Subgroups — C₃₁×#60469 #21517228

Order 7936.

2628 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#575	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#575	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3069	C₂	1	8	yes	maximal, complemented
5	C₃₁×#3068	C₂	1	8	yes	maximal, complemented
6	C₃₁×#3067	C₂	1	8	yes	maximal, complemented
7	C₂×C₃₁×#430	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#413	C₂	1	2	yes	maximal, complemented
9	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×#41×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#575	—	2	4	no	—
12	C₃₁×#575	—	2	4	no	—
13	C₃₁×#575	—	2	2	no	—
14	C₃₁×#575	—	2	2	no	—
15	C₃₁×#575	—	2	4	no	—
16	C₃₁×#575	C₂²	1	8	yes	—
17	C₃₁×#575	C₂²	1	8	yes	complemented
18	C₂×C₃₁×#137	—	2	2	no	—
19	C₂×C₃₁×#137	C₂²	1	4	yes	complemented
20	C₂×C₃₁×#137	C₂²	1	4	yes	complemented
21	C₂×C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
22	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
23	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
24	C₂×C₃₁×#135	—	2	4	no	—
25	C₂×C₃₁×#135	C₂²	1	4	yes	complemented
26	C₃₁×#568	—	2	4	no	—
27	C₃₁×#568	C₂²	1	8	yes	complemented
28	C₂×C₃₁×#131	—	2	2	no	—
29	C₂×C₃₁×#131	C₂²	1	4	yes	complemented
30	C₂²×M₁₆×C₃₁	—	2	2	no	—
31	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
32	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
33	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#473	C₂²	1	16	yes	complemented
35	C₃₁×#472	C₂²	1	16	yes	complemented
36	C₃₁×#471	C₂²	1	32	yes	complemented
37	C₃₁×#430	C₂²	1	8	yes	complemented
38	C₂×C₃₁×#108	C₂²	1	4	yes	complemented
39	C₃₁×#413	C₂²	1	8	yes	complemented
40	C₃₁×#413	C₂²	1	8	yes	complemented
41	C₂×C₃₁×#101	C₂²	1	4	yes	complemented
42	C₂×C₃₁×#100	C₂²	1	4	yes	complemented
43	C₂×#41×C₃₁	—	2	1	no	—
44	C₂×#41×C₃₁	—	2	2	no	—
45	C₂×#41×C₃₁	—	2	2	no	—
46	C₂×#41×C₃₁	—	2	1	no	—
47	C₂×#41×C₃₁	—	2	2	no	—
48	C₂×#41×C₃₁	—	2	2	no	—
49	C₂×#41×C₃₁	—	2	2	no	—
50	C₂×#41×C₃₁	—	2	2	no	—
51	C₂×#41×C₃₁	—	2	2	no	—
52	C₂×#41×C₃₁	—	2	2	no	—
53	C₂×#41×C₃₁	—	2	2	no	—
54	C₂×#41×C₃₁	—	2	2	no	—
55	C₂×#41×C₃₁	—	2	2	no	—
56	C₂×#41×C₃₁	—	2	2	no	—
57	C₂×#41×C₃₁	D₄	1	4	yes	—
58	C₂×#41×C₃₁	D₄	1	2	yes	—
59	C₂×#41×C₃₁	C₂³	1	2	yes	—
60	C₂×#41×C₃₁	C₂³	1	4	yes	—
61	C₂×#41×C₃₁	C₂³	1	4	yes	—
62	C₂×#41×C₃₁	C₂³	1	4	yes	—
63	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
64	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
65	C₂²×D₄×C₃₁	—	2	1	no	—
66	C₂²×D₄×C₃₁	—	2	2	no	—
67	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
68	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
69	C₂³×C₄×C₃₁	—	2	1	no	—
70	C₂³×C₄×C₃₁	—	2	2	no	—
71	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×#137	—	2	8	no	—
73	C₃₁×#137	—	2	8	no	—
74	C₃₁×#137	—	2	8	no	—
75	C₃₁×#137	—	2	8	no	—
76	C₃₁×#137	—	2	8	no	—
77	C₃₁×#137	C₂³	1	16	yes	—
78	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
79	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
