Subgroups — C₃₁×#22418 #21479177

Order 7936.

1542 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#550	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#416	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2354	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1982	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1893	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1875	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1788	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#103	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	2	yes	—
13	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#550	—	2	4	no	—
15	C₃₁×#550	C₄	1	8	yes	complemented
16	C₂×C₃₁×#127	—	2	1	no	—
17	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#121	—	2	2	no	—
19	C₂×C₃₁×#121	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₃₁	—	2	1	no	—
22	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#117	C₂²	1	2	yes	—
24	C₈×D₄×C₃₁	—	2	4	no	—
25	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₃₁×#416	—	2	8	no	—
27	C₂×C₃₁×#103	C₂²	1	2	yes	—
28	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#102	C₂²	1	2	yes	—
31	C₂×C₃₁×#102	C₂²	1	2	yes	—
32	C₂×C₃₁×#102	C₂²	1	2	yes	—
33	C₂×C₃₁×#102	C₂²	1	2	yes	—
34	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#390	—	2	1	no	—
37	C₃₁×#388	—	2	2	no	—
38	C₃₁×#384	C₂²	1	1	yes	characteristic
39	C₃₁×#380	C₂²	1	2	yes	—
40	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#340	C₂²	1	1	yes	characteristic
44	C₃₁×#340	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	—	2	4	no	—
46	C₂×#41×C₃₁	—	2	4	no	—
47	C₂×#41×C₃₁	—	2	4	no	—
48	C₂×#41×C₃₁	—	4	2	no	—
49	C₂×#41×C₃₁	—	4	2	no	—
50	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
51	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×D₄×C₃₁	—	2	2	no	—
53	C₂²×D₄×C₃₁	—	2	2	no	—
54	C₂²×D₄×C₃₁	—	2	1	no	—
55	C₂²×D₄×C₃₁	—	2	1	no	—
56	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
57	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
58	C₂³×C₄×C₃₁	—	2	2	no	—
59	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
60	C₂²×C₈×C₃₁	—	2	4	no	—
61	C₂²×C₈×C₃₁	—	2	4	no	—
62	C₂²×C₈×C₃₁	—	4	2	no	—
63	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
65	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×#127	—	2	8	no	—
67	C₃₁×#127	—	2	2	no	—
68	C₃₁×#127	C₂×C₄	1	4	yes	—
69	C₃₁×#121	—	2	4	no	—
70	C₃₁×#121	—	2	16	no	—
71	C₃₁×#121	C₂×C₄	1	8	yes	—
72	C₄×Q₈×C₃₁	—	2	2	no	—
73	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
74	C₄×D₄×C₃₁	—	2	8	no	—
75	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
76	C₄×D₄×C₃₁	D₄	1	4	yes	—
77	C₃₁×#117	—	2	4	no	—
78	C₃₁×#117	—	4	2	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
91	C₂×C₄²×C₃₁	—	2	1	no	—
92	C₂×C₄²×C₃₁	—	2	2	no	—
93	C₂×C₄²×C₃₁	—	2	1	no	—
94	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
95	C₃₁×C₄⋊C₈	—	2	2	no	—
96	C₃₁×#103	—	2	4	no	—
97	C₃₁×#103	—	2	2	no	—
98	C₃₁×#103	D₄	1	4	yes	complemented
99	C₃₁×#102	—	2	4	no	—
