Subgroups — C₃₁×#24190 #21480949

Order 7936.

1146 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#559	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#460	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2582	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2042	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2015	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1877	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1862	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#559	—	2	4	no	—
15	C₃₁×#559	—	2	4	no	—
16	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#124	—	2	1	no	—
18	C₂×C₃₁×#124	C₂²	1	2	yes	—
19	C₂×C₃₁×#123	—	2	2	no	—
20	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₃₁	—	2	1	no	—
24	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#460	—	2	4	no	—
26	C₄×Q₁₆×C₃₁	—	2	4	no	—
27	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#103	C₂²	1	2	yes	—
32	C₂×C₃₁×#102	C₂²	1	2	yes	—
33	C₂×C₃₁×#102	C₂²	1	2	yes	—
34	C₂×C₃₁×#102	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#398	—	2	2	no	—
37	C₃₁×#391	—	2	1	no	—
38	C₃₁×#391	C₂²	1	2	yes	—
39	C₃₁×#389	—	2	1	no	—
40	C₃₁×#389	C₂²	1	2	yes	—
41	C₃₁×#384	C₂²	1	1	yes	characteristic
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₃₁×#336	C₂²	1	2	yes	—
45	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
51	C₂³×C₄×C₃₁	—	2	2	no	—
52	C₂×Q₁₆×C₃₁	—	2	2	no	—
53	C₂×Q₁₆×C₃₁	—	2	2	no	—
54	C₂×Q₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	4	no	—
56	C₂×SD₁₆×C₃₁	—	2	4	no	—
57	C₂×SD₁₆×C₃₁	—	4	2	no	—
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×#127	—	2	2	no	—
62	C₃₁×#127	D₄	1	4	yes	—
63	C₃₁×#124	—	2	8	no	—
64	C₃₁×#124	—	2	4	no	—
65	C₃₁×#124	—	2	4	no	—
66	C₃₁×#123	—	2	16	no	—
67	C₄×Q₈×C₃₁	—	2	2	no	—
68	C₄×Q₈×C₃₁	—	2	2	no	—
69	C₄×Q₈×C₃₁	—	2	2	no	—
70	C₄×Q₈×C₃₁	—	2	2	no	—
71	C₄×Q₈×C₃₁	—	2	2	no	—
72	C₄×Q₈×C₃₁	D₄	1	4	yes	—
73	C₄×D₄×C₃₁	—	2	8	no	—
74	C₄×D₄×C₃₁	—	2	4	no	—
75	C₄×D₄×C₃₁	—	2	4	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
82	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
89	C₂×C₄²×C₃₁	—	2	1	no	—
90	C₂×C₄²×C₃₁	—	2	1	no	—
91	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
92	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
93	C₃₁×#107	—	2	2	no	—
94	C₃₁×C₄⋊C₈	—	2	2	no	—
95	C₃₁×#103	—	2	2	no	—
96	C₃₁×#103	—	2	2	no	—
97	C₃₁×#103	—	4	2	no	—
98	C₃₁×#102	—	2	4	no	—
99	C₃₁×#102	—	2	4	no	—
100	C₃₁×#102	—	4	2	no	—
101	C₄×C₈×C₃₁	—	2	2	no	—
102	C₃₁×#95	—	2	2	no	—
103	C₃₁×#95	—	2	2	no	—
104	C₃₁×#95	—	2	2	no	—
105	C₂⁴×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	—	2	1	no	—
110	C₂×Q₈×C₃₁	—	2	1	no	—
111	C₂×Q₈×C₃₁	—	2	1	no	—
112	C₂×Q₈×C₃₁	—	2	1	no	—
113	C₂×Q₈×C₃₁	—	2	1	no	—
114	C₂×Q₈×C₃₁	—	2	1	no	—
115	C₂×Q₈×C₃₁	—	2	1	no	—
116	C₂×Q₈×C₃₁	—	2	1	no	—
117	C₂×Q₈×C₃₁	—	2	1	no	—
118	C₂×Q₈×C₃₁	—	2	1	no	—
119	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
120	C₂×Q₈×C₃₁	#41	1	2	yes	—
121	C₂×D₄×C₃₁	—	2	4	no	—
122	C₂×D₄×C₃₁	—	2	4	no	—
123	C₂×D₄×C₃₁	—	2	2	no	—
124	C₂×D₄×C₃₁	—	2	2	no	—
125	C₂×D₄×C₃₁	—	2	2	no	—
126	C₂×D₄×C₃₁	—	2	2	no	—
127	C₂×D₄×C₃₁	—	2	2	no	—
