Subgroups — C₃₁×#17599 #21474358

Order 7936.

1046 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3519	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2974	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2319	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1519	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1516	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1513	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1510	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1509	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1505	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1502	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1500	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#555	C₄	1	1	yes	characteristic, complemented
13	C₃₁×#554	C₄	1	1	yes	characteristic, complemented
14	C₃₁×#553	—	2	1	no	—
15	C₃₁×#553	C₄	1	2	yes	complemented
16	C₃₁×#550	C₄	1	1	yes	characteristic, complemented
17	C₃₁×#546	—	2	2	no	—
18	C₃₁×#546	C₄	1	2	yes	complemented
19	D₄²×C₃₁	—	2	1	no	—
20	D₄²×C₃₁	C₄	1	1	yes	characteristic, complemented
21	C₃₁×#540	—	2	1	no	—
22	C₃₁×#538	—	2	2	no	—
23	C₃₁×#532	C₂²	1	1	yes	characteristic
24	C₃₁×#529	C₂²	1	1	yes	characteristic
25	C₃₁×#525	C₂²	1	2	yes	—
26	C₂×C₃₁×#121	C₂²	1	2	yes	—
27	C₃₁×#518	—	2	1	no	—
28	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#436	—	2	1	no	—
30	C₃₁×#435	—	2	2	no	—
31	C₈×D₄×C₃₁	—	2	1	no	—
32	C₃₁×#432	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#431	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×#407	C₂²	1	2	yes	complemented
35	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	complemented
36	C₃₁×#390	—	2	1	no	—
37	C₃₁×#389	—	2	1	no	—
38	C₃₁×#388	—	2	2	no	—
39	C₃₁×#381	C₂²	1	2	yes	complemented
40	C₃₁×#380	C₂²	1	2	yes	complemented
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
42	C₃₁×#376	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×#328	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×#328	C₂²	1	1	yes	characteristic, complemented
45	C₃₁×#328	C₂²	1	2	yes	complemented
46	C₃₁×#327	C₂²	1	1	yes	characteristic
47	C₃₁×#327	C₂²	1	1	yes	characteristic
48	C₃₁×#327	C₂²	1	2	yes	—
49	C₃₁×#327	C₂²	1	2	yes	complemented
50	C₃₁×#327	C₂²	1	2	yes	complemented
51	C₃₁×#327	C₂²	1	2	yes	complemented
52	C₃₁×#327	C₂²	1	2	yes	complemented
53	C₂×#41×C₃₁	—	2	1	no	—
54	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
55	C₂²×D₄×C₃₁	—	2	2	no	—
56	C₂²×D₄×C₃₁	—	2	1	no	—
57	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
58	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
60	C₂×M₁₆×C₃₁	—	2	2	no	—
61	C₂²×C₈×C₃₁	—	2	2	no	—
62	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
63	C₃₁×#127	—	2	1	no	—
64	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
65	C₃₁×#126	—	2	2	no	—
66	C₃₁×#126	—	2	1	no	—
67	C₃₁×#125	—	2	1	no	—
68	C₃₁×#125	C₂×C₄	1	1	yes	characteristic
69	C₃₁×#124	—	2	2	no	—
70	C₃₁×#124	—	2	1	no	—
71	C₃₁×#124	C₂×C₄	1	1	yes	characteristic
72	C₃₁×#123	—	2	2	no	—
73	C₃₁×#123	—	2	1	no	—
74	C₃₁×#123	—	2	2	no	—
75	C₃₁×#123	—	2	2	no	—
76	C₃₁×C₂²⋊Q₈	—	2	2	no	—
