Subgroups — C₃₁×#19885 #21476644

Order 7936.

830 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3391	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3274	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1712	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1687	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1668	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1621	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1619	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
13	C₂²×D₈×C₃₁	—	2	1	no	—
14	C₃₁×#550	C₂²	1	2	yes	complemented
15	D₄²×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#493	—	2	1	no	—
17	C₃₁×#493	C₂²	1	1	yes	characteristic
18	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#493	C₂²	1	2	yes	complemented
20	C₃₁×#493	C₂²	1	2	yes	complemented
21	C₃₁×#469	—	2	2	no	—
22	C₃₁×#466	—	2	1	no	—
23	C₃₁×#460	C₂²	1	2	yes	complemented
24	C₃₁×#459	—	2	2	no	—
25	C₃₁×#459	C₂²	1	2	yes	—
26	C₃₁×#459	C₂²	1	2	yes	—
27	C₃₁×#459	C₂²	1	2	yes	complemented
28	C₃₁×#449	—	2	2	no	—
29	C₃₁×#447	—	2	2	no	—
30	C₃₁×#447	—	2	2	no	—
31	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
32	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×D₈×C₃₁	—	2	1	no	—
34	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#436	C₂²	1	2	yes	complemented
36	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#335	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#334	C₂²	1	1	yes	characteristic, complemented
39	C₃₁×#331	C₂²	1	2	yes	complemented
40	C₃₁×#331	C₂²	1	2	yes	complemented
41	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
45	C₂×#41×C₃₁	—	2	2	no	—
46	C₂²×D₄×C₃₁	—	2	2	no	—
47	C₂²×D₄×C₃₁	—	2	2	no	—
48	C₃₁×AGL(1,ℤ/8)	—	4	8	no	—
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
51	C₂×D₈×C₃₁	—	2	2	no	—
52	C₂×D₈×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	2	2	no	—
54	C₂×D₈×C₃₁	—	2	1	no	—
55	C₂×D₈×C₃₁	—	2	1	no	—
56	C₂×D₈×C₃₁	—	2	2	no	—
57	C₂×D₈×C₃₁	—	2	1	no	—
58	C₂×D₈×C₃₁	—	2	2	no	—
59	C₂×D₈×C₃₁	—	2	1	no	—
60	C₂×D₈×C₃₁	—	4	2	no	—
61	C₂×D₈×C₃₁	—	4	4	no	—
62	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
63	C₂×M₁₆×C₃₁	—	2	2	no	—
64	C₂²×C₈×C₃₁	—	2	1	no	—
65	C₃₁×#127	—	2	2	no	—
66	C₃₁×#127	C₂³	1	2	yes	—
67	C₃₁×#127	C₂³	1	2	yes	—
68	C₃₁×#121	—	2	2	no	—
69	C₃₁×#121	—	2	2	no	—
70	C₃₁×#121	—	2	2	no	—
71	C₃₁×#121	—	4	2	no	—
72	C₃₁×#120	—	4	2	no	—
73	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₄×D₄×C₃₁	—	2	2	no	—
75	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₄×D₄×C₃₁	C₂³	1	2	yes	—
77	C₃₁×#107	D₄	1	1	yes	characteristic
78	C₃₁×#106	D₄	1	1	yes	characteristic
79	C₃₁×C₄⋊C₈	—	2	1	no	—
80	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
81	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
82	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
83	C₃₁×#103	D₄	1	1	yes	characteristic
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	—	2	2	no	—
88	C₃₁×#102	D₄	1	2	yes	—
89	C₃₁×#102	D₄	1	1	yes	characteristic
90	C₃₁×C₂²⋊C₈	—	2	1	no	—
