Subgroups — C₃₁×#60827 #21517586

Order 7936.

2154 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3224	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3219	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3217	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3153	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#496	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3138	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3135	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3043	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3041	C₂	1	2	yes	maximal, complemented
12	C₃₁×#3009	C₂	1	2	yes	maximal, complemented
13	C₃₁×#3008	C₂	1	2	yes	maximal, complemented
14	C₃₁×#2923	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	2	yes	—
16	C₂×C₃₁×#137	C₂²	1	2	yes	complemented
17	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
18	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
19	C₂²×D₈×C₃₁	C₂²	1	2	yes	complemented
20	C₃₁×#559	—	2	2	no	—
21	C₃₁×#556	—	2	2	no	—
22	C₃₁×#551	—	2	4	no	—
23	C₃₁×#550	—	2	2	no	—
24	D₄²×C₃₁	—	2	2	no	—
25	D₄²×C₃₁	—	2	4	no	—
26	C₃₁×#542	C₂²	1	4	yes	complemented
27	C₃₁×#536	C₂²	1	1	yes	characteristic
28	C₃₁×#535	C₂²	1	2	yes	—
29	C₃₁×#535	C₂²	1	1	yes	characteristic
30	C₃₁×#535	C₂²	1	4	yes	complemented
31	C₃₁×#533	C₂²	1	1	yes	characteristic, complemented
32	C₃₁×#532	C₂²	1	1	yes	characteristic
33	C₃₁×#532	C₂²	1	2	yes	complemented
34	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#127	C₂²	1	2	yes	complemented
36	C₃₁×#529	C₂²	1	4	yes	complemented
37	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#519	C₂²	1	1	yes	characteristic, complemented
39	C₃₁×#518	C₂²	1	1	yes	characteristic, complemented
40	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#497	C₂²	1	2	yes	complemented
42	C₃₁×#496	C₂²	1	4	yes	complemented
43	C₃₁×#496	C₂²	1	4	yes	complemented
44	C₃₁×#496	C₂²	1	2	yes	complemented
45	C₃₁×#495	C₂²	1	4	yes	complemented
46	C₃₁×#491	C₂²	1	2	yes	complemented
47	C₃₁×#490	C₂²	1	2	yes	complemented
48	C₃₁×#490	C₂²	1	4	yes	complemented
49	C₃₁×#489	C₂²	1	4	yes	complemented
50	C₃₁×#486	C₂²	1	4	yes	complemented
51	C₃₁×#464	C₂²	1	8	yes	complemented
52	C₃₁×#463	C₂²	1	8	yes	complemented
53	C₃₁×#460	C₂²	1	8	yes	complemented
54	C₃₁×#459	C₂²	1	8	yes	complemented
55	C₃₁×#452	C₂²	1	8	yes	complemented
56	C₃₁×#450	C₂²	1	8	yes	complemented
57	C₃₁×#449	C₂²	1	8	yes	complemented
58	C₃₁×#447	C₂²	1	8	yes	complemented
59	C₃₁×#433	C₂²	1	4	yes	complemented
60	C₃₁×#432	C₂²	1	4	yes	complemented
61	C₃₁×#424	C₂²	1	2	yes	complemented
62	C₃₁×#418	C₂²	1	2	yes	complemented
63	C₃₁×#418	C₂²	1	2	yes	complemented
64	C₃₁×#416	C₂²	1	2	yes	complemented
65	C₃₁×#416	C₂²	1	2	yes	complemented
66	C₃₁×#408	C₂²	1	2	yes	—
67	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
68	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic, complemented
69	C₃₁×H(5,𝔽₂)	—	4	4	no	—
70	C₃₁×H(5,𝔽₂)	—	4	4	no	—
71	C₂×#41×C₃₁	—	2	2	no	—
72	C₂×#41×C₃₁	C₂³	1	2	yes	—
73	C₂×#41×C₃₁	C₂³	1	4	yes	—
74	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
75	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×D₄×C₃₁	—	2	4	no	—
77	C₂²×D₄×C₃₁	—	2	2	no	—
78	C₂²×D₄×C₃₁	—	2	2	no	—
79	C₂²×D₄×C₃₁	—	2	4	no	—
