Subgroups — C₃₁×#61088 #21517847

Order 7936.

2714 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#577	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3500	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3282	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3277	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3273	C₂	1	4	yes	maximal, complemented
6	C₂×C₃₁×#495	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3057	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3040	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3011	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3008	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	2	yes	—
14	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	2	yes	complemented
15	C₃₁×#577	—	2	16	no	—
16	C₂×C₃₁×#137	C₂²	1	2	yes	complemented
17	C₂×C₃₁×#137	C₂²	1	4	yes	complemented
18	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
19	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
20	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#135	C₂²	1	4	yes	complemented
22	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
23	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
24	C₂²×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
25	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#131	C₂²	1	2	yes	complemented
27	C₃₁×#548	C₂²	1	8	yes	complemented
28	C₃₁×#546	—	2	8	no	—
29	C₃₁×#546	C₂²	1	8	yes	complemented
30	C₃₁×#536	—	2	2	no	—
31	C₃₁×#536	C₂²	1	2	yes	complemented
32	C₃₁×#535	—	2	2	no	—
33	C₃₁×#535	C₂²	1	2	yes	—
34	C₃₁×#532	—	2	2	no	—
35	C₃₁×#532	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#124	—	2	2	no	—
37	C₂×C₃₁×#124	C₂²	1	2	yes	complemented
38	C₄×#41×C₃₁	C₂²	1	2	yes	complemented
39	C₃₁×#495	C₂²	1	4	yes	complemented
40	C₃₁×#495	C₂²	1	4	yes	complemented
41	C₃₁×#468	C₂²	1	8	yes	complemented
42	C₃₁×#465	C₂²	1	8	yes	complemented
43	C₃₁×#464	C₂²	1	8	yes	complemented
44	C₃₁×#463	C₂²	1	8	yes	complemented
45	C₃₁×#452	C₂²	1	8	yes	complemented
46	C₃₁×#451	C₂²	1	8	yes	complemented
47	C₃₁×#449	C₂²	1	8	yes	complemented
48	C₃₁×#447	C₂²	1	8	yes	complemented
49	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
50	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
51	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
52	C₃₁×#436	C₂²	1	4	yes	complemented
53	C₈×D₄×C₃₁	C₂²	1	4	yes	complemented
54	C₃₁×#427	C₂²	1	1	yes	characteristic, complemented
55	C₃₁×#423	C₂²	1	1	yes	characteristic, complemented
56	C₃₁×#419	C₂²	1	2	yes	complemented
57	C₃₁×#418	C₂²	1	2	yes	complemented
58	C₃₁×#416	C₂²	1	2	yes	complemented
59	C₃₁×#408	C₂²	1	2	yes	complemented
60	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
61	C₃₁×2₋¹⁺⁴	—	2	8	no	—
62	C₃₁×2₋¹⁺⁴	—	4	4	no	—
63	C₃₁×H(5,𝔽₂)	—	2	8	no	—
64	C₃₁×H(5,𝔽₂)	—	4	4	no	—
65	C₂×#41×C₃₁	—	2	2	no	—
66	C₂×#41×C₃₁	—	2	2	no	—
67	C₂×#41×C₃₁	—	2	4	no	—
68	C₂×#41×C₃₁	C₂³	1	4	yes	—
69	C₂×#41×C₃₁	C₂³	1	4	yes	—
70	C₂×#41×C₃₁	C₂³	1	2	yes	—
71	C₂×#41×C₃₁	C₂³	1	2	yes	—
72	C₂²×Q₈×C₃₁	—	2	2	no	—
73	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