128	C₂×D₄×C₃₁	—	4	2	no	—
129	C₂×D₄×C₃₁	—	4	2	no	—
130	C₂×D₄×C₃₁	—	8	1	no	—
131	C₂×D₄×C₃₁	—	8	1	no	—
132	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
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	8	no	—
137	C₂²×C₄×C₃₁	—	2	2	no	—
138	C₂²×C₄×C₃₁	—	2	1	no	—
139	C₂²×C₄×C₃₁	—	8	2	no	—
140	C₂²×C₄×C₃₁	—	8	2	no	—
141	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
146	Q₁₆×C₃₁	—	2	8	no	—
147	SD₁₆×C₃₁	—	4	8	no	—
148	C₂×C₈×C₃₁	—	2	1	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	—	2	1	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₂×C₈×C₃₁	—	2	1	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	—	2	1	no	—
156	C₂×C₈×C₃₁	#41	1	2	yes	—
157	C₄⋊C₄×C₃₁	—	2	4	no	—
158	C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₄⋊C₄×C₃₁	—	2	4	no	—
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	2	no	—
164	C₄⋊C₄×C₃₁	—	2	2	no	—
165	C₄⋊C₄×C₃₁	—	4	2	no	—
166	C₄⋊C₄×C₃₁	—	4	2	no	—
167	C₄⋊C₄×C₃₁	—	8	1	no	—
168	C₄⋊C₄×C₃₁	—	8	2	no	—
169	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
170	C₂²⋊C₄×C₃₁	—	2	8	no	—
171	C₂²⋊C₄×C₃₁	—	2	8	no	—
172	C₂²⋊C₄×C₃₁	—	2	8	no	—
173	C₄²×C₃₁	—	2	4	no	—
174	C₄²×C₃₁	—	2	2	no	—
175	C₄²×C₃₁	—	2	2	no	—
176	C₄²×C₃₁	—	4	2	no	—
177	C₄²×C₃₁	C₂×D₄	1	2	yes	—
178	C₄²×C₃₁	C₂×D₄	1	2	yes	—
179	#24190	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂³×C₃₁	—	2	4	no	—
181	C₂³×C₃₁	—	4	2	no	—
182	C₂³×C₃₁	—	4	2	no	—
183	C₂³×C₃₁	—	4	2	no	—
184	C₂³×C₃₁	—	4	2	no	—
185	C₂³×C₃₁	—	4	2	no	—
186	C₂³×C₃₁	—	4	2	no	—
187	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
188	Q₈×C₃₁	—	2	2	no	—
189	Q₈×C₃₁	—	2	2	no	—
190	Q₈×C₃₁	—	2	2	no	—
191	Q₈×C₃₁	—	2	2	no	—
192	Q₈×C₃₁	—	4	1	no	—
193	Q₈×C₃₁	—	4	1	no	—
194	Q₈×C₃₁	—	4	1	no	—
195	Q₈×C₃₁	—	4	1	no	—
196	Q₈×C₃₁	—	4	4	no	—
197	D₄×C₃₁	—	2	4	no	—
198	D₄×C₃₁	—	2	4	no	—
199	D₄×C₃₁	—	4	2	no	—
200	D₄×C₃₁	—	4	2	no	—
201	D₄×C₃₁	—	4	8	no	—
202	D₄×C₃₁	—	8	4	no	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	2	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	2	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	4	1	no	—
215	C₂×C₄×C₃₁	—	4	1	no	—
216	C₂×C₄×C₃₁	—	4	1	no	—
217	C₂×C₄×C₃₁	—	4	1	no	—
218	C₂×C₄×C₃₁	—	4	1	no	—
219	C₂×C₄×C₃₁	—	4	1	no	—
220	C₂×C₄×C₃₁	—	4	1	no	—
221	C₂×C₄×C₃₁	—	4	2	no	—
222	C₂×C₄×C₃₁	—	4	2	no	—
223	C₂×C₄×C₃₁	—	4	2	no	—
224	C₂×C₄×C₃₁	—	4	1	no	—
225	C₂×C₄×C₃₁	—	4	1	no	—
226	C₂×C₄×C₃₁	—	8	8	no	—
227	C₂×C₄×C₃₁	#120	1	2	yes	—
228	C₂×C₄×C₃₁	#120	1	2	yes	—
229	C₂×C₄×C₃₁	#135	1	2	yes	—
230	C₂×C₄×C₃₁	#137	1	2	yes	—
231	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
234	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
235	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
236	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
237	C₈×C₃₁	—	2	2	no	—
238	C₈×C₃₁	—	4	1	no	—
239	C₈×C₃₁	—	4	2	no	—
240	C₂×#559	C₂×C₃₁	1	1	yes	characteristic
241	C₂×#460	C₂×C₃₁	1	1	yes	characteristic
242	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#2582	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#2042	C₂×C₃₁	1	2	yes	complemented
245	#2015	C₂×C₃₁	1	2	yes	complemented