77	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
78	C₃₁×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
79	C₃₁×#121	—	2	2	no	—
80	C₃₁×#121	—	2	1	no	—
81	C₃₁×#121	—	2	2	no	—
82	C₃₁×#121	—	2	2	no	—
83	C₃₁×#121	—	2	1	no	—
84	C₃₁×#121	—	2	1	no	—
85	C₃₁×#121	—	2	2	no	—
86	C₃₁×#121	—	2	2	no	—
87	C₃₁×#121	C₂×C₄	1	2	yes	—
88	C₃₁×#121	C₂×C₄	1	2	yes	—
89	C₃₁×#121	C₂×C₄	1	2	yes	—
90	C₃₁×#121	C₂³	1	1	yes	characteristic
91	C₃₁×#121	C₂³	1	2	yes	—
92	C₃₁×#120	—	2	2	no	—
93	C₃₁×#120	—	2	2	no	—
94	C₃₁×#120	—	2	2	no	—
95	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
96	C₄×D₄×C₃₁	—	2	1	no	—
97	C₄×D₄×C₃₁	—	2	2	no	—
98	C₄×D₄×C₃₁	—	2	2	no	—
99	C₄×D₄×C₃₁	—	2	1	no	—
100	C₄×D₄×C₃₁	—	2	1	no	—
101	C₄×D₄×C₃₁	—	2	2	no	—
102	C₄×D₄×C₃₁	—	2	1	no	—
103	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
104	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
105	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
106	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
107	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
108	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
109	C₄×D₄×C₃₁	D₄	1	2	yes	—
110	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
111	C₃₁×#117	—	2	1	no	—
112	C₃₁×#117	—	2	1	no	—
113	C₃₁×#117	C₂×C₄	1	2	yes	—
114	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
115	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
117	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
118	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
119	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
120	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
121	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
122	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
123	C₂×C₄²×C₃₁	—	2	1	no	—
124	C₂×C₄²×C₃₁	—	2	1	no	—
125	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
126	C₃₁×C₄⋊C₈	—	2	1	no	—
127	C₃₁×C₄⋊C₈	—	2	1	no	—
128	C₃₁×C₂²⋊C₈	—	2	2	no	—
129	C₃₁×C₂²⋊C₈	—	2	2	no	—
130	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
131	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
132	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
133	C₃₁×#97	—	2	1	no	—
134	C₄×C₈×C₃₁	—	2	1	no	—
135	C₃₁×#95	—	2	1	no	—
136	C₃₁×#95	—	2	2	no	—
137	C₃₁×#95	C₂³	1	1	yes	characteristic
138	C₃₁×#95	C₂³	1	1	yes	characteristic
139	C₃₁×#95	C₂³	1	2	yes	—
140	C₂⁴×C₃₁	—	2	2	no	—
141	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
142	#41×C₃₁	—	8	1	no	—
143	#41×C₃₁	—	8	1	no	—
144	#41×C₃₁	—	8	2	no	—
145	C₂×Q₈×C₃₁	—	2	1	no	—
146	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
147	C₂×D₄×C₃₁	—	2	2	no	—
148	C₂×D₄×C₃₁	—	2	2	no	—
149	C₂×D₄×C₃₁	—	2	1	no	—
150	C₂×D₄×C₃₁	—	2	2	no	—
151	C₂×D₄×C₃₁	—	2	2	no	—
152	C₂×D₄×C₃₁	—	2	2	no	—
153	C₂×D₄×C₃₁	—	2	1	no	—
154	C₂×D₄×C₃₁	—	2	1	no	—
155	C₂×D₄×C₃₁	—	2	2	no	—
156	C₂×D₄×C₃₁	—	4	2	no	—
157	C₂×D₄×C₃₁	—	4	4	no	—
158	C₂×D₄×C₃₁	—	8	2	no	—
159	C₂×D₄×C₃₁	—	8	2	no	—
160	C₂×D₄×C₃₁	—	8	2	no	—
161	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
162	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