80	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
81	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
83	C₃₁×#137	—	4	4	no	—
84	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
85	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
86	C₂×SD₁₆×C₃₁	—	4	4	no	—
87	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
88	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
89	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
90	C₂×D₈×C₃₁	—	4	4	no	—
91	C₂×D₈×C₃₁	C₂³	1	4	yes	—
92	C₂×D₈×C₃₁	C₂³	1	8	yes	—
93	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
94	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
95	C₃₁×C₄⋊Q₈	—	2	1	no	—
96	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
97	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
98	C₃₁×#127	—	2	1	no	—
99	C₃₁×#127	—	2	2	no	—
100	C₃₁×#127	—	2	4	no	—
101	C₃₁×#127	C₂³	1	4	yes	—
102	C₃₁×#127	C₂³	1	2	yes	—
103	C₃₁×#127	C₂³	1	2	yes	—
104	C₃₁×#127	C₂³	1	1	yes	characteristic
105	C₃₁×#126	—	2	4	no	—
106	C₃₁×#126	—	2	4	no	—
107	C₃₁×#125	C₂³	1	2	yes	—
108	C₃₁×#124	—	2	1	no	—
109	C₃₁×#124	—	2	1	no	—
110	C₃₁×#124	—	2	4	no	—
111	C₃₁×#124	—	2	2	no	—
112	C₃₁×#124	C₂³	1	4	yes	—
113	C₃₁×#124	C₂³	1	2	yes	—
114	C₃₁×#124	C₂³	1	2	yes	—
115	C₃₁×#124	C₂³	1	2	yes	—
116	C₃₁×#123	—	2	2	no	—
117	C₃₁×#123	—	2	4	no	—
118	C₃₁×#123	—	2	4	no	—
119	C₃₁×C₂²⋊Q₈	—	2	2	no	—
120	C₃₁×C₂²⋊Q₈	—	2	4	no	—
121	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
122	C₃₁×#121	—	2	2	no	—
123	C₃₁×#121	—	2	4	no	—
124	C₃₁×#121	—	2	4	no	—
125	C₃₁×#121	—	2	8	no	—
126	C₃₁×#121	C₂³	1	4	yes	—
127	C₃₁×#120	—	2	2	no	—
128	C₃₁×#120	—	2	4	no	—
129	C₃₁×#120	—	2	8	no	—
130	C₄×Q₈×C₃₁	—	2	2	no	—
131	C₄×Q₈×C₃₁	—	2	2	no	—
132	C₄×Q₈×C₃₁	D₄	1	2	yes	—
133	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
134	C₄×D₄×C₃₁	—	2	4	no	—
135	C₄×D₄×C₃₁	—	2	2	no	—
136	C₄×D₄×C₃₁	—	2	2	no	—
137	C₄×D₄×C₃₁	—	2	4	no	—
138	C₄×D₄×C₃₁	D₄	1	4	yes	—
139	C₄×D₄×C₃₁	D₄	1	2	yes	—
140	C₄×D₄×C₃₁	C₂³	1	4	yes	—
141	C₃₁×#117	—	2	2	no	—
142	C₃₁×#117	—	2	4	no	—
143	C₃₁×#117	C₂³	1	1	yes	characteristic
144	C₃₁×#117	C₂³	1	1	yes	characteristic
145	C₃₁×#117	C₂³	1	1	yes	characteristic
146	C₃₁×#117	C₂³	1	2	yes	—
147	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
148	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
149	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
150	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
151	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
152	C₃₁×#103	C₂³	1	4	yes	—
153	C₃₁×#103	C₂³	1	4	yes	—
154	C₃₁×#102	C₂³	1	4	yes	—
155	C₃₁×#102	C₂³	1	4	yes	—
156	C₃₁×#102	C₂³	1	8	yes	—
157	C₃₁×#102	C₂³	1	8	yes	—
158	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
159	C₃₁×#97	C₂³	1	2	yes	—
160	C₃₁×#97	C₂³	1	2	yes	—
161	C₃₁×#97	C₂³	1	2	yes	—
162	C₄×C₈×C₃₁	C₂³	1	2	yes	—
163	C₂⁴×C₃₁	—	2	1	no	—
164	C₂⁴×C₃₁	—	2	4	no	—
165	C₂⁴×C₃₁	—	2	2	no	—
166	#41×C₃₁	—	2	4	no	—
167	#41×C₃₁	—	2	4	no	—
168	#41×C₃₁	—	2	4	no	—
169	#41×C₃₁	—	4	4	no	—
170	#41×C₃₁	—	4	4	no	—
171	#41×C₃₁	—	4	8	no	—
172	C₂×Q₈×C₃₁	—	2	1	no	—
173	C₂×Q₈×C₃₁	—	2	2	no	—
174	C₂×Q₈×C₃₁	—	2	4	no	—
175	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
177	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
178	C₂×D₄×C₃₁	—	2	4	no	—
