Subgroups — C₃₁×#22465 #21479224

Order 7936.

1312 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#578	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2952	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2898	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1979	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1971	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1908	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1882	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1815	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1793	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1765	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	2	yes	—
13	C₃₁×#578	—	2	4	no	—
14	C₃₁×#578	C₄	1	8	yes	complemented
15	C₂×C₃₁×#137	—	2	2	no	—
16	C₂×C₃₁×#137	C₂²	1	2	yes	—
17	C₂×C₃₁×#135	—	2	1	no	—
18	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
19	C₂²×Q₁₆×C₃₁	—	2	1	no	—
20	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#131	C₂²	1	2	yes	—
22	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
23	C₃₁×#520	C₂²	1	2	yes	—
24	C₃₁×#444	—	2	4	no	—
25	C₃₁×#442	—	2	1	no	—
26	C₃₁×#441	—	2	1	no	—
27	C₄×SD₁₆×C₃₁	—	2	2	no	—
28	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#104	C₂²	1	2	yes	—
31	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
32	C₃₁×#419	—	2	2	no	—
33	C₃₁×#419	C₂²	1	2	yes	—
34	C₃₁×#417	—	2	2	no	—
35	C₃₁×#417	C₂²	1	2	yes	—
36	C₃₁×#417	C₂²	1	2	yes	—
37	C₃₁×#416	—	2	2	no	—
38	C₃₁×#416	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×#103	—	2	2	no	—
40	C₂×C₃₁×#103	C₂²	1	2	yes	—
41	C₂×C₃₁×#101	C₂²	1	2	yes	—
42	C₃₁×#405	C₂²	1	1	yes	characteristic
43	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
44	C₃₁×#391	C₂²	1	2	yes	—
45	C₃₁×#343	C₂²	1	2	yes	—
46	C₃₁×#341	C₂²	1	1	yes	characteristic
47	C₃₁×#339	C₂²	1	1	yes	characteristic
48	C₃₁×2₋¹⁺⁴	—	2	4	no	—
49	C₃₁×2₋¹⁺⁴	—	2	4	no	—
50	C₃₁×2₋¹⁺⁴	—	4	2	no	—
51	C₃₁×2₋¹⁺⁴	C₂×C₄	1	8	yes	complemented
52	C₂×#41×C₃₁	—	2	2	no	—
53	C₂×#41×C₃₁	—	2	2	no	—
54	C₂×#41×C₃₁	—	4	2	no	—
55	C₂×#41×C₃₁	C₂³	1	2	yes	—
56	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×Q₈×C₃₁	—	2	2	no	—
58	C₂²×Q₈×C₃₁	—	2	2	no	—
59	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
60	C₃₁×#137	—	2	4	no	—
61	C₃₁×#137	—	2	16	no	—
62	C₃₁×#137	C₂×C₄	1	8	yes	complemented
63	C₃₁×#135	—	2	8	no	—
64	C₃₁×#135	C₂×C₄	1	4	yes	—
65	C₃₁×#135	D₄	1	4	yes	—
66	C₂×Q₁₆×C₃₁	—	2	2	no	—
67	C₂×Q₁₆×C₃₁	—	2	8	no	—
68	C₂×Q₁₆×C₃₁	—	2	1	no	—
69	C₂×Q₁₆×C₃₁	—	2	1	no	—
70	C₂×Q₁₆×C₃₁	—	2	2	no	—
71	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
72	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
73	C₂×SD₁₆×C₃₁	—	2	2	no	—
74	C₂×SD₁₆×C₃₁	D₄	1	2	yes	—
75	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
76	C₃₁×#131	—	2	2	no	—
77	C₃₁×#131	—	2	4	no	—
78	C₃₁×#131	—	4	2	no	—
79	C₃₁×#131	C₂×C₄	1	4	yes	complemented
80	C₂×M₁₆×C₃₁	—	2	1	no	—
81	C₂×M₁₆×C₃₁	—	2	2	no	—