163	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
164	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
165	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
166	C₂²×C₄×C₃₁	—	2	2	no	—
167	C₂²×C₄×C₃₁	—	2	1	no	—
168	C₂²×C₄×C₃₁	—	2	2	no	—
169	C₂²×C₄×C₃₁	—	2	2	no	—
170	C₂²×C₄×C₃₁	—	2	1	no	—
171	C₂²×C₄×C₃₁	—	2	1	no	—
172	C₂²×C₄×C₃₁	—	2	2	no	—
173	C₂²×C₄×C₃₁	—	8	2	no	—
174	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
176	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
180	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
181	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
182	M₁₆×C₃₁	—	8	2	no	—
183	C₂×C₈×C₃₁	—	2	1	no	—
184	C₂×C₈×C₃₁	—	2	1	no	—
185	C₂×C₈×C₃₁	—	2	2	no	—
186	C₂×C₈×C₃₁	—	8	2	no	—
187	C₄⋊C₄×C₃₁	—	2	2	no	—
188	C₄⋊C₄×C₃₁	—	2	1	no	—
189	C₄⋊C₄×C₃₁	—	2	1	no	—
190	C₄⋊C₄×C₃₁	—	2	1	no	—
191	C₄⋊C₄×C₃₁	—	2	2	no	—
192	C₄⋊C₄×C₃₁	—	2	1	no	—
193	C₄⋊C₄×C₃₁	—	2	1	no	—
194	C₄⋊C₄×C₃₁	—	4	2	no	—
195	C₄⋊C₄×C₃₁	—	8	2	no	—
196	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
197	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
198	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
199	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
200	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
201	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
202	C₂²⋊C₄×C₃₁	—	2	2	no	—
203	C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂²⋊C₄×C₃₁	—	2	1	no	—
205	C₂²⋊C₄×C₃₁	—	2	1	no	—
206	C₂²⋊C₄×C₃₁	—	2	2	no	—
207	C₂²⋊C₄×C₃₁	—	2	2	no	—
208	C₂²⋊C₄×C₃₁	—	2	1	no	—
209	C₂²⋊C₄×C₃₁	—	2	2	no	—
210	C₂²⋊C₄×C₃₁	—	2	1	no	—
211	C₂²⋊C₄×C₃₁	—	2	2	no	—
212	C₂²⋊C₄×C₃₁	—	2	1	no	—
213	C₂²⋊C₄×C₃₁	—	2	2	no	—
214	C₂²⋊C₄×C₃₁	—	8	2	no	—
215	C₂²⋊C₄×C₃₁	—	8	2	no	—
216	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
217	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
218	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
219	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
220	C₄²×C₃₁	—	2	2	no	—
221	C₄²×C₃₁	—	2	1	no	—
222	C₄²×C₃₁	—	4	2	no	—
223	C₄²×C₃₁	—	8	1	no	—
224	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
225	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
226	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
227	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
228	#17599	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂³×C₃₁	—	2	2	no	—
230	C₂³×C₃₁	—	2	2	no	—
231	C₂³×C₃₁	—	2	1	no	—
232	C₂³×C₃₁	—	2	2	no	—
233	C₂³×C₃₁	—	4	2	no	—
234	C₂³×C₃₁	—	4	2	no	—
235	C₂³×C₃₁	—	4	4	no	—
236	C₂³×C₃₁	—	8	2	no	—
237	C₂³×C₃₁	—	8	2	no	—
238	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
239	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
240	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
241	Q₈×C₃₁	—	4	1	no	—
242	Q₈×C₃₁	—	8	1	no	—
243	D₄×C₃₁	—	2	2	no	—
244	D₄×C₃₁	—	2	2	no	—
245	D₄×C₃₁	—	4	2	no	—
246	D₄×C₃₁	—	4	1	no	—
247	D₄×C₃₁	—	4	2	no	—
248	D₄×C₃₁	—	4	4	no	—
249	D₄×C₃₁	—	4	1	no	—
250	D₄×C₃₁	—	4	2	no	—
251	D₄×C₃₁	—	4	1	no	—
252	D₄×C₃₁	—	4	2	no	—
253	D₄×C₃₁	—	4	2	no	—
254	D₄×C₃₁	—	8	2	no	—
255	D₄×C₃₁	—	8	1	no	—
256	D₄×C₃₁	—	8	2	no	—