80	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
81	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
82	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
83	C₃₁×AGL(1,ℤ/8)	C₂³	1	16	yes	—
84	C₃₁×#135	—	2	16	no	—
85	C₃₁×#135	—	2	16	no	—
86	C₃₁×#135	—	2	8	no	—
87	C₃₁×#135	—	2	8	no	—
88	C₂×Q₁₆×C₃₁	—	2	4	no	—
89	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	complemented
90	C₂×SD₁₆×C₃₁	—	2	4	no	—
91	C₂×SD₁₆×C₃₁	—	2	4	no	—
92	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	complemented
93	C₂×D₈×C₃₁	—	2	4	no	—
94	C₂×D₈×C₃₁	C₂³	1	4	yes	complemented
95	C₃₁×#131	—	2	16	no	—
96	C₃₁×#131	—	2	8	no	—
97	C₃₁×#131	C₂³	1	16	yes	—
98	C₂×M₁₆×C₃₁	—	2	1	no	—
99	C₂×M₁₆×C₃₁	—	2	2	no	—
100	C₂×M₁₆×C₃₁	—	2	2	no	—
101	C₂×M₁₆×C₃₁	—	2	2	no	—
102	C₂×M₁₆×C₃₁	—	2	2	no	—
103	C₂×M₁₆×C₃₁	—	2	16	no	—
104	C₂×M₁₆×C₃₁	—	2	4	no	—
105	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
106	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
107	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
108	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
109	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
110	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
111	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
112	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
113	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
114	C₂²×C₈×C₃₁	—	2	1	no	—
115	C₂²×C₈×C₃₁	—	2	2	no	—
116	C₂²×C₈×C₃₁	—	2	4	no	—
117	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
118	C₃₁×#108	C₂³	1	16	yes	complemented
119	C₃₁×#101	C₂³	1	16	yes	complemented
120	C₃₁×#100	C₂³	1	16	yes	complemented
121	C₂⁴×C₃₁	—	2	1	no	—
122	C₂⁴×C₃₁	—	2	2	no	—
123	C₂⁴×C₃₁	#41	1	1	yes	characteristic
124	#41×C₃₁	—	2	4	no	—
125	#41×C₃₁	—	2	4	no	—
126	#41×C₃₁	—	2	4	no	—
127	#41×C₃₁	—	2	4	no	—
128	#41×C₃₁	—	2	2	no	—
129	#41×C₃₁	—	2	4	no	—
130	#41×C₃₁	—	2	2	no	—
131	#41×C₃₁	—	2	4	no	—
132	#41×C₃₁	—	2	4	no	—
133	#41×C₃₁	—	2	4	no	—
134	#41×C₃₁	—	2	4	no	—
135	#41×C₃₁	—	2	4	no	—
136	#41×C₃₁	—	2	4	no	—
137	#41×C₃₁	—	2	4	no	—
138	#41×C₃₁	—	2	4	no	—
139	#41×C₃₁	—	4	2	no	—
140	#41×C₃₁	—	4	2	no	—
141	#41×C₃₁	—	4	2	no	—
142	#41×C₃₁	—	4	2	no	—
143	#41×C₃₁	—	4	2	no	—
144	#41×C₃₁	—	4	2	no	—
145	#41×C₃₁	—	4	2	no	—
146	#41×C₃₁	—	4	2	no	—
147	#41×C₃₁	C₂×D₄	1	4	yes	—
148	#41×C₃₁	C₂×D₄	1	8	yes	—
149	#41×C₃₁	C₂⁴	1	4	yes	—
150	C₂×Q₈×C₃₁	—	2	2	no	—
151	C₂×Q₈×C₃₁	—	2	1	no	—
152	C₂×Q₈×C₃₁	—	2	2	no	—
153	C₂×Q₈×C₃₁	—	2	2	no	—
154	C₂×Q₈×C₃₁	—	2	2	no	—
155	C₂×Q₈×C₃₁	—	2	2	no	—
156	C₂×Q₈×C₃₁	—	2	2	no	—
157	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
158	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
159	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
160	C₂×D₄×C₃₁	—	2	2	no	—
161	C₂×D₄×C₃₁	—	2	4	no	—
162	C₂×D₄×C₃₁	—	2	2	no	—
163	C₂×D₄×C₃₁	—	2	4	no	—
164	C₂×D₄×C₃₁	—	2	4	no	—
165	C₂×D₄×C₃₁	—	2	4	no	—
166	C₂×D₄×C₃₁	—	2	2	no	—
167	C₂×D₄×C₃₁	—	2	1	no	—
168	C₂×D₄×C₃₁	—	2	2	no	—