82	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
83	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
84	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₃₁	—	2	1	no	—
86	C₂²×C₈×C₃₁	—	2	2	no	—
87	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₄×Q₈×C₃₁	—	2	2	no	—
90	C₄×D₄×C₃₁	—	2	2	no	—
91	C₄×D₄×C₃₁	—	4	2	no	—
92	C₃₁×#117	—	2	2	no	—
93	C₃₁×#117	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
96	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
97	C₃₁×#108	—	2	2	no	—
98	C₃₁×#106	—	2	1	no	—
99	C₃₁×#106	—	2	1	no	—
100	C₃₁×#104	—	2	4	no	—
101	C₃₁×#104	—	2	2	no	—
102	C₃₁×#103	—	2	2	no	—
103	C₃₁×#103	—	2	2	no	—
104	C₃₁×#103	—	2	2	no	—
105	C₃₁×#103	—	2	2	no	—
106	C₃₁×#103	—	2	2	no	—
107	C₃₁×#103	—	2	2	no	—
108	C₃₁×#103	D₄	1	2	yes	—
109	C₃₁×#102	—	2	2	no	—
110	C₃₁×#102	—	2	2	no	—
111	C₃₁×#102	D₄	1	2	yes	—
112	C₃₁×#101	—	4	2	no	—
113	C₃₁×#97	—	2	2	no	—
114	C₃₁×#97	—	2	1	no	—
115	C₄×C₈×C₃₁	—	2	1	no	—
116	C₃₁×#95	—	4	2	no	—
117	#41×C₃₁	—	2	1	no	—
118	#41×C₃₁	—	2	2	no	—
119	#41×C₃₁	—	2	2	no	—
120	#41×C₃₁	—	2	1	no	—
121	#41×C₃₁	—	2	8	no	—
122	#41×C₃₁	—	2	8	no	—
123	#41×C₃₁	—	4	8	no	—
124	#41×C₃₁	—	4	4	no	—
125	#41×C₃₁	—	4	4	no	—
126	#41×C₃₁	—	4	8	no	—
127	#41×C₃₁	C₂²×C₄	1	4	yes	—
128	#41×C₃₁	C₂²×C₄	1	2	yes	—
129	#41×C₃₁	C₂×D₄	1	4	yes	—
130	#41×C₃₁	#41	1	2	yes	—
131	C₂×Q₈×C₃₁	—	2	2	no	—
132	C₂×Q₈×C₃₁	—	2	8	no	—
133	C₂×Q₈×C₃₁	—	2	8	no	—
134	C₂×Q₈×C₃₁	—	2	2	no	—
135	C₂×Q₈×C₃₁	—	2	2	no	—
136	C₂×Q₈×C₃₁	—	2	2	no	—
137	C₂×Q₈×C₃₁	—	2	2	no	—
138	C₂×Q₈×C₃₁	—	4	2	no	—
139	C₂×Q₈×C₃₁	—	4	4	no	—
140	C₂×Q₈×C₃₁	—	4	4	no	—
141	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
142	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
143	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂×D₄×C₃₁	—	2	2	no	—
145	C₂×D₄×C₃₁	—	2	1	no	—
146	C₂×D₄×C₃₁	—	4	2	no	—
147	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
148	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	—	2	2	no	—
150	C₂²×C₄×C₃₁	—	2	1	no	—
151	C₂²×C₄×C₃₁	—	2	2	no	—
152	C₂²×C₄×C₃₁	—	2	1	no	—
153	C₂²×C₄×C₃₁	—	2	2	no	—
154	C₂²×C₄×C₃₁	—	4	2	no	—
155	C₂²×C₄×C₃₁	—	4	2	no	—
156	C₂²×C₄×C₃₁	—	4	2	no	—
157	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
158	Q₁₆×C₃₁	—	2	8	no	—
159	Q₁₆×C₃₁	—	2	4	no	—
160	Q₁₆×C₃₁	—	2	4	no	—
161	Q₁₆×C₃₁	C₂²⋊C₄	1	4	yes	—
162	SD₁₆×C₃₁	—	2	8	no	—
163	SD₁₆×C₃₁	C₂²⋊C₄	1	8	yes	—
164	D₈×C₃₁	C₂²⋊C₄	1	4	yes	—
165	M₁₆×C₃₁	—	2	1	no	—
166	M₁₆×C₃₁	—	2	4	no	—
167	M₁₆×C₃₁	—	2	2	no	—
168	M₁₆×C₃₁	—	2	1	no	—
169	M₁₆×C₃₁	—	2	1	no	—
170	M₁₆×C₃₁	—	2	2	no	—
171	M₁₆×C₃₁	—	4	4	no	—
172	M₁₆×C₃₁	C₂²×C₄	1	2	yes	—
173	C₂×C₈×C₃₁	—	2	1	no	—
174	C₂×C₈×C₃₁	—	2	1	no	—
175	C₂×C₈×C₃₁	—	2	1	no	—
176	C₂×C₈×C₃₁	—	2	4	no	—
177	C₂×C₈×C₃₁	—	2	1	no	—
178	C₂×C₈×C₃₁	—	2	2	no	—
179	C₂×C₈×C₃₁	—	4	4	no	—
180	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
181	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
182	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
185	C₄⋊C₄×C₃₁	—	2	2	no	—
