Subgroups — C₃₁×#24855 #21481614

Order 7936.

752 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3079	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#474	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2906	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2089	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2063	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1997	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1907	C₂	1	2	yes	maximal
10	C₃₁×#1816	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1800	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#131	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#477	—	2	1	no	—
18	C₃₁×#476	—	2	1	no	—
19	C₃₁×#475	—	2	1	no	—
20	C₃₁×#474	—	2	1	no	—
21	C₃₁×#474	—	2	1	no	—
22	C₃₁×#474	—	2	1	no	—
23	C₃₁×#474	—	2	2	no	—
24	C₃₁×#442	—	2	1	no	—
25	C₃₁×#440	—	2	1	no	—
26	C₄×SD₁₆×C₃₁	—	2	1	no	—
27	C₄×D₈×C₃₁	—	2	1	no	—
28	C₂×C₃₁×#108	C₂²	1	2	yes	—
29	C₃₁×#428	C₂²	1	2	yes	—
30	C₃₁×#428	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#107	C₂²	1	2	yes	—
32	C₃₁×#424	—	2	2	no	—
33	C₃₁×#424	C₂²	1	2	yes	—
34	C₃₁×#423	—	2	1	no	—
35	C₃₁×#423	C₂²	1	1	yes	characteristic
36	C₃₁×#423	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
38	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₃₁×#418	C₂²	1	1	yes	characteristic
40	C₃₁×#417	C₂²	1	2	yes	—
41	C₃₁×#417	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#102	C₂²	1	2	yes	—
43	C₂×C₃₁×#100	C₂²	1	2	yes	—
44	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#343	C₂²	1	2	yes	—
46	C₃₁×#343	C₂²	1	2	yes	—
47	C₃₁×#343	C₂²	1	2	yes	—
48	C₃₁×#341	C₂²	1	2	yes	—
49	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×Q₈×C₃₁	—	2	1	no	—
51	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₃₁	—	2	1	no	—
53	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
54	C₃₁×AGL(1,ℤ/8)	Q₈	1	4	yes	complemented
55	C₂×SD₁₆×C₃₁	—	2	1	no	—
56	C₂×D₈×C₃₁	—	2	1	no	—
57	C₃₁×#131	—	2	4	no	—
58	C₃₁×#131	—	4	2	no	—
59	C₂×M₁₆×C₃₁	—	2	2	no	—
60	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
63	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	—	2	2	no	—
65	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
66	C₃₁×C₄⋊Q₈	—	2	1	no	—
67	C₃₁×C₄⋊Q₈	—	2	1	no	—
68	C₃₁×C₄⋊Q₈	—	2	1	no	—
69	C₃₁×C₄⋊Q₈	—	2	1	no	—
70	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
71	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
72	C₄×D₄×C₃₁	—	2	1	no	—
73	C₄×D₄×C₃₁	—	2	1	no	—
74	C₄×D₄×C₃₁	—	2	1	no	—
75	C₄×D₄×C₃₁	—	2	1	no	—
76	C₄×D₄×C₃₁	—	2	1	no	—
77	C₄×D₄×C₃₁	—	2	1	no	—
78	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
79	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
80	C₃₁×#117	—	2	1	no	—
81	C₃₁×#117	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₄²×C₃₁	—	2	1	no	—
87	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
88	C₃₁×#108	—	4	2	no	—
89	C₃₁×#107	—	2	2	no	—
90	C₃₁×#107	—	2	2	no	—
91	C₃₁×#107	—	2	1	no	—
92	C₃₁×#107	—	2	2	no	—
93	C₃₁×#106	—	2	1	no	—
94	C₃₁×#106	—	2	2	no	—
95	C₃₁×C₄⋊C₈	—	2	2	no	—
96	C₃₁×C₄⋊C₈	—	2	2	no	—
97	C₃₁×C₄⋊C₈	—	2	1	no	—
98	C₃₁×C₄⋊C₈	—	2	1	no	—
99	C₃₁×C₄⋊C₈	—	2	1	no	—
100	C₃₁×C₄⋊C₈	—	2	1	no	—
101	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
102	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
103	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