91	C₃₁×C₂²⋊C₈	—	2	2	no	—
92	C₄×C₈×C₃₁	—	2	1	no	—
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
96	C₂⁴×C₃₁	—	4	2	no	—
97	#41×C₃₁	—	4	4	no	—
98	#41×C₃₁	—	8	2	no	—
99	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	4	2	no	—
107	C₂×D₄×C₃₁	—	4	2	no	—
108	C₂×D₄×C₃₁	—	4	2	no	—
109	C₂×D₄×C₃₁	—	4	2	no	—
110	C₂×D₄×C₃₁	—	4	4	no	—
111	C₂×D₄×C₃₁	—	4	4	no	—
112	C₂×D₄×C₃₁	—	8	2	no	—
113	C₂×D₄×C₃₁	—	8	2	no	—
114	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
115	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	4	2	no	—
119	SD₁₆×C₃₁	—	4	4	no	—
120	SD₁₆×C₃₁	D₈	1	4	yes	complemented
121	D₈×C₃₁	—	2	2	no	—
122	D₈×C₃₁	—	4	4	no	—
123	D₈×C₃₁	—	4	4	no	—
124	D₈×C₃₁	—	4	4	no	—
125	D₈×C₃₁	—	4	2	no	—
126	D₈×C₃₁	—	4	2	no	—
127	D₈×C₃₁	—	8	1	no	—
128	D₈×C₃₁	—	8	1	no	—
129	D₈×C₃₁	—	8	2	no	—
130	D₈×C₃₁	—	8	4	no	—
131	D₈×C₃₁	—	8	2	no	—
132	M₁₆×C₃₁	—	4	4	no	—
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	4	2	no	—
138	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₄⋊C₄×C₃₁	—	2	1	no	—
143	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
145	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂²⋊C₄×C₃₁	—	2	1	no	—
147	C₂²⋊C₄×C₃₁	—	2	2	no	—
148	C₂²⋊C₄×C₃₁	—	4	2	no	—
149	C₂²⋊C₄×C₃₁	—	4	2	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
152	#19885	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂³×C₃₁	—	2	1	no	—
154	C₂³×C₃₁	—	2	2	no	—
155	C₂³×C₃₁	—	4	2	no	—
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	—	8	4	no	—
158	C₂³×C₃₁	—	8	4	no	—
159	C₂³×C₃₁	—	16	2	no	—
160	Q₈×C₃₁	—	2	1	no	—
161	Q₈×C₃₁	C₂×D₈	1	2	yes	complemented
162	D₄×C₃₁	—	2	2	no	—
163	D₄×C₃₁	—	2	1	no	—
164	D₄×C₃₁	—	2	2	no	—
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	4	2	no	—
167	D₄×C₃₁	—	4	2	no	—
168	D₄×C₃₁	—	4	2	no	—
169	D₄×C₃₁	—	4	2	no	—
170	D₄×C₃₁	—	4	2	no	—
171	D₄×C₃₁	—	4	2	no	—
172	D₄×C₃₁	—	4	2	no	—
173	D₄×C₃₁	—	8	4	no	—
174	D₄×C₃₁	—	8	2	no	—
175	D₄×C₃₁	—	8	2	no	—
176	D₄×C₃₁	—	8	4	no	—
177	D₄×C₃₁	—	8	4	no	—
178	D₄×C₃₁	—	16	2	no	—
179	D₄×C₃₁	C₂×D₈	1	2	yes	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	8	4	no	—
189	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
192	C₈×C₃₁	—	2	2	no	—
193	C₈×C₃₁	—	2	1	no	—
194	C₈×C₃₁	—	2	1	no	—
195	C₈×C₃₁	—	4	2	no	—
196	C₈×C₃₁	—	4	1	no	—
197	C₈×C₃₁	—	4	1	no	—
198	C₈×C₃₁	C₂×D₈	1	2	yes	complemented
199	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	complemented
200	#3391	C₂×C₃₁	1	1	yes	characteristic, complemented
201	D₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#3274	C₂×C₃₁	1	2	yes	complemented
203	#1712	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#1687	C₂×C₃₁	1	2	yes	complemented
205	#1668	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1667	C₂×C₃₁	1	1	yes	characteristic, complemented
207	#1621	C₂×C₃₁	1	2	yes	complemented
208	#1619	C₂×C₃₁	1	2	yes	complemented
209	#1589	C₂×C₃₁	1	1	yes	characteristic, complemented
210	C₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
211	C₂²×C₃₁	—	2	2	no	—