257	D₄×C₃₁	—	8	2	no	—
258	D₄×C₃₁	—	8	2	no	—
259	C₂×C₄×C₃₁	—	2	1	no	—
260	C₂×C₄×C₃₁	—	2	1	no	—
261	C₂×C₄×C₃₁	—	2	2	no	—
262	C₂×C₄×C₃₁	—	2	1	no	—
263	C₂×C₄×C₃₁	—	2	1	no	—
264	C₂×C₄×C₃₁	—	2	2	no	—
265	C₂×C₄×C₃₁	—	2	1	no	—
266	C₂×C₄×C₃₁	—	4	2	no	—
267	C₂×C₄×C₃₁	—	4	2	no	—
268	C₂×C₄×C₃₁	—	4	2	no	—
269	C₂×C₄×C₃₁	—	4	1	no	—
270	C₂×C₄×C₃₁	—	4	2	no	—
271	C₂×C₄×C₃₁	—	4	2	no	—
272	C₂×C₄×C₃₁	—	4	2	no	—
273	C₂×C₄×C₃₁	—	4	1	no	—
274	C₂×C₄×C₃₁	—	4	1	no	—
275	C₂×C₄×C₃₁	—	4	1	no	—
276	C₂×C₄×C₃₁	—	4	2	no	—
277	C₂×C₄×C₃₁	—	4	2	no	—
278	C₂×C₄×C₃₁	—	4	1	no	—
279	C₂×C₄×C₃₁	—	4	2	no	—
280	C₂×C₄×C₃₁	—	4	1	no	—
281	C₂×C₄×C₃₁	—	4	1	no	—
282	C₂×C₄×C₃₁	—	8	2	no	—
283	C₂×C₄×C₃₁	—	8	2	no	—
284	C₂×C₄×C₃₁	—	8	2	no	—
285	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
286	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
287	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
288	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
289	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
290	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
291	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
292	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
293	C₈×C₃₁	—	4	1	no	—
294	C₈×C₃₁	—	4	2	no	—
295	C₈×C₃₁	—	4	1	no	—
296	#3519	C₂×C₃₁	1	1	yes	characteristic
297	#2974	C₂×C₃₁	1	1	yes	characteristic, complemented
298	#2319	C₂×C₃₁	1	1	yes	characteristic, complemented
299	#1519	C₂×C₃₁	1	2	yes	complemented
300	#1516	C₂×C₃₁	1	2	yes	complemented
301	#1513	C₂×C₃₁	1	2	yes	complemented
302	#1510	C₂×C₃₁	1	2	yes	complemented
303	#1509	C₂×C₃₁	1	1	yes	characteristic, complemented
304	#1505	C₂×C₃₁	1	1	yes	characteristic, complemented
305	#1502	C₂×C₃₁	1	1	yes	characteristic, complemented
306	#1500	C₂×C₃₁	1	1	yes	characteristic, complemented
307	C₂²×C₃₁	—	2	1	no	—
308	C₂²×C₃₁	—	2	1	no	—
309	C₂²×C₃₁	—	2	2	no	—
310	C₂²×C₃₁	—	2	2	no	—
311	C₂²×C₃₁	—	2	2	no	—
312	C₂²×C₃₁	—	2	2	no	—
313	C₂²×C₃₁	—	2	2	no	—
314	C₂²×C₃₁	—	2	2	no	—
315	C₂²×C₃₁	—	4	2	no	—
316	C₂²×C₃₁	—	4	1	no	—
317	C₂²×C₃₁	—	4	2	no	—
318	C₂²×C₃₁	—	4	1	no	—
319	C₂²×C₃₁	—	4	1	no	—
320	C₂²×C₃₁	—	4	2	no	—
321	C₂²×C₃₁	—	8	2	no	—
322	C₂²×C₃₁	—	8	2	no	—
323	C₂²×C₃₁	—	16	4	no	—
324	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
325	C₂²×C₃₁	#576	1	1	yes	characteristic
326	C₂²×C₃₁	#577	1	1	yes	characteristic
327	C₄×C₃₁	—	2	1	no	—
328	C₄×C₃₁	—	2	1	no	—
329	C₄×C₃₁	—	2	2	no	—
330	C₄×C₃₁	—	2	2	no	—
331	C₄×C₃₁	—	2	1	no	—
332	C₄×C₃₁	—	2	1	no	—
333	C₄×C₃₁	—	2	1	no	—
334	C₄×C₃₁	—	4	1	no	—
335	C₄×C₃₁	—	4	1	no	—
336	C₄×C₃₁	—	4	2	no	—
337	C₄×C₃₁	—	4	1	no	—
338	C₄×C₃₁	—	8	1	no	—
339	C₄×C₃₁	—	8	2	no	—
340	C₄×C₃₁	—	8	1	no	—
341	C₄×C₃₁	#421	1	2	yes	—
342	#555	C₄×C₃₁	1	1	yes	characteristic, complemented
343	#554	C₄×C₃₁	1	1	yes	characteristic, complemented
344	#553	—	2	1	no	—
345	#553	C₄×C₃₁	1	2	yes	complemented
346	#550	C₄×C₃₁	1	1	yes	characteristic, complemented
347	#546	—	2	2	no	—
348	#546	C₄×C₃₁	1	2	yes	complemented
349	D₄²	—	2	1	no	—
350	D₄²	C₄×C₃₁	1	1	yes	characteristic, complemented
351	#540	—	2	1	no	—
352	#538	—	2	2	no	—
353	#532	C₂²×C₃₁	1	1	yes	characteristic
