Subgroups — C₃₁×#19170 #21475929

Order 7936.

766 conjugacy classes of subgroups.

The subgroup lattice has 524 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
2	C₃₁×#3103	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#460	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈×M₁₆×C₃₁	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×#1668	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1666	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1547	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1545	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	1	no	—
14	C₂²×SD₁₆×C₃₁	—	2	1	no	—
15	C₃₁×#532	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#496	—	2	2	no	—
19	C₃₁×#493	C₂²	1	2	yes	—
20	C₃₁×#492	C₂²	1	2	yes	—
21	C₃₁×#481	—	2	1	no	—
22	C₃₁×#475	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#475	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×#460	—	2	1	no	—
25	C₃₁×#460	—	2	1	no	—
26	C₃₁×#460	C₂²	1	2	yes	—
27	C₃₁×#460	C₂²	1	2	yes	—
28	C₃₁×#460	C₂²	1	1	yes	characteristic, complemented
29	C₃₁×#460	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×#448	—	2	1	no	—
31	C₃₁×#446	—	2	2	no	—
32	C₃₁×#446	C₂²	1	2	yes	complemented
33	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
34	C₄×SD₁₆×C₃₁	—	2	1	no	—
35	C₄×SD₁₆×C₃₁	—	2	1	no	—
36	C₃₁×#431	C₂²	1	1	yes	characteristic
37	C₃₁×#423	—	2	1	no	—
38	C₃₁×#423	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#102	—	2	1	no	—
41	C₄×M₁₆×C₃₁	—	2	1	no	—
42	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
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	C₂²×Q₈×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	4	no	—
55	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
56	C₂×SD₁₆×C₃₁	—	2	1	no	—
57	C₂×SD₁₆×C₃₁	—	2	1	no	—
58	C₂×SD₁₆×C₃₁	—	2	2	no	—
59	C₂×SD₁₆×C₃₁	—	2	2	no	—
60	C₂×SD₁₆×C₃₁	—	2	2	no	—
61	C₂×SD₁₆×C₃₁	—	2	2	no	—
62	C₂×SD₁₆×C₃₁	—	4	4	no	—
63	C₂×D₈×C₃₁	—	2	2	no	—
64	C₂×D₈×C₃₁	—	2	2	no	—
65	C₂×M₁₆×C₃₁	—	2	1	no	—
66	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
67	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
68	C₂²×C₈×C₃₁	—	2	1	no	—
69	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
70	C₃₁×#127	—	2	1	no	—
71	C₃₁×#127	—	2	1	no	—
72	C₃₁×#127	—	2	1	no	—
73	C₃₁×#127	C₂³	1	2	yes	—
74	C₃₁×#127	C₂³	1	1	yes	characteristic
75	C₃₁×#124	—	2	1	no	—
76	C₃₁×#121	—	2	1	no	—
77	C₃₁×#121	—	2	1	no	—
78	C₄×Q₈×C₃₁	—	2	1	no	—
79	C₄×Q₈×C₃₁	—	2	1	no	—
80	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
81	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
82	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₄×D₄×C₃₁	—	2	1	no	—
85	C₄×D₄×C₃₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
87	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
88	C₂×C₄²×C₃₁	—	2	1	no	—