74	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
75	C₂²×D₄×C₃₁	—	2	2	no	—
76	C₂²×D₄×C₃₁	—	2	4	no	—
77	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
78	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
79	C₃₁×#137	—	2	8	no	—
80	C₃₁×#137	—	2	16	no	—
81	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
82	C₃₁×AGL(1,ℤ/8)	—	2	16	no	—
83	C₃₁×#135	—	2	16	no	—
84	C₃₁×#135	D₄	1	8	yes	complemented
85	C₂×Q₁₆×C₃₁	—	4	2	no	—
86	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
87	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
88	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×SD₁₆×C₃₁	—	2	16	no	—
90	C₂×SD₁₆×C₃₁	—	2	8	no	—
91	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
92	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
93	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
94	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
95	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
96	C₂×D₈×C₃₁	—	4	2	no	—
97	C₂×D₈×C₃₁	C₂³	1	4	yes	—
98	C₂×D₈×C₃₁	C₂³	1	4	yes	—
99	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₃₁×#131	—	2	8	no	—
101	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
102	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
103	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
104	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
105	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
106	C₃₁×C₄⋊Q₈	—	2	2	no	—
107	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
108	C₃₁×#127	—	2	2	no	—
109	C₃₁×#127	D₄	1	1	yes	characteristic
110	C₃₁×#124	—	2	2	no	—
111	C₃₁×#124	—	2	2	no	—
112	C₃₁×#124	—	2	8	no	—
113	C₃₁×#124	—	2	2	no	—
114	C₃₁×#124	—	2	2	no	—
115	C₃₁×#124	D₄	1	2	yes	—
116	C₃₁×#124	C₂³	1	8	yes	—
117	C₃₁×#123	—	2	4	no	—
118	C₃₁×#123	—	2	8	no	—
119	C₃₁×C₂²⋊Q₈	—	2	8	no	—
120	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
121	C₃₁×#121	—	2	8	no	—
122	C₃₁×#121	—	2	8	no	—
123	C₃₁×#121	D₄	1	4	yes	—
124	C₃₁×#121	C₂³	1	8	yes	—
125	C₃₁×#120	—	2	8	no	—
126	C₃₁×#120	—	2	4	no	—
127	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
128	C₄×D₄×C₃₁	—	2	4	no	—
129	C₄×D₄×C₃₁	—	2	8	no	—
130	C₄×D₄×C₃₁	C₂³	1	4	yes	—
131	C₄×D₄×C₃₁	C₂³	1	4	yes	—
132	C₃₁×#117	—	2	2	no	—
133	C₃₁×#117	C₂³	1	2	yes	—
134	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
135	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
136	C₂×C₄²×C₃₁	—	2	2	no	—
137	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
138	C₃₁×#107	C₂³	1	2	yes	—
139	C₃₁×#106	C₂³	1	2	yes	—
140	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
141	C₃₁×#103	C₂³	1	4	yes	—
142	C₃₁×#103	C₂³	1	8	yes	—
143	C₃₁×#102	C₂³	1	4	yes	—
144	C₃₁×#102	C₂³	1	8	yes	—
145	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
146	C₄×C₈×C₃₁	C₂³	1	2	yes	—
147	C₄×C₈×C₃₁	C₂³	1	2	yes	—
148	C₂⁴×C₃₁	—	2	4	no	—
149	C₂⁴×C₃₁	—	2	2	no	—
150	#41×C₃₁	—	2	4	no	—
151	#41×C₃₁	—	2	4	no	—
152	#41×C₃₁	—	2	4	no	—
153	#41×C₃₁	—	2	8	no	—
154	#41×C₃₁	—	2	8	no	—
155	#41×C₃₁	—	4	4	no	—
156	#41×C₃₁	—	4	4	no	—
157	#41×C₃₁	—	4	8	no	—
158	#41×C₃₁	—	4	8	no	—
159	#41×C₃₁	—	4	16	no	—
160	#41×C₃₁	C₂×D₄	1	8	yes	—
161	C₂×Q₈×C₃₁	—	2	4	no	—