186	C₄⋊C₄×C₃₁	—	2	2	no	—
187	C₄⋊C₄×C₃₁	—	4	2	no	—
188	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
189	C₂²⋊C₄×C₃₁	—	2	1	no	—
190	C₂²⋊C₄×C₃₁	—	4	2	no	—
191	C₄²×C₃₁	—	2	1	no	—
192	C₄²×C₃₁	—	2	1	no	—
193	C₄²×C₃₁	—	2	1	no	—
194	C₄²×C₃₁	—	4	2	no	—
195	#22465	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂³×C₃₁	—	2	2	no	—
197	C₂³×C₃₁	—	2	1	no	—
198	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
199	Q₈×C₃₁	—	2	1	no	—
200	Q₈×C₃₁	—	2	2	no	—
201	Q₈×C₃₁	—	2	4	no	—
202	Q₈×C₃₁	—	2	4	no	—
203	Q₈×C₃₁	—	4	8	no	—
204	Q₈×C₃₁	—	4	2	no	—
205	Q₈×C₃₁	—	4	2	no	—
206	Q₈×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
207	Q₈×C₃₁	C₄×D₄	1	2	yes	—
208	D₄×C₃₁	—	2	2	no	—
209	D₄×C₃₁	—	2	4	no	—
210	D₄×C₃₁	—	2	2	no	—
211	D₄×C₃₁	—	2	4	no	—
212	D₄×C₃₁	—	2	1	no	—
213	D₄×C₃₁	—	2	2	no	—
214	D₄×C₃₁	—	4	8	no	—
215	D₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
216	D₄×C₃₁	C₄×D₄	1	2	yes	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	4	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	4	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	4	8	no	—
226	C₂×C₄×C₃₁	—	4	8	no	—
227	C₂×C₄×C₃₁	—	4	2	no	—
228	C₂×C₄×C₃₁	—	4	2	no	—
229	C₂×C₄×C₃₁	—	4	2	no	—
230	C₂×C₄×C₃₁	—	4	2	no	—
231	C₂×C₄×C₃₁	—	4	1	no	—
232	C₂×C₄×C₃₁	—	8	2	no	—
233	C₂×C₄×C₃₁	—	16	2	no	—
234	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
235	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
239	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
240	C₈×C₃₁	—	2	2	no	—
241	C₈×C₃₁	—	2	2	no	—
242	C₈×C₃₁	—	4	2	no	—
243	C₈×C₃₁	—	4	1	no	—
244	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
245	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
246	C₂×#578	C₂×C₃₁	1	1	yes	characteristic
247	C₂×#444	C₂×C₃₁	1	1	yes	characteristic, complemented
248	#2952	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#2898	C₂×C₃₁	1	2	yes	complemented
250	#1979	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#1971	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1908	C₂×C₃₁	1	2	yes	complemented
253	#1882	C₂×C₃₁	1	2	yes	complemented
254	#1815	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#1793	C₂×C₃₁	1	2	yes	complemented
256	#1765	C₂×C₃₁	1	1	yes	characteristic, complemented
257	C₂²×C₃₁	—	2	2	no	—
258	C₂²×C₃₁	—	2	2	no	—
259	C₂²×C₃₁	—	2	4	no	—
260	C₂²×C₃₁	—	4	2	no	—
261	C₂²×C₃₁	—	4	2	no	—
262	C₂²×C₃₁	—	4	2	no	—
263	C₂²×C₃₁	#518	1	1	yes	characteristic
264	C₂²×C₃₁	D₄²	1	1	yes	characteristic
265	C₂²×C₃₁	#546	1	1	yes	characteristic
266	C₄×C₃₁	—	2	2	no	—
267	C₄×C₃₁	—	2	4	no	—
268	C₄×C₃₁	—	4	4	no	—
269	C₄×C₃₁	—	4	2	no	—
270	C₄×C₃₁	—	8	1	no	—
271	C₄×C₃₁	—	8	2	no	—
272	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
273	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
274	C₄×C₃₁	D₄²	1	1	yes	characteristic
275	C₄×C₃₁	#546	1	1	yes	characteristic
276	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	2	yes	—
277	#578	—	2	4	no	—
278	#578	C₄×C₃₁	1	8	yes	complemented