169	C₂×D₄×C₃₁	—	2	2	no	—
170	C₂×D₄×C₃₁	—	2	2	no	—
171	C₂×D₄×C₃₁	—	2	2	no	—
172	C₂×D₄×C₃₁	—	2	2	no	—
173	C₂×D₄×C₃₁	—	4	2	no	—
174	C₂×D₄×C₃₁	—	4	2	no	—
175	C₂×D₄×C₃₁	—	4	2	no	—
176	C₂×D₄×C₃₁	—	4	2	no	—
177	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
178	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
179	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₃₁	—	2	1	no	—
181	C₂²×C₄×C₃₁	—	2	2	no	—
182	C₂²×C₄×C₃₁	—	2	4	no	—
183	C₂²×C₄×C₃₁	—	2	2	no	—
184	C₂²×C₄×C₃₁	—	2	4	no	—
185	C₂²×C₄×C₃₁	—	2	2	no	—
186	C₂²×C₄×C₃₁	—	2	4	no	—
187	C₂²×C₄×C₃₁	—	2	4	no	—
188	C₂²×C₄×C₃₁	—	4	2	no	—
189	C₂²×C₄×C₃₁	—	4	2	no	—
190	C₂²×C₄×C₃₁	—	4	2	no	—
191	C₂²×C₄×C₃₁	—	4	2	no	—
192	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
193	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
196	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
197	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
198	C₂²×C₄×C₃₁	#41	1	2	yes	—
199	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
200	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
201	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
202	Q₁₆×C₃₁	—	2	16	no	—
203	Q₁₆×C₃₁	—	2	8	no	—
204	SD₁₆×C₃₁	—	2	16	no	—
205	SD₁₆×C₃₁	—	2	16	no	—
206	SD₁₆×C₃₁	—	2	16	no	—
207	D₈×C₃₁	—	2	16	no	—
208	D₈×C₃₁	—	2	8	no	—
209	M₁₆×C₃₁	—	2	4	no	—
210	M₁₆×C₃₁	—	2	2	no	—
211	M₁₆×C₃₁	—	2	4	no	—
212	M₁₆×C₃₁	—	2	4	no	—
213	M₁₆×C₃₁	—	2	4	no	—
214	M₁₆×C₃₁	—	2	16	no	—
215	M₁₆×C₃₁	C₂×D₄	1	8	yes	—
216	M₁₆×C₃₁	C₂×D₄	1	8	yes	—
217	M₁₆×C₃₁	C₂⁴	1	4	yes	—
218	M₁₆×C₃₁	C₂⁴	1	8	yes	—
219	C₂×C₈×C₃₁	—	2	2	no	—
220	C₂×C₈×C₃₁	—	2	4	no	—
221	C₂×C₈×C₃₁	—	2	4	no	—
222	C₂×C₈×C₃₁	—	2	4	no	—
223	C₂×C₈×C₃₁	—	2	4	no	—
224	C₂×C₈×C₃₁	—	2	16	no	—
225	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
226	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
227	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
228	#60469	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂³×C₃₁	—	2	2	no	—
230	C₂³×C₃₁	—	2	4	no	—
231	C₂³×C₃₁	—	2	4	no	—
232	C₂³×C₃₁	—	4	2	no	—
233	C₂³×C₃₁	—	4	2	no	—
234	C₂³×C₃₁	—	8	1	no	—
235	C₂³×C₃₁	—	8	2	no	—
236	C₂³×C₃₁	—	8	4	no	—
237	C₂³×C₃₁	#121	1	1	yes	characteristic
238	C₂³×C₃₁	#121	1	2	yes	—
239	C₂³×C₃₁	#121	1	1	yes	characteristic
240	C₂³×C₃₁	C₂×#41	1	2	yes	—
241	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
242	Q₈×C₃₁	—	2	2	no	—
243	Q₈×C₃₁	—	2	4	no	—
244	Q₈×C₃₁	—	2	4	no	—
245	Q₈×C₃₁	—	2	4	no	—
246	Q₈×C₃₁	—	4	2	no	—
247	Q₈×C₃₁	—	4	2	no	—
248	Q₈×C₃₁	C₂²×D₄	1	4	yes	—
249	D₄×C₃₁	—	2	8	no	—
250	D₄×C₃₁	—	2	4	no	—
251	D₄×C₃₁	—	2	4	no	—
252	D₄×C₃₁	—	2	2	no	—
253	D₄×C₃₁	—	2	4	no	—
254	D₄×C₃₁	—	2	4	no	—
255	D₄×C₃₁	—	2	4	no	—
256	D₄×C₃₁	—	4	4	no	—
257	D₄×C₃₁	—	4	4	no	—
258	D₄×C₃₁	—	4	4	no	—
259	D₄×C₃₁	—	4	4	no	—
260	D₄×C₃₁	—	4	2	no	—
261	D₄×C₃₁	—	4	2	no	—
262	D₄×C₃₁	C₂²×D₄	1	4	yes	—
263	C₂×C₄×C₃₁	—	2	4	no	—
264	C₂×C₄×C₃₁	—	2	2	no	—
265	C₂×C₄×C₃₁	—	2	8	no	—
266	C₂×C₄×C₃₁	—	2	4	no	—
267	C₂×C₄×C₃₁	—	2	4	no	—