104	C₃₁×#103	—	2	1	no	—
105	C₃₁×#103	—	2	2	no	—
106	C₃₁×#102	—	2	1	no	—
107	C₃₁×#102	—	2	2	no	—
108	C₃₁×#102	—	2	2	no	—
109	C₃₁×#102	—	2	2	no	—
110	C₃₁×#102	—	2	1	no	—
111	C₃₁×#102	—	2	1	no	—
112	C₃₁×#100	—	4	2	no	—
113	C₃₁×#97	—	2	1	no	—
114	C₄×C₈×C₃₁	—	2	1	no	—
115	C₂⁴×C₃₁	—	2	1	no	—
116	#41×C₃₁	—	2	1	no	—
117	#41×C₃₁	—	2	1	no	—
118	#41×C₃₁	C₂×Q₈	1	2	yes	—
119	#41×C₃₁	#41	1	2	yes	—
120	C₂×Q₈×C₃₁	—	4	1	no	—
121	C₂×Q₈×C₃₁	—	4	1	no	—
122	C₂×Q₈×C₃₁	—	8	2	no	—
123	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×D₄×C₃₁	—	2	1	no	—
125	C₂×D₄×C₃₁	—	2	1	no	—
126	C₂×D₄×C₃₁	—	2	1	no	—
127	C₂×D₄×C₃₁	—	2	1	no	—
128	C₂×D₄×C₃₁	—	2	1	no	—
129	C₂×D₄×C₃₁	—	2	1	no	—
130	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
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	—	2	2	no	—
137	C₂²×C₄×C₃₁	—	4	1	no	—
138	C₂²×C₄×C₃₁	—	4	1	no	—
139	C₂²×C₄×C₃₁	—	8	2	no	—
140	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
143	SD₁₆×C₃₁	—	2	4	no	—
144	D₈×C₃₁	—	2	4	no	—
145	M₁₆×C₃₁	—	2	1	no	—
146	M₁₆×C₃₁	—	2	2	no	—
147	M₁₆×C₃₁	—	2	2	no	—
148	M₁₆×C₃₁	—	2	2	no	—
149	M₁₆×C₃₁	—	2	1	no	—
150	M₁₆×C₃₁	—	2	1	no	—
151	M₁₆×C₃₁	—	2	2	no	—
152	M₁₆×C₃₁	—	4	4	no	—
153	M₁₆×C₃₁	C₂×Q₈	1	2	yes	—
154	C₂×C₈×C₃₁	—	2	1	no	—
155	C₂×C₈×C₃₁	—	2	2	no	—
156	C₂×C₈×C₃₁	—	2	2	no	—
157	C₂×C₈×C₃₁	—	2	1	no	—
158	C₂×C₈×C₃₁	—	2	2	no	—
159	C₂×C₈×C₃₁	—	2	2	no	—
160	C₂×C₈×C₃₁	—	4	4	no	—
161	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
162	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
163	C₄⋊C₄×C₃₁	—	2	1	no	—
164	C₄⋊C₄×C₃₁	—	2	1	no	—
165	C₄⋊C₄×C₃₁	—	2	2	no	—
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	—	2	2	no	—
168	C₄⋊C₄×C₃₁	—	2	1	no	—
169	C₄⋊C₄×C₃₁	—	4	1	no	—
170	C₄⋊C₄×C₃₁	—	4	1	no	—
171	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₂²⋊C₄×C₃₁	—	2	1	no	—
174	C₂²⋊C₄×C₃₁	—	2	1	no	—
175	C₂²⋊C₄×C₃₁	—	2	1	no	—
176	C₂²⋊C₄×C₃₁	—	4	1	no	—
177	C₂²⋊C₄×C₃₁	—	4	1	no	—
178	C₄²×C₃₁	—	2	1	no	—
179	C₄²×C₃₁	—	2	1	no	—
180	C₄²×C₃₁	—	2	1	no	—
181	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
185	#24855	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂³×C₃₁	—	2	1	no	—
187	C₂³×C₃₁	—	2	2	no	—
188	C₂³×C₃₁	—	4	1	no	—
189	C₂³×C₃₁	—	4	1	no	—
190	C₂³×C₃₁	—	8	1	no	—
191	C₂³×C₃₁	—	8	1	no	—
192	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
193	Q₈×C₃₁	—	2	1	no	—
194	Q₈×C₃₁	—	8	2	no	—
195	Q₈×C₃₁	—	8	2	no	—
196	Q₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
197	D₄×C₃₁	—	2	2	no	—
198	D₄×C₃₁	—	2	2	no	—
199	D₄×C₃₁	—	2	2	no	—
200	D₄×C₃₁	—	2	1	no	—
201	D₄×C₃₁	—	2	2	no	—
202	D₄×C₃₁	—	4	1	no	—
203	D₄×C₃₁	—	4	1	no	—
204	D₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	1	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	8	1	no	—
216	C₂×C₄×C₃₁	—	8	2	no	—
217	C₂×C₄×C₃₁	—	8	2	no	—
218	C₂×C₄×C₃₁	—	8	1	no	—
219	C₂×C₄×C₃₁	—	8	2	no	—
220	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
221	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
222	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
223	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
227	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
228	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
229	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
230	C₈×C₃₁	—	2	2	no	—
231	C₈×C₃₁	—	2	2	no	—
232	C₈×C₃₁	—	2	2	no	—
233	C₈×C₃₁	—	4	1	no	—
234	C₈×C₃₁	—	4	2	no	—
