Subgroups — C₃₁×#19312 #21476071

Order 7936.

768 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3151	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3115	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3038	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2970	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1688	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1686	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1667	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1665	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1569	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1557	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1555	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
13	C₂×C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
16	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#496	—	2	1	no	—
18	C₃₁×#496	—	2	1	no	—
19	C₃₁×#493	C₂²	1	2	yes	—
20	C₃₁×#492	C₂²	1	2	yes	—
21	C₃₁×#482	—	2	1	no	—
22	C₃₁×#475	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#474	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×#460	—	2	1	no	—
25	C₃₁×#460	C₂²	1	2	yes	—
26	C₃₁×#460	C₂²	1	1	yes	characteristic, complemented
27	C₃₁×#459	—	2	1	no	—
28	C₃₁×#459	C₂²	1	2	yes	—
29	C₃₁×#459	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×#449	—	2	1	no	—
31	C₃₁×#446	C₂²	1	2	yes	complemented
32	C₃₁×#442	—	2	1	no	—
33	C₃₁×#440	—	2	1	no	—
34	C₃₁×#435	—	2	2	no	—
35	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
36	C₃₁×#432	—	2	1	no	—
37	C₃₁×#432	C₂²	1	1	yes	characteristic
38	C₃₁×#432	C₂²	1	1	yes	characteristic
39	C₃₁×#423	C₂²	1	1	yes	characteristic
40	C₃₁×#418	—	2	1	no	—
41	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#97	—	2	1	no	—
43	C₃₁×#335	C₂²	1	2	yes	complemented
44	C₃₁×#334	C₂²	1	2	yes	complemented
45	C₃₁×#332	C₂²	1	2	yes	complemented
46	C₃₁×#331	C₂²	1	2	yes	—
47	C₃₁×#331	C₂²	1	2	yes	—
48	C₃₁×#331	C₂²	1	2	yes	complemented
49	C₂×#41×C₃₁	—	2	1	no	—
50	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	—	2	1	no	—
53	C₂²×D₄×C₃₁	—	2	1	no	—
54	C₃₁×AGL(1,ℤ/8)	—	4	8	no	—
55	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
56	C₂×SD₁₆×C₃₁	—	2	1	no	—
57	C₂×SD₁₆×C₃₁	—	2	2	no	—
58	C₂×SD₁₆×C₃₁	—	2	2	no	—
59	C₂×SD₁₆×C₃₁	—	2	2	no	—
60	C₂×D₈×C₃₁	—	2	1	no	—
61	C₂×D₈×C₃₁	—	2	2	no	—
62	C₂×D₈×C₃₁	—	2	2	no	—
63	C₂×D₈×C₃₁	—	2	2	no	—
64	C₂×M₁₆×C₃₁	—	2	1	no	—
65	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
66	C₂²×C₈×C₃₁	—	2	2	no	—
67	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
68	C₃₁×#127	—	2	1	no	—
69	C₃₁×#127	—	2	1	no	—
70	C₃₁×#127	—	2	1	no	—
71	C₃₁×#127	C₂³	1	2	yes	—
72	C₃₁×#127	C₂³	1	1	yes	characteristic
73	C₃₁×#124	—	2	1	no	—
74	C₃₁×#121	—	2	1	no	—
75	C₃₁×#121	—	2	1	no	—
76	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