89	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
90	C₃₁×#106	—	2	1	no	—
91	C₃₁×#106	—	2	1	no	—
92	C₃₁×C₄⋊C₈	—	2	2	no	—
93	C₃₁×C₄⋊C₈	—	2	2	no	—
94	C₃₁×C₄⋊C₈	—	2	1	no	—
95	C₃₁×C₄⋊C₈	—	2	1	no	—
96	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
97	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
98	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
99	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
100	C₃₁×#103	—	2	1	no	—
101	C₃₁×#103	—	2	1	no	—
102	C₃₁×#102	—	2	2	no	—
103	C₃₁×#102	—	2	2	no	—
104	C₃₁×#102	—	2	1	no	—
105	C₃₁×#102	—	2	1	no	—
106	C₃₁×C₂²⋊C₈	—	2	1	no	—
107	C₃₁×#97	—	2	2	no	—
108	C₃₁×#97	D₄	1	2	yes	—
109	C₄×C₈×C₃₁	—	2	2	no	—
110	C₄×C₈×C₃₁	—	2	1	no	—
111	C₄×C₈×C₃₁	C₂³	1	2	yes	—
112	C₂⁴×C₃₁	—	4	1	no	—
113	#41×C₃₁	—	4	2	no	—
114	#41×C₃₁	—	4	2	no	—
115	C₂×Q₈×C₃₁	—	2	1	no	—
116	C₂×Q₈×C₃₁	—	2	2	no	—
117	C₂×Q₈×C₃₁	—	2	1	no	—
118	C₂×Q₈×C₃₁	—	4	1	no	—
119	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×D₄×C₃₁	—	2	1	no	—
122	C₂×D₄×C₃₁	—	2	2	no	—
123	C₂×D₄×C₃₁	—	2	1	no	—
124	C₂×D₄×C₃₁	—	2	2	no	—
125	C₂×D₄×C₃₁	—	2	1	no	—
126	C₂×D₄×C₃₁	—	2	2	no	—
127	C₂×D₄×C₃₁	—	4	2	no	—
128	C₂×D₄×C₃₁	—	4	2	no	—
129	C₂×D₄×C₃₁	—	4	1	no	—
130	C₂×D₄×C₃₁	—	4	1	no	—
131	C₂×D₄×C₃₁	—	4	1	no	—
132	C₂×D₄×C₃₁	—	4	1	no	—
133	C₂×D₄×C₃₁	—	4	2	no	—
134	C₂×D₄×C₃₁	—	4	2	no	—
135	C₂×D₄×C₃₁	—	8	2	no	—
136	C₂²×C₄×C₃₁	—	2	1	no	—
137	C₂²×C₄×C₃₁	—	2	1	no	—
138	C₂²×C₄×C₃₁	—	4	1	no	—
139	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
142	SD₁₆×C₃₁	—	4	4	no	—
143	SD₁₆×C₃₁	—	4	2	no	—
144	SD₁₆×C₃₁	—	4	4	no	—
145	SD₁₆×C₃₁	—	4	2	no	—
146	SD₁₆×C₃₁	—	4	4	no	—
147	SD₁₆×C₃₁	—	4	4	no	—
148	D₈×C₃₁	—	4	4	no	—
149	D₈×C₃₁	—	4	4	no	—
150	M₁₆×C₃₁	—	2	2	no	—
151	M₁₆×C₃₁	—	4	2	no	—
152	M₁₆×C₃₁	SD₁₆	1	4	yes	complemented
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	2	1	no	—
155	C₂×C₈×C₃₁	—	2	1	no	—
156	C₂×C₈×C₃₁	—	2	2	no	—
157	C₂×C₈×C₃₁	—	4	2	no	—
158	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
159	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
160	C₄⋊C₄×C₃₁	—	2	1	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₄⋊C₄×C₃₁	—	2	1	no	—
164	C₄⋊C₄×C₃₁	—	2	1	no	—
165	C₄⋊C₄×C₃₁	—	2	1	no	—
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
168	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₂²⋊C₄×C₃₁	—	4	1	no	—
170	C₂²⋊C₄×C₃₁	—	4	1	no	—
171	C₄²×C₃₁	—	2	1	no	—
172	C₄²×C₃₁	—	2	1	no	—
173	C₄²×C₃₁	—	2	1	no	—
174	C₄²×C₃₁	—	2	1	no	—
175	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₄²×C₃₁	#41	1	1	yes	characteristic
178	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
179	#19170	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂³×C₃₁	—	4	1	no	—
181	C₂³×C₃₁	—	4	2	no	—
182	C₂³×C₃₁	—	8	2	no	—
183	C₂³×C₃₁	—	8	2	no	—
184	C₂³×C₃₁	—	8	2	no	—
185	C₂³×C₃₁	#121	1	1	yes	characteristic