268	C₂×C₄×C₃₁	—	2	4	no	—
269	C₂×C₄×C₃₁	—	4	2	no	—
270	C₂×C₄×C₃₁	—	4	2	no	—
271	C₂×C₄×C₃₁	—	4	4	no	—
272	C₂×C₄×C₃₁	—	4	4	no	—
273	C₂×C₄×C₃₁	—	4	4	no	—
274	C₂×C₄×C₃₁	—	4	4	no	—
275	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
276	C₂×C₄×C₃₁	#121	1	2	yes	—
277	C₂×C₄×C₃₁	#121	1	2	yes	—
278	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
279	C₂×C₄×C₃₁	#121	1	2	yes	—
280	C₂×C₄×C₃₁	#121	1	2	yes	—
281	C₂×C₄×C₃₁	#121	1	2	yes	—
282	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
283	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
284	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
285	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
288	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
289	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
290	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
291	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
292	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
293	C₈×C₃₁	—	2	4	no	—
294	C₈×C₃₁	—	2	8	no	—
295	C₈×C₃₁	C₂²×D₄	1	8	yes	—
296	C₂×#575	C₂×C₃₁	1	2	yes	complemented
297	C₂×#575	C₂×C₃₁	1	1	yes	characteristic, complemented
298	C₂×#568	C₂×C₃₁	1	1	yes	characteristic, complemented
299	#3069	C₂×C₃₁	1	8	yes	complemented
300	#3068	C₂×C₃₁	1	8	yes	complemented
301	#3067	C₂×C₃₁	1	8	yes	complemented
302	C₂×#430	C₂×C₃₁	1	1	yes	characteristic, complemented
303	C₂×#413	C₂×C₃₁	1	2	yes	complemented
304	C₂²×C₃₁	—	2	4	no	—
305	C₂²×C₃₁	—	4	1	no	—
306	C₂²×C₃₁	—	4	2	no	—
307	C₂²×C₃₁	—	4	1	no	—
308	C₂²×C₃₁	—	4	2	no	—
309	C₂²×C₃₁	—	4	1	no	—
310	C₂²×C₃₁	—	4	4	no	—
311	C₂²×C₃₁	—	4	4	no	—
312	C₂²×C₃₁	—	8	4	no	—
313	C₂²×C₃₁	—	8	4	no	—
314	C₂²×C₃₁	—	8	4	no	—
315	C₂²×C₃₁	—	8	4	no	—
316	C₂²×C₃₁	C₂×#121	1	2	yes	—
317	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
318	C₂²×C₃₁	C₂×#121	1	2	yes	—
319	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
320	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
321	C₄×C₃₁	—	2	4	no	—
322	C₄×C₃₁	—	4	4	no	—
323	C₄×C₃₁	C₂×#121	1	2	yes	—
324	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
325	C₄×C₃₁	C₂×#121	1	2	yes	—
326	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
327	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
328	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
329	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
330	C₂²×#41	C₂²×C₃₁	1	2	yes	—
331	#575	—	2	4	no	—
332	#575	—	2	4	no	—
333	#575	—	2	2	no	—
334	#575	—	2	2	no	—
335	#575	—	2	4	no	—
336	#575	C₂²×C₃₁	1	8	yes	—
337	#575	C₂²×C₃₁	1	8	yes	complemented
338	C₂×#137	—	2	2	no	—
339	C₂×#137	C₂²×C₃₁	1	4	yes	complemented
340	C₂×#137	C₂²×C₃₁	1	4	yes	complemented
341	C₂×AGL(1,ℤ/8)	—	2	2	no	—
342	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	complemented
343	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	complemented
344	C₂×#135	—	2	4	no	—
345	C₂×#135	C₂²×C₃₁	1	4	yes	complemented
346	#568	—	2	4	no	—
347	#568	C₂²×C₃₁	1	8	yes	complemented