162	C₂×Q₈×C₃₁	—	2	2	no	—
163	C₂×Q₈×C₃₁	—	2	8	no	—
164	C₂×Q₈×C₃₁	—	2	4	no	—
165	C₂×Q₈×C₃₁	—	2	4	no	—
166	C₂×Q₈×C₃₁	—	4	4	no	—
167	C₂×Q₈×C₃₁	—	4	8	no	—
168	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
169	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
170	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
171	C₂×D₄×C₃₁	—	2	2	no	—
172	C₂×D₄×C₃₁	—	2	8	no	—
173	C₂×D₄×C₃₁	—	2	4	no	—
174	C₂×D₄×C₃₁	—	2	4	no	—
175	C₂×D₄×C₃₁	—	2	8	no	—
176	C₂×D₄×C₃₁	—	2	4	no	—
177	C₂×D₄×C₃₁	—	2	4	no	—
178	C₂×D₄×C₃₁	—	2	2	no	—
179	C₂×D₄×C₃₁	—	2	4	no	—
180	C₂×D₄×C₃₁	—	4	4	no	—
181	C₂×D₄×C₃₁	—	4	8	no	—
182	C₂×D₄×C₃₁	—	4	16	no	—
183	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
184	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
185	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
186	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
187	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₃₁	—	2	2	no	—
189	C₂²×C₄×C₃₁	—	2	2	no	—
190	C₂²×C₄×C₃₁	—	2	4	no	—
191	C₂²×C₄×C₃₁	—	2	4	no	—
192	C₂²×C₄×C₃₁	—	2	4	no	—
193	C₂²×C₄×C₃₁	—	2	4	no	—
194	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
195	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
196	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
197	Q₁₆×C₃₁	—	2	8	no	—
198	Q₁₆×C₃₁	—	4	4	no	—
199	Q₁₆×C₃₁	C₂×D₄	1	4	yes	—
200	SD₁₆×C₃₁	—	2	8	no	—
201	SD₁₆×C₃₁	—	2	8	no	—
202	SD₁₆×C₃₁	—	2	8	no	—
203	SD₁₆×C₃₁	—	2	8	no	—
204	SD₁₆×C₃₁	C₂×D₄	1	8	yes	—
205	D₈×C₃₁	—	2	8	no	—
206	D₈×C₃₁	—	4	4	no	—
207	D₈×C₃₁	C₂×D₄	1	4	yes	—
208	M₁₆×C₃₁	—	2	4	no	—
209	M₁₆×C₃₁	—	2	8	no	—
210	C₂×C₈×C₃₁	—	2	2	no	—
211	C₂×C₈×C₃₁	—	2	8	no	—
212	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
213	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
214	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
215	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
216	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
217	C₄⋊C₄×C₃₁	—	2	4	no	—
218	C₄⋊C₄×C₃₁	—	2	8	no	—
219	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
220	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
221	C₂²⋊C₄×C₃₁	—	2	8	no	—
222	C₂²⋊C₄×C₃₁	—	2	2	no	—
223	C₂²⋊C₄×C₃₁	—	2	8	no	—
224	C₂²⋊C₄×C₃₁	—	2	8	no	—
225	C₂²⋊C₄×C₃₁	—	2	8	no	—
226	C₂²⋊C₄×C₃₁	—	2	4	no	—
227	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
228	C₄²×C₃₁	—	2	2	no	—
229	C₄²×C₃₁	—	2	4	no	—
230	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
231	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
232	C₄²×C₃₁	C₂⁴	1	2	yes	—
233	#61088	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₂³×C₃₁	—	2	2	no	—
235	C₂³×C₃₁	—	2	4	no	—
236	C₂³×C₃₁	—	4	8	no	—
237	C₂³×C₃₁	—	4	4	no	—
238	C₂³×C₃₁	—	8	4	no	—
239	C₂³×C₃₁	—	8	8	no	—
240	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
241	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
242	Q₈×C₃₁	—	2	4	no	—
243	Q₈×C₃₁	—	2	2	no	—
244	Q₈×C₃₁	—	2	4	no	—
245	Q₈×C₃₁	—	2	4	no	—
246	Q₈×C₃₁	—	4	8	no	—
247	Q₈×C₃₁	—	4	4	no	—
248	Q₈×C₃₁	C₂²×D₄	1	4	yes	—
249	D₄×C₃₁	—	2	8	no	—
250	D₄×C₃₁	—	2	8	no	—
251	D₄×C₃₁	—	2	4	no	—
252	D₄×C₃₁	—	2	4	no	—