77	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₄×D₄×C₃₁	—	2	1	no	—
79	C₄×D₄×C₃₁	—	2	1	no	—
80	C₄×D₄×C₃₁	—	2	1	no	—
81	C₄×D₄×C₃₁	—	2	1	no	—
82	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
83	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#117	—	2	1	no	—
85	C₃₁×#117	D₄	1	1	yes	characteristic
86	C₂×C₄²×C₃₁	—	2	1	no	—
87	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
88	C₃₁×#107	—	2	1	no	—
89	C₃₁×#106	—	2	1	no	—
90	C₃₁×C₄⋊C₈	—	2	2	no	—
91	C₃₁×C₄⋊C₈	—	2	1	no	—
92	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
93	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
94	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
95	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
96	C₃₁×#103	—	2	1	no	—
97	C₃₁×#102	—	2	1	no	—
98	C₃₁×#102	—	2	2	no	—
99	C₃₁×#102	—	2	2	no	—
100	C₃₁×#102	—	2	1	no	—
101	C₃₁×#102	—	2	1	no	—
102	C₃₁×C₂²⋊C₈	—	2	1	no	—
103	C₃₁×C₂²⋊C₈	—	2	1	no	—
104	C₃₁×C₂²⋊C₈	—	2	2	no	—
105	C₃₁×#97	—	2	2	no	—
106	C₃₁×#97	—	2	1	no	—
107	C₃₁×#97	—	2	1	no	—
108	C₃₁×#97	D₄	1	1	yes	characteristic
109	C₃₁×#97	D₄	1	1	yes	characteristic
110	C₄×C₈×C₃₁	C₂³	1	2	yes	—
111	C₂⁴×C₃₁	—	4	1	no	—
112	#41×C₃₁	—	2	2	no	—
113	#41×C₃₁	—	4	1	no	—
114	#41×C₃₁	—	4	4	no	—
115	C₂×Q₈×C₃₁	—	2	1	no	—
116	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	1	no	—
119	C₂×D₄×C₃₁	—	2	2	no	—
120	C₂×D₄×C₃₁	—	2	1	no	—
121	C₂×D₄×C₃₁	—	2	2	no	—
122	C₂×D₄×C₃₁	—	2	1	no	—
123	C₂×D₄×C₃₁	—	2	2	no	—
124	C₂×D₄×C₃₁	—	4	4	no	—
125	C₂×D₄×C₃₁	—	4	2	no	—
126	C₂×D₄×C₃₁	—	4	2	no	—
127	C₂×D₄×C₃₁	—	4	1	no	—
128	C₂×D₄×C₃₁	—	4	1	no	—
129	C₂×D₄×C₃₁	—	4	1	no	—
130	C₂×D₄×C₃₁	—	4	1	no	—
131	C₂×D₄×C₃₁	—	8	2	no	—
132	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	—	2	1	no	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	2	1	no	—
136	C₂²×C₄×C₃₁	—	4	1	no	—
137	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
140	SD₁₆×C₃₁	—	4	4	no	—
141	SD₁₆×C₃₁	—	4	2	no	—
142	SD₁₆×C₃₁	—	4	4	no	—
143	SD₁₆×C₃₁	—	4	4	no	—
144	D₈×C₃₁	—	4	4	no	—
145	D₈×C₃₁	—	4	2	no	—
146	D₈×C₃₁	—	4	4	no	—
147	D₈×C₃₁	—	4	4	no	—
148	M₁₆×C₃₁	—	2	4	no	—
149	M₁₆×C₃₁	—	4	2	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	1	no	—
152	C₂×C₈×C₃₁	—	2	1	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	4	4	no	—
155	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
156	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
157	C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₄⋊C₄×C₃₁	—	2	1	no	—
160	C₄⋊C₄×C₃₁	—	2	1	no	—
161	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂²⋊C₄×C₃₁	—	2	1	no	—
164	C₂²⋊C₄×C₃₁	—	2	1	no	—
165	C₂²⋊C₄×C₃₁	—	2	1	no	—
166	C₂²⋊C₄×C₃₁	—	4	1	no	—
167	C₂²⋊C₄×C₃₁	—	4	1	no	—
168	C₄²×C₃₁	—	2	1	no	—
169	C₄²×C₃₁	—	2	1	no	—
170	C₄²×C₃₁	—	2	1	no	—
171	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₃₁	#41	1	1	yes	characteristic