354	#529	C₂²×C₃₁	1	1	yes	characteristic
355	#525	C₂²×C₃₁	1	2	yes	—
356	C₂×#121	C₂²×C₃₁	1	2	yes	—
357	#518	—	2	1	no	—
358	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
359	#436	—	2	1	no	—
360	#435	—	2	2	no	—
361	C₈×D₄	—	2	1	no	—
362	#432	C₂²×C₃₁	1	1	yes	characteristic, complemented
363	#431	C₂²×C₃₁	1	1	yes	characteristic, complemented
364	#407	C₂²×C₃₁	1	2	yes	complemented
365	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	complemented
366	#390	—	2	1	no	—
367	#389	—	2	1	no	—
368	#388	—	2	2	no	—
369	#381	C₂²×C₃₁	1	2	yes	complemented
370	#380	C₂²×C₃₁	1	2	yes	complemented
371	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
372	#376	C₂²×C₃₁	1	1	yes	characteristic, complemented
373	#328	C₂²×C₃₁	1	1	yes	characteristic, complemented
374	#328	C₂²×C₃₁	1	1	yes	characteristic, complemented
375	#328	C₂²×C₃₁	1	2	yes	complemented
376	#327	C₂²×C₃₁	1	1	yes	characteristic
377	#327	C₂²×C₃₁	1	1	yes	characteristic
378	#327	C₂²×C₃₁	1	2	yes	—
379	#327	C₂²×C₃₁	1	2	yes	complemented
380	#327	C₂²×C₃₁	1	2	yes	complemented
381	#327	C₂²×C₃₁	1	2	yes	complemented
382	#327	C₂²×C₃₁	1	2	yes	complemented
383	C₂×C₃₁	—	2	1	no	—
384	C₂×C₃₁	—	2	1	no	—
385	C₂×C₃₁	—	4	2	no	—
386	C₂×C₃₁	—	4	2	no	—
387	C₂×C₃₁	—	8	2	no	—
388	C₂×C₃₁	—	8	1	no	—
389	C₂×C₃₁	#2883	1	1	yes	characteristic
390	C₂×C₃₁	#2897	1	1	yes	characteristic
391	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
392	C₂×#41	—	2	1	no	—
393	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
394	C₂²×D₄	—	2	2	no	—
395	C₂²×D₄	—	2	1	no	—
396	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
397	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
398	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
399	C₂×M₁₆	—	2	2	no	—
400	C₂²×C₈	—	2	2	no	—
401	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
402	#127	—	2	1	no	—
403	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
404	#126	—	2	2	no	—
405	#126	—	2	1	no	—
406	#125	—	2	1	no	—
407	#125	C₂×C₄×C₃₁	1	1	yes	characteristic
408	#124	—	2	2	no	—
409	#124	—	2	1	no	—
410	#124	C₂×C₄×C₃₁	1	1	yes	characteristic
411	#123	—	2	2	no	—
412	#123	—	2	1	no	—
413	#123	—	2	2	no	—
414	#123	—	2	2	no	—
415	C₂²⋊Q₈	—	2	2	no	—
416	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
417	C₂²⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
418	#121	—	2	2	no	—
419	#121	—	2	1	no	—
420	#121	—	2	2	no	—
421	#121	—	2	2	no	—
422	#121	—	2	1	no	—
423	#121	—	2	1	no	—
424	#121	—	2	2	no	—
425	#121	—	2	2	no	—
426	#121	C₂×C₄×C₃₁	1	2	yes	—
427	#121	C₂×C₄×C₃₁	1	2	yes	—
428	#121	C₂×C₄×C₃₁	1	2	yes	—
429	#121	C₂³×C₃₁	1	1	yes	characteristic
430	#121	C₂³×C₃₁	1	2	yes	—
431	#120	—	2	2	no	—
432	#120	—	2	2	no	—
433	#120	—	2	2	no	—
434	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
435	C₄×D₄	—	2	1	no	—
436	C₄×D₄	—	2	2	no	—
437	C₄×D₄	—	2	2	no	—
438	C₄×D₄	—	2	1	no	—
439	C₄×D₄	—	2	1	no	—
440	C₄×D₄	—	2	2	no	—
441	C₄×D₄	—	2	1	no	—
442	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
443	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
444	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