186	Q₈×C₃₁	—	2	1	no	—
187	Q₈×C₃₁	—	2	1	no	—
188	Q₈×C₃₁	—	2	1	no	—
189	Q₈×C₃₁	—	4	2	no	—
190	Q₈×C₃₁	—	4	1	no	—
191	Q₈×C₃₁	—	4	1	no	—
192	Q₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	complemented
193	D₄×C₃₁	—	4	1	no	—
194	D₄×C₃₁	—	4	2	no	—
195	D₄×C₃₁	—	4	1	no	—
196	D₄×C₃₁	—	4	2	no	—
197	D₄×C₃₁	—	4	2	no	—
198	D₄×C₃₁	—	4	1	no	—
199	D₄×C₃₁	—	4	1	no	—
200	D₄×C₃₁	—	4	2	no	—
201	D₄×C₃₁	—	8	1	no	—
202	D₄×C₃₁	—	8	2	no	—
203	D₄×C₃₁	—	8	2	no	—
204	D₄×C₃₁	—	8	2	no	—
205	D₄×C₃₁	—	8	2	no	—
206	D₄×C₃₁	—	8	2	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	1	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	1	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	1	no	—
216	C₂×C₄×C₃₁	—	8	1	no	—
217	C₂×C₄×C₃₁	—	8	2	no	—
218	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
219	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
220	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
222	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
223	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
226	C₈×C₃₁	—	2	2	no	—
227	C₈×C₃₁	—	4	2	no	—
228	C₈×C₃₁	—	4	2	no	—
229	C₈×C₃₁	—	4	1	no	—
230	C₈×C₃₁	—	4	1	no	—
231	C₈×C₃₁	C₂×SD₁₆	1	4	yes	complemented
232	#3151	C₂×C₃₁	1	1	yes	characteristic
233	#3103	C₂×C₃₁	1	1	yes	characteristic, complemented
234	C₂×#460	C₂×C₃₁	1	1	yes	characteristic, complemented
235	Q₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#1688	C₂×C₃₁	1	2	yes	complemented
237	#1686	C₂×C₃₁	1	2	yes	complemented
238	#1668	C₂×C₃₁	1	2	yes	complemented
239	#1666	C₂×C₃₁	1	2	yes	complemented
240	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#1547	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#1545	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂²×C₃₁	—	2	1	no	—
244	C₂²×C₃₁	—	2	1	no	—
245	C₂²×C₃₁	—	8	1	no	—
246	C₂²×C₃₁	—	8	2	no	—
247	C₂²×C₃₁	—	8	2	no	—
248	C₂²×C₃₁	—	8	1	no	—
249	C₂²×C₃₁	—	8	2	no	—
250	C₂²×C₃₁	—	8	1	no	—
251	C₂²×C₃₁	—	16	4	no	—
252	C₂²×C₃₁	#460	1	2	yes	—
253	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
254	C₄×C₃₁	—	2	1	no	—
255	C₄×C₃₁	—	2	2	no	—
256	C₄×C₃₁	—	2	1	no	—
257	C₄×C₃₁	—	2	1	no	—
258	C₄×C₃₁	—	4	1	no	—
259	C₄×C₃₁	—	4	1	no	—
260	C₄×C₃₁	—	4	1	no	—
261	C₄×C₃₁	—	8	1	no	—
262	C₄×C₃₁	#460	1	2	yes	—
263	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
264	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
265	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
266	C₄×C₃₁	#575	1	1	yes	characteristic
267	C₂×AGL(1,ℤ/8)	—	2	1	no	—
268	C₂×AGL(1,ℤ/8)	—	2	1	no	—
269	C₂²×SD₁₆	—	2	1	no	—
270	#532	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
273	#496	—	2	2	no	—
274	#493	C₂²×C₃₁	1	2	yes	—
275	#492	C₂²×C₃₁	1	2	yes	—
276	#481	—	2	1	no	—
277	#475	C₂²×C₃₁	1	1	yes	characteristic, complemented
278	#475	C₂²×C₃₁	1	1	yes	characteristic, complemented