100	C₃₁×#102	—	2	4	no	—
101	C₃₁×#102	—	4	2	no	—
102	C₃₁×#102	—	4	2	no	—
103	C₃₁×C₂²⋊C₈	—	2	4	no	—
104	C₄×C₈×C₃₁	—	2	2	no	—
105	C₃₁×#95	—	2	2	no	—
106	C₃₁×#95	—	2	1	no	—
107	C₂⁴×C₃₁	—	2	2	no	—
108	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
109	#41×C₃₁	—	2	16	no	—
110	#41×C₃₁	—	4	8	no	—
111	#41×C₃₁	—	4	8	no	—
112	#41×C₃₁	—	4	8	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	C₂²×C₄	1	2	yes	—
117	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
118	C₂×Q₈×C₃₁	#41	1	2	yes	—
119	C₂×D₄×C₃₁	—	2	8	no	—
120	C₂×D₄×C₃₁	—	2	8	no	—
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	—	2	2	no	—
129	C₂×D₄×C₃₁	—	2	2	no	—
130	C₂×D₄×C₃₁	—	4	4	no	—
131	C₂×D₄×C₃₁	—	4	4	no	—
132	C₂×D₄×C₃₁	—	4	2	no	—
133	C₂×D₄×C₃₁	—	4	2	no	—
134	C₂×D₄×C₃₁	—	8	1	no	—
135	C₂×D₄×C₃₁	—	8	1	no	—
136	C₂×D₄×C₃₁	—	8	2	no	—
137	C₂×D₄×C₃₁	—	8	2	no	—
138	C₂×D₄×C₃₁	C₂²⋊C₄	1	4	yes	—
139	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
140	C₂²×C₄×C₃₁	—	2	1	no	—
141	C₂²×C₄×C₃₁	—	2	2	no	—
142	C₂²×C₄×C₃₁	—	2	2	no	—
143	C₂²×C₄×C₃₁	—	2	2	no	—
144	C₂²×C₄×C₃₁	—	2	2	no	—
145	C₂²×C₄×C₃₁	—	2	1	no	—
146	C₂²×C₄×C₃₁	—	2	1	no	—
147	C₂²×C₄×C₃₁	—	2	8	no	—
148	C₂²×C₄×C₃₁	—	2	1	no	—
149	C₂²×C₄×C₃₁	—	2	2	no	—
150	C₂²×C₄×C₃₁	—	4	4	no	—
151	C₂²×C₄×C₃₁	—	4	4	no	—
152	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
153	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
158	C₂×C₈×C₃₁	—	2	1	no	—
159	C₂×C₈×C₃₁	—	2	1	no	—
160	C₂×C₈×C₃₁	—	2	1	no	—
161	C₂×C₈×C₃₁	—	2	2	no	—
162	C₂×C₈×C₃₁	—	2	2	no	—
163	C₂×C₈×C₃₁	—	2	1	no	—
164	C₂×C₈×C₃₁	—	2	1	no	—
165	C₂×C₈×C₃₁	—	2	2	no	—
166	C₂×C₈×C₃₁	—	4	8	no	—
167	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
168	C₄⋊C₄×C₃₁	—	2	4	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	2	2	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	2	no	—
175	C₄⋊C₄×C₃₁	—	4	2	no	—
176	C₄⋊C₄×C₃₁	—	8	1	no	—
177	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
178	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
179	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
180	C₂²⋊C₄×C₃₁	—	2	8	no	—
181	C₂²⋊C₄×C₃₁	—	4	4	no	—
182	C₂²⋊C₄×C₃₁	—	8	2	no	—
183	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
184	C₄²×C₃₁	—	2	4	no	—
185	C₄²×C₃₁	—	4	4	no	—
186	C₄²×C₃₁	—	4	2	no	—
187	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
188	C₄²×C₃₁	C₂×D₄	1	2	yes	—
189	#22418	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂³×C₃₁	—	2	2	no	—
191	C₂³×C₃₁	—	2	2	no	—
192	C₂³×C₃₁	—	2	2	no	—
193	C₂³×C₃₁	—	2	2	no	—
194	C₂³×C₃₁	—	2	2	no	—
195	C₂³×C₃₁	—	2	2	no	—
196	C₂³×C₃₁	—	2	4	no	—
197	C₂³×C₃₁	—	4	2	no	—
198	C₂³×C₃₁	—	4	2	no	—
199	C₂³×C₃₁	—	4	2	no	—
200	C₂³×C₃₁	—	4	2	no	—
201	C₂³×C₃₁	—	4	2	no	—
202	C₂³×C₃₁	—	4	2	no	—
203	C₂³×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
204	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
205	Q₈×C₃₁	—	2	2	no	—
206	Q₈×C₃₁	—	2	2	no	—
207	Q₈×C₃₁	—	2	2	no	—
208	Q₈×C₃₁	C₄×D₄	1	4	yes	—
209	D₄×C₃₁	—	2	4	no	—
210	D₄×C₃₁	—	2	4	no	—
211	D₄×C₃₁	—	2	8	no	—
212	D₄×C₃₁	—	4	16	no	—