445	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
446	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
447	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
448	C₄×D₄	D₄×C₃₁	1	2	yes	—
449	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
450	#117	—	2	1	no	—
451	#117	—	2	1	no	—
452	#117	C₂×C₄×C₃₁	1	2	yes	—
453	C₂×C₄⋊C₄	—	2	1	no	—
454	C₂×C₄⋊C₄	—	2	2	no	—
455	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
456	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
457	C₂×C₂²⋊C₄	—	2	2	no	—
458	C₂×C₂²⋊C₄	—	2	2	no	—
459	C₂×C₂²⋊C₄	—	2	2	no	—
460	C₂×C₂²⋊C₄	—	2	2	no	—
461	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
462	C₂×C₄²	—	2	1	no	—
463	C₂×C₄²	—	2	1	no	—
464	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
465	C₄⋊C₈	—	2	1	no	—
466	C₄⋊C₈	—	2	1	no	—
467	C₂²⋊C₈	—	2	2	no	—
468	C₂²⋊C₈	—	2	2	no	—
469	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
470	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
471	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
472	#97	—	2	1	no	—
473	C₄×C₈	—	2	1	no	—
474	#95	—	2	1	no	—
475	#95	—	2	2	no	—
476	#95	C₂³×C₃₁	1	1	yes	characteristic
477	#95	C₂³×C₃₁	1	1	yes	characteristic
478	#95	C₂³×C₃₁	1	2	yes	—
479	C₃₁	#17599	1	1	yes	characteristic, direct product factor, Sylow
480	C₂⁴	—	2	2	no	—
481	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
482	#41	—	8	1	no	—
483	#41	—	8	1	no	—
484	#41	—	8	2	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
487	C₂×D₄	—	2	2	no	—
488	C₂×D₄	—	2	2	no	—
489	C₂×D₄	—	2	1	no	—
490	C₂×D₄	—	2	2	no	—
491	C₂×D₄	—	2	2	no	—
492	C₂×D₄	—	2	2	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	—	2	1	no	—
495	C₂×D₄	—	2	2	no	—
496	C₂×D₄	—	4	2	no	—
497	C₂×D₄	—	4	4	no	—
498	C₂×D₄	—	8	2	no	—
499	C₂×D₄	—	8	2	no	—
500	C₂×D₄	—	8	2	no	—
501	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
502	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
503	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
504	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
505	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
506	C₂²×C₄	—	2	2	no	—
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	2	2	no	—
509	C₂²×C₄	—	2	2	no	—
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	2	2	no	—
513	C₂²×C₄	—	8	2	no	—
514	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
515	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
516	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
517	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
518	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
519	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
520	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
521	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
522	M₁₆	—	8	2	no	—
523	C₂×C₈	—	2	1	no	—
524	C₂×C₈	—	2	1	no	—
525	C₂×C₈	—	2	2	no	—
526	C₂×C₈	—	8	2	no	—
527	C₄⋊C₄	—	2	2	no	—
528	C₄⋊C₄	—	2	1	no	—
529	C₄⋊C₄	—	2	1	no	—
530	C₄⋊C₄	—	2	1	no	—
531	C₄⋊C₄	—	2	2	no	—
532	C₄⋊C₄	—	2	1	no	—
533	C₄⋊C₄	—	2	1	no	—
534	C₄⋊C₄	—	4	2	no	—
535	C₄⋊C₄	—	8	2	no	—
536	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
