Subgroups — C₃₁×#17466 #21474225

Order 7936.

1066 conjugacy classes of subgroups.

The subgroup lattice has 620 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	C₂²×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1483	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1481	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1480	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1477	C₂	1	4	yes	maximal, complemented
8	C₂³×C₈×C₃₁	—	2	1	no	—
9	C₃₁×#533	C₄	1	1	yes	characteristic
10	C₃₁×#533	C₄	1	1	yes	characteristic
11	C₃₁×#533	C₂²	1	1	yes	characteristic
12	C₃₁×#532	C₄	1	1	yes	characteristic
13	C₃₁×#532	C₄	1	1	yes	characteristic
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₃₁×#529	—	2	1	no	—
16	C₃₁×#529	—	2	1	no	—
17	C₃₁×#529	—	2	1	no	—
18	C₃₁×#529	—	2	1	no	—
19	C₃₁×#529	C₄	1	2	yes	—
20	C₃₁×#529	C₂²	1	2	yes	—
21	C₃₁×#525	—	2	1	no	—
22	C₃₁×#525	—	2	1	no	—
23	C₃₁×#525	C₄	1	2	yes	—
24	C₄×#41×C₃₁	—	2	1	no	—
25	C₄×#41×C₃₁	—	2	1	no	—
26	C₄×#41×C₃₁	C₂²	1	2	yes	—
27	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#432	—	2	1	no	—
29	C₃₁×#432	—	2	1	no	—
30	C₃₁×#431	C₂²	1	2	yes	—
31	C₃₁×#431	C₂²	1	2	yes	—
32	C₃₁×#423	—	2	1	no	—
33	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
34	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
35	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
36	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
37	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
38	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
39	C₃₁×#407	—	2	1	no	—
40	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
41	C₃₁×Q₈⋊C₈	C₂²	1	8	yes	complemented
42	C₃₁×D₄⋊C₈	C₂²	1	8	yes	complemented
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	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂×M₁₆×C₃₁	—	2	2	no	—
50	C₂²×C₈×C₃₁	—	2	2	no	—
51	C₂²×C₈×C₃₁	—	2	2	no	—
52	C₂²×C₈×C₃₁	—	2	2	no	—
53	C₂²×C₈×C₃₁	—	4	4	no	—
54	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
55	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
56	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
57	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
58	C₃₁×#126	—	2	2	no	—
59	C₃₁×#126	—	2	2	no	—
60	C₃₁×#125	—	2	1	no	—
61	C₃₁×#125	—	2	1	no	—
62	C₃₁×#125	C₂×C₄	1	2	yes	—
63	C₃₁×#124	—	2	1	no	—
64	C₃₁×#124	—	2	1	no	—
65	C₃₁×#124	C₂×C₄	1	2	yes	—
66	C₃₁×#123	—	2	2	no	—
67	C₃₁×#123	—	2	2	no	—
68	C₃₁×#123	—	2	2	no	—
69	C₃₁×C₂²⋊Q₈	—	2	1	no	—
70	C₃₁×C₂²⋊Q₈	—	2	1	no	—
71	C₃₁×C₂²⋊Q₈	—	2	1	no	—
72	C₃₁×C₂²⋊Q₈	—	2	1	no	—
73	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
74	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
75	C₃₁×#121	—	2	1	no	—
76	C₃₁×#121	—	2	1	no	—
77	C₃₁×#121	—	2	1	no	—
78	C₃₁×#121	—	2	1	no	—
79	C₃₁×#121	C₂×C₄	1	2	yes	—
80	C₃₁×#121	C₂×C₄	1	2	yes	—
81	C₃₁×#120	—	2	2	no	—
82	C₄×Q₈×C₃₁	—	2	1	no	—
83	C₄×Q₈×C₃₁	—	2	1	no	—
84	C₄×Q₈×C₃₁	—	2	1	no	—
85	C₄×Q₈×C₃₁	—	2	1	no	—
86	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
87	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
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	2	no	—
92	C₄×D₄×C₃₁	—	2	2	no	—
93	C₄×D₄×C₃₁	—	2	2	no	—
94	C₄×D₄×C₃₁	—	2	1	no	—
95	C₄×D₄×C₃₁	—	2	1	no	—
96	C₄×D₄×C₃₁	—	2	1	no	—
97	C₄×D₄×C₃₁	—	2	1	no	—
98	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
99	C₄×D₄×C₃₁	C₂³	1	2	yes	—
100	C₃₁×#117	—	2	1	no	—
101	C₃₁×#117	—	2	1	no	—
102	C₃₁×#117	—	2	2	no	—
