Subgroups — C₃₁×#24771 #21481530

Order 7936.

872 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#471	C₂	1	2	yes	maximal
2	C₃₁×#3039	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₃₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2906	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2064	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2046	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1966	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1908	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1800	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₂²	1	2	yes	—
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×#131	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#471	—	2	8	no	—
20	C₃₁×#462	—	2	1	no	—
21	C₃₁×#462	—	2	1	no	—
22	C₃₁×#461	—	2	1	no	—
23	C₃₁×#461	—	2	1	no	—
24	C₃₁×#461	—	2	1	no	—
25	C₃₁×#461	—	2	1	no	—
26	C₃₁×#461	—	2	2	no	—
27	C₃₁×#442	—	2	1	no	—
28	C₃₁×#440	—	2	1	no	—
29	C₄×SD₁₆×C₃₁	—	2	1	no	—
30	C₄×D₈×C₃₁	—	2	1	no	—
31	C₂×C₃₁×#108	C₂²	1	2	yes	—
32	C₃₁×#428	C₂²	1	1	yes	characteristic
33	C₃₁×#424	—	2	2	no	—
34	C₃₁×#424	C₂²	1	2	yes	—
35	C₃₁×#423	—	2	1	no	—
36	C₃₁×#423	C₂²	1	1	yes	characteristic
37	C₃₁×#423	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₃₁×#419	C₂²	1	2	yes	—
41	C₃₁×#418	C₂²	1	1	yes	characteristic
42	C₃₁×#417	C₂²	1	1	yes	characteristic
43	C₂×C₃₁×#103	C₂²	1	2	yes	—
44	C₂×C₃₁×#101	C₂²	1	2	yes	—
45	C₂×C₃₁×#101	C₂²	1	2	yes	—
46	C₂×C₃₁×#100	C₂²	1	2	yes	—
47	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
48	C₃₁×#343	C₂²	1	2	yes	—
49	C₃₁×#341	C₂²	1	2	yes	—
50	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₃₁	—	2	1	no	—
54	C₃₁×#137	—	2	4	no	—
55	C₃₁×#137	—	2	4	no	—
56	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
57	C₃₁×AGL(1,ℤ/8)	D₄	1	4	yes	—
58	C₂×Q₁₆×C₃₁	—	2	2	no	—
59	C₂×SD₁₆×C₃₁	—	2	1	no	—
60	C₂×SD₁₆×C₃₁	—	2	2	no	—
61	C₂×SD₁₆×C₃₁	—	2	4	no	—
62	C₂×SD₁₆×C₃₁	—	2	2	no	—
63	C₂×SD₁₆×C₃₁	—	2	2	no	—
64	C₂×SD₁₆×C₃₁	—	4	2	no	—
65	C₂×D₈×C₃₁	—	2	1	no	—
66	C₃₁×#131	—	2	4	no	—
67	C₃₁×#131	—	4	2	no	—
68	C₂×M₁₆×C₃₁	—	2	2	no	—
69	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
71	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
72	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₃₁	—	2	2	no	—
74	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
75	C₃₁×C₄⋊Q₈	—	2	1	no	—
76	C₃₁×C₄⋊Q₈	—	2	1	no	—
77	C₃₁×C₄⋊Q₈	—	2	1	no	—
78	C₃₁×C₄⋊Q₈	—	2	1	no	—
79	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
80	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
81	C₄×D₄×C₃₁	—	2	1	no	—
82	C₄×D₄×C₃₁	—	2	1	no	—
83	C₄×D₄×C₃₁	—	2	1	no	—
84	C₄×D₄×C₃₁	—	2	1	no	—
85	C₄×D₄×C₃₁	—	2	1	no	—
86	C₄×D₄×C₃₁	—	2	1	no	—
87	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
88	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
89	C₃₁×#117	—	2	1	no	—
90	C₃₁×#117	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