179	C₂×D₄×C₃₁	—	2	2	no	—
180	C₂×D₄×C₃₁	—	2	4	no	—
181	C₂×D₄×C₃₁	—	2	1	no	—
182	C₂×D₄×C₃₁	—	2	2	no	—
183	C₂×D₄×C₃₁	—	2	4	no	—
184	C₂×D₄×C₃₁	—	2	4	no	—
185	C₂×D₄×C₃₁	—	2	2	no	—
186	C₂×D₄×C₃₁	—	2	8	no	—
187	C₂×D₄×C₃₁	—	2	4	no	—
188	C₂×D₄×C₃₁	—	2	4	no	—
189	C₂×D₄×C₃₁	—	2	4	no	—
190	C₂×D₄×C₃₁	—	2	4	no	—
191	C₂×D₄×C₃₁	—	4	4	no	—
192	C₂×D₄×C₃₁	—	4	4	no	—
193	C₂×D₄×C₃₁	—	4	16	no	—
194	C₂×D₄×C₃₁	—	4	8	no	—
195	C₂×D₄×C₃₁	—	8	4	no	—
196	C₂×D₄×C₃₁	—	8	8	no	—
197	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
198	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
199	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
200	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
201	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
202	C₂²×C₄×C₃₁	—	2	1	no	—
203	C₂²×C₄×C₃₁	—	2	4	no	—
204	C₂²×C₄×C₃₁	—	2	2	no	—
205	C₂²×C₄×C₃₁	—	2	4	no	—
206	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
207	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
208	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
209	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
210	Q₁₆×C₃₁	—	4	4	no	—
211	SD₁₆×C₃₁	—	4	8	no	—
212	SD₁₆×C₃₁	—	4	4	no	—
213	SD₁₆×C₃₁	—	4	4	no	—
214	D₈×C₃₁	—	4	8	no	—
215	D₈×C₃₁	—	4	4	no	—
216	M₁₆×C₃₁	—	2	4	no	—
217	C₂×C₈×C₃₁	—	2	4	no	—
218	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
219	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
220	C₄⋊C₄×C₃₁	—	2	2	no	—
221	C₄⋊C₄×C₃₁	—	2	4	no	—
222	C₄⋊C₄×C₃₁	—	2	4	no	—
223	C₄⋊C₄×C₃₁	—	2	2	no	—
224	C₄⋊C₄×C₃₁	—	2	2	no	—
225	C₄⋊C₄×C₃₁	—	2	2	no	—
226	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
227	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
228	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
229	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
230	C₂²⋊C₄×C₃₁	—	2	1	no	—
231	C₂²⋊C₄×C₃₁	—	2	1	no	—
232	C₂²⋊C₄×C₃₁	—	2	4	no	—
233	C₂²⋊C₄×C₃₁	—	2	4	no	—
234	C₂²⋊C₄×C₃₁	—	2	4	no	—
235	C₂²⋊C₄×C₃₁	—	2	4	no	—
236	C₂²⋊C₄×C₃₁	—	2	4	no	—
237	C₂²⋊C₄×C₃₁	—	2	8	no	—
238	C₂²⋊C₄×C₃₁	—	2	4	no	—
239	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
240	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
241	C₄²×C₃₁	—	2	4	no	—
242	C₄²×C₃₁	—	2	1	no	—
243	C₄²×C₃₁	—	2	1	no	—
244	C₄²×C₃₁	C₂×D₄	1	4	yes	—
245	C₄²×C₃₁	C₂×D₄	1	2	yes	—
246	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
247	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
248	C₄²×C₃₁	C₂⁴	1	2	yes	—
249	#60827	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
250	C₂³×C₃₁	—	2	2	no	—
251	C₂³×C₃₁	—	2	4	no	—
252	C₂³×C₃₁	—	2	2	no	—
253	C₂³×C₃₁	—	4	8	no	—
254	C₂³×C₃₁	—	4	4	no	—
255	C₂³×C₃₁	—	4	2	no	—
256	C₂³×C₃₁	—	8	4	no	—
257	C₂³×C₃₁	—	8	8	no	—
258	C₂³×C₃₁	—	8	2	no	—
259	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
260	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
261	Q₈×C₃₁	—	2	2	no	—
262	Q₈×C₃₁	—	2	4	no	—
263	Q₈×C₃₁	—	2	2	no	—
264	Q₈×C₃₁	—	2	2	no	—
265	Q₈×C₃₁	—	4	2	no	—
266	Q₈×C₃₁	—	8	2	no	—
267	D₄×C₃₁	—	2	4	no	—
268	D₄×C₃₁	—	2	2	no	—
269	D₄×C₃₁	—	2	4	no	—
270	D₄×C₃₁	—	2	4	no	—
271	D₄×C₃₁	—	2	2	no	—
272	D₄×C₃₁	—	2	4	no	—
273	D₄×C₃₁	—	2	2	no	—
274	D₄×C₃₁	—	4	8	no	—
275	D₄×C₃₁	—	4	4	no	—
276	D₄×C₃₁	—	4	8	no	—
277	D₄×C₃₁	—	4	8	no	—
278	D₄×C₃₁	—	4	4	no	—