213	D₄×C₃₁	—	4	2	no	—
214	D₄×C₃₁	—	4	2	no	—
215	D₄×C₃₁	—	4	8	no	—
216	D₄×C₃₁	—	8	4	no	—
217	D₄×C₃₁	—	8	8	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	1	no	—
224	C₂×C₄×C₃₁	—	2	1	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	2	no	—
231	C₂×C₄×C₃₁	—	4	16	no	—
232	C₂×C₄×C₃₁	—	4	1	no	—
233	C₂×C₄×C₃₁	—	4	1	no	—
234	C₂×C₄×C₃₁	—	4	1	no	—
235	C₂×C₄×C₃₁	—	4	1	no	—
236	C₂×C₄×C₃₁	—	4	2	no	—
237	C₂×C₄×C₃₁	—	4	1	no	—
238	C₂×C₄×C₃₁	—	4	1	no	—
239	C₂×C₄×C₃₁	—	4	2	no	—
240	C₂×C₄×C₃₁	—	4	1	no	—
241	C₂×C₄×C₃₁	—	4	2	no	—
242	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
243	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
244	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
245	C₂×C₄×C₃₁	#135	1	2	yes	—
246	C₂×C₄×C₃₁	#135	1	2	yes	—
247	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
248	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
251	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
252	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
253	C₈×C₃₁	—	2	2	no	—
254	C₈×C₃₁	—	4	2	no	—
255	C₈×C₃₁	—	4	1	no	—
256	C₂×#550	C₂×C₃₁	1	1	yes	characteristic
257	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
258	C₂×#416	C₂×C₃₁	1	2	yes	complemented
259	#2354	C₂×C₃₁	1	1	yes	characteristic, complemented
260	#1982	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#1957	C₂×C₃₁	1	1	yes	characteristic, complemented
262	#1893	C₂×C₃₁	1	2	yes	complemented
263	#1875	C₂×C₃₁	1	2	yes	complemented
264	#1803	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#1788	C₂×C₃₁	1	2	yes	complemented
266	C₄×#103	C₂×C₃₁	1	1	yes	characteristic, complemented
267	C₂²×C₃₁	—	2	4	no	—
268	C₂²×C₃₁	—	2	4	no	—
269	C₂²×C₃₁	—	2	4	no	—
270	C₂²×C₃₁	—	4	2	no	—
271	C₂²×C₃₁	—	4	2	no	—
272	C₂²×C₃₁	—	4	2	no	—
273	C₂²×C₃₁	—	4	2	no	—
274	C₂²×C₃₁	—	4	4	no	—
275	C₂²×C₃₁	—	4	4	no	—
276	C₂²×C₃₁	—	4	4	no	—
277	C₂²×C₃₁	—	8	2	no	—
278	C₂²×C₃₁	—	8	2	no	—
279	C₂²×C₃₁	—	8	2	no	—
280	C₂²×C₃₁	—	8	2	no	—
281	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
282	C₂²×C₃₁	D₄²	1	1	yes	characteristic
283	C₂²×C₃₁	#546	1	1	yes	characteristic
284	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
285	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
286	C₂²×C₃₁	#576	1	1	yes	characteristic
287	C₂²×C₃₁	#577	1	1	yes	characteristic
288	C₄×C₃₁	—	2	4	no	—
289	C₄×C₃₁	—	2	4	no	—
290	C₄×C₃₁	—	2	2	no	—
291	C₄×C₃₁	—	4	2	no	—
292	C₄×C₃₁	—	4	2	no	—
293	C₄×C₃₁	—	8	2	no	—
294	C₄×C₃₁	—	8	1	no	—
295	C₄×C₃₁	#416	1	4	yes	complemented
296	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
297	C₄×C₃₁	#518	1	1	yes	characteristic
298	C₄×C₃₁	D₄²	1	1	yes	characteristic
299	C₄×C₃₁	#546	1	1	yes	characteristic
300	C₂²×#41	C₂²×C₃₁	1	2	yes	—
301	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
302	#550	—	2	4	no	—
303	#550	C₄×C₃₁	1	8	yes	complemented
304	C₂×#127	—	2	1	no	—
305	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
306	C₂×#121	—	2	2	no	—
307	C₂×#121	C₂²×C₃₁	1	2	yes	—