253	D₄×C₃₁	—	2	2	no	—
254	D₄×C₃₁	—	2	4	no	—
255	D₄×C₃₁	—	4	4	no	—
256	D₄×C₃₁	—	4	8	no	—
257	D₄×C₃₁	—	4	8	no	—
258	D₄×C₃₁	—	4	8	no	—
259	D₄×C₃₁	—	4	4	no	—
260	D₄×C₃₁	—	8	8	no	—
261	D₄×C₃₁	C₂²×D₄	1	4	yes	—
262	C₂×C₄×C₃₁	—	2	2	no	—
263	C₂×C₄×C₃₁	—	2	8	no	—
264	C₂×C₄×C₃₁	—	2	4	no	—
265	C₂×C₄×C₃₁	—	2	4	no	—
266	C₂×C₄×C₃₁	—	2	4	no	—
267	C₂×C₄×C₃₁	—	2	4	no	—
268	C₂×C₄×C₃₁	—	4	2	no	—
269	C₂×C₄×C₃₁	—	4	8	no	—
270	C₂×C₄×C₃₁	—	4	8	no	—
271	C₂×C₄×C₃₁	—	4	8	no	—
272	C₂×C₄×C₃₁	—	4	4	no	—
273	C₂×C₄×C₃₁	—	8	4	no	—
274	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
275	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
276	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
277	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
278	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
279	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
280	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
281	C₈×C₃₁	—	2	4	no	—
282	C₈×C₃₁	—	2	2	no	—
283	C₈×C₃₁	C₂²×D₄	1	2	yes	—
284	C₈×C₃₁	C₂²×D₄	1	2	yes	—
285	C₂×#577	C₂×C₃₁	1	2	yes	complemented
286	#3500	C₂×C₃₁	1	2	yes	complemented
287	#3282	C₂×C₃₁	1	8	yes	complemented
288	#3277	C₂×C₃₁	1	4	yes	complemented
289	#3273	C₂×C₃₁	1	4	yes	complemented
290	C₂×#495	C₂×C₃₁	1	1	yes	characteristic, complemented
291	#3057	C₂×C₃₁	1	2	yes	complemented
292	#3040	C₂×C₃₁	1	2	yes	complemented
293	#3011	C₂×C₃₁	1	2	yes	complemented
294	#3008	C₂×C₃₁	1	2	yes	complemented
295	C₄×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
296	#2931	C₂×C₃₁	1	1	yes	characteristic, complemented
297	C₂²×C₃₁	—	2	2	no	—
298	C₂²×C₃₁	—	2	8	no	—
299	C₂²×C₃₁	—	2	4	no	—
300	C₂²×C₃₁	—	2	4	no	—
301	C₂²×C₃₁	—	4	4	no	—
302	C₂²×C₃₁	—	4	8	no	—
303	C₂²×C₃₁	—	4	4	no	—
304	C₂²×C₃₁	—	16	4	no	—
305	C₂²×C₃₁	—	16	8	no	—
306	C₂²×C₃₁	D₄²	1	2	yes	—
307	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
308	C₄×C₃₁	—	2	2	no	—
309	C₄×C₃₁	—	2	2	no	—
310	C₄×C₃₁	—	2	4	no	—
311	C₄×C₃₁	—	2	4	no	—
312	C₄×C₃₁	—	4	4	no	—
313	C₄×C₃₁	—	4	4	no	—
314	C₄×C₃₁	D₄²	1	2	yes	—
315	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
316	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
317	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	2	yes	—
318	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	2	yes	complemented
319	#577	—	2	16	no	—
320	C₂×#137	C₂²×C₃₁	1	2	yes	complemented
321	C₂×#137	C₂²×C₃₁	1	4	yes	complemented
322	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
323	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	complemented
324	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
325	C₂×#135	C₂²×C₃₁	1	4	yes	complemented
326	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
327	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
328	C₂²×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
329	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
330	C₂×#131	C₂²×C₃₁	1	2	yes	complemented
331	#548	C₂²×C₃₁	1	8	yes	complemented
332	#546	—	2	8	no	—
333	#546	C₂²×C₃₁	1	8	yes	complemented
334	#536	—	2	2	no	—
335	#536	C₂²×C₃₁	1	2	yes	complemented
336	#535	—	2	2	no	—
337	#535	C₂²×C₃₁	1	2	yes	—