348	C₂×#131	—	2	2	no	—
349	C₂×#131	C₂²×C₃₁	1	4	yes	complemented
350	C₂²×M₁₆	—	2	2	no	—
351	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
352	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
353	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
354	#473	C₂²×C₃₁	1	16	yes	complemented
355	#472	C₂²×C₃₁	1	16	yes	complemented
356	#471	C₂²×C₃₁	1	32	yes	complemented
357	#430	C₂²×C₃₁	1	8	yes	complemented
358	C₂×#108	C₂²×C₃₁	1	4	yes	complemented
359	#413	C₂²×C₃₁	1	8	yes	complemented
360	#413	C₂²×C₃₁	1	8	yes	complemented
361	C₂×#101	C₂²×C₃₁	1	4	yes	complemented
362	C₂×#100	C₂²×C₃₁	1	4	yes	complemented
363	C₂×C₃₁	—	2	1	no	—
364	C₂×C₃₁	—	2	2	no	—
365	C₂×C₃₁	—	2	1	no	—
366	C₂×C₃₁	—	2	2	no	—
367	C₂×C₃₁	—	2	1	no	—
368	C₂×C₃₁	—	4	4	no	—
369	C₂×C₃₁	—	8	4	no	—
370	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
371	C₂×#41	—	2	1	no	—
372	C₂×#41	—	2	2	no	—
373	C₂×#41	—	2	2	no	—
374	C₂×#41	—	2	1	no	—
375	C₂×#41	—	2	2	no	—
376	C₂×#41	—	2	2	no	—
377	C₂×#41	—	2	2	no	—
378	C₂×#41	—	2	2	no	—
379	C₂×#41	—	2	2	no	—
380	C₂×#41	—	2	2	no	—
381	C₂×#41	—	2	2	no	—
382	C₂×#41	—	2	2	no	—
383	C₂×#41	—	2	2	no	—
384	C₂×#41	—	2	2	no	—
385	C₂×#41	D₄×C₃₁	1	4	yes	—
386	C₂×#41	D₄×C₃₁	1	2	yes	—
387	C₂×#41	C₂³×C₃₁	1	2	yes	—
388	C₂×#41	C₂³×C₃₁	1	4	yes	—
389	C₂×#41	C₂³×C₃₁	1	4	yes	—
390	C₂×#41	C₂³×C₃₁	1	4	yes	—
391	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
392	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
393	C₂²×D₄	—	2	1	no	—
394	C₂²×D₄	—	2	2	no	—
395	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
396	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
397	C₂³×C₄	—	2	1	no	—
398	C₂³×C₄	—	2	2	no	—
399	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
400	#137	—	2	8	no	—
401	#137	—	2	8	no	—
402	#137	—	2	8	no	—
403	#137	—	2	8	no	—
404	#137	—	2	8	no	—
405	#137	C₂³×C₃₁	1	16	yes	—
406	AGL(1,ℤ/8)	—	2	8	no	—
407	AGL(1,ℤ/8)	—	2	8	no	—
408	AGL(1,ℤ/8)	—	2	8	no	—
409	AGL(1,ℤ/8)	—	2	8	no	—
410	AGL(1,ℤ/8)	—	2	8	no	—
411	AGL(1,ℤ/8)	C₂³×C₃₁	1	16	yes	—
412	#135	—	2	16	no	—
413	#135	—	2	16	no	—
414	#135	—	2	8	no	—
415	#135	—	2	8	no	—
416	C₂×Q₁₆	—	2	4	no	—
417	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	complemented
418	C₂×SD₁₆	—	2	4	no	—
419	C₂×SD₁₆	—	2	4	no	—
420	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	complemented
421	C₂×D₈	—	2	4	no	—
422	C₂×D₈	C₂³×C₃₁	1	4	yes	complemented
423	#131	—	2	16	no	—
424	#131	—	2	8	no	—
425	#131	C₂³×C₃₁	1	16	yes	—
426	C₂×M₁₆	—	2	1	no	—
427	C₂×M₁₆	—	2	2	no	—
428	C₂×M₁₆	—	2	2	no	—
429	C₂×M₁₆	—	2	2	no	—
430	C₂×M₁₆	—	2	2	no	—
431	C₂×M₁₆	—	2	16	no	—
432	C₂×M₁₆	—	2	4	no	—
433	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
434	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
435	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
436	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
437	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