537	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
538	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
539	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
540	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
541	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
542	C₂²⋊C₄	—	2	2	no	—
543	C₂²⋊C₄	—	2	2	no	—
544	C₂²⋊C₄	—	2	1	no	—
545	C₂²⋊C₄	—	2	1	no	—
546	C₂²⋊C₄	—	2	2	no	—
547	C₂²⋊C₄	—	2	2	no	—
548	C₂²⋊C₄	—	2	1	no	—
549	C₂²⋊C₄	—	2	2	no	—
550	C₂²⋊C₄	—	2	1	no	—
551	C₂²⋊C₄	—	2	2	no	—
552	C₂²⋊C₄	—	2	1	no	—
553	C₂²⋊C₄	—	2	2	no	—
554	C₂²⋊C₄	—	8	2	no	—
555	C₂²⋊C₄	—	8	2	no	—
556	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
557	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
558	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
559	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
560	C₄²	—	2	2	no	—
561	C₄²	—	2	1	no	—
562	C₄²	—	4	2	no	—
563	C₄²	—	8	1	no	—
564	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
565	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
566	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
567	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
568	C₂³	—	2	2	no	—
569	C₂³	—	2	2	no	—
570	C₂³	—	2	1	no	—
571	C₂³	—	2	2	no	—
572	C₂³	—	4	2	no	—
573	C₂³	—	4	2	no	—
574	C₂³	—	4	4	no	—
575	C₂³	—	8	2	no	—
576	C₂³	—	8	2	no	—
577	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
578	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
579	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
580	Q₈	—	4	1	no	—
581	Q₈	—	8	1	no	—
582	D₄	—	2	2	no	—
583	D₄	—	2	2	no	—
584	D₄	—	4	2	no	—
585	D₄	—	4	1	no	—
586	D₄	—	4	2	no	—
587	D₄	—	4	4	no	—
588	D₄	—	4	1	no	—
589	D₄	—	4	2	no	—
590	D₄	—	4	1	no	—
591	D₄	—	4	2	no	—
592	D₄	—	4	2	no	—
593	D₄	—	8	2	no	—
594	D₄	—	8	1	no	—
595	D₄	—	8	2	no	—
596	D₄	—	8	2	no	—
597	D₄	—	8	2	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	4	2	no	—
606	C₂×C₄	—	4	2	no	—
607	C₂×C₄	—	4	2	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	2	no	—
610	C₂×C₄	—	4	2	no	—
611	C₂×C₄	—	4	2	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	1	no	—
615	C₂×C₄	—	4	2	no	—
616	C₂×C₄	—	4	2	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	2	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	8	2	no	—
622	C₂×C₄	—	8	2	no	—
623	C₂×C₄	—	8	2	no	—
624	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
625	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
626	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
627	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
628	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
629	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
630	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
631	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
632	C₈	—	4	1	no	—
633	C₈	—	4	2	no	—
634	C₈	—	4	1	no	—
635	C₂²	—	2	1	no	—
636	C₂²	—	2	1	no	—
637	C₂²	—	2	2	no	—
638	C₂²	—	2	2	no	—
639	C₂²	—	2	2	no	—
640	C₂²	—	2	2	no	—
641	C₂²	—	2	2	no	—
642	C₂²	—	2	2	no	—
643	C₂²	—	4	2	no	—
644	C₂²	—	4	1	no	—
645	C₂²	—	4	2	no	—
646	C₂²	—	4	1	no	—