279	C₂×#137	—	2	2	no	—
280	C₂×#137	C₂²×C₃₁	1	2	yes	—
281	C₂×#135	—	2	1	no	—
282	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
283	C₂²×Q₁₆	—	2	1	no	—
284	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
285	C₂×#131	C₂²×C₃₁	1	2	yes	—
286	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
287	#520	C₂²×C₃₁	1	2	yes	—
288	#444	—	2	4	no	—
289	#442	—	2	1	no	—
290	#441	—	2	1	no	—
291	C₄×SD₁₆	—	2	2	no	—
292	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
293	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
294	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×#104	C₂²×C₃₁	1	2	yes	—
296	#419	—	2	2	no	—
297	#419	C₂²×C₃₁	1	2	yes	—
298	#417	—	2	2	no	—
299	#417	C₂²×C₃₁	1	2	yes	—
300	#417	C₂²×C₃₁	1	2	yes	—
301	#416	—	2	2	no	—
302	#416	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×#103	—	2	2	no	—
304	C₂×#103	C₂²×C₃₁	1	2	yes	—
305	C₂×#101	C₂²×C₃₁	1	2	yes	—
306	#405	C₂²×C₃₁	1	1	yes	characteristic
307	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
308	#391	C₂²×C₃₁	1	2	yes	—
309	#343	C₂²×C₃₁	1	2	yes	—
310	#341	C₂²×C₃₁	1	1	yes	characteristic
311	#339	C₂²×C₃₁	1	1	yes	characteristic
312	C₂×C₃₁	—	2	1	no	—
313	C₂×C₃₁	—	2	1	no	—
314	C₂×C₃₁	—	4	2	no	—
315	C₂×C₃₁	—	4	4	no	—
316	C₂×C₃₁	D₄×C₂²⋊C₄	1	1	yes	characteristic
317	C₂×C₃₁	#3289	1	2	yes	—
318	2₋¹⁺⁴	—	2	4	no	—
319	2₋¹⁺⁴	—	2	4	no	—
320	2₋¹⁺⁴	—	4	2	no	—
321	2₋¹⁺⁴	C₂×C₄×C₃₁	1	8	yes	complemented
322	C₂×#41	—	2	2	no	—
323	C₂×#41	—	2	2	no	—
324	C₂×#41	—	4	2	no	—
325	C₂×#41	C₂³×C₃₁	1	2	yes	—
326	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
327	C₂²×Q₈	—	2	2	no	—
328	C₂²×Q₈	—	2	2	no	—
329	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
330	#137	—	2	4	no	—
331	#137	—	2	16	no	—
332	#137	C₂×C₄×C₃₁	1	8	yes	complemented
333	#135	—	2	8	no	—
334	#135	C₂×C₄×C₃₁	1	4	yes	—
335	#135	D₄×C₃₁	1	4	yes	—
336	C₂×Q₁₆	—	2	2	no	—
337	C₂×Q₁₆	—	2	8	no	—
338	C₂×Q₁₆	—	2	1	no	—
339	C₂×Q₁₆	—	2	1	no	—
340	C₂×Q₁₆	—	2	2	no	—
341	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
342	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
343	C₂×SD₁₆	—	2	2	no	—
344	C₂×SD₁₆	D₄×C₃₁	1	2	yes	—
345	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
346	#131	—	2	2	no	—
347	#131	—	2	4	no	—
348	#131	—	4	2	no	—
349	#131	C₂×C₄×C₃₁	1	4	yes	complemented
350	C₂×M₁₆	—	2	1	no	—
351	C₂×M₁₆	—	2	2	no	—
352	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
353	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
354	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
355	C₂²×C₈	—	2	1	no	—
356	C₂²×C₈	—	2	2	no	—
357	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
358	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
359	C₄×Q₈	—	2	2	no	—
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	—	4	2	no	—
362	#117	—	2	2	no	—
363	#117	C₂³×C₃₁	1	1	yes	characteristic
364	C₂×C₄⋊C₄	—	2	2	no	—
365	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
366	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
