Subgroups — C₃₁×#17588 #21474347

Order 7936.

1086 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3444	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#382	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1518	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1517	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1512	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1504	C₂	1	4	yes	maximal, complemented
8	C₃₁×#533	C₄	1	1	yes	characteristic, complemented
9	C₃₁×#533	C₄	1	1	yes	characteristic, complemented
10	C₃₁×#533	C₂²	1	1	yes	characteristic
11	C₃₁×#532	C₄	1	1	yes	characteristic, complemented
12	C₃₁×#532	C₄	1	1	yes	characteristic, complemented
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₃₁×#529	—	2	1	no	—
15	C₃₁×#529	—	2	1	no	—
16	C₃₁×#529	—	2	1	no	—
17	C₃₁×#529	—	2	1	no	—
18	C₃₁×#529	C₄	1	2	yes	complemented
19	C₃₁×#529	C₂²	1	2	yes	—
20	C₃₁×#525	—	2	1	no	—
21	C₃₁×#525	—	2	1	no	—
22	C₃₁×#525	C₂²	1	2	yes	—
23	C₄×#41×C₃₁	—	2	1	no	—
24	C₄×#41×C₃₁	—	2	1	no	—
25	C₄×#41×C₃₁	C₄	1	2	yes	complemented
26	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
27	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#432	—	2	1	no	—
29	C₃₁×#432	C₂²	1	2	yes	—
30	C₃₁×#431	—	2	1	no	—
31	C₃₁×#431	C₂²	1	2	yes	—
32	C₃₁×#423	—	2	1	no	—
33	C₃₁×#407	C₂²	1	2	yes	—
34	C₄×M₁₆×C₃₁	—	2	1	no	—
35	C₃₁×#382	—	2	1	no	—
36	C₃₁×#382	—	2	1	no	—
37	C₃₁×#382	C₂²	1	2	yes	—
38	C₃₁×#382	C₂²	1	2	yes	—
39	C₂×C₃₁×#95	—	2	1	no	—
40	C₂×C₃₁×#95	C₂²	1	2	yes	—
41	C₃₁×#328	C₂²	1	8	yes	—
42	C₃₁×#327	C₂²	1	8	yes	—
43	C₂×#41×C₃₁	—	2	1	no	—
44	C₂×#41×C₃₁	—	2	1	no	—
45	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₃₁	—	2	2	no	—
47	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
52	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
53	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
54	C₃₁×#127	C₂×C₄	1	1	yes	characteristic, complemented
55	C₃₁×#126	—	2	2	no	—
56	C₃₁×#126	—	2	2	no	—
57	C₃₁×#125	—	2	1	no	—
58	C₃₁×#125	—	2	1	no	—
59	C₃₁×#125	C₂×C₄	1	2	yes	complemented
60	C₃₁×#124	—	2	1	no	—
61	C₃₁×#124	—	2	1	no	—
62	C₃₁×#124	C₂×C₄	1	2	yes	complemented
63	C₃₁×#123	—	2	2	no	—
64	C₃₁×#123	—	2	2	no	—
65	C₃₁×#123	—	2	2	no	—
66	C₃₁×C₂²⋊Q₈	—	2	1	no	—
67	C₃₁×C₂²⋊Q₈	—	2	1	no	—
68	C₃₁×C₂²⋊Q₈	—	2	1	no	—
69	C₃₁×C₂²⋊Q₈	—	2	1	no	—
70	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
71	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
72	C₃₁×#121	—	2	1	no	—
73	C₃₁×#121	—	2	1	no	—
74	C₃₁×#121	—	2	1	no	—
75	C₃₁×#121	—	2	1	no	—
76	C₃₁×#121	C₂×C₄	1	2	yes	—
77	C₃₁×#121	C₂³	1	2	yes	—
78	C₃₁×#120	—	2	2	no	—
79	C₄×Q₈×C₃₁	—	2	1	no	—
80	C₄×Q₈×C₃₁	—	2	1	no	—
81	C₄×Q₈×C₃₁	—	2	1	no	—
82	C₄×Q₈×C₃₁	—	2	1	no	—
83	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	complemented
84	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	complemented
85	C₄×D₄×C₃₁	—	2	2	no	—
86	C₄×D₄×C₃₁	—	2	2	no	—
87	C₄×D₄×C₃₁	—	2	2	no	—
88	C₄×D₄×C₃₁	—	2	2	no	—
89	C₄×D₄×C₃₁	—	2	2	no	—
90	C₄×D₄×C₃₁	—	2	2	no	—
91	C₄×D₄×C₃₁	—	2	1	no	—
92	C₄×D₄×C₃₁	—	2	1	no	—
93	C₄×D₄×C₃₁	—	2	1	no	—
94	C₄×D₄×C₃₁	—	2	1	no	—
95	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	complemented
96	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	complemented
97	C₃₁×#117	—	2	1	no	—
98	C₃₁×#117	—	2	1	no	—
99	C₃₁×#117	—	2	2	no	—
100	C₃₁×#117	—	2	2	no	—
101	C₃₁×#117	—	2	2	no	—
102	C₃₁×#117	C₂×C₄	1	2	yes	—
103	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