92	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
94	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
95	C₂×C₄²×C₃₁	—	2	1	no	—
96	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
97	C₃₁×#108	—	2	4	no	—
98	C₃₁×#107	—	2	1	no	—
99	C₃₁×#106	—	2	1	no	—
100	C₃₁×C₄⋊C₈	—	2	2	no	—
101	C₃₁×C₄⋊C₈	—	2	2	no	—
102	C₃₁×C₄⋊C₈	—	2	1	no	—
103	C₃₁×C₄⋊C₈	—	2	1	no	—
104	C₃₁×C₄⋊C₈	—	2	1	no	—
105	C₃₁×C₄⋊C₈	—	2	1	no	—
106	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
107	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
108	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
109	C₃₁×#103	—	2	1	no	—
110	C₃₁×#103	—	2	2	no	—
111	C₃₁×#103	—	2	2	no	—
112	C₃₁×#103	—	2	2	no	—
113	C₃₁×#103	—	2	2	no	—
114	C₃₁×#102	—	2	1	no	—
115	C₃₁×#102	—	2	1	no	—
116	C₃₁×#102	—	2	1	no	—
117	C₃₁×#101	—	2	4	no	—
118	C₃₁×#101	—	4	2	no	—
119	C₃₁×#100	—	2	4	no	—
120	C₃₁×#97	—	2	1	no	—
121	C₄×C₈×C₃₁	—	2	1	no	—
122	C₂⁴×C₃₁	—	2	1	no	—
123	#41×C₃₁	—	2	1	no	—
124	#41×C₃₁	—	2	1	no	—
125	#41×C₃₁	C₂×D₄	1	2	yes	—
126	#41×C₃₁	#41	1	2	yes	—
127	C₂×Q₈×C₃₁	—	2	2	no	—
128	C₂×Q₈×C₃₁	—	2	2	no	—
129	C₂×Q₈×C₃₁	—	2	2	no	—
130	C₂×Q₈×C₃₁	—	2	2	no	—
131	C₂×Q₈×C₃₁	—	2	2	no	—
132	C₂×Q₈×C₃₁	—	2	2	no	—
133	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂×D₄×C₃₁	—	2	1	no	—
135	C₂×D₄×C₃₁	—	2	1	no	—
136	C₂×D₄×C₃₁	—	2	1	no	—
137	C₂×D₄×C₃₁	—	2	1	no	—
138	C₂×D₄×C₃₁	—	2	1	no	—
139	C₂×D₄×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
141	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	—	2	1	no	—
143	C₂²×C₄×C₃₁	—	2	1	no	—
144	C₂²×C₄×C₃₁	—	2	1	no	—
145	C₂²×C₄×C₃₁	—	2	2	no	—
146	C₂²×C₄×C₃₁	—	2	1	no	—
147	C₂²×C₄×C₃₁	—	4	1	no	—
148	C₂²×C₄×C₃₁	—	4	1	no	—
149	C₂²×C₄×C₃₁	—	8	2	no	—
150	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
152	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
153	Q₁₆×C₃₁	—	4	4	no	—
154	SD₁₆×C₃₁	—	2	4	no	—
155	SD₁₆×C₃₁	—	4	4	no	—
156	SD₁₆×C₃₁	—	4	4	no	—
157	SD₁₆×C₃₁	—	4	4	no	—
158	D₈×C₃₁	—	2	4	no	—
159	M₁₆×C₃₁	—	2	1	no	—
160	M₁₆×C₃₁	—	2	2	no	—
161	M₁₆×C₃₁	—	2	2	no	—
162	M₁₆×C₃₁	—	2	2	no	—
163	M₁₆×C₃₁	—	2	1	no	—
164	M₁₆×C₃₁	—	2	2	no	—
165	M₁₆×C₃₁	—	2	1	no	—
166	M₁₆×C₃₁	—	4	4	no	—
167	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
168	C₂×C₈×C₃₁	—	2	1	no	—
169	C₂×C₈×C₃₁	—	2	2	no	—
170	C₂×C₈×C₃₁	—	2	2	no	—
171	C₂×C₈×C₃₁	—	2	1	no	—
172	C₂×C₈×C₃₁	—	2	2	no	—
173	C₂×C₈×C₃₁	—	2	2	no	—
174	C₂×C₈×C₃₁	—	4	4	no	—
175	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
176	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	2	1	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	—	2	1	no	—
181	C₄⋊C₄×C₃₁	—	4	1	no	—
182	C₄⋊C₄×C₃₁	—	4	1	no	—
183	C₄⋊C₄×C₃₁	—	4	1	no	—
184	C₄⋊C₄×C₃₁	—	4	1	no	—
185	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
186	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
187	C₂²⋊C₄×C₃₁	—	2	1	no	—
188	C₂²⋊C₄×C₃₁	—	2	1	no	—
189	C₂²⋊C₄×C₃₁	—	2	1	no	—
190	C₂²⋊C₄×C₃₁	—	4	1	no	—
191	C₂²⋊C₄×C₃₁	—	4	1	no	—