647	C₂²	—	4	1	no	—
648	C₂²	—	4	2	no	—
649	C₂²	—	8	2	no	—
650	C₂²	—	8	2	no	—
651	C₂²	—	16	4	no	—
652	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
653	C₂²	C₃₁×#576	1	1	yes	characteristic
654	C₂²	C₃₁×#577	1	1	yes	characteristic
655	C₄	—	2	1	no	—
656	C₄	—	2	1	no	—
657	C₄	—	2	2	no	—
658	C₄	—	2	2	no	—
659	C₄	—	2	1	no	—
660	C₄	—	2	1	no	—
661	C₄	—	2	1	no	—
662	C₄	—	4	1	no	—
663	C₄	—	4	1	no	—
664	C₄	—	4	2	no	—
665	C₄	—	4	1	no	—
666	C₄	—	8	1	no	—
667	C₄	—	8	2	no	—
668	C₄	—	8	1	no	—
669	C₄	C₃₁×#421	1	2	yes	—
670	C₂	—	2	1	no	—
671	C₂	—	2	1	no	—
672	C₂	—	4	2	no	—
673	C₂	—	4	2	no	—
674	C₂	—	8	1	no	—
675	C₂	—	8	2	no	—
676	C₂	C₃₁×#2883	1	1	yes	characteristic
677	C₂	C₃₁×#2897	1	1	yes	characteristic
678	C₂	C₂×C₃₁×#421	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3519	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2974	C₂	semidirect
3. C₃₁×#2319	C₂	semidirect
4. C₃₁×#1519	C₂	semidirect
5. C₃₁×#1516	C₂	semidirect
6. C₃₁×#1513	C₂	semidirect
7. C₃₁×#1510	C₂	semidirect
8. C₃₁×#1509	C₂	semidirect
9. C₃₁×#1505	C₂	semidirect
10. C₃₁×#1502	C₂	semidirect
11. C₃₁×#1500	C₂	semidirect
12. C₃₁×#555	C₄	semidirect
13. C₃₁×#554	C₄	semidirect
15. C₃₁×#553	C₄	semidirect
16. C₃₁×#550	C₄	semidirect
18. C₃₁×#546	C₄	semidirect
20. D₄²×C₃₁	C₄	semidirect
32. C₃₁×#432	C₂²	semidirect
33. C₃₁×#431	C₂²	semidirect
34. C₃₁×#407	C₂²	semidirect
35. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
39. C₃₁×#381	C₂²	semidirect
40. C₃₁×#380	C₂²	semidirect
41. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
42. C₃₁×#376	C₂²	semidirect
43. C₃₁×#328	C₂²	semidirect
44. C₃₁×#328	C₂²	semidirect
45. C₃₁×#328	C₂²	semidirect
49. C₃₁×#327	C₂²	semidirect
50. C₃₁×#327	C₂²	semidirect
51. C₃₁×#327	C₂²	semidirect
52. C₃₁×#327	C₂²	semidirect
228. #17599	C₃₁	direct (N × H)
297. #2974	C₂×C₃₁	semidirect
298. #2319	C₂×C₃₁	semidirect
299. #1519	C₂×C₃₁	semidirect
300. #1516	C₂×C₃₁	semidirect
301. #1513	C₂×C₃₁	semidirect
302. #1510	C₂×C₃₁	semidirect
303. #1509	C₂×C₃₁	semidirect
304. #1505	C₂×C₃₁	semidirect
305. #1502	C₂×C₃₁	semidirect
306. #1500	C₂×C₃₁	semidirect
342. #555	C₄×C₃₁	semidirect
343. #554	C₄×C₃₁	semidirect
345. #553	C₄×C₃₁	semidirect
346. #550	C₄×C₃₁	semidirect
348. #546	C₄×C₃₁	semidirect
350. D₄²	C₄×C₃₁	semidirect
362. #432	C₂²×C₃₁	semidirect
363. #431	C₂²×C₃₁	semidirect
364. #407	C₂²×C₃₁	semidirect
365. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
369. #381	C₂²×C₃₁	semidirect
370. #380	C₂²×C₃₁	semidirect
371. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
372. #376	C₂²×C₃₁	semidirect
373. #328	C₂²×C₃₁	semidirect
374. #328	C₂²×C₃₁	semidirect
375. #328	C₂²×C₃₁	semidirect
379. #327	C₂²×C₃₁	semidirect
380. #327	C₂²×C₃₁	semidirect
381. #327	C₂²×C₃₁	semidirect
382. #327	C₂²×C₃₁	semidirect
479. C₃₁	#17599	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×#3519	2
C₃₁×#2974	2
C₃₁×#2319	2
C₃₁×#1519	2
C₃₁×#1516	2
C₃₁×#1513	2
C₃₁×#1510	2
C₃₁×#1509	2
C₃₁×#1505	2
C₃₁×#1502	2
C₃₁×#1500	2
#17599	31

Subgroups — C31×#17599 #21474358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17599 #21474358