106	C₂×C₄²×C₃₁	—	2	2	no	—
107	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
108	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
109	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
110	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
112	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
113	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
117	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
119	C₃₁×C₄⋊C₈	—	2	2	no	—
120	C₃₁×C₄⋊C₈	—	2	2	no	—
121	C₃₁×C₂²⋊C₈	—	2	2	no	—
122	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
123	C₃₁×#97	—	2	2	no	—
124	C₄×C₈×C₃₁	—	2	2	no	—
125	C₃₁×#95	—	2	2	no	—
126	C₃₁×#95	—	2	2	no	—
127	C₃₁×#95	—	2	2	no	—
128	C₃₁×#95	C₂³	1	4	yes	—
129	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
130	#41×C₃₁	—	4	4	no	—
131	#41×C₃₁	—	4	4	no	—
132	#41×C₃₁	—	4	2	no	—
133	#41×C₃₁	—	4	2	no	—
134	C₂×Q₈×C₃₁	—	2	1	no	—
135	C₂×Q₈×C₃₁	—	2	1	no	—
136	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
137	C₂×D₄×C₃₁	—	2	2	no	—
138	C₂×D₄×C₃₁	—	2	2	no	—
139	C₂×D₄×C₃₁	—	2	2	no	—
140	C₂×D₄×C₃₁	—	2	1	no	—
141	C₂×D₄×C₃₁	—	2	1	no	—
142	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
143	C₂²×C₄×C₃₁	—	2	2	no	—
144	C₂²×C₄×C₃₁	—	2	2	no	—
145	C₂²×C₄×C₃₁	—	2	2	no	—
146	C₂²×C₄×C₃₁	—	2	2	no	—
147	C₂²×C₄×C₃₁	—	2	2	no	—
148	C₂²×C₄×C₃₁	—	2	4	no	—
149	C₂²×C₄×C₃₁	—	4	2	no	—
150	C₂²×C₄×C₃₁	—	4	2	no	—
151	C₂²×C₄×C₃₁	—	4	4	no	—
152	C₂²×C₄×C₃₁	—	4	4	no	—
153	C₂²×C₄×C₃₁	—	4	4	no	—
154	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
172	M₁₆×C₃₁	—	4	4	no	—
173	C₂×C₈×C₃₁	—	2	4	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	2	1	no	—
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	2	2	no	—
179	C₄⋊C₄×C₃₁	—	2	2	no	—
180	C₄⋊C₄×C₃₁	—	2	2	no	—
181	C₄⋊C₄×C₃₁	—	2	2	no	—
182	C₄⋊C₄×C₃₁	—	2	2	no	—
183	C₄⋊C₄×C₃₁	—	2	2	no	—
184	C₄⋊C₄×C₃₁	—	4	4	no	—
185	C₄⋊C₄×C₃₁	—	4	4	no	—
186	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
187	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
188	C₂²⋊C₄×C₃₁	—	2	2	no	—
189	C₂²⋊C₄×C₃₁	—	2	2	no	—
190	C₂²⋊C₄×C₃₁	—	2	2	no	—
191	C₂²⋊C₄×C₃₁	—	2	2	no	—
192	C₂²⋊C₄×C₃₁	—	2	2	no	—
193	C₂²⋊C₄×C₃₁	—	2	2	no	—
194	C₂²⋊C₄×C₃₁	—	2	2	no	—
195	C₂²⋊C₄×C₃₁	—	2	2	no	—
196	C₂²⋊C₄×C₃₁	—	2	2	no	—
197	C₄²×C₃₁	—	2	2	no	—
198	C₄²×C₃₁	—	2	2	no	—
199	C₄²×C₃₁	—	2	2	no	—
200	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
201	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
202	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
203	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
204	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
205	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
206	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
207	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
208	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
209	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
210	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
211	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
212	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
213	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
214	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
216	#17588	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂³×C₃₁	—	2	1	no	—
218	C₂³×C₃₁	—	2	1	no	—
219	C₂³×C₃₁	—	2	2	no	—
220	C₂³×C₃₁	—	2	2	no	—
221	C₂³×C₃₁	—	2	2	no	—
222	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