103	C₃₁×#117	—	2	2	no	—
104	C₃₁×#117	—	2	2	no	—
105	C₃₁×#117	C₂×C₄	1	2	yes	—
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	D₄	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	C₂³	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×C₄⋊C₈	—	2	2	no	—
122	C₃₁×C₄⋊C₈	—	2	2	no	—
123	C₃₁×C₄⋊C₈	—	2	2	no	—
124	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
125	C₃₁×C₂²⋊C₈	—	2	2	no	—
126	C₃₁×C₂²⋊C₈	—	2	2	no	—
127	C₃₁×#97	—	2	2	no	—
128	C₄×C₈×C₃₁	C₂³	1	4	yes	—
129	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
130	#41×C₃₁	—	2	8	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	4	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	2	no	—
149	C₂²×C₄×C₃₁	—	4	2	no	—
150	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂×D₄	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	M₁₆×C₃₁	—	4	4	no	—
169	C₂×C₈×C₃₁	—	2	4	no	—
170	C₂×C₈×C₃₁	—	2	4	no	—
171	C₂×C₈×C₃₁	—	4	4	no	—
172	C₂×C₈×C₃₁	—	4	4	no	—
173	C₂×C₈×C₃₁	—	4	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	C₂²×C₄	1	2	yes	—
185	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
186	C₂²⋊C₄×C₃₁	—	2	2	no	—
187	C₂²⋊C₄×C₃₁	—	2	2	no	—
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	C₄²×C₃₁	—	2	2	no	—
196	C₄²×C₃₁	—	2	2	no	—
197	C₄²×C₃₁	—	2	2	no	—
198	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
199	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
209	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
211	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
213	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
214	#17466	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂³×C₃₁	—	2	2	no	—
216	C₂³×C₃₁	—	2	2	no	—
217	C₂³×C₃₁	—	2	2	no	—
218	C₂³×C₃₁	—	2	2	no	—
219	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
220	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
221	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
222	Q₈×C₃₁	—	2	4	no	—
223	Q₈×C₃₁	—	4	2	no	—
224	Q₈×C₃₁	—	4	2	no	—
225	D₄×C₃₁	—	2	4	no	—
226	D₄×C₃₁	—	4	2	no	—
227	D₄×C₃₁	—	4	4	no	—
228	D₄×C₃₁	—	4	4	no	—
229	D₄×C₃₁	—	4	4	no	—
230	D₄×C₃₁	—	4	2	no	—
231	C₂×C₄×C₃₁	—	2	2	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	2	2	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	1	no	—
238	C₂×C₄×C₃₁	—	2	1	no	—
239	C₂×C₄×C₃₁	—	2	1	no	—
240	C₂×C₄×C₃₁	—	2	1	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	4	no	—
253	C₂×C₄×C₃₁	—	4	2	no	—
254	C₂×C₄×C₃₁	—	4	2	no	—
255	C₂×C₄×C₃₁	—	4	4	no	—
256	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	#131	1	4	yes	—
266	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
267	C₂×C₄×C₃₁	#137	1	2	yes	—
268	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
269	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
271	C₈×C₃₁	—	4	4	no	—
272	C₈×C₃₁	—	4	4	no	—
273	#3444	C₂×C₃₁	1	1	yes	characteristic
274	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
275	C₂²×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#1483	C₂×C₃₁	1	4	yes	complemented
277	#1481	C₂×C₃₁	1	2	yes	complemented
278	#1480	C₂×C₃₁	1	2	yes	complemented
279	#1477	C₂×C₃₁	1	4	yes	complemented
280	C₂²×C₃₁	—	2	8	no	—
281	C₂²×C₃₁	—	2	1	no	—
282	C₂²×C₃₁	—	2	1	no	—
283	C₂²×C₃₁	—	2	1	no	—
284	C₂²×C₃₁	—	2	1	no	—
285	C₂²×C₃₁	—	2	1	no	—
286	C₂²×C₃₁	—	2	1	no	—
287	C₂²×C₃₁	—	4	2	no	—
288	C₂²×C₃₁	—	4	2	no	—
289	C₂²×C₃₁	—	4	2	no	—
290	C₂²×C₃₁	#408	1	2	yes	—
291	C₂²×C₃₁	#417	1	2	yes	—
292	C₂²×C₃₁	#421	1	2	yes	—