438	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
439	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
440	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
441	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
442	C₂²×C₈	—	2	1	no	—
443	C₂²×C₈	—	2	2	no	—
444	C₂²×C₈	—	2	4	no	—
445	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
446	#108	C₂³×C₃₁	1	16	yes	complemented
447	#101	C₂³×C₃₁	1	16	yes	complemented
448	#100	C₂³×C₃₁	1	16	yes	complemented
449	C₃₁	#60469	1	1	yes	characteristic, direct product factor, Sylow
450	C₂⁴	—	2	1	no	—
451	C₂⁴	—	2	2	no	—
452	C₂⁴	#41×C₃₁	1	1	yes	characteristic
453	#41	—	2	4	no	—
454	#41	—	2	4	no	—
455	#41	—	2	4	no	—
456	#41	—	2	4	no	—
457	#41	—	2	2	no	—
458	#41	—	2	2	no	—
459	#41	—	2	4	no	—
460	#41	—	2	4	no	—
461	#41	—	2	4	no	—
462	#41	—	2	4	no	—
463	#41	—	2	4	no	—
464	#41	—	2	4	no	—
465	#41	—	2	4	no	—
466	#41	—	2	4	no	—
467	#41	—	2	4	no	—
468	#41	—	4	2	no	—
469	#41	—	4	2	no	—
470	#41	—	4	2	no	—
471	#41	—	4	2	no	—
472	#41	—	4	2	no	—
473	#41	—	4	2	no	—
474	#41	—	4	2	no	—
475	#41	—	4	2	no	—
476	#41	C₂×D₄×C₃₁	1	4	yes	—
477	#41	C₂×D₄×C₃₁	1	8	yes	—
478	#41	C₂⁴×C₃₁	1	4	yes	—
479	C₂×Q₈	—	2	2	no	—
480	C₂×Q₈	—	2	1	no	—
481	C₂×Q₈	—	2	2	no	—
482	C₂×Q₈	—	2	2	no	—
483	C₂×Q₈	—	2	2	no	—
484	C₂×Q₈	—	2	2	no	—
485	C₂×Q₈	—	2	2	no	—
486	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
487	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
488	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
489	C₂×D₄	—	2	2	no	—
490	C₂×D₄	—	2	4	no	—
491	C₂×D₄	—	2	2	no	—
492	C₂×D₄	—	2	4	no	—
493	C₂×D₄	—	2	4	no	—
494	C₂×D₄	—	2	4	no	—
495	C₂×D₄	—	2	2	no	—
496	C₂×D₄	—	2	1	no	—
497	C₂×D₄	—	2	2	no	—
498	C₂×D₄	—	2	2	no	—
499	C₂×D₄	—	2	2	no	—
500	C₂×D₄	—	2	2	no	—
501	C₂×D₄	—	2	2	no	—
502	C₂×D₄	—	4	2	no	—
503	C₂×D₄	—	4	2	no	—
504	C₂×D₄	—	4	2	no	—
505	C₂×D₄	—	4	2	no	—
506	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
507	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
508	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
509	C₂²×C₄	—	2	1	no	—
510	C₂²×C₄	—	2	4	no	—
511	C₂²×C₄	—	2	2	no	—
512	C₂²×C₄	—	2	2	no	—
513	C₂²×C₄	—	2	4	no	—
514	C₂²×C₄	—	2	2	no	—
515	C₂²×C₄	—	2	4	no	—
516	C₂²×C₄	—	2	4	no	—
517	C₂²×C₄	—	4	2	no	—
518	C₂²×C₄	—	4	2	no	—
519	C₂²×C₄	—	4	2	no	—
520	C₂²×C₄	—	4	2	no	—
521	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
522	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
523	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
524	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
525	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
526	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
527	C₂²×C₄	#41×C₃₁	1	2	yes	—
528	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
529	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
530	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
531	Q₁₆	—	2	16	no	—
532	Q₁₆	—	2	8	no	—
533	SD₁₆	—	2	16	no	—
534	SD₁₆	—	2	16	no	—
535	SD₁₆	—	2	16	no	—