192	C₄²×C₃₁	—	2	1	no	—
193	C₄²×C₃₁	—	2	1	no	—
194	C₄²×C₃₁	—	2	1	no	—
195	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
196	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
199	#24771	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂³×C₃₁	—	2	1	no	—
201	C₂³×C₃₁	—	2	2	no	—
202	C₂³×C₃₁	—	4	1	no	—
203	C₂³×C₃₁	—	4	1	no	—
204	C₂³×C₃₁	—	8	1	no	—
205	C₂³×C₃₁	—	8	1	no	—
206	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
207	Q₈×C₃₁	—	2	1	no	—
208	Q₈×C₃₁	—	4	2	no	—
209	Q₈×C₃₁	—	4	2	no	—
210	Q₈×C₃₁	—	4	2	no	—
211	Q₈×C₃₁	—	4	2	no	—
212	Q₈×C₃₁	#121	1	2	yes	—
213	D₄×C₃₁	—	2	2	no	—
214	D₄×C₃₁	—	2	2	no	—
215	D₄×C₃₁	—	2	2	no	—
216	D₄×C₃₁	—	2	1	no	—
217	D₄×C₃₁	—	2	2	no	—
218	D₄×C₃₁	—	4	1	no	—
219	D₄×C₃₁	—	4	1	no	—
220	D₄×C₃₁	#121	1	2	yes	—
221	C₂×C₄×C₃₁	—	2	1	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	1	no	—
226	C₂×C₄×C₃₁	—	2	1	no	—
227	C₂×C₄×C₃₁	—	4	1	no	—
228	C₂×C₄×C₃₁	—	4	2	no	—
229	C₂×C₄×C₃₁	—	4	1	no	—
230	C₂×C₄×C₃₁	—	4	2	no	—
231	C₂×C₄×C₃₁	—	4	2	no	—
232	C₂×C₄×C₃₁	—	4	2	no	—
233	C₂×C₄×C₃₁	—	8	1	no	—
234	C₂×C₄×C₃₁	—	8	2	no	—
235	C₂×C₄×C₃₁	—	8	2	no	—
236	C₂×C₄×C₃₁	—	8	1	no	—
237	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
238	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
239	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
240	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
241	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
245	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
246	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
247	C₈×C₃₁	—	2	2	no	—
248	C₈×C₃₁	—	2	2	no	—
249	C₈×C₃₁	—	2	2	no	—
250	C₈×C₃₁	—	4	2	no	—
251	C₈×C₃₁	—	4	1	no	—
252	C₈×C₃₁	—	4	2	no	—
253	C₂×#471	C₂×C₃₁	1	2	yes	—
254	#3039	C₂×C₃₁	1	1	yes	characteristic, complemented
255	C₂×#461	C₂×C₃₁	1	1	yes	characteristic, complemented
256	C₄×AGL(1,ℤ/8)	C₂×C₃₁	1	1	yes	characteristic
257	#2906	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#2064	C₂×C₃₁	1	2	yes	complemented
259	#2046	C₂×C₃₁	1	2	yes	complemented
260	#1966	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
262	#1908	C₂×C₃₁	1	2	yes	complemented
263	#1800	C₂×C₃₁	1	1	yes	characteristic, complemented
264	C₂²×C₃₁	—	2	1	no	—
265	C₂²×C₃₁	—	2	1	no	—
266	C₂²×C₃₁	—	2	2	no	—
267	C₂²×C₃₁	—	4	1	no	—
268	C₂²×C₃₁	—	4	1	no	—
269	C₂²×C₃₁	—	4	2	no	—
270	C₂²×C₃₁	—	8	2	no	—
271	C₂²×C₃₁	—	8	2	no	—
272	C₂²×C₃₁	—	8	1	no	—
273	C₂²×C₃₁	—	8	1	no	—
274	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
275	C₂²×C₃₁	#547	1	1	yes	characteristic
276	C₂²×C₃₁	#553	1	1	yes	characteristic
277	C₄×C₃₁	—	2	2	no	—
278	C₄×C₃₁	—	2	1	no	—
279	C₄×C₃₁	—	2	1	no	—
280	C₄×C₃₁	—	2	1	no	—
281	C₄×C₃₁	—	4	1	no	—
282	C₄×C₃₁	—	4	1	no	—
283	C₄×C₃₁	—	8	2	no	—
284	C₄×C₃₁	—	8	1	no	—
285	C₄×C₃₁	#471	1	2	yes	complemented
286	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
287	C₄×C₃₁	#537	1	1	yes	characteristic
288	C₄×C₃₁	D₄²	1	1	yes	characteristic