338	#532	—	2	2	no	—
339	#532	C₂²×C₃₁	1	1	yes	characteristic
340	C₂×#124	—	2	2	no	—
341	C₂×#124	C₂²×C₃₁	1	2	yes	complemented
342	C₄×#41	C₂²×C₃₁	1	2	yes	complemented
343	#495	C₂²×C₃₁	1	4	yes	complemented
344	#495	C₂²×C₃₁	1	4	yes	complemented
345	#468	C₂²×C₃₁	1	8	yes	complemented
346	#465	C₂²×C₃₁	1	8	yes	complemented
347	#464	C₂²×C₃₁	1	8	yes	complemented
348	#463	C₂²×C₃₁	1	8	yes	complemented
349	#452	C₂²×C₃₁	1	8	yes	complemented
350	#451	C₂²×C₃₁	1	8	yes	complemented
351	#449	C₂²×C₃₁	1	8	yes	complemented
352	#447	C₂²×C₃₁	1	8	yes	complemented
353	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
354	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
355	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
356	#436	C₂²×C₃₁	1	4	yes	complemented
357	C₈×D₄	C₂²×C₃₁	1	4	yes	complemented
358	#427	C₂²×C₃₁	1	1	yes	characteristic, complemented
359	#423	C₂²×C₃₁	1	1	yes	characteristic, complemented
360	#419	C₂²×C₃₁	1	2	yes	complemented
361	#418	C₂²×C₃₁	1	2	yes	complemented
362	#416	C₂²×C₃₁	1	2	yes	complemented
363	#408	C₂²×C₃₁	1	2	yes	complemented
364	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
365	C₂×C₃₁	—	2	2	no	—
366	C₂×C₃₁	—	4	4	no	—
367	C₂×C₃₁	—	4	4	no	—
368	C₂×C₃₁	—	8	4	no	—
369	C₂×C₃₁	C₂×D₄²	1	1	yes	characteristic
370	C₂×C₃₁	#3584	1	2	yes	—
371	2₋¹⁺⁴	—	2	8	no	—
372	2₋¹⁺⁴	—	4	4	no	—
373	H(5,𝔽₂)	—	2	8	no	—
374	H(5,𝔽₂)	—	4	4	no	—
375	C₂×#41	—	2	2	no	—
376	C₂×#41	—	2	2	no	—
377	C₂×#41	—	2	4	no	—
378	C₂×#41	C₂³×C₃₁	1	4	yes	—
379	C₂×#41	C₂³×C₃₁	1	4	yes	—
380	C₂×#41	C₂³×C₃₁	1	2	yes	—
381	C₂×#41	C₂³×C₃₁	1	2	yes	—
382	C₂²×Q₈	—	2	2	no	—
383	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
384	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
385	C₂²×D₄	—	2	2	no	—
386	C₂²×D₄	—	2	4	no	—
387	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
388	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
389	#137	—	2	8	no	—
390	#137	—	2	16	no	—
391	AGL(1,ℤ/8)	—	2	8	no	—
392	AGL(1,ℤ/8)	—	2	16	no	—
393	#135	—	2	16	no	—
394	#135	D₄×C₃₁	1	8	yes	complemented
395	C₂×Q₁₆	—	4	2	no	—
396	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
397	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
398	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
399	C₂×SD₁₆	—	2	16	no	—
400	C₂×SD₁₆	—	2	8	no	—
401	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
402	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
403	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
404	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
405	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
406	C₂×D₈	—	4	2	no	—
407	C₂×D₈	C₂³×C₃₁	1	4	yes	—
408	C₂×D₈	C₂³×C₃₁	1	4	yes	—
409	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
410	#131	—	2	8	no	—
411	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
412	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
413	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
414	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
415	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
416	C₄⋊Q₈	—	2	2	no	—
417	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