279	D₄×C₃₁	—	4	2	no	—
280	D₄×C₃₁	—	4	4	no	—
281	D₄×C₃₁	—	8	8	no	—
282	D₄×C₃₁	—	8	4	no	—
283	D₄×C₃₁	—	8	4	no	—
284	C₂×C₄×C₃₁	—	2	4	no	—
285	C₂×C₄×C₃₁	—	2	2	no	—
286	C₂×C₄×C₃₁	—	2	2	no	—
287	C₂×C₄×C₃₁	—	2	2	no	—
288	C₂×C₄×C₃₁	—	2	2	no	—
289	C₂×C₄×C₃₁	—	4	4	no	—
290	C₂×C₄×C₃₁	—	4	2	no	—
291	C₂×C₄×C₃₁	—	4	8	no	—
292	C₂×C₄×C₃₁	—	4	2	no	—
293	C₂×C₄×C₃₁	—	4	4	no	—
294	C₂×C₄×C₃₁	—	8	4	no	—
295	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
296	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
297	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
298	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
299	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
300	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
301	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
302	C₈×C₃₁	—	2	4	no	—
303	C₈×C₃₁	—	2	4	no	—
304	#3510	C₂×C₃₁	1	1	yes	characteristic, complemented
305	#3498	C₂×C₃₁	1	1	yes	characteristic, complemented
306	#3224	C₂×C₃₁	1	8	yes	complemented
307	#3219	C₂×C₃₁	1	4	yes	complemented
308	#3217	C₂×C₃₁	1	4	yes	complemented
309	#3153	C₂×C₃₁	1	1	yes	characteristic, complemented
310	C₂×#496	C₂×C₃₁	1	1	yes	characteristic, complemented
311	#3138	C₂×C₃₁	1	1	yes	characteristic, complemented
312	#3135	C₂×C₃₁	1	1	yes	characteristic, complemented
313	#3043	C₂×C₃₁	1	2	yes	complemented
314	#3041	C₂×C₃₁	1	2	yes	complemented
315	#3009	C₂×C₃₁	1	2	yes	complemented
316	#3008	C₂×C₃₁	1	2	yes	complemented
317	#2923	C₂×C₃₁	1	1	yes	characteristic, complemented
318	C₂²×C₃₁	—	2	2	no	—
319	C₂²×C₃₁	—	2	4	no	—
320	C₂²×C₃₁	—	2	2	no	—
321	C₂²×C₃₁	—	4	4	no	—
322	C₂²×C₃₁	—	4	8	no	—
323	C₂²×C₃₁	—	4	4	no	—
324	C₂²×C₃₁	—	4	4	no	—
325	C₂²×C₃₁	—	4	2	no	—
326	C₂²×C₃₁	—	4	2	no	—
327	C₂²×C₃₁	—	16	2	no	—
328	C₂²×C₃₁	—	16	4	no	—
329	C₂²×C₃₁	—	16	8	no	—
330	C₂²×C₃₁	#535	1	2	yes	—
331	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
332	C₄×C₃₁	—	2	2	no	—
333	C₄×C₃₁	—	2	4	no	—
334	C₄×C₃₁	—	2	2	no	—
335	C₄×C₃₁	—	2	2	no	—
336	C₄×C₃₁	—	4	2	no	—
337	C₄×C₃₁	—	4	2	no	—
338	C₄×C₃₁	—	4	2	no	—
339	C₄×C₃₁	—	4	2	no	—
340	C₄×C₃₁	#535	1	2	yes	—
341	C₄×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
342	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	2	yes	—
343	C₂×#137	C₂²×C₃₁	1	2	yes	complemented
344	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	complemented
345	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
346	C₂²×D₈	C₂²×C₃₁	1	2	yes	complemented
347	#559	—	2	2	no	—
348	#556	—	2	2	no	—
349	#551	—	2	4	no	—
350	#550	—	2	2	no	—
351	D₄²	—	2	2	no	—
352	D₄²	—	2	4	no	—
353	#542	C₂²×C₃₁	1	4	yes	complemented
354	#536	C₂²×C₃₁	1	1	yes	characteristic
355	#535	C₂²×C₃₁	1	2	yes	—
356	#535	C₂²×C₃₁	1	1	yes	characteristic
357	#535	C₂²×C₃₁	1	4	yes	complemented
358	#533	C₂²×C₃₁	1	1	yes	characteristic, complemented
359	#532	C₂²×C₃₁	1	1	yes	characteristic
360	#532	C₂²×C₃₁	1	2	yes	complemented
361	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
362	C₂×#127	C₂²×C₃₁	1	2	yes	complemented
363	#529	C₂²×C₃₁	1	4	yes	complemented
364	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic, complemented
365	#519	C₂²×C₃₁	1	1	yes	characteristic, complemented
366	#518	C₂²×C₃₁	1	1	yes	characteristic, complemented