367	#108	—	2	2	no	—
368	#106	—	2	1	no	—
369	#106	—	2	1	no	—
370	#104	—	2	4	no	—
371	#104	—	2	2	no	—
372	#103	—	2	2	no	—
373	#103	—	2	2	no	—
374	#103	—	2	2	no	—
375	#103	—	2	2	no	—
376	#103	—	2	2	no	—
377	#103	—	2	2	no	—
378	#103	D₄×C₃₁	1	2	yes	—
379	#102	—	2	2	no	—
380	#102	—	2	2	no	—
381	#102	D₄×C₃₁	1	2	yes	—
382	#101	—	4	2	no	—
383	#97	—	2	2	no	—
384	#97	—	2	1	no	—
385	C₄×C₈	—	2	1	no	—
386	#95	—	4	2	no	—
387	C₃₁	#22465	1	1	yes	characteristic, direct product factor, Sylow
388	#41	—	2	1	no	—
389	#41	—	2	2	no	—
390	#41	—	2	2	no	—
391	#41	—	2	1	no	—
392	#41	—	2	8	no	—
393	#41	—	2	8	no	—
394	#41	—	4	8	no	—
395	#41	—	4	4	no	—
396	#41	—	4	4	no	—
397	#41	—	4	8	no	—
398	#41	C₂²×C₄×C₃₁	1	4	yes	—
399	#41	C₂²×C₄×C₃₁	1	2	yes	—
400	#41	C₂×D₄×C₃₁	1	4	yes	—
401	#41	#41×C₃₁	1	2	yes	—
402	C₂×Q₈	—	2	2	no	—
403	C₂×Q₈	—	2	8	no	—
404	C₂×Q₈	—	2	8	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	2	2	no	—
408	C₂×Q₈	—	2	2	no	—
409	C₂×Q₈	—	4	2	no	—
410	C₂×Q₈	—	4	4	no	—
411	C₂×Q₈	—	4	4	no	—
412	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
413	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
414	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
415	C₂×D₄	—	2	2	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	—	4	2	no	—
418	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
419	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	2	no	—
425	C₂²×C₄	—	4	2	no	—
426	C₂²×C₄	—	4	2	no	—
427	C₂²×C₄	—	4	2	no	—
428	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
429	Q₁₆	—	2	8	no	—
430	Q₁₆	—	2	4	no	—
431	Q₁₆	—	2	4	no	—
432	Q₁₆	C₂²⋊C₄×C₃₁	1	4	yes	—
433	SD₁₆	—	2	8	no	—
434	SD₁₆	C₂²⋊C₄×C₃₁	1	8	yes	—
435	D₈	C₂²⋊C₄×C₃₁	1	4	yes	—
436	M₁₆	—	2	1	no	—
437	M₁₆	—	2	4	no	—
438	M₁₆	—	2	2	no	—
439	M₁₆	—	2	1	no	—
440	M₁₆	—	2	1	no	—
441	M₁₆	—	2	2	no	—
442	M₁₆	—	4	4	no	—
443	M₁₆	C₂²×C₄×C₃₁	1	2	yes	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	1	no	—
446	C₂×C₈	—	2	1	no	—
447	C₂×C₈	—	2	4	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	4	4	no	—
451	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
452	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
453	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
454	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
455	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	4	2	no	—
459	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
460	C₂²⋊C₄	—	2	1	no	—
461	C₂²⋊C₄	—	4	2	no	—
462	C₄²	—	2	1	no	—
463	C₄²	—	2	1	no	—
464	C₄²	—	2	1	no	—
465	C₄²	—	4	2	no	—
466	C₂³	—	2	2	no	—
467	C₂³	—	2	1	no	—
468	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
469	Q₈	—	2	1	no	—
470	Q₈	—	2	2	no	—
471	Q₈	—	2	4	no	—
472	Q₈	—	2	4	no	—
473	Q₈	—	4	8	no	—
474	Q₈	—	4	2	no	—
475	Q₈	—	4	2	no	—
476	Q₈	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
477	Q₈	C₄×D₄×C₃₁	1	2	yes	—
478	D₄	—	2	2	no	—
479	D₄	—	2	4	no	—
480	D₄	—	2	2	no	—
481	D₄	—	2	4	no	—
482	D₄	—	2	1	no	—