308	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×C₄×D₄	—	2	1	no	—
310	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
311	C₂×#117	C₂²×C₃₁	1	2	yes	—
312	C₈×D₄	—	2	4	no	—
313	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
314	#416	—	2	8	no	—
315	C₂×#103	C₂²×C₃₁	1	2	yes	—
316	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
317	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
318	C₂×#102	C₂²×C₃₁	1	2	yes	—
319	C₂×#102	C₂²×C₃₁	1	2	yes	—
320	C₂×#102	C₂²×C₃₁	1	2	yes	—
321	C₂×#102	C₂²×C₃₁	1	2	yes	—
322	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
323	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
324	#390	—	2	1	no	—
325	#388	—	2	2	no	—
326	#384	C₂²×C₃₁	1	1	yes	characteristic
327	#380	C₂²×C₃₁	1	2	yes	—
328	C₄×C₄⋊C₄	—	2	1	no	—
329	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
330	#340	C₂²×C₃₁	1	2	yes	—
331	#340	C₂²×C₃₁	1	1	yes	characteristic
332	#340	C₂²×C₃₁	1	1	yes	characteristic
333	C₂×C₃₁	—	4	4	no	—
334	C₂×C₃₁	—	8	4	no	—
335	C₂×C₃₁	D₄×C₂²⋊C₄	1	1	yes	characteristic
336	C₂×C₃₁	C₂×#416	1	1	yes	characteristic
337	C₂×C₃₁	#2897	1	1	yes	characteristic
338	C₂×C₃₁	#3284	1	1	yes	characteristic
339	C₂×C₃₁	#3287	1	1	yes	characteristic
340	C₂×C₃₁	#3304	1	1	yes	characteristic
341	C₂×C₃₁	#3305	1	1	yes	characteristic
342	C₂×#41	—	2	4	no	—
343	C₂×#41	—	2	4	no	—
344	C₂×#41	—	2	4	no	—
345	C₂×#41	—	4	2	no	—
346	C₂×#41	—	4	2	no	—
347	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
348	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
349	C₂²×D₄	—	2	2	no	—
350	C₂²×D₄	—	2	2	no	—
351	C₂²×D₄	—	2	1	no	—
352	C₂²×D₄	—	2	1	no	—
353	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
354	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
355	C₂³×C₄	—	2	2	no	—
356	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
357	C₂²×C₈	—	2	4	no	—
358	C₂²×C₈	—	2	4	no	—
359	C₂²×C₈	—	4	2	no	—
360	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
361	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
362	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
363	#127	—	2	8	no	—
364	#127	—	2	2	no	—
365	#127	C₂×C₄×C₃₁	1	4	yes	—
366	#121	—	2	4	no	—
367	#121	—	2	16	no	—
368	#121	C₂×C₄×C₃₁	1	8	yes	—
369	C₄×Q₈	—	2	2	no	—
370	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
371	C₄×D₄	—	2	8	no	—
372	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
373	C₄×D₄	D₄×C₃₁	1	4	yes	—
374	#117	—	2	4	no	—
375	#117	—	4	2	no	—
376	C₂×C₄⋊C₄	—	2	1	no	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
381	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
383	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
384	C₂×C₂²⋊C₄	—	2	2	no	—
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₂²⋊C₄	—	2	2	no	—
387	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
388	C₂×C₄²	—	2	1	no	—
389	C₂×C₄²	—	2	2	no	—
390	C₂×C₄²	—	2	1	no	—
391	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
392	C₄⋊C₈	—	2	2	no	—
393	#103	—	2	4	no	—
394	#103	—	2	2	no	—
395	#103	D₄×C₃₁	1	4	yes	complemented
396	#102	—	2	4	no	—