174	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
175	#19312	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂³×C₃₁	—	2	1	no	—
177	C₂³×C₃₁	—	4	1	no	—
178	C₂³×C₃₁	—	4	2	no	—
179	C₂³×C₃₁	—	8	2	no	—
180	C₂³×C₃₁	—	8	2	no	—
181	C₂³×C₃₁	—	8	2	no	—
182	C₂³×C₃₁	#121	1	1	yes	characteristic
183	Q₈×C₃₁	—	2	1	no	—
184	Q₈×C₃₁	—	2	1	no	—
185	Q₈×C₃₁	—	4	2	no	—
186	D₄×C₃₁	—	2	1	no	—
187	D₄×C₃₁	—	4	2	no	—
188	D₄×C₃₁	—	4	2	no	—
189	D₄×C₃₁	—	4	2	no	—
190	D₄×C₃₁	—	4	2	no	—
191	D₄×C₃₁	—	4	2	no	—
192	D₄×C₃₁	—	4	1	no	—
193	D₄×C₃₁	—	4	1	no	—
194	D₄×C₃₁	—	4	2	no	—
195	D₄×C₃₁	—	8	1	no	—
196	D₄×C₃₁	—	8	2	no	—
197	D₄×C₃₁	—	8	2	no	—
198	D₄×C₃₁	—	8	2	no	—
199	D₄×C₃₁	—	8	2	no	—
200	D₄×C₃₁	—	8	2	no	—
201	D₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	complemented
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	1	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	8	1	no	—
213	C₂×C₄×C₃₁	—	8	2	no	—
214	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
215	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
216	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
217	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
220	C₈×C₃₁	—	2	2	no	—
221	C₈×C₃₁	—	4	2	no	—
222	C₈×C₃₁	—	4	2	no	—
223	C₈×C₃₁	—	4	1	no	—
224	C₈×C₃₁	—	4	1	no	—
225	C₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	complemented
226	#3151	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#3115	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#3038	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#2970	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1688	C₂×C₃₁	1	2	yes	complemented
231	#1686	C₂×C₃₁	1	2	yes	complemented
232	#1667	C₂×C₃₁	1	2	yes	complemented
233	#1665	C₂×C₃₁	1	2	yes	complemented
234	#1569	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#1557	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#1555	C₂×C₃₁	1	1	yes	characteristic, complemented
237	C₂²×C₃₁	—	2	1	no	—
238	C₂²×C₃₁	—	2	1	no	—
239	C₂²×C₃₁	—	2	2	no	—
240	C₂²×C₃₁	—	2	1	no	—
241	C₂²×C₃₁	—	4	1	no	—
242	C₂²×C₃₁	—	4	1	no	—
243	C₂²×C₃₁	—	8	1	no	—
244	C₂²×C₃₁	—	8	2	no	—
245	C₂²×C₃₁	—	8	2	no	—
246	C₂²×C₃₁	—	8	1	no	—
247	C₂²×C₃₁	—	8	2	no	—
248	C₂²×C₃₁	—	8	1	no	—
249	C₂²×C₃₁	—	16	4	no	—
250	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
251	C₄×C₃₁	—	2	1	no	—
252	C₄×C₃₁	—	2	1	no	—
253	C₄×C₃₁	—	2	1	no	—
254	C₄×C₃₁	—	2	1	no	—
255	C₄×C₃₁	—	4	1	no	—
256	C₄×C₃₁	—	4	1	no	—
257	C₄×C₃₁	—	4	1	no	—
258	C₄×C₃₁	—	8	1	no	—
259	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
260	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
261	C₄×C₃₁	#575	1	1	yes	characteristic
262	C₂×AGL(1,ℤ/8)	—	2	1	no	—
263	C₂×AGL(1,ℤ/8)	—	2	2	no	—
264	#532	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
266	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