223	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
224	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	Q₈×C₃₁	—	4	2	no	—
226	Q₈×C₃₁	—	4	2	no	—
227	Q₈×C₃₁	—	4	2	no	—
228	D₄×C₃₁	—	4	2	no	—
229	D₄×C₃₁	—	4	2	no	—
230	D₄×C₃₁	—	4	2	no	—
231	D₄×C₃₁	—	4	2	no	—
232	D₄×C₃₁	—	4	4	no	—
233	D₄×C₃₁	—	4	4	no	—
234	D₄×C₃₁	—	4	2	no	—
235	C₂×C₄×C₃₁	—	2	1	no	—
236	C₂×C₄×C₃₁	—	2	1	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	2	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	2	no	—
243	C₂×C₄×C₃₁	—	2	2	no	—
244	C₂×C₄×C₃₁	—	2	2	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	2	no	—
247	C₂×C₄×C₃₁	—	2	2	no	—
248	C₂×C₄×C₃₁	—	4	1	no	—
249	C₂×C₄×C₃₁	—	4	1	no	—
250	C₂×C₄×C₃₁	—	4	1	no	—
251	C₂×C₄×C₃₁	—	4	1	no	—
252	C₂×C₄×C₃₁	—	4	1	no	—
253	C₂×C₄×C₃₁	—	4	1	no	—
254	C₂×C₄×C₃₁	—	4	1	no	—
255	C₂×C₄×C₃₁	—	4	1	no	—
256	C₂×C₄×C₃₁	—	4	4	no	—
257	C₂×C₄×C₃₁	—	4	4	no	—
258	C₂×C₄×C₃₁	—	4	4	no	—
259	C₂×C₄×C₃₁	—	4	4	no	—
260	C₂×C₄×C₃₁	—	4	4	no	—
261	C₂×C₄×C₃₁	—	4	4	no	—
262	C₂×C₄×C₃₁	—	8	8	no	—
263	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
272	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
274	C₂×C₄×C₃₁	#137	1	2	yes	—
275	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
276	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
277	C₈×C₃₁	—	4	4	no	—
278	#3444	C₂×C₃₁	1	1	yes	characteristic
279	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
280	C₂×#382	C₂×C₃₁	1	1	yes	characteristic, complemented
281	#1518	C₂×C₃₁	1	2	yes	complemented
282	#1517	C₂×C₃₁	1	2	yes	complemented
283	#1512	C₂×C₃₁	1	4	yes	complemented
284	#1504	C₂×C₃₁	1	4	yes	complemented
285	C₂²×C₃₁	—	2	4	no	—
286	C₂²×C₃₁	—	2	4	no	—
287	C₂²×C₃₁	—	2	1	no	—
288	C₂²×C₃₁	—	2	1	no	—
289	C₂²×C₃₁	—	2	1	no	—
290	C₂²×C₃₁	—	2	1	no	—
291	C₂²×C₃₁	—	2	1	no	—
292	C₂²×C₃₁	—	2	1	no	—
293	C₂²×C₃₁	—	4	2	no	—
294	C₂²×C₃₁	—	4	2	no	—
295	C₂²×C₃₁	—	4	2	no	—
296	C₂²×C₃₁	#410	1	2	yes	—
297	C₂²×C₃₁	#421	1	2	yes	—
298	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
299	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
300	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
301	C₄×C₃₁	—	2	1	no	—
302	C₄×C₃₁	—	2	1	no	—
303	C₄×C₃₁	—	2	1	no	—
304	C₄×C₃₁	—	2	1	no	—
305	C₄×C₃₁	—	2	1	no	—
306	C₄×C₃₁	—	2	1	no	—
307	C₄×C₃₁	—	2	1	no	—
308	C₄×C₃₁	—	2	1	no	—
309	C₄×C₃₁	—	2	1	no	—
310	C₄×C₃₁	—	2	1	no	—
311	C₄×C₃₁	—	4	2	no	—
312	C₄×C₃₁	—	4	2	no	—
313	C₄×C₃₁	—	4	2	no	—
314	C₄×C₃₁	—	8	4	no	—
315	C₄×C₃₁	#417	1	2	yes	—
316	C₄×C₃₁	#417	1	2	yes	—
317	#533	C₄×C₃₁	1	1	yes	characteristic, complemented
318	#533	C₄×C₃₁	1	1	yes	characteristic, complemented
319	#533	C₂²×C₃₁	1	1	yes	characteristic
320	#532	C₄×C₃₁	1	1	yes	characteristic, complemented
321	#532	C₄×C₃₁	1	1	yes	characteristic, complemented
322	#532	C₂²×C₃₁	1	1	yes	characteristic
323	#529	—	2	1	no	—
324	#529	—	2	1	no	—
325	#529	—	2	1	no	—
326	#529	—	2	1	no	—
327	#529	C₄×C₃₁	1	2	yes	complemented
328	#529	C₂²×C₃₁	1	2	yes	—
329	#525	—	2	1	no	—
330	#525	—	2	1	no	—
331	#525	C₂²×C₃₁	1	2	yes	—
332	C₄×#41	—	2	1	no	—
333	C₄×#41	—	2	1	no	—
334	C₄×#41	C₄×C₃₁	1	2	yes	complemented
335	C₂²×C₄⋊C₄	—	2	1	no	—
336	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
337	#432	—	2	1	no	—
338	#432	C₂²×C₃₁	1	2	yes	—
339	#431	—	2	1	no	—
340	#431	C₂²×C₃₁	1	2	yes	—
341	#423	—	2	1	no	—
342	#407	C₂²×C₃₁	1	2	yes	—
343	C₄×M₁₆	—	2	1	no	—
344	#382	—	2	1	no	—
345	#382	—	2	1	no	—
346	#382	C₂²×C₃₁	1	2	yes	—
347	#382	C₂²×C₃₁	1	2	yes	—