212	C₂²×C₃₁	—	4	1	no	—
213	C₂²×C₃₁	—	4	1	no	—
214	C₂²×C₃₁	—	4	2	no	—
215	C₂²×C₃₁	—	4	2	no	—
216	C₂²×C₃₁	—	4	2	no	—
217	C₂²×C₃₁	—	8	2	no	—
218	C₂²×C₃₁	—	8	2	no	—
219	C₂²×C₃₁	—	8	2	no	—
220	C₂²×C₃₁	—	8	2	no	—
221	C₂²×C₃₁	—	8	2	no	—
222	C₂²×C₃₁	—	8	2	no	—
223	C₂²×C₃₁	—	16	8	no	—
224	C₂²×C₃₁	D₄²	1	1	yes	characteristic
225	C₄×C₃₁	—	2	2	no	—
226	C₄×C₃₁	—	2	1	no	—
227	C₄×C₃₁	—	4	1	no	—
228	C₄×C₃₁	—	4	2	no	—
229	C₄×C₃₁	—	4	1	no	—
230	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
231	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
232	C₄×C₃₁	#576	1	1	yes	characteristic
233	C₄×C₃₁	#576	1	1	yes	characteristic
234	C₂×AGL(1,ℤ/8)	—	2	2	no	—
235	C₂²×D₈	—	2	1	no	—
236	#550	C₂²×C₃₁	1	2	yes	complemented
237	D₄²	C₂²×C₃₁	1	2	yes	—
238	#493	—	2	1	no	—
239	#493	C₂²×C₃₁	1	1	yes	characteristic
240	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
241	#493	C₂²×C₃₁	1	2	yes	complemented
242	#493	C₂²×C₃₁	1	2	yes	complemented
243	#469	—	2	2	no	—
244	#466	—	2	1	no	—
245	#460	C₂²×C₃₁	1	2	yes	complemented
246	#459	—	2	2	no	—
247	#459	C₂²×C₃₁	1	2	yes	—
248	#459	C₂²×C₃₁	1	2	yes	—
249	#459	C₂²×C₃₁	1	2	yes	complemented
250	#449	—	2	2	no	—
251	#447	—	2	2	no	—
252	#447	—	2	2	no	—
253	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
254	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
255	C₄×D₈	—	2	1	no	—
256	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
257	#436	C₂²×C₃₁	1	2	yes	complemented
258	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
259	#335	C₂²×C₃₁	1	1	yes	characteristic, complemented
260	#334	C₂²×C₃₁	1	1	yes	characteristic, complemented
261	#331	C₂²×C₃₁	1	2	yes	complemented
262	#331	C₂²×C₃₁	1	2	yes	complemented
263	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
264	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
265	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
266	C₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
267	C₂×C₃₁	—	4	2	no	—
268	C₂×C₃₁	—	8	2	no	—
269	C₂×C₃₁	—	16	2	no	—
270	C₂×C₃₁	—	16	2	no	—
271	C₂×C₃₁	#3274	1	1	yes	characteristic
272	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
273	C₂×C₃₁	#3292	1	1	yes	characteristic
274	C₂×#41	—	2	2	no	—
275	C₂²×D₄	—	2	2	no	—
276	C₂²×D₄	—	2	2	no	—
277	AGL(1,ℤ/8)	—	4	8	no	—
278	C₂×SD₁₆	—	2	2	no	—
279	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
280	C₂×D₈	—	2	2	no	—
281	C₂×D₈	—	2	2	no	—
282	C₂×D₈	—	2	2	no	—
283	C₂×D₈	—	2	1	no	—
284	C₂×D₈	—	2	2	no	—
285	C₂×D₈	—	2	1	no	—
286	C₂×D₈	—	2	1	no	—
287	C₂×D₈	—	2	2	no	—
288	C₂×D₈	—	2	1	no	—
289	C₂×D₈	—	4	2	no	—
290	C₂×D₈	—	4	4	no	—
291	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
292	C₂×M₁₆	—	2	2	no	—
293	C₂²×C₈	—	2	1	no	—
294	#127	—	2	2	no	—
295	#127	C₂³×C₃₁	1	2	yes	—
296	#127	C₂³×C₃₁	1	2	yes	—
297	#121	—	2	2	no	—
298	#121	—	2	2	no	—
299	#121	—	2	2	no	—
300	#121	—	4	2	no	—
301	#120	—	4	2	no	—
302	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
303	C₄×D₄	—	2	2	no	—
304	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