235	C₈×C₃₁	—	4	2	no	—
236	#3079	C₂×C₃₁	1	1	yes	characteristic, complemented
237	C₂×#474	C₂×C₃₁	1	1	yes	characteristic, complemented
238	C₄×AGL(1,ℤ/8)	C₂×C₃₁	1	1	yes	characteristic
239	#2906	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#2089	C₂×C₃₁	1	2	yes	complemented
241	#2063	C₂×C₃₁	1	2	yes	complemented
242	#1997	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#1907	C₂×C₃₁	1	2	yes	—
245	#1816	C₂×C₃₁	1	2	yes	complemented
246	#1800	C₂×C₃₁	1	1	yes	characteristic, complemented
247	C₂²×C₃₁	—	2	1	no	—
248	C₂²×C₃₁	—	2	1	no	—
249	C₂²×C₃₁	—	2	2	no	—
250	C₂²×C₃₁	—	4	1	no	—
251	C₂²×C₃₁	—	4	1	no	—
252	C₂²×C₃₁	—	4	2	no	—
253	C₂²×C₃₁	—	8	2	no	—
254	C₂²×C₃₁	—	8	2	no	—
255	C₂²×C₃₁	—	8	1	no	—
256	C₂²×C₃₁	—	8	1	no	—
257	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
258	C₂²×C₃₁	#552	1	1	yes	characteristic
259	C₂²×C₃₁	#554	1	1	yes	characteristic
260	C₄×C₃₁	—	2	2	no	—
261	C₄×C₃₁	—	2	1	no	—
262	C₄×C₃₁	—	2	1	no	—
263	C₄×C₃₁	—	2	1	no	—
264	C₄×C₃₁	—	4	1	no	—
265	C₄×C₃₁	—	4	1	no	—
266	C₄×C₃₁	—	8	1	no	—
267	C₄×C₃₁	—	8	2	no	—
268	C₄×C₃₁	#472	1	2	yes	—
269	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
270	C₄×C₃₁	#537	1	1	yes	characteristic
271	C₄×C₃₁	#546	1	1	yes	characteristic
272	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
273	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×#131	C₂²×C₃₁	1	2	yes	—
275	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
276	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
278	#477	—	2	1	no	—
279	#476	—	2	1	no	—
280	#475	—	2	1	no	—
281	#474	—	2	1	no	—
282	#474	—	2	1	no	—
283	#474	—	2	1	no	—
284	#474	—	2	2	no	—
285	#442	—	2	1	no	—
286	#440	—	2	1	no	—
287	C₄×SD₁₆	—	2	1	no	—
288	C₄×D₈	—	2	1	no	—
289	C₂×#108	C₂²×C₃₁	1	2	yes	—
290	#428	C₂²×C₃₁	1	2	yes	—
291	#428	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×#107	C₂²×C₃₁	1	2	yes	—
293	#424	—	2	2	no	—
294	#424	C₂²×C₃₁	1	2	yes	—
295	#423	—	2	1	no	—
296	#423	C₂²×C₃₁	1	1	yes	characteristic
297	#423	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×C₄⋊C₈	—	2	1	no	—
299	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
300	#418	C₂²×C₃₁	1	1	yes	characteristic
301	#417	C₂²×C₃₁	1	2	yes	—
302	#417	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×#102	C₂²×C₃₁	1	2	yes	—
304	C₂×#100	C₂²×C₃₁	1	2	yes	—
305	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
306	#343	C₂²×C₃₁	1	2	yes	—
307	#343	C₂²×C₃₁	1	2	yes	—
308	#343	C₂²×C₃₁	1	2	yes	—
309	#341	C₂²×C₃₁	1	2	yes	—
310	C₂×C₃₁	—	2	1	no	—
311	C₂×C₃₁	—	2	1	no	—
312	C₂×C₃₁	—	4	2	no	—
313	C₂×C₃₁	—	8	2	no	—
314	C₂×C₃₁	#2417	1	1	yes	characteristic
315	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
316	C₂×C₃₁	#3069	1	1	yes	characteristic
317	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
318	C₂²×Q₈	—	2	1	no	—
319	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
320	C₂³×C₄	—	2	1	no	—
321	AGL(1,ℤ/8)	—	2	2	no	—
322	AGL(1,ℤ/8)	Q₈×C₃₁	1	4	yes	complemented
323	C₂×SD₁₆	—	2	1	no	—
324	C₂×D₈	—	2	1	no	—
325	#131	—	2	4	no	—
326	#131	—	4	2	no	—
327	C₂×M₁₆	—	2	2	no	—
328	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
329	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
330	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
331	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
332	C₂²×C₈	—	2	2	no	—