246	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#1957	C₂×C₃₁	1	1	yes	characteristic
248	#1877	C₂×C₃₁	1	2	yes	complemented
249	#1862	C₂×C₃₁	1	2	yes	complemented
250	#1803	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₂²×C₃₁	—	4	4	no	—
252	C₂²×C₃₁	—	4	4	no	—
253	C₂²×C₃₁	—	4	4	no	—
254	C₂²×C₃₁	—	8	2	no	—
255	C₂²×C₃₁	—	8	2	no	—
256	C₂²×C₃₁	—	8	2	no	—
257	C₂²×C₃₁	—	8	2	no	—
258	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
259	C₂²×C₃₁	#548	1	1	yes	characteristic
260	C₂²×C₃₁	#550	1	1	yes	characteristic
261	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
262	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
263	C₂²×C₃₁	#576	1	1	yes	characteristic
264	C₂²×C₃₁	#577	1	1	yes	characteristic
265	C₄×C₃₁	—	2	2	no	—
266	C₄×C₃₁	—	4	2	no	—
267	C₄×C₃₁	—	4	2	no	—
268	C₄×C₃₁	—	4	2	no	—
269	C₄×C₃₁	—	4	2	no	—
270	C₄×C₃₁	—	4	2	no	—
271	C₄×C₃₁	—	8	2	no	—
272	C₄×C₃₁	—	8	1	no	—
273	C₄×C₃₁	#452	1	4	yes	—
274	C₄×C₃₁	#525	1	1	yes	characteristic
275	C₄×C₃₁	#537	1	1	yes	characteristic
276	C₄×C₃₁	#546	1	1	yes	characteristic
277	C₄×C₃₁	#547	1	1	yes	characteristic
278	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
279	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
280	#559	—	2	4	no	—
281	#559	—	2	4	no	—
282	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×#124	—	2	1	no	—
284	C₂×#124	C₂²×C₃₁	1	2	yes	—
285	C₂×#123	—	2	2	no	—
286	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×C₄×D₄	—	2	1	no	—
290	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
291	#460	—	2	4	no	—
292	C₄×Q₁₆	—	2	4	no	—
293	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
294	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
297	C₂×#103	C₂²×C₃₁	1	2	yes	—
298	C₂×#102	C₂²×C₃₁	1	2	yes	—
299	C₂×#102	C₂²×C₃₁	1	2	yes	—
300	C₂×#102	C₂²×C₃₁	1	2	yes	—
301	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
302	#398	—	2	2	no	—
303	#391	—	2	1	no	—
304	#391	C₂²×C₃₁	1	2	yes	—
305	#389	—	2	1	no	—
306	#389	C₂²×C₃₁	1	2	yes	—
307	#384	C₂²×C₃₁	1	1	yes	characteristic
308	#340	C₂²×C₃₁	1	2	yes	—
309	#336	C₂²×C₃₁	1	2	yes	—
310	#336	C₂²×C₃₁	1	2	yes	—
311	C₂×C₃₁	—	8	4	no	—
312	C₂×C₃₁	#2407	1	1	yes	characteristic
313	C₂×C₃₁	C₂×#452	1	1	yes	characteristic
314	C₂×C₃₁	#3012	1	1	yes	characteristic
315	C₂×C₃₁	#3365	1	1	yes	characteristic
316	C₂×C₃₁	#3375	1	1	yes	characteristic
317	C₂×C₃₁	#3396	1	1	yes	characteristic
318	C₂×C₃₁	#3400	1	1	yes	characteristic
319	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
320	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
321	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
322	C₂²×D₄	—	2	1	no	—
323	C₂²×D₄	—	2	1	no	—
324	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
325	C₂³×C₄	—	2	2	no	—
326	C₂×Q₁₆	—	2	2	no	—
327	C₂×Q₁₆	—	2	2	no	—
328	C₂×Q₁₆	—	2	2	no	—
329	C₂×SD₁₆	—	2	4	no	—
330	C₂×SD₁₆	—	2	4	no	—
331	C₂×SD₁₆	—	4	2	no	—
332	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
333	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
334	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
335	#127	—	2	2	no	—
336	#127	D₄×C₃₁	1	4	yes	—
337	#124	—	2	8	no	—