305	C₄×D₄	C₂³×C₃₁	1	2	yes	—
306	#107	D₄×C₃₁	1	1	yes	characteristic
307	#106	D₄×C₃₁	1	1	yes	characteristic
308	C₄⋊C₈	—	2	1	no	—
309	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
310	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
311	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
312	#103	D₄×C₃₁	1	1	yes	characteristic
313	#102	—	2	2	no	—
314	#102	—	2	2	no	—
315	#102	—	2	2	no	—
316	#102	—	2	2	no	—
317	#102	D₄×C₃₁	1	2	yes	—
318	#102	D₄×C₃₁	1	1	yes	characteristic
319	C₂²⋊C₈	—	2	1	no	—
320	C₂²⋊C₈	—	2	2	no	—
321	C₄×C₈	—	2	1	no	—
322	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
323	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
324	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
325	C₃₁	#19885	1	1	yes	characteristic, direct product factor, Sylow
326	C₂⁴	—	4	2	no	—
327	#41	—	4	4	no	—
328	#41	—	8	2	no	—
329	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	4	2	no	—
337	C₂×D₄	—	4	2	no	—
338	C₂×D₄	—	4	2	no	—
339	C₂×D₄	—	4	2	no	—
340	C₂×D₄	—	4	4	no	—
341	C₂×D₄	—	4	4	no	—
342	C₂×D₄	—	8	2	no	—
343	C₂×D₄	—	8	2	no	—
344	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
345	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	4	2	no	—
349	SD₁₆	—	4	4	no	—
350	SD₁₆	D₈×C₃₁	1	4	yes	complemented
351	D₈	—	2	2	no	—
352	D₈	—	4	4	no	—
353	D₈	—	4	4	no	—
354	D₈	—	4	4	no	—
355	D₈	—	4	2	no	—
356	D₈	—	4	2	no	—
357	D₈	—	8	1	no	—
358	D₈	—	8	1	no	—
359	D₈	—	8	2	no	—
360	D₈	—	8	4	no	—
361	D₈	—	8	2	no	—
362	M₁₆	—	4	4	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	4	2	no	—
368	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
369	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
370	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
371	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
375	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
376	C₂²⋊C₄	—	2	1	no	—
377	C₂²⋊C₄	—	2	2	no	—
378	C₂²⋊C₄	—	4	2	no	—
379	C₂²⋊C₄	—	4	2	no	—
380	C₄²	—	2	1	no	—
381	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
382	C₂³	—	2	1	no	—
383	C₂³	—	2	2	no	—
384	C₂³	—	4	2	no	—
385	C₂³	—	4	2	no	—
386	C₂³	—	8	4	no	—
387	C₂³	—	8	4	no	—
388	C₂³	—	16	2	no	—
389	Q₈	—	2	1	no	—
390	Q₈	C₂×D₈×C₃₁	1	2	yes	complemented
391	D₄	—	2	2	no	—
392	D₄	—	2	1	no	—
393	D₄	—	2	2	no	—
394	D₄	—	4	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	2	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	2	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	2	no	—
401	D₄	—	4	2	no	—
402	D₄	—	8	4	no	—
403	D₄	—	8	2	no	—
404	D₄	—	8	2	no	—
405	D₄	—	8	4	no	—
406	D₄	—	8	4	no	—
407	D₄	—	16	2	no	—
408	D₄	C₂×D₈×C₃₁	1	2	yes	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	8	4	no	—
418	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
419	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
420	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
421	C₈	—	2	2	no	—
422	C₈	—	2	1	no	—
423	C₈	—	2	1	no	—
424	C₈	—	4	2	no	—
425	C₈	—	4	1	no	—
426	C₈	—	4	1	no	—
427	C₈	C₂×D₈×C₃₁	1	2	yes	complemented