267	#496	—	2	1	no	—
268	#496	—	2	1	no	—
269	#493	C₂²×C₃₁	1	2	yes	—
270	#492	C₂²×C₃₁	1	2	yes	—
271	#482	—	2	1	no	—
272	#475	C₂²×C₃₁	1	1	yes	characteristic, complemented
273	#474	C₂²×C₃₁	1	1	yes	characteristic, complemented
274	#460	—	2	1	no	—
275	#460	C₂²×C₃₁	1	2	yes	—
276	#460	C₂²×C₃₁	1	1	yes	characteristic, complemented
277	#459	—	2	1	no	—
278	#459	C₂²×C₃₁	1	2	yes	—
279	#459	C₂²×C₃₁	1	1	yes	characteristic, complemented
280	#449	—	2	1	no	—
281	#446	C₂²×C₃₁	1	2	yes	complemented
282	#442	—	2	1	no	—
283	#440	—	2	1	no	—
284	#435	—	2	2	no	—
285	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
286	#432	—	2	1	no	—
287	#432	C₂²×C₃₁	1	1	yes	characteristic
288	#432	C₂²×C₃₁	1	1	yes	characteristic
289	#423	C₂²×C₃₁	1	1	yes	characteristic
290	#418	—	2	1	no	—
291	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×#97	—	2	1	no	—
293	#335	C₂²×C₃₁	1	2	yes	complemented
294	#334	C₂²×C₃₁	1	2	yes	complemented
295	#332	C₂²×C₃₁	1	2	yes	complemented
296	#331	C₂²×C₃₁	1	2	yes	—
297	#331	C₂²×C₃₁	1	2	yes	—
298	#331	C₂²×C₃₁	1	2	yes	complemented
299	C₂×C₃₁	—	4	1	no	—
300	C₂×C₃₁	—	4	2	no	—
301	C₂×C₃₁	—	16	1	no	—
302	C₂×C₃₁	—	16	2	no	—
303	C₂×C₃₁	#3038	1	1	yes	characteristic
304	C₂×C₃₁	#3055	1	1	yes	characteristic
305	C₂×C₃₁	#3068	1	1	yes	characteristic
306	C₂×#41	—	2	1	no	—
307	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
308	C₂²×D₄	—	2	1	no	—
309	C₂²×D₄	—	2	1	no	—
310	C₂²×D₄	—	2	1	no	—
311	AGL(1,ℤ/8)	—	4	8	no	—
312	AGL(1,ℤ/8)	—	4	4	no	—
313	C₂×SD₁₆	—	2	2	no	—
314	C₂×SD₁₆	—	2	1	no	—
315	C₂×SD₁₆	—	2	2	no	—
316	C₂×SD₁₆	—	2	2	no	—
317	C₂×D₈	—	2	2	no	—
318	C₂×D₈	—	2	1	no	—
319	C₂×D₈	—	2	2	no	—
320	C₂×D₈	—	2	2	no	—
321	C₂×M₁₆	—	2	1	no	—
322	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
323	C₂²×C₈	—	2	2	no	—
324	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
325	#127	—	2	1	no	—
326	#127	—	2	1	no	—
327	#127	—	2	1	no	—
328	#127	C₂³×C₃₁	1	2	yes	—
329	#127	C₂³×C₃₁	1	1	yes	characteristic
330	#124	—	2	1	no	—
331	#121	—	2	1	no	—
332	#121	—	2	1	no	—
333	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
334	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
335	C₄×D₄	—	2	1	no	—
336	C₄×D₄	—	2	1	no	—
337	C₄×D₄	—	2	1	no	—
338	C₄×D₄	—	2	1	no	—
339	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
340	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
341	#117	—	2	1	no	—
342	#117	D₄×C₃₁	1	1	yes	characteristic
343	C₂×C₄²	—	2	1	no	—
344	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
345	#107	—	2	1	no	—
346	#106	—	2	1	no	—
347	C₄⋊C₈	—	2	2	no	—
348	C₄⋊C₈	—	2	1	no	—
349	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
350	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
351	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
352	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
353	#103	—	2	1	no	—