348	C₂×#95	—	2	1	no	—
349	C₂×#95	C₂²×C₃₁	1	2	yes	—
350	#328	C₂²×C₃₁	1	8	yes	—
351	#327	C₂²×C₃₁	1	8	yes	—
352	C₂×C₃₁	—	2	1	no	—
353	C₂×C₃₁	—	2	1	no	—
354	C₂×C₃₁	—	2	2	no	—
355	C₂×C₃₁	—	2	2	no	—
356	C₂×C₃₁	—	8	2	no	—
357	C₂×C₃₁	C₂×#410	1	1	yes	characteristic
358	C₂×C₃₁	C₂×#417	1	1	yes	characteristic
359	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
360	C₂×#41	—	2	1	no	—
361	C₂×#41	—	2	1	no	—
362	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
363	C₂³×C₄	—	2	2	no	—
364	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
365	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
366	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
367	C₂×M₁₆	—	2	2	no	—
368	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
369	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
370	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
371	#127	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
372	#126	—	2	2	no	—
373	#126	—	2	2	no	—
374	#125	—	2	1	no	—
375	#125	—	2	1	no	—
376	#125	C₂×C₄×C₃₁	1	2	yes	complemented
377	#124	—	2	1	no	—
378	#124	—	2	1	no	—
379	#124	C₂×C₄×C₃₁	1	2	yes	complemented
380	#123	—	2	2	no	—
381	#123	—	2	2	no	—
382	#123	—	2	2	no	—
383	C₂²⋊Q₈	—	2	1	no	—
384	C₂²⋊Q₈	—	2	1	no	—
385	C₂²⋊Q₈	—	2	1	no	—
386	C₂²⋊Q₈	—	2	1	no	—
387	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
388	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
389	#121	—	2	1	no	—
390	#121	—	2	1	no	—
391	#121	—	2	1	no	—
392	#121	—	2	1	no	—
393	#121	C₂×C₄×C₃₁	1	2	yes	—
394	#121	C₂³×C₃₁	1	2	yes	—
395	#120	—	2	2	no	—
396	C₄×Q₈	—	2	1	no	—
397	C₄×Q₈	—	2	1	no	—
398	C₄×Q₈	—	2	1	no	—
399	C₄×Q₈	—	2	1	no	—
400	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
401	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
402	C₄×D₄	—	2	2	no	—
403	C₄×D₄	—	2	2	no	—
404	C₄×D₄	—	2	2	no	—
405	C₄×D₄	—	2	2	no	—
406	C₄×D₄	—	2	2	no	—
407	C₄×D₄	—	2	2	no	—
408	C₄×D₄	—	2	1	no	—
409	C₄×D₄	—	2	1	no	—
410	C₄×D₄	—	2	1	no	—
411	C₄×D₄	—	2	1	no	—
412	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	complemented
413	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	complemented
414	#117	—	2	1	no	—
415	#117	—	2	1	no	—
416	#117	—	2	2	no	—
417	#117	—	2	2	no	—
418	#117	—	2	2	no	—
419	#117	C₂×C₄×C₃₁	1	2	yes	—
420	C₂×C₄⋊C₄	—	2	2	no	—
421	C₂×C₄⋊C₄	—	2	2	no	—
422	C₂×C₄⋊C₄	—	4	4	no	—
423	C₂×C₄²	—	2	2	no	—
424	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
425	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
426	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
427	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
428	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
429	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
431	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
432	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
433	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
434	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
435	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
436	C₄⋊C₈	—	2	2	no	—
437	C₄⋊C₈	—	2	2	no	—
438	C₂²⋊C₈	—	2	2	no	—
439	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
440	#97	—	2	2	no	—
441	C₄×C₈	—	2	2	no	—
442	#95	—	2	2	no	—
443	#95	—	2	2	no	—
444	#95	—	2	2	no	—
445	#95	C₂³×C₃₁	1	4	yes	—
446	C₃₁	#17588	1	1	yes	characteristic, direct product factor, Sylow
447	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
448	#41	—	4	4	no	—
449	#41	—	4	4	no	—
450	#41	—	4	2	no	—