279	#460	—	2	1	no	—
280	#460	—	2	1	no	—
281	#460	C₂²×C₃₁	1	2	yes	—
282	#460	C₂²×C₃₁	1	2	yes	—
283	#460	C₂²×C₃₁	1	1	yes	characteristic, complemented
284	#460	C₂²×C₃₁	1	1	yes	characteristic, complemented
285	#448	—	2	1	no	—
286	#446	—	2	2	no	—
287	#446	C₂²×C₃₁	1	2	yes	complemented
288	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
289	C₄×SD₁₆	—	2	1	no	—
290	C₄×SD₁₆	—	2	1	no	—
291	#431	C₂²×C₃₁	1	1	yes	characteristic
292	#423	—	2	1	no	—
293	#423	C₂²×C₃₁	1	1	yes	characteristic
294	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×#102	—	2	1	no	—
296	C₄×M₁₆	—	2	1	no	—
297	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
298	#335	C₂²×C₃₁	1	2	yes	complemented
299	#334	C₂²×C₃₁	1	2	yes	complemented
300	#332	C₂²×C₃₁	1	2	yes	complemented
301	#331	C₂²×C₃₁	1	2	yes	—
302	#331	C₂²×C₃₁	1	2	yes	—
303	#331	C₂²×C₃₁	1	2	yes	complemented
304	C₂×C₃₁	—	2	2	no	—
305	C₂×C₃₁	—	16	1	no	—
306	C₂×C₃₁	—	16	2	no	—
307	C₂×C₃₁	C₂×#460	1	1	yes	characteristic
308	C₂×C₃₁	#3055	1	1	yes	characteristic
309	C₂×C₃₁	#3068	1	1	yes	characteristic
310	C₂×#41	—	2	1	no	—
311	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
312	C₂²×D₄	—	2	1	no	—
313	C₂²×D₄	—	2	1	no	—
314	C₂²×D₄	—	2	1	no	—
315	AGL(1,ℤ/8)	—	4	4	no	—
316	AGL(1,ℤ/8)	—	4	4	no	—
317	C₂×SD₁₆	—	2	2	no	—
318	C₂×SD₁₆	—	2	2	no	—
319	C₂×SD₁₆	—	2	1	no	—
320	C₂×SD₁₆	—	2	1	no	—
321	C₂×SD₁₆	—	2	2	no	—
322	C₂×SD₁₆	—	2	2	no	—
323	C₂×SD₁₆	—	4	4	no	—
324	C₂×D₈	—	2	2	no	—
325	C₂×D₈	—	2	2	no	—
326	C₂×M₁₆	—	2	1	no	—
327	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
328	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
329	C₂²×C₈	—	2	1	no	—
330	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
331	#127	—	2	1	no	—
332	#127	—	2	1	no	—
333	#127	—	2	1	no	—
334	#127	C₂³×C₃₁	1	2	yes	—
335	#127	C₂³×C₃₁	1	1	yes	characteristic
336	#124	—	2	1	no	—
337	#121	—	2	1	no	—
338	#121	—	2	1	no	—
339	C₄×Q₈	—	2	1	no	—
340	C₄×Q₈	—	2	1	no	—
341	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
342	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
343	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
344	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
345	C₄×D₄	—	2	1	no	—
346	C₄×D₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	1	no	—
348	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
349	C₂×C₄²	—	2	1	no	—
350	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
351	#106	—	2	1	no	—
352	#106	—	2	1	no	—
353	C₄⋊C₈	—	2	2	no	—
354	C₄⋊C₈	—	2	2	no	—
355	C₄⋊C₈	—	2	1	no	—
356	C₄⋊C₈	—	2	1	no	—
357	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
358	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
359	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
360	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
361	#103	—	2	1	no	—
362	#103	—	2	1	no	—