289	C₄×C₃₁	#548	1	1	yes	characteristic
290	C₂×#137	C₂²×C₃₁	1	2	yes	—
291	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
292	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
293	C₂×#131	C₂²×C₃₁	1	2	yes	—
294	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
295	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
297	#471	—	2	8	no	—
298	#462	—	2	1	no	—
299	#462	—	2	1	no	—
300	#461	—	2	1	no	—
301	#461	—	2	1	no	—
302	#461	—	2	1	no	—
303	#461	—	2	1	no	—
304	#461	—	2	2	no	—
305	#442	—	2	1	no	—
306	#440	—	2	1	no	—
307	C₄×SD₁₆	—	2	1	no	—
308	C₄×D₈	—	2	1	no	—
309	C₂×#108	C₂²×C₃₁	1	2	yes	—
310	#428	C₂²×C₃₁	1	1	yes	characteristic
311	#424	—	2	2	no	—
312	#424	C₂²×C₃₁	1	2	yes	—
313	#423	—	2	1	no	—
314	#423	C₂²×C₃₁	1	1	yes	characteristic
315	#423	C₂²×C₃₁	1	1	yes	characteristic
316	C₂×C₄⋊C₈	—	2	1	no	—
317	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
318	#419	C₂²×C₃₁	1	2	yes	—
319	#418	C₂²×C₃₁	1	1	yes	characteristic
320	#417	C₂²×C₃₁	1	1	yes	characteristic
321	C₂×#103	C₂²×C₃₁	1	2	yes	—
322	C₂×#101	C₂²×C₃₁	1	2	yes	—
323	C₂×#101	C₂²×C₃₁	1	2	yes	—
324	C₂×#100	C₂²×C₃₁	1	2	yes	—
325	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
326	#343	C₂²×C₃₁	1	2	yes	—
327	#341	C₂²×C₃₁	1	2	yes	—
328	C₂×C₃₁	—	2	1	no	—
329	C₂×C₃₁	—	2	1	no	—
330	C₂×C₃₁	—	4	2	no	—
331	C₂×C₃₁	—	8	2	no	—
332	C₂×C₃₁	#2416	1	1	yes	characteristic
333	C₂×C₃₁	C₂×#471	1	1	yes	characteristic
334	C₂×C₃₁	#3069	1	1	yes	characteristic
335	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
336	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
337	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
338	C₂³×C₄	—	2	1	no	—
339	#137	—	2	4	no	—
340	#137	—	2	4	no	—
341	AGL(1,ℤ/8)	—	2	2	no	—
342	AGL(1,ℤ/8)	D₄×C₃₁	1	4	yes	—
343	C₂×Q₁₆	—	2	2	no	—
344	C₂×SD₁₆	—	2	1	no	—
345	C₂×SD₁₆	—	2	4	no	—
346	C₂×SD₁₆	—	2	2	no	—
347	C₂×SD₁₆	—	2	2	no	—
348	C₂×SD₁₆	—	2	2	no	—
349	C₂×SD₁₆	—	4	2	no	—
350	C₂×D₈	—	2	1	no	—
351	#131	—	2	4	no	—
352	#131	—	4	2	no	—
353	C₂×M₁₆	—	2	2	no	—
354	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
355	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
356	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
357	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
358	C₂²×C₈	—	2	2	no	—
359	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
360	C₄⋊Q₈	—	2	1	no	—
361	C₄⋊Q₈	—	2	1	no	—
362	C₄⋊Q₈	—	2	1	no	—
363	C₄⋊Q₈	—	2	1	no	—
364	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
365	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
366	C₄×D₄	—	2	1	no	—
367	C₄×D₄	—	2	1	no	—
368	C₄×D₄	—	2	1	no	—
369	C₄×D₄	—	2	1	no	—
370	C₄×D₄	—	2	1	no	—
371	C₄×D₄	—	2	1	no	—
372	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
373	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
374	#117	—	2	1	no	—
375	#117	C₂³×C₃₁	1	1	yes	characteristic
376	C₂×C₄⋊C₄	—	2	1	no	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