367	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
368	#497	C₂²×C₃₁	1	2	yes	complemented
369	#496	C₂²×C₃₁	1	4	yes	complemented
370	#496	C₂²×C₃₁	1	4	yes	complemented
371	#496	C₂²×C₃₁	1	2	yes	complemented
372	#495	C₂²×C₃₁	1	4	yes	complemented
373	#491	C₂²×C₃₁	1	2	yes	complemented
374	#490	C₂²×C₃₁	1	2	yes	complemented
375	#490	C₂²×C₃₁	1	4	yes	complemented
376	#489	C₂²×C₃₁	1	4	yes	complemented
377	#486	C₂²×C₃₁	1	4	yes	complemented
378	#464	C₂²×C₃₁	1	8	yes	complemented
379	#463	C₂²×C₃₁	1	8	yes	complemented
380	#460	C₂²×C₃₁	1	8	yes	complemented
381	#459	C₂²×C₃₁	1	8	yes	complemented
382	#452	C₂²×C₃₁	1	8	yes	complemented
383	#450	C₂²×C₃₁	1	8	yes	complemented
384	#449	C₂²×C₃₁	1	8	yes	complemented
385	#447	C₂²×C₃₁	1	8	yes	complemented
386	#433	C₂²×C₃₁	1	4	yes	complemented
387	#432	C₂²×C₃₁	1	4	yes	complemented
388	#424	C₂²×C₃₁	1	2	yes	complemented
389	#418	C₂²×C₃₁	1	2	yes	complemented
390	#418	C₂²×C₃₁	1	2	yes	complemented
391	#416	C₂²×C₃₁	1	2	yes	complemented
392	#416	C₂²×C₃₁	1	2	yes	complemented
393	#408	C₂²×C₃₁	1	2	yes	—
394	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
395	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic, complemented
396	C₂×C₃₁	—	2	2	no	—
397	C₂×C₃₁	—	4	2	no	—
398	C₂×C₃₁	—	8	2	no	—
399	C₂×C₃₁	—	8	4	no	—
400	C₂×C₃₁	—	8	2	no	—
401	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
402	C₂×C₃₁	#3585	1	2	yes	—
403	H(5,𝔽₂)	—	4	4	no	—
404	H(5,𝔽₂)	—	4	4	no	—
405	C₂×#41	—	2	2	no	—
406	C₂×#41	C₂³×C₃₁	1	2	yes	—
407	C₂×#41	C₂³×C₃₁	1	4	yes	—
408	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
409	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
410	C₂²×D₄	—	2	4	no	—
411	C₂²×D₄	—	2	2	no	—
412	C₂²×D₄	—	2	2	no	—
413	C₂²×D₄	—	2	4	no	—
414	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
415	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
416	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
417	#137	—	4	4	no	—
418	AGL(1,ℤ/8)	—	4	4	no	—
419	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
420	C₂×SD₁₆	—	4	4	no	—
421	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
422	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
423	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
424	C₂×D₈	—	4	4	no	—
425	C₂×D₈	C₂³×C₃₁	1	8	yes	—
426	C₂×D₈	C₂³×C₃₁	1	4	yes	—
427	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
428	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
429	C₄⋊Q₈	—	2	1	no	—
430	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
431	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
432	#127	—	2	1	no	—
433	#127	—	2	2	no	—
434	#127	—	2	4	no	—
435	#127	C₂³×C₃₁	1	2	yes	—
436	#127	C₂³×C₃₁	1	2	yes	—
437	#127	C₂³×C₃₁	1	1	yes	characteristic
438	#127	C₂³×C₃₁	1	4	yes	—
439	#126	—	2	4	no	—
440	#126	—	2	4	no	—
441	#125	C₂³×C₃₁	1	2	yes	—
442	#124	—	2	1	no	—
443	#124	—	2	1	no	—
444	#124	—	2	4	no	—
445	#124	—	2	2	no	—
446	#124	C₂³×C₃₁	1	2	yes	—
447	#124	C₂³×C₃₁	1	2	yes	—
448	#124	C₂³×C₃₁	1	2	yes	—
449	#124	C₂³×C₃₁	1	4	yes	—
450	#123	—	2	2	no	—
451	#123	—	2	4	no	—
452	#123	—	2	4	no	—
453	C₂²⋊Q₈	—	2	2	no	—
454	C₂²⋊Q₈	—	2	4	no	—
455	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
456	#121	—	2	2	no	—
457	#121	—	2	4	no	—
458	#121	—	2	4	no	—
459	#121	—	2	8	no	—
460	#121	C₂³×C₃₁	1	4	yes	—