536	D₈	—	2	16	no	—
537	D₈	—	2	8	no	—
538	M₁₆	—	2	4	no	—
539	M₁₆	—	2	2	no	—
540	M₁₆	—	2	4	no	—
541	M₁₆	—	2	4	no	—
542	M₁₆	—	2	4	no	—
543	M₁₆	—	2	16	no	—
544	M₁₆	C₂×D₄×C₃₁	1	8	yes	—
545	M₁₆	C₂×D₄×C₃₁	1	8	yes	—
546	M₁₆	C₂⁴×C₃₁	1	4	yes	—
547	M₁₆	C₂⁴×C₃₁	1	8	yes	—
548	C₂×C₈	—	2	2	no	—
549	C₂×C₈	—	2	4	no	—
550	C₂×C₈	—	2	4	no	—
551	C₂×C₈	—	2	4	no	—
552	C₂×C₈	—	2	4	no	—
553	C₂×C₈	—	2	16	no	—
554	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
555	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
556	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
557	C₂³	—	2	2	no	—
558	C₂³	—	2	4	no	—
559	C₂³	—	2	4	no	—
560	C₂³	—	4	2	no	—
561	C₂³	—	4	2	no	—
562	C₂³	—	8	1	no	—
563	C₂³	—	8	2	no	—
564	C₂³	—	8	4	no	—
565	C₂³	C₃₁×#121	1	1	yes	characteristic
566	C₂³	C₃₁×#121	1	2	yes	—
567	C₂³	C₃₁×#121	1	1	yes	characteristic
568	C₂³	C₂×#41×C₃₁	1	2	yes	—
569	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
570	Q₈	—	2	2	no	—
571	Q₈	—	2	4	no	—
572	Q₈	—	2	4	no	—
573	Q₈	—	2	4	no	—
574	Q₈	—	4	2	no	—
575	Q₈	—	4	2	no	—
576	Q₈	C₂²×D₄×C₃₁	1	4	yes	—
577	D₄	—	2	8	no	—
578	D₄	—	2	4	no	—
579	D₄	—	2	4	no	—
580	D₄	—	2	2	no	—
581	D₄	—	2	4	no	—
582	D₄	—	2	4	no	—
583	D₄	—	2	4	no	—
584	D₄	—	4	4	no	—
585	D₄	—	4	4	no	—
586	D₄	—	4	4	no	—
587	D₄	—	4	4	no	—
588	D₄	—	4	2	no	—
589	D₄	—	4	2	no	—
590	D₄	C₂²×D₄×C₃₁	1	4	yes	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	4	no	—
594	C₂×C₄	—	2	8	no	—
595	C₂×C₄	—	2	4	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	4	4	no	—
600	C₂×C₄	—	4	4	no	—
601	C₂×C₄	—	4	4	no	—
602	C₂×C₄	—	4	4	no	—
603	C₂×C₄	C₃₁×#121	1	2	yes	—
604	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
605	C₂×C₄	C₃₁×#121	1	2	yes	—
606	C₂×C₄	C₃₁×#121	1	2	yes	—
607	C₂×C₄	C₃₁×#121	1	2	yes	—
608	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
609	C₂×C₄	C₃₁×#121	1	2	yes	—
610	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
611	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
612	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
613	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
614	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
615	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
616	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
617	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
618	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
619	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
620	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
621	C₈	—	2	4	no	—
622	C₈	—	2	8	no	—
623	C₈	C₂²×D₄×C₃₁	1	8	yes	—
624	C₂²	—	2	4	no	—
625	C₂²	—	4	1	no	—
626	C₂²	—	4	2	no	—
627	C₂²	—	4	1	no	—
628	C₂²	—	4	2	no	—
629	C₂²	—	4	1	no	—
630	C₂²	—	4	4	no	—
631	C₂²	—	4	4	no	—
632	C₂²	—	8	4	no	—
633	C₂²	—	8	4	no	—
634	C₂²	—	8	4	no	—
635	C₂²	—	8	4	no	—
636	C₂²	C₂×C₃₁×#121	1	2	yes	—
637	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
638	C₂²	C₂×C₃₁×#121	1	2	yes	—
639	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