354	#102	—	2	1	no	—
355	#102	—	2	2	no	—
356	#102	—	2	2	no	—
357	#102	—	2	1	no	—
358	#102	—	2	1	no	—
359	C₂²⋊C₈	—	2	1	no	—
360	C₂²⋊C₈	—	2	1	no	—
361	C₂²⋊C₈	—	2	2	no	—
362	#97	—	2	2	no	—
363	#97	—	2	1	no	—
364	#97	—	2	1	no	—
365	#97	D₄×C₃₁	1	1	yes	characteristic
366	#97	D₄×C₃₁	1	1	yes	characteristic
367	C₄×C₈	C₂³×C₃₁	1	2	yes	—
368	C₃₁	#19312	1	1	yes	characteristic, direct product factor, Sylow
369	C₂⁴	—	4	1	no	—
370	#41	—	2	2	no	—
371	#41	—	4	1	no	—
372	#41	—	4	4	no	—
373	C₂×Q₈	—	2	1	no	—
374	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	1	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	4	4	no	—
383	C₂×D₄	—	4	2	no	—
384	C₂×D₄	—	4	2	no	—
385	C₂×D₄	—	4	1	no	—
386	C₂×D₄	—	4	1	no	—
387	C₂×D₄	—	4	1	no	—
388	C₂×D₄	—	4	1	no	—
389	C₂×D₄	—	8	2	no	—
390	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	4	1	no	—
395	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
397	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
398	SD₁₆	—	4	4	no	—
399	SD₁₆	—	4	2	no	—
400	SD₁₆	—	4	4	no	—
401	SD₁₆	—	4	4	no	—
402	D₈	—	4	4	no	—
403	D₈	—	4	2	no	—
404	D₈	—	4	4	no	—
405	D₈	—	4	4	no	—
406	M₁₆	—	2	4	no	—
407	M₁₆	—	4	2	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	4	4	no	—
413	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
414	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
420	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
421	C₂²⋊C₄	—	2	1	no	—
422	C₂²⋊C₄	—	2	1	no	—
423	C₂²⋊C₄	—	2	1	no	—
424	C₂²⋊C₄	—	4	1	no	—
425	C₂²⋊C₄	—	4	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	1	no	—
428	C₄²	—	2	1	no	—
429	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
430	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
431	C₄²	#41×C₃₁	1	1	yes	characteristic
432	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
433	C₂³	—	2	1	no	—
434	C₂³	—	4	1	no	—
435	C₂³	—	4	2	no	—
436	C₂³	—	8	2	no	—
437	C₂³	—	8	2	no	—
438	C₂³	—	8	2	no	—
439	C₂³	C₃₁×#121	1	1	yes	characteristic
440	Q₈	—	2	1	no	—
441	Q₈	—	2	1	no	—
442	Q₈	—	4	2	no	—
443	D₄	—	2	1	no	—
444	D₄	—	4	2	no	—
445	D₄	—	4	2	no	—
446	D₄	—	4	2	no	—
447	D₄	—	4	2	no	—
448	D₄	—	4	2	no	—
449	D₄	—	4	1	no	—
450	D₄	—	4	1	no	—
451	D₄	—	4	2	no	—
452	D₄	—	8	1	no	—
453	D₄	—	8	2	no	—
454	D₄	—	8	2	no	—
455	D₄	—	8	2	no	—
456	D₄	—	8	2	no	—
457	D₄	—	8	2	no	—
458	D₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	complemented
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	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	8	1	no	—
470	C₂×C₄	—	8	2	no	—
471	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
472	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
473	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
474	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
475	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