461	#120	—	2	2	no	—
462	#120	—	2	4	no	—
463	#120	—	2	8	no	—
464	C₄×Q₈	—	2	2	no	—
465	C₄×Q₈	—	2	2	no	—
466	C₄×Q₈	D₄×C₃₁	1	2	yes	—
467	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
468	C₄×D₄	—	2	4	no	—
469	C₄×D₄	—	2	2	no	—
470	C₄×D₄	—	2	2	no	—
471	C₄×D₄	—	2	4	no	—
472	C₄×D₄	D₄×C₃₁	1	4	yes	—
473	C₄×D₄	D₄×C₃₁	1	2	yes	—
474	C₄×D₄	C₂³×C₃₁	1	4	yes	—
475	#117	—	2	2	no	—
476	#117	—	2	4	no	—
477	#117	C₂³×C₃₁	1	1	yes	characteristic
478	#117	C₂³×C₃₁	1	1	yes	characteristic
479	#117	C₂³×C₃₁	1	1	yes	characteristic
480	#117	C₂³×C₃₁	1	2	yes	—
481	C₂×C₂²⋊C₄	—	2	2	no	—
482	C₂×C₂²⋊C₄	—	2	2	no	—
483	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
484	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
485	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
486	#103	C₂³×C₃₁	1	4	yes	—
487	#103	C₂³×C₃₁	1	4	yes	—
488	#102	C₂³×C₃₁	1	4	yes	—
489	#102	C₂³×C₃₁	1	4	yes	—
490	#102	C₂³×C₃₁	1	8	yes	—
491	#102	C₂³×C₃₁	1	8	yes	—
492	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
493	#97	C₂³×C₃₁	1	2	yes	—
494	#97	C₂³×C₃₁	1	2	yes	—
495	#97	C₂³×C₃₁	1	2	yes	—
496	C₄×C₈	C₂³×C₃₁	1	2	yes	—
497	C₃₁	#60827	1	1	yes	characteristic, direct product factor, Sylow
498	C₂⁴	—	2	1	no	—
499	C₂⁴	—	2	4	no	—
500	C₂⁴	—	2	2	no	—
501	#41	—	2	4	no	—
502	#41	—	2	4	no	—
503	#41	—	2	4	no	—
504	#41	—	4	4	no	—
505	#41	—	4	4	no	—
506	#41	—	4	8	no	—
507	C₂×Q₈	—	2	1	no	—
508	C₂×Q₈	—	2	2	no	—
509	C₂×Q₈	—	2	4	no	—
510	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
511	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
512	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
513	C₂×D₄	—	2	4	no	—
514	C₂×D₄	—	2	2	no	—
515	C₂×D₄	—	2	4	no	—
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	2	2	no	—
518	C₂×D₄	—	2	4	no	—
519	C₂×D₄	—	2	4	no	—
520	C₂×D₄	—	2	2	no	—
521	C₂×D₄	—	2	8	no	—
522	C₂×D₄	—	2	4	no	—
523	C₂×D₄	—	2	4	no	—
524	C₂×D₄	—	2	4	no	—
525	C₂×D₄	—	2	4	no	—
526	C₂×D₄	—	4	4	no	—
527	C₂×D₄	—	4	4	no	—
528	C₂×D₄	—	4	16	no	—
529	C₂×D₄	—	4	8	no	—
530	C₂×D₄	—	8	4	no	—
531	C₂×D₄	—	8	8	no	—
532	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
533	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
534	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
535	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
536	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
537	C₂²×C₄	—	2	1	no	—
538	C₂²×C₄	—	2	4	no	—
539	C₂²×C₄	—	2	2	no	—
540	C₂²×C₄	—	2	4	no	—
541	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
542	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
543	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
544	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
545	Q₁₆	—	4	4	no	—
546	SD₁₆	—	4	8	no	—
547	SD₁₆	—	4	4	no	—
548	SD₁₆	—	4	4	no	—
549	D₈	—	4	8	no	—
550	D₈	—	4	4	no	—
551	M₁₆	—	2	4	no	—
552	C₂×C₈	—	2	4	no	—
553	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
554	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
555	C₄⋊C₄	—	2	2	no	—
556	C₄⋊C₄	—	2	4	no	—
557	C₄⋊C₄	—	2	4	no	—
558	C₄⋊C₄	—	2	2	no	—
559	C₄⋊C₄	—	2	2	no	—
560	C₄⋊C₄	—	2	2	no	—
561	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
562	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
563	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