418	#127	—	2	2	no	—
419	#127	D₄×C₃₁	1	1	yes	characteristic
420	#124	—	2	2	no	—
421	#124	—	2	2	no	—
422	#124	—	2	8	no	—
423	#124	—	2	2	no	—
424	#124	—	2	2	no	—
425	#124	D₄×C₃₁	1	2	yes	—
426	#124	C₂³×C₃₁	1	8	yes	—
427	#123	—	2	4	no	—
428	#123	—	2	8	no	—
429	C₂²⋊Q₈	—	2	8	no	—
430	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
431	#121	—	2	8	no	—
432	#121	—	2	8	no	—
433	#121	D₄×C₃₁	1	4	yes	—
434	#121	C₂³×C₃₁	1	8	yes	—
435	#120	—	2	8	no	—
436	#120	—	2	4	no	—
437	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
438	C₄×D₄	—	2	4	no	—
439	C₄×D₄	—	2	8	no	—
440	C₄×D₄	C₂³×C₃₁	1	4	yes	—
441	C₄×D₄	C₂³×C₃₁	1	4	yes	—
442	#117	—	2	2	no	—
443	#117	C₂³×C₃₁	1	2	yes	—
444	C₂×C₂²⋊C₄	—	2	4	no	—
445	C₂×C₂²⋊C₄	—	2	8	no	—
446	C₂×C₄²	—	2	2	no	—
447	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
448	#107	C₂³×C₃₁	1	2	yes	—
449	#106	C₂³×C₃₁	1	2	yes	—
450	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
451	#103	C₂³×C₃₁	1	4	yes	—
452	#103	C₂³×C₃₁	1	8	yes	—
453	#102	C₂³×C₃₁	1	4	yes	—
454	#102	C₂³×C₃₁	1	8	yes	—
455	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
456	C₄×C₈	C₂³×C₃₁	1	2	yes	—
457	C₄×C₈	C₂³×C₃₁	1	2	yes	—
458	C₃₁	#61088	1	1	yes	characteristic, direct product factor, Sylow
459	C₂⁴	—	2	4	no	—
460	C₂⁴	—	2	2	no	—
461	#41	—	2	4	no	—
462	#41	—	2	4	no	—
463	#41	—	2	4	no	—
464	#41	—	2	8	no	—
465	#41	—	2	8	no	—
466	#41	—	4	4	no	—
467	#41	—	4	4	no	—
468	#41	—	4	8	no	—
469	#41	—	4	8	no	—
470	#41	—	4	16	no	—
471	#41	C₂×D₄×C₃₁	1	8	yes	—
472	C₂×Q₈	—	2	4	no	—
473	C₂×Q₈	—	2	2	no	—
474	C₂×Q₈	—	2	8	no	—
475	C₂×Q₈	—	2	4	no	—
476	C₂×Q₈	—	2	4	no	—
477	C₂×Q₈	—	4	4	no	—
478	C₂×Q₈	—	4	8	no	—
479	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
480	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
481	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
482	C₂×D₄	—	2	2	no	—
483	C₂×D₄	—	2	8	no	—
484	C₂×D₄	—	2	4	no	—
485	C₂×D₄	—	2	4	no	—
486	C₂×D₄	—	2	8	no	—
487	C₂×D₄	—	2	4	no	—
488	C₂×D₄	—	2	4	no	—
489	C₂×D₄	—	2	2	no	—
490	C₂×D₄	—	2	4	no	—
491	C₂×D₄	—	4	4	no	—
492	C₂×D₄	—	4	8	no	—
493	C₂×D₄	—	4	16	no	—
494	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
495	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
496	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
497	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
498	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
499	C₂²×C₄	—	2	2	no	—
500	C₂²×C₄	—	2	2	no	—
501	C₂²×C₄	—	2	4	no	—
502	C₂²×C₄	—	2	4	no	—
503	C₂²×C₄	—	2	4	no	—
504	C₂²×C₄	—	2	4	no	—
505	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
506	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
507	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
508	Q₁₆	—	2	8	no	—
509	Q₁₆	—	4	4	no	—
510	Q₁₆	C₂×D₄×C₃₁	1	4	yes	—
511	SD₁₆	—	2	8	no	—
512	SD₁₆	—	2	8	no	—
513	SD₁₆	—	2	8	no	—
514	SD₁₆	—	2	8	no	—
515	SD₁₆	C₂×D₄×C₃₁	1	8	yes	—
516	D₈	—	2	8	no	—
517	D₈	—	4	4	no	—
518	D₈	C₂×D₄×C₃₁	1	4	yes	—
519	M₁₆	—	2	4	no	—
520	M₁₆	—	2	8	no	—
521	C₂×C₈	—	2	2	no	—
522	C₂×C₈	—	2	8	no	—
523	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