476	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
477	C₈	—	2	2	no	—
478	C₈	—	4	2	no	—
479	C₈	—	4	2	no	—
480	C₈	—	4	1	no	—
481	C₈	—	4	1	no	—
482	C₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	complemented
483	C₂²	—	2	1	no	—
484	C₂²	—	2	1	no	—
485	C₂²	—	2	2	no	—
486	C₂²	—	2	1	no	—
487	C₂²	—	4	1	no	—
488	C₂²	—	4	1	no	—
489	C₂²	—	8	1	no	—
490	C₂²	—	8	2	no	—
491	C₂²	—	8	2	no	—
492	C₂²	—	8	1	no	—
493	C₂²	—	8	2	no	—
494	C₂²	—	8	1	no	—
495	C₂²	—	16	4	no	—
496	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
497	C₄	—	2	1	no	—
498	C₄	—	2	1	no	—
499	C₄	—	2	1	no	—
500	C₄	—	2	1	no	—
501	C₄	—	4	1	no	—
502	C₄	—	4	1	no	—
503	C₄	—	4	1	no	—
504	C₄	—	8	1	no	—
505	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
506	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
507	C₄	C₃₁×#575	1	1	yes	characteristic
508	C₂	—	4	1	no	—
509	C₂	—	4	2	no	—
510	C₂	—	16	2	no	—
511	C₂	—	16	1	no	—
512	C₂	C₃₁×#3038	1	1	yes	characteristic
513	C₂	C₃₁×#3055	1	1	yes	characteristic
514	C₂	C₃₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ 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
1. C₃₁×#3151	C₂	semidirect
2. C₃₁×#3115	C₂	semidirect
3. C₃₁×#3038	C₂	semidirect
4. C₃₁×#2970	C₂	semidirect
5. C₃₁×#1688	C₂	semidirect
6. C₃₁×#1686	C₂	semidirect
7. C₃₁×#1667	C₂	semidirect
8. C₃₁×#1665	C₂	semidirect
9. C₃₁×#1569	C₂	semidirect
10. C₃₁×#1557	C₂	semidirect
11. C₃₁×#1555	C₂	semidirect
22. C₃₁×#475	C₂²	semidirect
23. C₃₁×#474	C₂²	semidirect
26. C₃₁×#460	C₂²	semidirect
29. C₃₁×#459	C₂²	semidirect
31. C₃₁×#446	C₂²	semidirect
35. C₈×D₄×C₃₁	C₂²	semidirect
43. C₃₁×#335	C₂²	semidirect
44. C₃₁×#334	C₂²	semidirect
45. C₃₁×#332	C₂²	semidirect
48. C₃₁×#331	C₂²	semidirect
175. #19312	C₃₁	direct (N × H)
201. D₄×C₃₁	AGL(1,ℤ/8)	semidirect
225. C₈×C₃₁	AGL(1,ℤ/8)	semidirect
226. #3151	C₂×C₃₁	semidirect
227. #3115	C₂×C₃₁	semidirect
228. #3038	C₂×C₃₁	semidirect
229. #2970	C₂×C₃₁	semidirect
230. #1688	C₂×C₃₁	semidirect
231. #1686	C₂×C₃₁	semidirect
232. #1667	C₂×C₃₁	semidirect
233. #1665	C₂×C₃₁	semidirect
234. #1569	C₂×C₃₁	semidirect
235. #1557	C₂×C₃₁	semidirect
236. #1555	C₂×C₃₁	semidirect
272. #475	C₂²×C₃₁	semidirect
273. #474	C₂²×C₃₁	semidirect
276. #460	C₂²×C₃₁	semidirect
279. #459	C₂²×C₃₁	semidirect
281. #446	C₂²×C₃₁	semidirect
285. C₈×D₄	C₂²×C₃₁	semidirect
293. #335	C₂²×C₃₁	semidirect
294. #334	C₂²×C₃₁	semidirect
295. #332	C₂²×C₃₁	semidirect
298. #331	C₂²×C₃₁	semidirect
368. C₃₁	#19312	direct (N × H)
458. D₄	C₃₁×AGL(1,ℤ/8)	semidirect
482. C₈	C₃₁×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3151	2
C₃₁×#3115	2
C₃₁×#3038	2
C₃₁×#2970	2
C₃₁×#1688	2
C₃₁×#1686	2
C₃₁×#1667	2
C₃₁×#1665	2
C₃₁×#1569	2
C₃₁×#1557	2
C₃₁×#1555	2
#19312	31

Subgroups — C31×#19312 #21476071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19312 #21476071