293	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
294	C₄×C₃₁	—	2	1	no	—
295	C₄×C₃₁	—	2	1	no	—
296	C₄×C₃₁	—	2	1	no	—
297	C₄×C₃₁	—	2	1	no	—
298	C₄×C₃₁	—	2	1	no	—
299	C₄×C₃₁	—	2	1	no	—
300	C₄×C₃₁	—	2	1	no	—
301	C₄×C₃₁	—	2	1	no	—
302	C₄×C₃₁	—	4	2	no	—
303	C₄×C₃₁	—	4	2	no	—
304	C₄×C₃₁	—	4	2	no	—
305	C₄×C₃₁	#408	1	2	yes	—
306	C₄×C₃₁	#417	1	2	yes	—
307	C₄×C₃₁	C₂×#131	1	2	yes	—
308	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
309	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
310	C₂³×C₈	—	2	1	no	—
311	#533	C₄×C₃₁	1	1	yes	characteristic
312	#533	C₄×C₃₁	1	1	yes	characteristic
313	#533	C₂²×C₃₁	1	1	yes	characteristic
314	#532	C₄×C₃₁	1	1	yes	characteristic
315	#532	C₄×C₃₁	1	1	yes	characteristic
316	#532	C₂²×C₃₁	1	1	yes	characteristic
317	#529	—	2	1	no	—
318	#529	—	2	1	no	—
319	#529	—	2	1	no	—
320	#529	—	2	1	no	—
321	#529	C₄×C₃₁	1	2	yes	—
322	#529	C₂²×C₃₁	1	2	yes	—
323	#525	—	2	1	no	—
324	#525	—	2	1	no	—
325	#525	C₄×C₃₁	1	2	yes	—
326	C₄×#41	—	2	1	no	—
327	C₄×#41	—	2	1	no	—
328	C₄×#41	C₂²×C₃₁	1	2	yes	—
329	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
330	#432	—	2	1	no	—
331	#432	—	2	1	no	—
332	#431	C₂²×C₃₁	1	2	yes	—
333	#431	C₂²×C₃₁	1	2	yes	—
334	#423	—	2	1	no	—
335	C₂×C₄⋊C₈	—	2	1	no	—
336	C₂×C₄⋊C₈	—	2	1	no	—
337	C₂×C₄⋊C₈	—	2	1	no	—
338	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
339	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
340	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
341	#407	—	2	1	no	—
342	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
343	Q₈⋊C₈	C₂²×C₃₁	1	8	yes	complemented
344	D₄⋊C₈	C₂²×C₃₁	1	8	yes	complemented
345	C₂×C₃₁	—	2	2	no	—
346	C₂×C₃₁	—	2	2	no	—
347	C₂×C₃₁	—	2	2	no	—
348	C₂×C₃₁	—	8	2	no	—
349	C₂×C₃₁	C₂×#408	1	1	yes	characteristic
350	C₂×C₃₁	C₂×#417	1	1	yes	characteristic
351	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
352	C₂×#41	—	2	1	no	—
353	C₂×#41	—	2	1	no	—
354	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
355	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
356	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
357	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
358	C₂×M₁₆	—	2	2	no	—
359	C₂²×C₈	—	2	2	no	—
360	C₂²×C₈	—	2	2	no	—
361	C₂²×C₈	—	2	2	no	—
362	C₂²×C₈	—	4	4	no	—
363	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
364	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
365	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
366	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
367	#126	—	2	2	no	—
368	#126	—	2	2	no	—
369	#125	—	2	1	no	—
370	#125	—	2	1	no	—
371	#125	C₂×C₄×C₃₁	1	2	yes	—
372	#124	—	2	1	no	—
373	#124	—	2	1	no	—
374	#124	C₂×C₄×C₃₁	1	2	yes	—
375	#123	—	2	2	no	—
376	#123	—	2	2	no	—
377	#123	—	2	2	no	—
378	C₂²⋊Q₈	—	2	1	no	—
379	C₂²⋊Q₈	—	2	1	no	—
380	C₂²⋊Q₈	—	2	1	no	—
381	C₂²⋊Q₈	—	2	1	no	—
382	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
383	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
384	#121	—	2	1	no	—
385	#121	—	2	1	no	—
386	#121	—	2	1	no	—
387	#121	—	2	1	no	—
388	#121	C₂×C₄×C₃₁	1	2	yes	—
389	#121	C₂×C₄×C₃₁	1	2	yes	—
390	#120	—	2	2	no	—
391	C₄×Q₈	—	2	1	no	—
392	C₄×Q₈	—	2	1	no	—
393	C₄×Q₈	—	2	1	no	—
394	C₄×Q₈	—	2	1	no	—
395	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
396	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	—	2	2	no	—
401	C₄×D₄	—	2	2	no	—
402	C₄×D₄	—	2	2	no	—
403	C₄×D₄	—	2	1	no	—
404	C₄×D₄	—	2	1	no	—