338	#124	—	2	4	no	—
339	#124	—	2	4	no	—
340	#123	—	2	16	no	—
341	C₄×Q₈	—	2	2	no	—
342	C₄×Q₈	—	2	2	no	—
343	C₄×Q₈	—	2	2	no	—
344	C₄×Q₈	—	2	2	no	—
345	C₄×Q₈	—	2	2	no	—
346	C₄×Q₈	D₄×C₃₁	1	4	yes	—
347	C₄×D₄	—	2	8	no	—
348	C₄×D₄	—	2	4	no	—
349	C₄×D₄	—	2	4	no	—
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	—	2	1	no	—
352	C₂×C₄⋊C₄	—	2	1	no	—
353	C₂×C₄⋊C₄	—	2	1	no	—
354	C₂×C₄⋊C₄	—	2	2	no	—
355	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
356	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
357	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
358	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
359	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
360	C₂×C₂²⋊C₄	—	2	2	no	—
361	C₂×C₂²⋊C₄	—	2	2	no	—
362	C₂×C₂²⋊C₄	—	2	2	no	—
363	C₂×C₄²	—	2	1	no	—
364	C₂×C₄²	—	2	1	no	—
365	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
366	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
367	#107	—	2	2	no	—
368	C₄⋊C₈	—	2	2	no	—
369	#103	—	2	2	no	—
370	#103	—	2	2	no	—
371	#103	—	4	2	no	—
372	#102	—	2	4	no	—
373	#102	—	2	4	no	—
374	#102	—	4	2	no	—
375	C₄×C₈	—	2	2	no	—
376	#95	—	2	2	no	—
377	#95	—	2	2	no	—
378	#95	—	2	2	no	—
379	C₃₁	#24190	1	1	yes	characteristic, direct product factor, Sylow
380	C₂⁴	—	2	2	no	—
381	C₂×Q₈	—	2	2	no	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	—	2	2	no	—
384	C₂×Q₈	—	2	1	no	—
385	C₂×Q₈	—	2	1	no	—
386	C₂×Q₈	—	2	1	no	—
387	C₂×Q₈	—	2	1	no	—
388	C₂×Q₈	—	2	1	no	—
389	C₂×Q₈	—	2	1	no	—
390	C₂×Q₈	—	2	1	no	—
391	C₂×Q₈	—	2	1	no	—
392	C₂×Q₈	—	2	1	no	—
393	C₂×Q₈	—	2	1	no	—
394	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
395	C₂×Q₈	#41×C₃₁	1	2	yes	—
396	C₂×D₄	—	2	4	no	—
397	C₂×D₄	—	2	4	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	4	2	no	—
404	C₂×D₄	—	4	2	no	—
405	C₂×D₄	—	8	1	no	—
406	C₂×D₄	—	8	1	no	—
407	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
408	C₂²×C₄	—	2	1	no	—
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	8	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	8	2	no	—
415	C₂²×C₄	—	8	2	no	—
416	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
417	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
418	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
419	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
420	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
421	Q₁₆	—	2	8	no	—
422	SD₁₆	—	4	8	no	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	1	no	—
429	C₂×C₈	—	2	2	no	—
430	C₂×C₈	—	2	1	no	—
431	C₂×C₈	#41×C₃₁	1	2	yes	—
432	C₄⋊C₄	—	2	4	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	2	4	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	4	2	no	—
441	C₄⋊C₄	—	4	2	no	—
442	C₄⋊C₄	—	8	1	no	—
443	C₄⋊C₄	—	8	2	no	—
444	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
445	C₂²⋊C₄	—	2	8	no	—
446	C₂²⋊C₄	—	2	8	no	—
447	C₂²⋊C₄	—	2	8	no	—
448	C₄²	—	2	4	no	—
449	C₄²	—	2	2	no	—
450	C₄²	—	2	2	no	—
451	C₄²	—	4	2	no	—
452	C₄²	C₂×D₄×C₃₁	1	2	yes	—
453	C₄²	C₂×D₄×C₃₁	1	2	yes	—
454	C₂³	—	2	4	no	—
455	C₂³	—	4	2	no	—
456	C₂³	—	4	2	no	—
457	C₂³	—	4	2	no	—
458	C₂³	—	4	2	no	—