379	C₂×C₂²⋊C₄	—	2	1	no	—
380	C₂×C₄²	—	2	1	no	—
381	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
382	#108	—	2	4	no	—
383	#107	—	2	1	no	—
384	#106	—	2	1	no	—
385	C₄⋊C₈	—	2	2	no	—
386	C₄⋊C₈	—	2	2	no	—
387	C₄⋊C₈	—	2	1	no	—
388	C₄⋊C₈	—	2	1	no	—
389	C₄⋊C₈	—	2	1	no	—
390	C₄⋊C₈	—	2	1	no	—
391	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
392	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
393	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
394	#103	—	2	1	no	—
395	#103	—	2	2	no	—
396	#103	—	2	2	no	—
397	#103	—	2	2	no	—
398	#103	—	2	2	no	—
399	#102	—	2	1	no	—
400	#102	—	2	1	no	—
401	#102	—	2	1	no	—
402	#101	—	2	4	no	—
403	#101	—	4	2	no	—
404	#100	—	2	4	no	—
405	#97	—	2	1	no	—
406	C₄×C₈	—	2	1	no	—
407	C₃₁	#24771	1	1	yes	characteristic, direct product factor, Sylow
408	C₂⁴	—	2	1	no	—
409	#41	—	2	1	no	—
410	#41	—	2	1	no	—
411	#41	C₂×D₄×C₃₁	1	2	yes	—
412	#41	#41×C₃₁	1	2	yes	—
413	C₂×Q₈	—	2	2	no	—
414	C₂×Q₈	—	2	2	no	—
415	C₂×Q₈	—	2	2	no	—
416	C₂×Q₈	—	2	2	no	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	—	2	1	no	—
423	C₂×D₄	—	2	1	no	—
424	C₂×D₄	—	2	1	no	—
425	C₂×D₄	—	2	1	no	—
426	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
427	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	4	1	no	—
434	C₂²×C₄	—	4	1	no	—
435	C₂²×C₄	—	8	2	no	—
436	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
437	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
438	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
439	Q₁₆	—	4	4	no	—
440	SD₁₆	—	2	4	no	—
441	SD₁₆	—	4	4	no	—
442	SD₁₆	—	4	4	no	—
443	SD₁₆	—	4	4	no	—
444	D₈	—	2	4	no	—
445	M₁₆	—	2	1	no	—
446	M₁₆	—	2	2	no	—
447	M₁₆	—	2	2	no	—
448	M₁₆	—	2	2	no	—
449	M₁₆	—	2	1	no	—
450	M₁₆	—	2	2	no	—
451	M₁₆	—	2	1	no	—
452	M₁₆	—	4	4	no	—
453	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	2	no	—
456	C₂×C₈	—	2	2	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	—	2	2	no	—
460	C₂×C₈	—	4	4	no	—
461	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
462	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	4	1	no	—
468	C₄⋊C₄	—	4	1	no	—
469	C₄⋊C₄	—	4	1	no	—
470	C₄⋊C₄	—	4	1	no	—
471	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
472	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
473	C₂²⋊C₄	—	2	1	no	—
474	C₂²⋊C₄	—	2	1	no	—
475	C₂²⋊C₄	—	2	1	no	—
476	C₂²⋊C₄	—	4	1	no	—
477	C₂²⋊C₄	—	4	1	no	—
478	C₄²	—	2	1	no	—
479	C₄²	—	2	1	no	—
480	C₄²	—	2	1	no	—
481	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
482	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
483	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
484	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
485	C₂³	—	2	1	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	4	1	no	—
488	C₂³	—	4	1	no	—
489	C₂³	—	8	1	no	—
490	C₂³	—	8	1	no	—
491	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
492	Q₈	—	2	1	no	—
493	Q₈	—	4	2	no	—
494	Q₈	—	4	2	no	—
495	Q₈	—	4	2	no	—
