₃₁	semidirect
315. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
321. C₄×Q₈	C₂×C₄×C₃₁	semidirect
349. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
353. #102	C₂×C₄×C₃₁	semidirect
362. C₃₁	#22147	direct (N × H)
429. C₄²	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#460	2
C₄×C₃₁×#127	2
C₃₁×#2295	2
C₃₁×#1982	2
C₃₁×#1968	2
C₃₁×#1948	2
C₃₁×#1934	2
C₃₁×#1877	2
C₃₁×#1864	2
C₃₁×#1803	2
C₄×C₃₁×C₄⋊C₈	2
#22147	31

Subgroups — C31×#22147 #21478906

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22147 #21478906