363	#102	—	2	2	no	—
364	#102	—	2	2	no	—
365	#102	—	2	1	no	—
366	#102	—	2	1	no	—
367	C₂²⋊C₈	—	2	1	no	—
368	#97	—	2	2	no	—
369	#97	D₄×C₃₁	1	2	yes	—
370	C₄×C₈	—	2	2	no	—
371	C₄×C₈	—	2	1	no	—
372	C₄×C₈	C₂³×C₃₁	1	2	yes	—
373	C₃₁	#19170	1	1	yes	characteristic, direct product factor, Sylow
374	C₂⁴	—	4	1	no	—
375	#41	—	4	2	no	—
376	#41	—	4	2	no	—
377	C₂×Q₈	—	2	1	no	—
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	—	2	1	no	—
380	C₂×Q₈	—	4	1	no	—
381	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
383	C₂×D₄	—	2	1	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	4	2	no	—
390	C₂×D₄	—	4	2	no	—
391	C₂×D₄	—	4	1	no	—
392	C₂×D₄	—	4	1	no	—
393	C₂×D₄	—	4	1	no	—
394	C₂×D₄	—	4	1	no	—
395	C₂×D₄	—	4	2	no	—
396	C₂×D₄	—	4	2	no	—
397	C₂×D₄	—	8	2	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	4	1	no	—
401	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
403	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
404	SD₁₆	—	4	4	no	—
405	SD₁₆	—	4	2	no	—
406	SD₁₆	—	4	4	no	—
407	SD₁₆	—	4	2	no	—
408	SD₁₆	—	4	4	no	—
409	SD₁₆	—	4	4	no	—
410	D₈	—	4	4	no	—
411	D₈	—	4	4	no	—
412	M₁₆	—	2	2	no	—
413	M₁₆	—	4	2	no	—
414	M₁₆	SD₁₆×C₃₁	1	4	yes	complemented
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	4	2	no	—
420	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
421	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
430	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
431	C₂²⋊C₄	—	4	1	no	—
432	C₂²⋊C₄	—	4	1	no	—
433	C₄²	—	2	1	no	—
434	C₄²	—	2	1	no	—
435	C₄²	—	2	1	no	—
436	C₄²	—	2	1	no	—
437	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
438	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
439	C₄²	#41×C₃₁	1	1	yes	characteristic
440	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
441	C₂³	—	4	1	no	—
442	C₂³	—	4	2	no	—
443	C₂³	—	8	2	no	—
444	C₂³	—	8	2	no	—
445	C₂³	—	8	2	no	—
446	C₂³	C₃₁×#121	1	1	yes	characteristic
447	Q₈	—	2	1	no	—
448	Q₈	—	2	1	no	—
449	Q₈	—	2	1	no	—
450	Q₈	—	4	2	no	—
451	Q₈	—	4	1	no	—
452	Q₈	—	4	1	no	—
453	Q₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	complemented
454	D₄	—	4	1	no	—
455	D₄	—	4	2	no	—
456	D₄	—	4	1	no	—
457	D₄	—	4	2	no	—
458	D₄	—	4	2	no	—
459	D₄	—	4	1	no	—
460	D₄	—	4	1	no	—
461	D₄	—	4	2	no	—
462	D₄	—	8	1	no	—
463	D₄	—	8	2	no	—
464	D₄	—	8	2	no	—
465	D₄	—	8	2	no	—
466	D₄	—	8	2	no	—
467	D₄	—	8	2	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	8	1	no	—
478	C₂×C₄	—	8	2	no	—
479	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
480	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
481	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
482	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
483	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
484	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