451	#41	—	4	2	no	—
452	C₂×Q₈	—	2	1	no	—
453	C₂×Q₈	—	2	1	no	—
454	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	1	no	—
459	C₂×D₄	—	2	1	no	—
460	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	4	2	no	—
468	C₂²×C₄	—	4	2	no	—
469	C₂²×C₄	—	4	4	no	—
470	C₂²×C₄	—	4	4	no	—
471	C₂²×C₄	—	4	4	no	—
472	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
473	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
474	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
475	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
476	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
478	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
479	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
480	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
481	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
483	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
484	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
485	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
486	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
487	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
488	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
489	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
490	M₁₆	—	4	4	no	—
491	C₂×C₈	—	2	4	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	4	4	no	—
503	C₄⋊C₄	—	4	4	no	—
504	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
505	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	2	no	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₄²	—	2	2	no	—
516	C₄²	—	2	2	no	—
517	C₄²	—	2	2	no	—
518	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
519	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
520	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
521	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
522	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
523	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
524	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
525	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
526	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
527	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
528	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
529	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
530	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
531	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
532	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
533	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
534	C₂³	—	2	2	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	1	no	—
537	C₂³	—	2	1	no	—
538	C₂³	—	2	2	no	—
539	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
540	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
541	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
542	Q₈	—	4	2	no	—
543	Q₈	—	4	2	no	—
544	Q₈	—	4	2	no	—
545	D₄	—	4	2	no	—
546	D₄	—	4	2	no	—
547	D₄	—	4	4	no	—
548	D₄	—	4	4	no	—
549	D₄	—	4	2	no	—
550	D₄	—	4	2	no	—
551	D₄	—	4	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	4	no	—
574	C₂×C₄	—	4	4	no	—
575	C₂×C₄	—	4	4	no	—
576	C₂×C₄	—	4	4	no	—
577	C₂×C₄	—	4	4	no	—
578	C₂×C₄	—	4	4	no	—
579	C₂×C₄	—	8	8	no	—
580	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
581	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
582	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
583	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