640	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
641	C₄	—	2	4	no	—
642	C₄	—	4	4	no	—
643	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
644	C₄	C₂×C₃₁×#121	1	2	yes	—
645	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
646	C₄	C₂×C₃₁×#121	1	2	yes	—
647	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
648	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
649	C₂	—	2	1	no	—
650	C₂	—	2	2	no	—
651	C₂	—	2	1	no	—
652	C₂	—	2	2	no	—
653	C₂	—	2	1	no	—
654	C₂	—	4	4	no	—
655	C₂	—	8	4	no	—
656	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂×C₃₁	C₂²×#121
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#575	C₂	semidirect
2. C₂×C₃₁×#575	C₂	semidirect
3. C₂×C₃₁×#568	C₂	semidirect
4. C₃₁×#3069	C₂	semidirect
5. C₃₁×#3068	C₂	semidirect
6. C₃₁×#3067	C₂	semidirect
7. C₂×C₃₁×#430	C₂	semidirect
8. C₂×C₃₁×#413	C₂	semidirect
17. C₃₁×#575	C₂²	semidirect
19. C₂×C₃₁×#137	C₂²	semidirect
20. C₂×C₃₁×#137	C₂²	semidirect
22. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
23. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
25. C₂×C₃₁×#135	C₂²	semidirect
27. C₃₁×#568	C₂²	semidirect
29. C₂×C₃₁×#131	C₂²	semidirect
34. C₃₁×#473	C₂²	semidirect
35. C₃₁×#472	C₂²	semidirect
36. C₃₁×#471	C₂²	semidirect
37. C₃₁×#430	C₂²	semidirect
38. C₂×C₃₁×#108	C₂²	semidirect
39. C₃₁×#413	C₂²	semidirect
40. C₃₁×#413	C₂²	semidirect
41. C₂×C₃₁×#101	C₂²	semidirect
42. C₂×C₃₁×#100	C₂²	semidirect
89. C₂×Q₁₆×C₃₁	C₂³	semidirect
92. C₂×SD₁₆×C₃₁	C₂³	semidirect
94. C₂×D₈×C₃₁	C₂³	semidirect
118. C₃₁×#108	C₂³	semidirect
119. C₃₁×#101	C₂³	semidirect
120. C₃₁×#100	C₂³	semidirect
228. #60469	C₃₁	direct (N × H)
296. C₂×#575	C₂×C₃₁	semidirect
297. C₂×#575	C₂×C₃₁	semidirect
298. C₂×#568	C₂×C₃₁	semidirect
299. #3069	C₂×C₃₁	semidirect
300. #3068	C₂×C₃₁	semidirect
301. #3067	C₂×C₃₁	semidirect
302. C₂×#430	C₂×C₃₁	semidirect
303. C₂×#413	C₂×C₃₁	semidirect
337. #575	C₂²×C₃₁	semidirect
339. C₂×#137	C₂²×C₃₁	semidirect
340. C₂×#137	C₂²×C₃₁	semidirect
342. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
343. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
345. C₂×#135	C₂²×C₃₁	semidirect
347. #568	C₂²×C₃₁	semidirect
349. C₂×#131	C₂²×C₃₁	semidirect
354. #473	C₂²×C₃₁	semidirect
355. #472	C₂²×C₃₁	semidirect
356. #471	C₂²×C₃₁	semidirect
357. #430	C₂²×C₃₁	semidirect
358. C₂×#108	C₂²×C₃₁	semidirect
359. #413	C₂²×C₃₁	semidirect
360. #413	C₂²×C₃₁	semidirect
361. C₂×#101	C₂²×C₃₁	semidirect
362. C₂×#100	C₂²×C₃₁	semidirect
417. C₂×Q₁₆	C₂³×C₃₁	semidirect
420. C₂×SD₁₆	C₂³×C₃₁	semidirect
422. C₂×D₈	C₂³×C₃₁	semidirect
446. #108	C₂³×C₃₁	semidirect
447. #101	C₂³×C₃₁	semidirect
448. #100	C₂³×C₃₁	semidirect
449. C₃₁	#60469	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×#575	2
C₂×C₃₁×#568	2
C₃₁×#3069	2
C₃₁×#3068	2
C₃₁×#3067	2
C₂×C₃₁×#430	2
C₂×C₃₁×#413	2
#60469	31

Subgroups — C31×#60469 #21517228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60469 #21517228