564	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
565	C₂²⋊C₄	—	2	1	no	—
566	C₂²⋊C₄	—	2	1	no	—
567	C₂²⋊C₄	—	2	4	no	—
568	C₂²⋊C₄	—	2	4	no	—
569	C₂²⋊C₄	—	2	4	no	—
570	C₂²⋊C₄	—	2	4	no	—
571	C₂²⋊C₄	—	2	4	no	—
572	C₂²⋊C₄	—	2	8	no	—
573	C₂²⋊C₄	—	2	4	no	—
574	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
575	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
576	C₄²	—	2	4	no	—
577	C₄²	—	2	1	no	—
578	C₄²	—	2	1	no	—
579	C₄²	C₂×D₄×C₃₁	1	4	yes	—
580	C₄²	C₂×D₄×C₃₁	1	2	yes	—
581	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
582	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
583	C₄²	C₂⁴×C₃₁	1	2	yes	—
584	C₂³	—	2	2	no	—
585	C₂³	—	2	4	no	—
586	C₂³	—	2	2	no	—
587	C₂³	—	4	8	no	—
588	C₂³	—	4	4	no	—
589	C₂³	—	4	2	no	—
590	C₂³	—	8	4	no	—
591	C₂³	—	8	8	no	—
592	C₂³	—	8	2	no	—
593	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
594	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
595	Q₈	—	2	2	no	—
596	Q₈	—	2	4	no	—
597	Q₈	—	2	2	no	—
598	Q₈	—	2	2	no	—
599	Q₈	—	4	2	no	—
600	Q₈	—	8	2	no	—
601	D₄	—	2	4	no	—
602	D₄	—	2	2	no	—
603	D₄	—	2	4	no	—
604	D₄	—	2	4	no	—
605	D₄	—	2	2	no	—
606	D₄	—	2	4	no	—
607	D₄	—	2	2	no	—
608	D₄	—	4	8	no	—
609	D₄	—	4	4	no	—
610	D₄	—	4	8	no	—
611	D₄	—	4	8	no	—
612	D₄	—	4	4	no	—
613	D₄	—	4	2	no	—
614	D₄	—	4	4	no	—
615	D₄	—	8	8	no	—
616	D₄	—	8	4	no	—
617	D₄	—	8	4	no	—
618	C₂×C₄	—	2	4	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	4	4	no	—
624	C₂×C₄	—	4	2	no	—
625	C₂×C₄	—	4	8	no	—
626	C₂×C₄	—	4	2	no	—
627	C₂×C₄	—	4	4	no	—
628	C₂×C₄	—	8	4	no	—
629	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
630	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
631	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
632	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
633	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
634	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
635	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
636	C₈	—	2	4	no	—
637	C₈	—	2	4	no	—
638	C₂²	—	2	2	no	—
639	C₂²	—	2	2	no	—
640	C₂²	—	2	4	no	—
641	C₂²	—	4	4	no	—
642	C₂²	—	4	2	no	—
643	C₂²	—	4	2	no	—
644	C₂²	—	4	4	no	—
645	C₂²	—	4	8	no	—
646	C₂²	—	4	4	no	—
647	C₂²	—	16	4	no	—
648	C₂²	—	16	2	no	—
649	C₂²	—	16	8	no	—
650	C₂²	C₃₁×#535	1	2	yes	—
651	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
652	C₄	—	2	2	no	—
653	C₄	—	2	2	no	—
654	C₄	—	2	4	no	—
655	C₄	—	2	2	no	—
656	C₄	—	4	2	no	—
657	C₄	—	4	2	no	—
658	C₄	—	4	2	no	—
659	C₄	—	4	2	no	—
660	C₄	C₃₁×#535	1	2	yes	—
661	C₄	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
662	C₂	—	2	2	no	—
663	C₂	—	4	2	no	—
664	C₂	—	8	4	no	—
665	C₂	—	8	2	no	—
666	C₂	—	8	2	no	—
667	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
668	C₂	C₃₁×#3585	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 ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3510	C₂	semidirect
2. C₃₁×#3498	C₂	semidirect
3. C₃₁×#3224	C₂	semidirect
4. C₃₁×#3219	C₂	semidirect
5. C₃₁×#3217	C₂	semidirect
6. C₃₁×#3153	C₂	semidirect
7. C₂×C₃₁×#496	C₂	semidirect
8. C₃₁×#3138	C₂	semidirect
9. C₃₁×#3135	C₂	semidirect
10. C₃₁×#3043	C₂	semidirect
11. C₃₁×#3041	C₂	semidirect
12. C₃₁×#3009	C₂	semidirect
13. C₃₁×#3008	C₂	semidirect
14. C₃₁×#2923	C₂	semidirect