483	D₄	—	2	2	no	—
484	D₄	—	4	8	no	—
485	D₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
486	D₄	C₄×D₄×C₃₁	1	2	yes	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	4	8	no	—
496	C₂×C₄	—	4	8	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	4	2	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	8	2	no	—
503	C₂×C₄	—	16	2	no	—
504	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
505	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
506	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
508	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
509	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
510	C₈	—	2	2	no	—
511	C₈	—	2	2	no	—
512	C₈	—	4	2	no	—
513	C₈	—	4	1	no	—
514	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
515	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
516	C₂²	—	2	2	no	—
517	C₂²	—	2	2	no	—
518	C₂²	—	2	4	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	2	no	—
521	C₂²	—	4	2	no	—
522	C₂²	C₃₁×#518	1	1	yes	characteristic
523	C₂²	D₄²×C₃₁	1	1	yes	characteristic
524	C₂²	C₃₁×#546	1	1	yes	characteristic
525	C₄	—	2	2	no	—
526	C₄	—	2	4	no	—
527	C₄	—	4	4	no	—
528	C₄	—	4	2	no	—
529	C₄	—	8	1	no	—
530	C₄	—	8	2	no	—
531	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
532	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
533	C₄	D₄²×C₃₁	1	1	yes	characteristic
534	C₄	C₃₁×#546	1	1	yes	characteristic
535	C₂	—	2	1	no	—
536	C₂	—	2	1	no	—
537	C₂	—	4	2	no	—
538	C₂	—	4	4	no	—
539	C₂	D₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
540	C₂	C₃₁×#3289	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#578	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#444	C₂	semidirect
3. C₃₁×#2952	C₂	semidirect
4. C₃₁×#2898	C₂	semidirect
5. C₃₁×#1979	C₂	semidirect
6. C₃₁×#1971	C₂	semidirect
7. C₃₁×#1908	C₂	semidirect
8. C₃₁×#1882	C₂	semidirect
9. C₃₁×#1815	C₂	semidirect
10. C₃₁×#1793	C₂	semidirect
11. C₃₁×#1765	C₂	semidirect
14. C₃₁×#578	C₄	semidirect
51. C₃₁×2₋¹⁺⁴	C₂×C₄	semidirect
62. C₃₁×#137	C₂×C₄	semidirect
71. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
79. C₃₁×#131	C₂×C₄	semidirect
195. #22465	C₃₁	direct (N × H)
247. C₂×#444	C₂×C₃₁	semidirect
248. #2952	C₂×C₃₁	semidirect
249. #2898	C₂×C₃₁	semidirect
250. #1979	C₂×C₃₁	semidirect
251. #1971	C₂×C₃₁	semidirect
252. #1908	C₂×C₃₁	semidirect
253. #1882	C₂×C₃₁	semidirect
254. #1815	C₂×C₃₁	semidirect
255. #1793	C₂×C₃₁	semidirect
256. #1765	C₂×C₃₁	semidirect
278. #578	C₄×C₃₁	semidirect
321. 2₋¹⁺⁴	C₂×C₄×C₃₁	semidirect
332. #137	C₂×C₄×C₃₁	semidirect
341. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
349. #131	C₂×C₄×C₃₁	semidirect
387. C₃₁	#22465	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#578	2
C₂×C₃₁×#444	2
C₃₁×#2952	2
C₃₁×#2898	2
C₃₁×#1979	2
C₃₁×#1971	2
C₃₁×#1908	2
C₃₁×#1882	2
C₃₁×#1815	2
C₃₁×#1793	2
C₃₁×#1765	2
#22465	31

Subgroups — C31×#22465 #21479224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22465 #21479224