486	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
487	C₈	—	2	2	no	—
488	C₈	—	4	2	no	—
489	C₈	—	4	2	no	—
490	C₈	—	4	1	no	—
491	C₈	—	4	1	no	—
492	C₈	C₂×SD₁₆×C₃₁	1	4	yes	complemented
493	C₂²	—	2	1	no	—
494	C₂²	—	2	1	no	—
495	C₂²	—	8	1	no	—
496	C₂²	—	8	2	no	—
497	C₂²	—	8	2	no	—
498	C₂²	—	8	1	no	—
499	C₂²	—	8	2	no	—
500	C₂²	—	8	1	no	—
501	C₂²	—	16	4	no	—
502	C₂²	C₃₁×#460	1	2	yes	—
503	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
504	C₄	—	2	1	no	—
505	C₄	—	2	2	no	—
506	C₄	—	2	1	no	—
507	C₄	—	2	1	no	—
508	C₄	—	4	1	no	—
509	C₄	—	4	1	no	—
510	C₄	—	4	1	no	—
511	C₄	—	8	1	no	—
512	C₄	C₃₁×#460	1	2	yes	—
513	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
514	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
515	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
516	C₄	C₃₁×#575	1	1	yes	characteristic
517	C₂	—	2	2	no	—
518	C₂	—	16	2	no	—
519	C₂	—	16	1	no	—
520	C₂	C₂×C₃₁×#460	1	1	yes	characteristic
521	C₂	C₃₁×#3055	1	1	yes	characteristic
522	C₂	C₃₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3151	C₂
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
2. C₃₁×#3103	C₂	semidirect
3. C₂×C₃₁×#460	C₂	semidirect
4. Q₈×M₁₆×C₃₁	C₂	semidirect
5. C₃₁×#1688	C₂	semidirect
6. C₃₁×#1686	C₂	semidirect
7. C₃₁×#1668	C₂	semidirect
8. C₃₁×#1666	C₂	semidirect
9. C₃₁×#1565	C₂	semidirect
10. C₃₁×#1547	C₂	semidirect
11. C₃₁×#1545	C₂	semidirect
22. C₃₁×#475	C₂²	semidirect
23. C₃₁×#475	C₂²	semidirect
28. C₃₁×#460	C₂²	semidirect
29. C₃₁×#460	C₂²	semidirect
32. C₃₁×#446	C₂²	semidirect
33. C₈×Q₈×C₃₁	C₂²	semidirect
43. C₃₁×#335	C₂²	semidirect
44. C₃₁×#334	C₂²	semidirect
45. C₃₁×#332	C₂²	semidirect
48. C₃₁×#331	C₂²	semidirect
152. M₁₆×C₃₁	SD₁₆	semidirect
179. #19170	C₃₁	direct (N × H)
192. Q₈×C₃₁	AGL(1,ℤ/8)	semidirect
231. C₈×C₃₁	C₂×SD₁₆	semidirect
233. #3103	C₂×C₃₁	semidirect
234. C₂×#460	C₂×C₃₁	semidirect
235. Q₈×M₁₆	C₂×C₃₁	semidirect
236. #1688	C₂×C₃₁	semidirect
237. #1686	C₂×C₃₁	semidirect
238. #1668	C₂×C₃₁	semidirect
239. #1666	C₂×C₃₁	semidirect
240. #1565	C₂×C₃₁	semidirect
241. #1547	C₂×C₃₁	semidirect
242. #1545	C₂×C₃₁	semidirect
277. #475	C₂²×C₃₁	semidirect
278. #475	C₂²×C₃₁	semidirect
283. #460	C₂²×C₃₁	semidirect
284. #460	C₂²×C₃₁	semidirect
287. #446	C₂²×C₃₁	semidirect
288. C₈×Q₈	C₂²×C₃₁	semidirect
298. #335	C₂²×C₃₁	semidirect
299. #334	C₂²×C₃₁	semidirect
300. #332	C₂²×C₃₁	semidirect
303. #331	C₂²×C₃₁	semidirect
373. C₃₁	#19170	direct (N × H)
414. M₁₆	SD₁₆×C₃₁	semidirect
453. Q₈	C₃₁×AGL(1,ℤ/8)	semidirect
492. C₈	C₂×SD₁₆×C₃₁	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×#3103	2
C₂×C₃₁×#460	2
Q₈×M₁₆×C₃₁	2
C₃₁×#1688	2
C₃₁×#1686	2
C₃₁×#1668	2
C₃₁×#1666	2
C₃₁×#1565	2
C₃₁×#1547	2
C₃₁×#1545	2
#19170	31

Subgroups — C31×#19170 #21475929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19170 #21475929