16. C₂×C₃₁×#137	C₂²	semidirect
17. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
18. C₂²×SD₁₆×C₃₁	C₂²	semidirect
19. C₂²×D₈×C₃₁	C₂²	semidirect
26. C₃₁×#542	C₂²	semidirect
30. C₃₁×#535	C₂²	semidirect
31. C₃₁×#533	C₂²	semidirect
33. C₃₁×#532	C₂²	semidirect
35. C₂×C₃₁×#127	C₂²	semidirect
36. C₃₁×#529	C₂²	semidirect
37. C₂×C₃₁×#124	C₂²	semidirect
38. C₃₁×#519	C₂²	semidirect
39. C₃₁×#518	C₂²	semidirect
41. C₃₁×#497	C₂²	semidirect
42. C₃₁×#496	C₂²	semidirect
43. C₃₁×#496	C₂²	semidirect
44. C₃₁×#496	C₂²	semidirect
45. C₃₁×#495	C₂²	semidirect
46. C₃₁×#491	C₂²	semidirect
47. C₃₁×#490	C₂²	semidirect
48. C₃₁×#490	C₂²	semidirect
49. C₃₁×#489	C₂²	semidirect
50. C₃₁×#486	C₂²	semidirect
51. C₃₁×#464	C₂²	semidirect
52. C₃₁×#463	C₂²	semidirect
53. C₃₁×#460	C₂²	semidirect
54. C₃₁×#459	C₂²	semidirect
55. C₃₁×#452	C₂²	semidirect
56. C₃₁×#450	C₂²	semidirect
57. C₃₁×#449	C₂²	semidirect
58. C₃₁×#447	C₂²	semidirect
59. C₃₁×#433	C₂²	semidirect
60. C₃₁×#432	C₂²	semidirect
61. C₃₁×#424	C₂²	semidirect
62. C₃₁×#418	C₂²	semidirect
63. C₃₁×#418	C₂²	semidirect
64. C₃₁×#416	C₂²	semidirect
65. C₃₁×#416	C₂²	semidirect
67. C₄×M₁₆×C₃₁	C₂²	semidirect
68. C₂×C₃₁×#97	C₂²	semidirect
249. #60827	C₃₁	direct (N × H)
304. #3510	C₂×C₃₁	semidirect
305. #3498	C₂×C₃₁	semidirect
306. #3224	C₂×C₃₁	semidirect
307. #3219	C₂×C₃₁	semidirect
308. #3217	C₂×C₃₁	semidirect
309. #3153	C₂×C₃₁	semidirect
310. C₂×#496	C₂×C₃₁	semidirect
311. #3138	C₂×C₃₁	semidirect
312. #3135	C₂×C₃₁	semidirect
313. #3043	C₂×C₃₁	semidirect
314. #3041	C₂×C₃₁	semidirect
315. #3009	C₂×C₃₁	semidirect
316. #3008	C₂×C₃₁	semidirect
317. #2923	C₂×C₃₁	semidirect
343. C₂×#137	C₂²×C₃₁	semidirect
344. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
345. C₂²×SD₁₆	C₂²×C₃₁	semidirect
346. C₂²×D₈	C₂²×C₃₁	semidirect
353. #542	C₂²×C₃₁	semidirect
357. #535	C₂²×C₃₁	semidirect
358. #533	C₂²×C₃₁	semidirect
360. #532	C₂²×C₃₁	semidirect
362. C₂×#127	C₂²×C₃₁	semidirect
363. #529	C₂²×C₃₁	semidirect
364. C₂×#124	C₂²×C₃₁	semidirect
365. #519	C₂²×C₃₁	semidirect
366. #518	C₂²×C₃₁	semidirect
368. #497	C₂²×C₃₁	semidirect
369. #496	C₂²×C₃₁	semidirect
370. #496	C₂²×C₃₁	semidirect
371. #496	C₂²×C₃₁	semidirect
372. #495	C₂²×C₃₁	semidirect
373. #491	C₂²×C₃₁	semidirect
374. #490	C₂²×C₃₁	semidirect
375. #490	C₂²×C₃₁	semidirect
376. #489	C₂²×C₃₁	semidirect
377. #486	C₂²×C₃₁	semidirect
378. #464	C₂²×C₃₁	semidirect
379. #463	C₂²×C₃₁	semidirect
380. #460	C₂²×C₃₁	semidirect
381. #459	C₂²×C₃₁	semidirect
382. #452	C₂²×C₃₁	semidirect
383. #450	C₂²×C₃₁	semidirect
384. #449	C₂²×C₃₁	semidirect
385. #447	C₂²×C₃₁	semidirect
386. #433	C₂²×C₃₁	semidirect
387. #432	C₂²×C₃₁	semidirect
388. #424	C₂²×C₃₁	semidirect
389. #418	C₂²×C₃₁	semidirect
390. #418	C₂²×C₃₁	semidirect
391. #416	C₂²×C₃₁	semidirect
392. #416	C₂²×C₃₁	semidirect
394. C₄×M₁₆	C₂²×C₃₁	semidirect
395. C₂×#97	C₂²×C₃₁	semidirect
497. C₃₁	#60827	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×#3510	2
C₃₁×#3498	2
C₃₁×#3224	2
C₃₁×#3219	2
C₃₁×#3217	2
C₃₁×#3153	2
C₂×C₃₁×#496	2
C₃₁×#3138	2
C₃₁×#3135	2
C₃₁×#3043	2
C₃₁×#3041	2
C₃₁×#3009	2
C₃₁×#3008	2
C₃₁×#2923	2
#60827	31

Subgroups — C31×#60827 #21517586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60827 #21517586