397	#102	—	2	4	no	—
398	#102	—	4	2	no	—
399	#102	—	4	2	no	—
400	C₂²⋊C₈	—	2	4	no	—
401	C₄×C₈	—	2	2	no	—
402	#95	—	2	2	no	—
403	#95	—	2	1	no	—
404	C₃₁	#22418	1	1	yes	characteristic, direct product factor, Sylow
405	C₂⁴	—	2	2	no	—
406	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
407	#41	—	2	16	no	—
408	#41	—	4	8	no	—
409	#41	—	4	8	no	—
410	#41	—	4	8	no	—
411	C₂×Q₈	—	2	1	no	—
412	C₂×Q₈	—	2	1	no	—
413	C₂×Q₈	—	2	1	no	—
414	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
415	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
416	C₂×Q₈	#41×C₃₁	1	2	yes	—
417	C₂×D₄	—	2	8	no	—
418	C₂×D₄	—	2	8	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	4	4	no	—
429	C₂×D₄	—	4	4	no	—
430	C₂×D₄	—	4	2	no	—
431	C₂×D₄	—	4	2	no	—
432	C₂×D₄	—	8	1	no	—
433	C₂×D₄	—	8	1	no	—
434	C₂×D₄	—	8	2	no	—
435	C₂×D₄	—	8	2	no	—
436	C₂×D₄	C₂²⋊C₄×C₃₁	1	4	yes	—
437	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	8	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	4	4	no	—
449	C₂²×C₄	—	4	4	no	—
450	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
451	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
452	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
454	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
455	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	2	no	—
460	C₂×C₈	—	2	2	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	2	no	—
464	C₂×C₈	—	4	8	no	—
465	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	2	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	4	2	no	—
474	C₄⋊C₄	—	8	1	no	—
475	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
476	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
477	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
478	C₂²⋊C₄	—	2	8	no	—
479	C₂²⋊C₄	—	4	4	no	—
480	C₂²⋊C₄	—	8	2	no	—
481	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
482	C₄²	—	2	4	no	—
483	C₄²	—	4	4	no	—
484	C₄²	—	4	2	no	—
485	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
486	C₄²	C₂×D₄×C₃₁	1	2	yes	—
487	C₂³	—	2	2	no	—
488	C₂³	—	2	2	no	—
489	C₂³	—	2	2	no	—
490	C₂³	—	2	2	no	—
491	C₂³	—	2	2	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	4	no	—
494	C₂³	—	4	2	no	—
495	C₂³	—	4	2	no	—
496	C₂³	—	4	2	no	—
497	C₂³	—	4	2	no	—
498	C₂³	—	4	2	no	—
499	C₂³	—	4	2	no	—
500	C₂³	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
501	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
502	Q₈	—	2	2	no	—
503	Q₈	—	2	2	no	—
504	Q₈	—	2	2	no	—
505	Q₈	C₄×D₄×C₃₁	1	4	yes	—
506	D₄	—	2	8	no	—
507	D₄	—	2	4	no	—
508	D₄	—	2	4	no	—
509	D₄	—	4	16	no	—
510	D₄	—	4	8	no	—
511	D₄	—	4	2	no	—
512	D₄	—	4	2	no	—
513	D₄	—	8	4	no	—
514	D₄	—	8	8	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	4	16	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	2	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	2	no	—
539	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
540	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
541	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