405	C₄×D₄	—	2	1	no	—
406	C₄×D₄	—	2	1	no	—
407	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
408	C₄×D₄	C₂³×C₃₁	1	2	yes	—
409	#117	—	2	1	no	—
410	#117	—	2	1	no	—
411	#117	—	2	2	no	—
412	#117	—	2	2	no	—
413	#117	—	2	2	no	—
414	#117	C₂×C₄×C₃₁	1	2	yes	—
415	C₂×C₄²	—	2	2	no	—
416	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
417	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
418	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
419	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
420	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
421	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
422	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
423	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
424	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
425	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
426	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
427	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
428	C₄⋊C₈	—	2	2	no	—
429	C₄⋊C₈	—	2	2	no	—
430	C₄⋊C₈	—	2	2	no	—
431	C₄⋊C₈	—	2	2	no	—
432	C₄⋊C₈	—	2	2	no	—
433	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
434	C₂²⋊C₈	—	2	2	no	—
435	C₂²⋊C₈	—	2	2	no	—
436	#97	—	2	2	no	—
437	C₄×C₈	C₂³×C₃₁	1	4	yes	—
438	C₃₁	#17466	1	1	yes	characteristic, direct product factor, Sylow
439	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
440	#41	—	2	8	no	—
441	#41	—	4	4	no	—
442	#41	—	4	2	no	—
443	#41	—	4	2	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	1	no	—
452	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
453	C₂²×C₄	—	2	4	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	4	2	no	—
460	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
461	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
462	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
463	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
464	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
465	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
466	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
467	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
468	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
469	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
470	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
471	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
474	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
478	M₁₆	—	4	4	no	—
479	C₂×C₈	—	2	4	no	—
480	C₂×C₈	—	2	4	no	—
481	C₂×C₈	—	4	4	no	—
482	C₂×C₈	—	4	4	no	—
483	C₂×C₈	—	4	4	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	2	2	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
495	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₂²⋊C₄	—	2	2	no	—
499	C₂²⋊C₄	—	2	2	no	—
500	C₂²⋊C₄	—	2	2	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₂²⋊C₄	—	2	2	no	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₄²	—	2	2	no	—
506	C₄²	—	2	2	no	—
507	C₄²	—	2	2	no	—
508	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
509	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
510	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
511	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
512	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
513	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
514	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
515	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