333	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
334	C₄⋊Q₈	—	2	1	no	—
335	C₄⋊Q₈	—	2	1	no	—
336	C₄⋊Q₈	—	2	1	no	—
337	C₄⋊Q₈	—	2	1	no	—
338	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
339	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
340	C₄×D₄	—	2	1	no	—
341	C₄×D₄	—	2	1	no	—
342	C₄×D₄	—	2	1	no	—
343	C₄×D₄	—	2	1	no	—
344	C₄×D₄	—	2	1	no	—
345	C₄×D₄	—	2	1	no	—
346	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
347	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
348	#117	—	2	1	no	—
349	#117	C₂³×C₃₁	1	1	yes	characteristic
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
352	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
353	C₂×C₂²⋊C₄	—	2	1	no	—
354	C₂×C₄²	—	2	1	no	—
355	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
356	#108	—	4	2	no	—
357	#107	—	2	2	no	—
358	#107	—	2	2	no	—
359	#107	—	2	1	no	—
360	#107	—	2	2	no	—
361	#106	—	2	1	no	—
362	#106	—	2	2	no	—
363	C₄⋊C₈	—	2	2	no	—
364	C₄⋊C₈	—	2	2	no	—
365	C₄⋊C₈	—	2	1	no	—
366	C₄⋊C₈	—	2	1	no	—
367	C₄⋊C₈	—	2	1	no	—
368	C₄⋊C₈	—	2	1	no	—
369	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
370	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
371	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
372	#103	—	2	1	no	—
373	#103	—	2	2	no	—
374	#102	—	2	1	no	—
375	#102	—	2	2	no	—
376	#102	—	2	2	no	—
377	#102	—	2	2	no	—
378	#102	—	2	1	no	—
379	#102	—	2	1	no	—
380	#100	—	4	2	no	—
381	#97	—	2	1	no	—
382	C₄×C₈	—	2	1	no	—
383	C₃₁	#24855	1	1	yes	characteristic, direct product factor, Sylow
384	C₂⁴	—	2	1	no	—
385	#41	—	2	1	no	—
386	#41	—	2	1	no	—
387	#41	C₂×Q₈×C₃₁	1	2	yes	—
388	#41	#41×C₃₁	1	2	yes	—
389	C₂×Q₈	—	4	1	no	—
390	C₂×Q₈	—	4	1	no	—
391	C₂×Q₈	—	8	2	no	—
392	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	—	2	1	no	—
396	C₂×D₄	—	2	1	no	—
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	2	1	no	—
399	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
400	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	1	no	—
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	4	1	no	—
407	C₂²×C₄	—	4	1	no	—
408	C₂²×C₄	—	8	2	no	—
409	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
410	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
411	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
412	SD₁₆	—	2	4	no	—
413	D₈	—	2	4	no	—
414	M₁₆	—	2	1	no	—
415	M₁₆	—	2	2	no	—
416	M₁₆	—	2	2	no	—
417	M₁₆	—	2	2	no	—
418	M₁₆	—	2	1	no	—
419	M₁₆	—	2	2	no	—
420	M₁₆	—	2	1	no	—
421	M₁₆	—	4	4	no	—
422	M₁₆	C₂×Q₈×C₃₁	1	2	yes	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	2	1	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	2	no	—
429	C₂×C₈	—	4	4	no	—
430	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
431	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
432	C₄⋊C₄	—	2	1	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	2	2	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	2	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	4	1	no	—
439	C₄⋊C₄	—	4	1	no	—
440	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
441	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
442	C₂²⋊C₄	—	2	1	no	—
443	C₂²⋊C₄	—	2	1	no	—
444	C₂²⋊C₄	—	2	1	no	—
445	C₂²⋊C₄	—	4	1	no	—
446	C₂²⋊C₄	—	4	1	no	—
447	C₄²	—	2	1	no	—
448	C₄²	—	2	1	no	—
449	C₄²	—	2	1	no	—
450	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
451	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