542	C₂×C₄	C₃₁×#135	1	2	yes	—
543	C₂×C₄	C₃₁×#135	1	2	yes	—
544	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
545	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
546	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
547	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
548	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
549	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
550	C₈	—	2	2	no	—
551	C₈	—	4	2	no	—
552	C₈	—	4	1	no	—
553	C₂²	—	2	4	no	—
554	C₂²	—	2	4	no	—
555	C₂²	—	2	4	no	—
556	C₂²	—	4	2	no	—
557	C₂²	—	4	2	no	—
558	C₂²	—	4	2	no	—
559	C₂²	—	4	2	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	4	4	no	—
562	C₂²	—	4	4	no	—
563	C₂²	—	8	2	no	—
564	C₂²	—	8	2	no	—
565	C₂²	—	8	2	no	—
566	C₂²	—	8	2	no	—
567	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
568	C₂²	D₄²×C₃₁	1	1	yes	characteristic
569	C₂²	C₃₁×#546	1	1	yes	characteristic
570	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
571	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
572	C₂²	C₃₁×#576	1	1	yes	characteristic
573	C₂²	C₃₁×#577	1	1	yes	characteristic
574	C₄	—	2	4	no	—
575	C₄	—	2	4	no	—
576	C₄	—	2	2	no	—
577	C₄	—	4	2	no	—
578	C₄	—	4	2	no	—
579	C₄	—	8	2	no	—
580	C₄	—	8	1	no	—
581	C₄	C₃₁×#416	1	4	yes	complemented
582	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
583	C₄	C₃₁×#518	1	1	yes	characteristic
584	C₄	D₄²×C₃₁	1	1	yes	characteristic
585	C₄	C₃₁×#546	1	1	yes	characteristic
586	C₂	—	4	4	no	—
587	C₂	—	8	4	no	—
588	C₂	D₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
589	C₂	C₂×C₃₁×#416	1	1	yes	characteristic
590	C₂	C₃₁×#2897	1	1	yes	characteristic
591	C₂	C₃₁×#3284	1	1	yes	characteristic
592	C₂	C₃₁×#3287	1	1	yes	characteristic
593	C₂	C₃₁×#3304	1	1	yes	characteristic
594	C₂	C₃₁×#3305	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#550	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×D₄×C₃₁	C₂	semidirect
3. C₂×C₃₁×#416	C₂	semidirect
4. C₃₁×#2354	C₂	semidirect
5. C₃₁×#1982	C₂	semidirect
6. C₃₁×#1957	C₂	semidirect
7. C₃₁×#1893	C₂	semidirect
8. C₃₁×#1875	C₂	semidirect
9. C₃₁×#1803	C₂	semidirect
10. C₃₁×#1788	C₂	semidirect
11. C₄×C₃₁×#103	C₂	semidirect
15. C₃₁×#550	C₄	semidirect
98. C₃₁×#103	D₄	semidirect
189. #22418	C₃₁	direct (N × H)
257. C₂×C₈×D₄	C₂×C₃₁	semidirect
258. C₂×#416	C₂×C₃₁	semidirect
259. #2354	C₂×C₃₁	semidirect
260. #1982	C₂×C₃₁	semidirect
261. #1957	C₂×C₃₁	semidirect
262. #1893	C₂×C₃₁	semidirect
263. #1875	C₂×C₃₁	semidirect
264. #1803	C₂×C₃₁	semidirect
265. #1788	C₂×C₃₁	semidirect
266. C₄×#103	C₂×C₃₁	semidirect
295. C₄×C₃₁	#416	semidirect
303. #550	C₄×C₃₁	semidirect
395. #103	D₄×C₃₁	semidirect
404. C₃₁	#22418	direct (N × H)
581. C₄	C₃₁×#416	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#550	2
C₂×C₈×D₄×C₃₁	2
C₂×C₃₁×#416	2
C₃₁×#2354	2
C₃₁×#1982	2
C₃₁×#1957	2
C₃₁×#1893	2
C₃₁×#1875	2
C₃₁×#1803	2
C₃₁×#1788	2
C₄×C₃₁×#103	2
#22418	31

Subgroups — C31×#22418 #21479177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22418 #21479177