428	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	complemented
429	C₂²	—	2	2	no	—
430	C₂²	—	4	1	no	—
431	C₂²	—	4	1	no	—
432	C₂²	—	4	2	no	—
433	C₂²	—	4	2	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	8	2	no	—
436	C₂²	—	8	2	no	—
437	C₂²	—	8	2	no	—
438	C₂²	—	8	2	no	—
439	C₂²	—	8	2	no	—
440	C₂²	—	8	2	no	—
441	C₂²	—	16	8	no	—
442	C₂²	D₄²×C₃₁	1	1	yes	characteristic
443	C₄	—	2	2	no	—
444	C₄	—	2	1	no	—
445	C₄	—	4	1	no	—
446	C₄	—	4	2	no	—
447	C₄	—	4	1	no	—
448	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
449	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
450	C₄	C₃₁×#576	1	1	yes	characteristic
451	C₄	C₃₁×#576	1	1	yes	characteristic
452	C₂	—	4	2	no	—
453	C₂	—	8	2	no	—
454	C₂	—	16	2	no	—
455	C₂	—	16	2	no	—
456	C₂	C₃₁×#3274	1	1	yes	characteristic
457	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
458	C₂	C₃₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3391	C₂	semidirect
2. D₄×D₈×C₃₁	C₂	semidirect
3. C₃₁×#3274	C₂	semidirect
4. C₃₁×#1712	C₂	semidirect
5. C₃₁×#1687	C₂	semidirect
6. C₃₁×#1668	C₂	semidirect
7. C₃₁×#1667	C₂	semidirect
8. C₃₁×#1621	C₂	semidirect
9. C₃₁×#1619	C₂	semidirect
10. C₃₁×#1589	C₂	semidirect
11. C₈×SD₁₆×C₃₁	C₂	semidirect
14. C₃₁×#550	C₂²	semidirect
18. C₃₁×#493	C₂²	semidirect
19. C₃₁×#493	C₂²	semidirect
20. C₃₁×#493	C₂²	semidirect
23. C₃₁×#460	C₂²	semidirect
27. C₃₁×#459	C₂²	semidirect
31. C₈×Q₈×C₃₁	C₂²	semidirect
35. C₃₁×#436	C₂²	semidirect
37. C₃₁×#335	C₂²	semidirect
38. C₃₁×#334	C₂²	semidirect
39. C₃₁×#331	C₂²	semidirect
40. C₃₁×#331	C₂²	semidirect
41. C₃₁×Q₈⋊C₈	C₂²	semidirect
43. C₃₁×D₄⋊C₈	C₂²	semidirect
44. C₈²×C₃₁	C₂²	semidirect
120. SD₁₆×C₃₁	D₈	semidirect
152. #19885	C₃₁	direct (N × H)
161. Q₈×C₃₁	C₂×D₈	semidirect
198. C₈×C₃₁	C₂×D₈	semidirect
199. C₈×C₃₁	AGL(1,ℤ/8)	semidirect
200. #3391	C₂×C₃₁	semidirect
201. D₄×D₈	C₂×C₃₁	semidirect
202. #3274	C₂×C₃₁	semidirect
203. #1712	C₂×C₃₁	semidirect
204. #1687	C₂×C₃₁	semidirect
205. #1668	C₂×C₃₁	semidirect
206. #1667	C₂×C₃₁	semidirect
207. #1621	C₂×C₃₁	semidirect
208. #1619	C₂×C₃₁	semidirect
209. #1589	C₂×C₃₁	semidirect
210. C₈×SD₁₆	C₂×C₃₁	semidirect
236. #550	C₂²×C₃₁	semidirect
240. #493	C₂²×C₃₁	semidirect
241. #493	C₂²×C₃₁	semidirect
242. #493	C₂²×C₃₁	semidirect
245. #460	C₂²×C₃₁	semidirect
249. #459	C₂²×C₃₁	semidirect
253. C₈×Q₈	C₂²×C₃₁	semidirect
257. #436	C₂²×C₃₁	semidirect
259. #335	C₂²×C₃₁	semidirect
260. #334	C₂²×C₃₁	semidirect
261. #331	C₂²×C₃₁	semidirect
262. #331	C₂²×C₃₁	semidirect
263. Q₈⋊C₈	C₂²×C₃₁	semidirect
265. D₄⋊C₈	C₂²×C₃₁	semidirect
266. C₈²	C₂²×C₃₁	semidirect
325. C₃₁	#19885	direct (N × H)
350. SD₁₆	D₈×C₃₁	semidirect
390. Q₈	C₂×D₈×C₃₁	semidirect
427. C₈	C₂×D₈×C₃₁	semidirect
428. 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×#3391	2
D₄×D₈×C₃₁	2
C₃₁×#3274	2
C₃₁×#1712	2
C₃₁×#1687	2
C₃₁×#1668	2
C₃₁×#1667	2
C₃₁×#1621	2
C₃₁×#1619	2
C₃₁×#1589	2
C₈×SD₁₆×C₃₁	2
#19885	31

Subgroups — C31×#19885 #21476644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19885 #21476644