459	C₂³	—	4	2	no	—
460	C₂³	—	4	2	no	—
461	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
462	Q₈	—	2	2	no	—
463	Q₈	—	2	2	no	—
464	Q₈	—	2	2	no	—
465	Q₈	—	2	2	no	—
466	Q₈	—	4	4	no	—
467	Q₈	—	4	1	no	—
468	Q₈	—	4	1	no	—
469	Q₈	—	4	1	no	—
470	Q₈	—	4	1	no	—
471	D₄	—	2	4	no	—
472	D₄	—	2	4	no	—
473	D₄	—	4	8	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	2	no	—
476	D₄	—	8	4	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	4	1	no	—
493	C₂×C₄	—	4	1	no	—
494	C₂×C₄	—	4	1	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	4	2	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	4	1	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	8	8	no	—
501	C₂×C₄	C₃₁×#120	1	2	yes	—
502	C₂×C₄	C₃₁×#120	1	2	yes	—
503	C₂×C₄	C₃₁×#135	1	2	yes	—
504	C₂×C₄	C₃₁×#137	1	2	yes	—
505	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
506	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
507	C₂×C₄	C₂×#41×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	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
511	C₈	—	2	2	no	—
512	C₈	—	4	1	no	—
513	C₈	—	4	2	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	4	4	no	—
516	C₂²	—	4	4	no	—
517	C₂²	—	8	2	no	—
518	C₂²	—	8	2	no	—
519	C₂²	—	8	2	no	—
520	C₂²	—	8	2	no	—
521	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
522	C₂²	C₃₁×#548	1	1	yes	characteristic
523	C₂²	C₃₁×#550	1	1	yes	characteristic
524	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
525	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
526	C₂²	C₃₁×#576	1	1	yes	characteristic
527	C₂²	C₃₁×#577	1	1	yes	characteristic
528	C₄	—	2	2	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	2	no	—
532	C₄	—	4	2	no	—
533	C₄	—	4	2	no	—
534	C₄	—	8	2	no	—
535	C₄	—	8	1	no	—
536	C₄	C₃₁×#452	1	4	yes	—
537	C₄	C₃₁×#525	1	1	yes	characteristic
538	C₄	C₃₁×#537	1	1	yes	characteristic
539	C₄	C₃₁×#546	1	1	yes	characteristic
540	C₄	C₃₁×#547	1	1	yes	characteristic
541	C₂	—	8	4	no	—
542	C₂	C₃₁×#2407	1	1	yes	characteristic
543	C₂	C₂×C₃₁×#452	1	1	yes	characteristic
544	C₂	C₃₁×#3012	1	1	yes	characteristic
545	C₂	C₃₁×#3365	1	1	yes	characteristic
546	C₂	C₃₁×#3375	1	1	yes	characteristic
547	C₂	C₃₁×#3396	1	1	yes	characteristic
548	C₂	C₃₁×#3400	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Q₁₆×C₃₁	C₂	semidirect
4. C₃₁×#2582	C₂	semidirect
5. C₃₁×#2042	C₂	semidirect
6. C₃₁×#2015	C₂	semidirect
7. C₃₁×#1984	C₂	semidirect
9. C₃₁×#1877	C₂	semidirect
10. C₃₁×#1862	C₂	semidirect
11. C₃₁×#1803	C₂	semidirect
179. #24190	C₃₁	direct (N × H)
242. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
243. #2582	C₂×C₃₁	semidirect
244. #2042	C₂×C₃₁	semidirect
245. #2015	C₂×C₃₁	semidirect
246. #1984	C₂×C₃₁	semidirect
248. #1877	C₂×C₃₁	semidirect
249. #1862	C₂×C₃₁	semidirect
250. #1803	C₂×C₃₁	semidirect
379. C₃₁	#24190	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×#559	2
C₂×C₃₁×#460	2
C₂×C₄×Q₁₆×C₃₁	2
C₃₁×#2582	2
C₃₁×#2042	2
C₃₁×#2015	2
C₃₁×#1984	2
C₃₁×#1957	2
C₃₁×#1877	2
C₃₁×#1862	2
C₃₁×#1803	2
#24190	31

Subgroups — C31×#24190 #21480949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24190 #21480949