452	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
453	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
454	C₂³	—	2	1	no	—
455	C₂³	—	2	2	no	—
456	C₂³	—	4	1	no	—
457	C₂³	—	4	1	no	—
458	C₂³	—	8	1	no	—
459	C₂³	—	8	1	no	—
460	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
461	Q₈	—	2	1	no	—
462	Q₈	—	8	2	no	—
463	Q₈	—	8	2	no	—
464	Q₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
465	D₄	—	2	2	no	—
466	D₄	—	2	2	no	—
467	D₄	—	2	2	no	—
468	D₄	—	2	1	no	—
469	D₄	—	2	2	no	—
470	D₄	—	4	1	no	—
471	D₄	—	4	1	no	—
472	D₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	8	1	no	—
484	C₂×C₄	—	8	2	no	—
485	C₂×C₄	—	8	2	no	—
486	C₂×C₄	—	8	1	no	—
487	C₂×C₄	—	8	2	no	—
488	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
489	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
490	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
491	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
492	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
493	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
494	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
495	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
496	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
497	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
498	C₈	—	2	2	no	—
499	C₈	—	2	2	no	—
500	C₈	—	2	2	no	—
501	C₈	—	4	1	no	—
502	C₈	—	4	2	no	—
503	C₈	—	4	2	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	2	1	no	—
506	C₂²	—	2	2	no	—
507	C₂²	—	4	1	no	—
508	C₂²	—	4	1	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	8	2	no	—
511	C₂²	—	8	2	no	—
512	C₂²	—	8	1	no	—
513	C₂²	—	8	1	no	—
514	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
515	C₂²	C₃₁×#552	1	1	yes	characteristic
516	C₂²	C₃₁×#554	1	1	yes	characteristic
517	C₄	—	2	2	no	—
518	C₄	—	2	1	no	—
519	C₄	—	2	1	no	—
520	C₄	—	2	1	no	—
521	C₄	—	4	1	no	—
522	C₄	—	4	1	no	—
523	C₄	—	8	1	no	—
524	C₄	—	8	2	no	—
525	C₄	C₃₁×#472	1	2	yes	—
526	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
527	C₄	C₃₁×#537	1	1	yes	characteristic
528	C₄	C₃₁×#546	1	1	yes	characteristic
529	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
530	C₂	—	2	1	no	—
531	C₂	—	2	1	no	—
532	C₂	—	4	2	no	—
533	C₂	—	8	2	no	—
534	C₂	C₃₁×#2417	1	1	yes	characteristic
535	C₂	C₂×C₃₁×#472	1	1	yes	characteristic
536	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
1. C₃₁×#3079	C₂	semidirect
2. C₂×C₃₁×#474	C₂	semidirect
4. C₃₁×#2906	C₂	semidirect
5. C₃₁×#2089	C₂	semidirect
6. C₃₁×#2063	C₂	semidirect
7. C₃₁×#1997	C₂	semidirect
8. C₃₁×#1950	C₂	semidirect
10. C₃₁×#1816	C₂	semidirect
11. C₃₁×#1800	C₂	semidirect
54. C₃₁×AGL(1,ℤ/8)	Q₈	semidirect
185. #24855	C₃₁	direct (N × H)
236. #3079	C₂×C₃₁	semidirect
237. C₂×#474	C₂×C₃₁	semidirect
239. #2906	C₂×C₃₁	semidirect
240. #2089	C₂×C₃₁	semidirect
241. #2063	C₂×C₃₁	semidirect
242. #1997	C₂×C₃₁	semidirect
243. #1950	C₂×C₃₁	semidirect
245. #1816	C₂×C₃₁	semidirect
246. #1800	C₂×C₃₁	semidirect
322. AGL(1,ℤ/8)	Q₈×C₃₁	semidirect
383. C₃₁	#24855	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3079	2
C₂×C₃₁×#474	2
C₄×C₃₁×AGL(1,ℤ/8)	2
C₃₁×#2906	2
C₃₁×#2089	2
C₃₁×#2063	2
C₃₁×#1997	2
C₃₁×#1950	2
C₃₁×#1907	2
C₃₁×#1816	2
C₃₁×#1800	2
#24855	31

Subgroups — C31×#24855 #21481614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24855 #21481614