584	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
585	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
586	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
587	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
588	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
589	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
590	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
591	C₂×C₄	C₃₁×#137	1	2	yes	—
592	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
593	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
594	C₈	—	4	4	no	—
595	C₂²	—	2	4	no	—
596	C₂²	—	2	4	no	—
597	C₂²	—	2	1	no	—
598	C₂²	—	2	1	no	—
599	C₂²	—	2	1	no	—
600	C₂²	—	2	1	no	—
601	C₂²	—	2	1	no	—
602	C₂²	—	2	1	no	—
603	C₂²	—	4	2	no	—
604	C₂²	—	4	2	no	—
605	C₂²	—	4	2	no	—
606	C₂²	C₃₁×#410	1	2	yes	—
607	C₂²	C₃₁×#421	1	2	yes	—
608	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
609	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
610	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
611	C₄	—	2	1	no	—
612	C₄	—	2	1	no	—
613	C₄	—	2	1	no	—
614	C₄	—	2	1	no	—
615	C₄	—	2	1	no	—
616	C₄	—	2	1	no	—
617	C₄	—	2	1	no	—
618	C₄	—	2	1	no	—
619	C₄	—	2	1	no	—
620	C₄	—	2	1	no	—
621	C₄	—	4	2	no	—
622	C₄	—	4	2	no	—
623	C₄	—	4	2	no	—
624	C₄	—	8	4	no	—
625	C₄	C₃₁×#417	1	2	yes	—
626	C₄	C₃₁×#417	1	2	yes	—
627	C₂	—	2	1	no	—
628	C₂	—	2	2	no	—
629	C₂	—	2	2	no	—
630	C₂	—	2	1	no	—
631	C₂	—	8	2	no	—
632	C₂	C₂×C₃₁×#410	1	1	yes	characteristic
633	C₂	C₂×C₃₁×#417	1	1	yes	characteristic
634	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×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2920	C₂	semidirect
3. C₂×C₃₁×#382	C₂	semidirect
4. C₃₁×#1518	C₂	semidirect
5. C₃₁×#1517	C₂	semidirect
6. C₃₁×#1512	C₂	semidirect
7. C₃₁×#1504	C₂	semidirect
8. C₃₁×#533	C₄	semidirect
9. C₃₁×#533	C₄	semidirect
11. C₃₁×#532	C₄	semidirect
12. C₃₁×#532	C₄	semidirect
18. C₃₁×#529	C₄	semidirect
25. C₄×#41×C₃₁	C₄	semidirect
51. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
52. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
53. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
54. C₃₁×#127	C₂×C₄	semidirect
59. C₃₁×#125	C₂×C₄	semidirect
62. C₃₁×#124	C₂×C₄	semidirect
83. C₄×Q₈×C₃₁	C₂×C₄	semidirect
84. C₄×Q₈×C₃₁	C₂×C₄	semidirect
95. C₄×D₄×C₃₁	C₂×C₄	semidirect
96. C₄×D₄×C₃₁	C₂×C₄	semidirect
216. #17588	C₃₁	direct (N × H)
279. #2920	C₂×C₃₁	semidirect
280. C₂×#382	C₂×C₃₁	semidirect
281. #1518	C₂×C₃₁	semidirect
282. #1517	C₂×C₃₁	semidirect
283. #1512	C₂×C₃₁	semidirect
284. #1504	C₂×C₃₁	semidirect
317. #533	C₄×C₃₁	semidirect
318. #533	C₄×C₃₁	semidirect
320. #532	C₄×C₃₁	semidirect
321. #532	C₄×C₃₁	semidirect
327. #529	C₄×C₃₁	semidirect
334. C₄×#41	C₄×C₃₁	semidirect
368. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
369. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
370. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
371. #127	C₂×C₄×C₃₁	semidirect
376. #125	C₂×C₄×C₃₁	semidirect
379. #124	C₂×C₄×C₃₁	semidirect
400. C₄×Q₈	C₂×C₄×C₃₁	semidirect
401. C₄×Q₈	C₂×C₄×C₃₁	semidirect
412. C₄×D₄	C₂×C₄×C₃₁	semidirect
413. C₄×D₄	C₂×C₄×C₃₁	semidirect
446. C₃₁	#17588	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×#3444	2
C₃₁×#2920	2
C₂×C₃₁×#382	2
C₃₁×#1518	2
C₃₁×#1517	2
C₃₁×#1512	2
C₃₁×#1504	2
#17588	31

Subgroups — C31×#17588 #21474347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17588 #21474347