524	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
525	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
526	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
527	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
528	C₄⋊C₄	—	2	4	no	—
529	C₄⋊C₄	—	2	8	no	—
530	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
531	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
532	C₂²⋊C₄	—	2	8	no	—
533	C₂²⋊C₄	—	2	2	no	—
534	C₂²⋊C₄	—	2	8	no	—
535	C₂²⋊C₄	—	2	8	no	—
536	C₂²⋊C₄	—	2	8	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
539	C₄²	—	2	2	no	—
540	C₄²	—	2	4	no	—
541	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
542	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
543	C₄²	C₂⁴×C₃₁	1	2	yes	—
544	C₂³	—	2	2	no	—
545	C₂³	—	2	4	no	—
546	C₂³	—	4	8	no	—
547	C₂³	—	4	4	no	—
548	C₂³	—	8	4	no	—
549	C₂³	—	8	8	no	—
550	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
551	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
552	Q₈	—	2	4	no	—
553	Q₈	—	2	2	no	—
554	Q₈	—	2	4	no	—
555	Q₈	—	2	4	no	—
556	Q₈	—	4	8	no	—
557	Q₈	—	4	4	no	—
558	Q₈	C₂²×D₄×C₃₁	1	4	yes	—
559	D₄	—	2	8	no	—
560	D₄	—	2	8	no	—
561	D₄	—	2	4	no	—
562	D₄	—	2	4	no	—
563	D₄	—	2	2	no	—
564	D₄	—	2	4	no	—
565	D₄	—	4	4	no	—
566	D₄	—	4	8	no	—
567	D₄	—	4	8	no	—
568	D₄	—	4	8	no	—
569	D₄	—	4	4	no	—
570	D₄	—	8	8	no	—
571	D₄	C₂²×D₄×C₃₁	1	4	yes	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	8	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	4	2	no	—
579	C₂×C₄	—	4	8	no	—
580	C₂×C₄	—	4	8	no	—
581	C₂×C₄	—	4	8	no	—
582	C₂×C₄	—	4	4	no	—
583	C₂×C₄	—	8	4	no	—
584	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
585	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
586	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
587	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
588	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
589	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
590	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
591	C₈	—	2	4	no	—
592	C₈	—	2	2	no	—
593	C₈	C₂²×D₄×C₃₁	1	2	yes	—
594	C₈	C₂²×D₄×C₃₁	1	2	yes	—
595	C₂²	—	2	2	no	—
596	C₂²	—	2	4	no	—
597	C₂²	—	2	4	no	—
598	C₂²	—	2	8	no	—
599	C₂²	—	4	4	no	—
600	C₂²	—	4	4	no	—
601	C₂²	—	4	8	no	—
602	C₂²	—	16	4	no	—
603	C₂²	—	16	8	no	—
604	C₂²	D₄²×C₃₁	1	2	yes	—
605	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
606	C₄	—	2	4	no	—
607	C₄	—	2	4	no	—
608	C₄	—	2	2	no	—
609	C₄	—	2	2	no	—
610	C₄	—	4	4	no	—
611	C₄	—	4	4	no	—
612	C₄	D₄²×C₃₁	1	2	yes	—
613	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
614	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
615	C₂	—	2	2	no	—
616	C₂	—	4	4	no	—
617	C₂	—	4	4	no	—
618	C₂	—	8	4	no	—
619	C₂	C₂×D₄²×C₃₁	1	1	yes	characteristic
620	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#532 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#577	C₂	semidirect
2. C₃₁×#3500	C₂	semidirect
3. C₃₁×#3282	C₂	semidirect
4. C₃₁×#3277	C₂	semidirect
5. C₃₁×#3273	C₂	semidirect
6. C₂×C₃₁×#495	C₂	semidirect
7. C₃₁×#3057	C₂	semidirect
8. C₃₁×#3040	C₂	semidirect
9. C₃₁×#3011	C₂	semidirect
10. C₃₁×#3008	C₂	semidirect