496	Q₈	—	4	2	no	—
497	Q₈	C₃₁×#121	1	2	yes	—
498	D₄	—	2	2	no	—
499	D₄	—	2	2	no	—
500	D₄	—	2	2	no	—
501	D₄	—	2	1	no	—
502	D₄	—	2	2	no	—
503	D₄	—	4	1	no	—
504	D₄	—	4	1	no	—
505	D₄	C₃₁×#121	1	2	yes	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	2	no	—
514	C₂×C₄	—	4	1	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	8	1	no	—
519	C₂×C₄	—	8	2	no	—
520	C₂×C₄	—	8	2	no	—
521	C₂×C₄	—	8	1	no	—
522	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
523	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
524	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
525	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
526	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
528	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
529	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
530	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
531	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
532	C₈	—	2	2	no	—
533	C₈	—	2	2	no	—
534	C₈	—	2	2	no	—
535	C₈	—	4	2	no	—
536	C₈	—	4	1	no	—
537	C₈	—	4	2	no	—
538	C₂²	—	2	1	no	—
539	C₂²	—	2	1	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	4	1	no	—
542	C₂²	—	4	1	no	—
543	C₂²	—	4	2	no	—
544	C₂²	—	8	2	no	—
545	C₂²	—	8	2	no	—
546	C₂²	—	8	1	no	—
547	C₂²	—	8	1	no	—
548	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
549	C₂²	C₃₁×#547	1	1	yes	characteristic
550	C₂²	C₃₁×#553	1	1	yes	characteristic
551	C₄	—	2	2	no	—
552	C₄	—	2	1	no	—
553	C₄	—	2	1	no	—
554	C₄	—	2	1	no	—
555	C₄	—	4	1	no	—
556	C₄	—	4	1	no	—
557	C₄	—	8	2	no	—
558	C₄	—	8	1	no	—
559	C₄	C₃₁×#471	1	2	yes	complemented
560	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
561	C₄	C₃₁×#537	1	1	yes	characteristic
562	C₄	D₄²×C₃₁	1	1	yes	characteristic
563	C₄	C₃₁×#548	1	1	yes	characteristic
564	C₂	—	2	1	no	—
565	C₂	—	2	1	no	—
566	C₂	—	4	2	no	—
567	C₂	—	8	2	no	—
568	C₂	C₃₁×#2416	1	1	yes	characteristic
569	C₂	C₂×C₃₁×#471	1	1	yes	characteristic
570	C₂	C₃₁×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3039	C₂	semidirect
3. C₂×C₃₁×#461	C₂	semidirect
5. C₃₁×#2906	C₂	semidirect
6. C₃₁×#2064	C₂	semidirect
7. C₃₁×#2046	C₂	semidirect
8. C₃₁×#1966	C₂	semidirect
9. C₃₁×#1950	C₂	semidirect
10. C₃₁×#1908	C₂	semidirect
11. C₃₁×#1800	C₂	semidirect
199. #24771	C₃₁	direct (N × H)
254. #3039	C₂×C₃₁	semidirect
255. C₂×#461	C₂×C₃₁	semidirect
257. #2906	C₂×C₃₁	semidirect
258. #2064	C₂×C₃₁	semidirect
259. #2046	C₂×C₃₁	semidirect
260. #1966	C₂×C₃₁	semidirect
261. #1950	C₂×C₃₁	semidirect
262. #1908	C₂×C₃₁	semidirect
263. #1800	C₂×C₃₁	semidirect
285. C₄×C₃₁	#471	semidirect
407. C₃₁	#24771	direct (N × H)
559. C₄	C₃₁×#471	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#471	2
C₃₁×#3039	2
C₂×C₃₁×#461	2
C₄×C₃₁×AGL(1,ℤ/8)	2
C₃₁×#2906	2
C₃₁×#2064	2
C₃₁×#2046	2
C₃₁×#1966	2
C₃₁×#1950	2
C₃₁×#1908	2
C₃₁×#1800	2
#24771	31

Subgroups — C31×#24771 #21481530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24771 #21481530