11. C₄×C₃₁×#135	C₂	semidirect
12. C₃₁×#2931	C₂	semidirect
14. C₂×C₃₁×H(5,𝔽₂)	C₂²	semidirect
16. C₂×C₃₁×#137	C₂²	semidirect
17. C₂×C₃₁×#137	C₂²	semidirect
19. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
21. C₂×C₃₁×#135	C₂²	semidirect
22. C₂²×Q₁₆×C₃₁	C₂²	semidirect
23. C₂²×SD₁₆×C₃₁	C₂²	semidirect
24. C₂²×SD₁₆×C₃₁	C₂²	semidirect
26. C₂×C₃₁×#131	C₂²	semidirect
27. C₃₁×#548	C₂²	semidirect
29. C₃₁×#546	C₂²	semidirect
31. C₃₁×#536	C₂²	semidirect
37. C₂×C₃₁×#124	C₂²	semidirect
38. C₄×#41×C₃₁	C₂²	semidirect
39. C₃₁×#495	C₂²	semidirect
40. C₃₁×#495	C₂²	semidirect
41. C₃₁×#468	C₂²	semidirect
42. C₃₁×#465	C₂²	semidirect
43. C₃₁×#464	C₂²	semidirect
44. C₃₁×#463	C₂²	semidirect
45. C₃₁×#452	C₂²	semidirect
46. C₃₁×#451	C₂²	semidirect
47. C₃₁×#449	C₂²	semidirect
48. C₃₁×#447	C₂²	semidirect
49. C₄×Q₁₆×C₃₁	C₂²	semidirect
50. C₄×SD₁₆×C₃₁	C₂²	semidirect
51. C₄×D₈×C₃₁	C₂²	semidirect
52. C₃₁×#436	C₂²	semidirect
53. C₈×D₄×C₃₁	C₂²	semidirect
54. C₃₁×#427	C₂²	semidirect
55. C₃₁×#423	C₂²	semidirect
56. C₃₁×#419	C₂²	semidirect
57. C₃₁×#418	C₂²	semidirect
58. C₃₁×#416	C₂²	semidirect
59. C₃₁×#408	C₂²	semidirect
60. C₂×C₄×C₈×C₃₁	C₂²	semidirect
84. C₃₁×#135	D₄	semidirect
233. #61088	C₃₁	direct (N × H)
285. C₂×#577	C₂×C₃₁	semidirect
286. #3500	C₂×C₃₁	semidirect
287. #3282	C₂×C₃₁	semidirect
288. #3277	C₂×C₃₁	semidirect
289. #3273	C₂×C₃₁	semidirect
290. C₂×#495	C₂×C₃₁	semidirect
291. #3057	C₂×C₃₁	semidirect
292. #3040	C₂×C₃₁	semidirect
293. #3011	C₂×C₃₁	semidirect
294. #3008	C₂×C₃₁	semidirect
295. C₄×#135	C₂×C₃₁	semidirect
296. #2931	C₂×C₃₁	semidirect
318. C₂×H(5,𝔽₂)	C₂²×C₃₁	semidirect
320. C₂×#137	C₂²×C₃₁	semidirect
321. C₂×#137	C₂²×C₃₁	semidirect
323. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
325. C₂×#135	C₂²×C₃₁	semidirect
326. C₂²×Q₁₆	C₂²×C₃₁	semidirect
327. C₂²×SD₁₆	C₂²×C₃₁	semidirect
328. C₂²×SD₁₆	C₂²×C₃₁	semidirect
330. C₂×#131	C₂²×C₃₁	semidirect
331. #548	C₂²×C₃₁	semidirect
333. #546	C₂²×C₃₁	semidirect
335. #536	C₂²×C₃₁	semidirect
341. C₂×#124	C₂²×C₃₁	semidirect
342. C₄×#41	C₂²×C₃₁	semidirect
343. #495	C₂²×C₃₁	semidirect
344. #495	C₂²×C₃₁	semidirect
345. #468	C₂²×C₃₁	semidirect
346. #465	C₂²×C₃₁	semidirect
347. #464	C₂²×C₃₁	semidirect
348. #463	C₂²×C₃₁	semidirect
349. #452	C₂²×C₃₁	semidirect
350. #451	C₂²×C₃₁	semidirect
351. #449	C₂²×C₃₁	semidirect
352. #447	C₂²×C₃₁	semidirect
353. C₄×Q₁₆	C₂²×C₃₁	semidirect
354. C₄×SD₁₆	C₂²×C₃₁	semidirect
355. C₄×D₈	C₂²×C₃₁	semidirect
356. #436	C₂²×C₃₁	semidirect
357. C₈×D₄	C₂²×C₃₁	semidirect
358. #427	C₂²×C₃₁	semidirect
359. #423	C₂²×C₃₁	semidirect
360. #419	C₂²×C₃₁	semidirect
361. #418	C₂²×C₃₁	semidirect
362. #416	C₂²×C₃₁	semidirect
363. #408	C₂²×C₃₁	semidirect
364. C₂×C₄×C₈	C₂²×C₃₁	semidirect
394. #135	D₄×C₃₁	semidirect
458. C₃₁	#61088	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×#577	2
C₃₁×#3500	2
C₃₁×#3282	2
C₃₁×#3277	2
C₃₁×#3273	2
C₂×C₃₁×#495	2
C₃₁×#3057	2
C₃₁×#3040	2
C₃₁×#3011	2
C₃₁×#3008	2
C₄×C₃₁×#135	2
C₃₁×#2931	2
#61088	31

Subgroups — C31×#61088 #21517847

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61088 #21517847