516	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
517	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
518	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
519	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
520	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
521	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
522	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
523	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
524	C₂³	—	2	2	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	2	2	no	—
528	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
529	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
530	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
531	Q₈	—	2	4	no	—
532	Q₈	—	4	2	no	—
533	Q₈	—	4	2	no	—
534	D₄	—	2	4	no	—
535	D₄	—	4	2	no	—
536	D₄	—	4	4	no	—
537	D₄	—	4	4	no	—
538	D₄	—	4	4	no	—
539	D₄	—	4	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	4	no	—
562	C₂×C₄	—	4	4	no	—
563	C₂×C₄	—	4	2	no	—
564	C₂×C₄	—	4	2	no	—
565	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
566	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
567	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
568	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
569	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
570	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
571	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
572	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
573	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
574	C₂×C₄	C₃₁×#131	1	4	yes	—
575	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
576	C₂×C₄	C₃₁×#137	1	2	yes	—
577	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
578	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
579	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
580	C₈	—	4	4	no	—
581	C₈	—	4	4	no	—
582	C₂²	—	2	8	no	—
583	C₂²	—	2	1	no	—
584	C₂²	—	2	1	no	—
585	C₂²	—	2	1	no	—
586	C₂²	—	2	1	no	—
587	C₂²	—	2	1	no	—
588	C₂²	—	2	1	no	—
589	C₂²	—	4	2	no	—
590	C₂²	—	4	2	no	—
591	C₂²	—	4	2	no	—
592	C₂²	C₃₁×#408	1	2	yes	—
593	C₂²	C₃₁×#417	1	2	yes	—
594	C₂²	C₃₁×#421	1	2	yes	—
595	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
596	C₄	—	2	1	no	—
597	C₄	—	2	1	no	—
598	C₄	—	2	1	no	—
599	C₄	—	2	1	no	—
600	C₄	—	2	1	no	—
601	C₄	—	2	1	no	—
602	C₄	—	2	1	no	—
603	C₄	—	2	1	no	—
604	C₄	—	4	2	no	—
605	C₄	—	4	2	no	—
606	C₄	—	4	2	no	—
607	C₄	C₃₁×#408	1	2	yes	—
608	C₄	C₃₁×#417	1	2	yes	—
609	C₄	C₂×C₃₁×#131	1	2	yes	—
610	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
611	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
612	C₂	—	2	2	no	—
613	C₂	—	2	2	no	—
614	C₂	—	2	2	no	—
615	C₂	—	8	2	no	—
616	C₂	C₂×C₃₁×#408	1	1	yes	characteristic
617	C₂	C₂×C₃₁×#417	1	1	yes	characteristic
618	C₂	C₂×C₃₁×#421	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₃₁×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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. C₂²×C₃₁×C₄⋊C₈	C₂	semidirect
4. C₃₁×#1483	C₂	semidirect
5. C₃₁×#1481	C₂	semidirect
6. C₃₁×#1480	C₂	semidirect
7. C₃₁×#1477	C₂	semidirect
41. C₃₁×Q₈⋊C₈	C₂²	semidirect
42. C₃₁×D₄⋊C₈	C₂²	semidirect
214. #17466	C₃₁	direct (N × H)
274. #2920	C₂×C₃₁	semidirect
275. C₂²×C₄⋊C₈	C₂×C₃₁	semidirect
276. #1483	C₂×C₃₁	semidirect
277. #1481	C₂×C₃₁	semidirect
278. #1480	C₂×C₃₁	semidirect
279. #1477	C₂×C₃₁	semidirect
343. Q₈⋊C₈	C₂²×C₃₁	semidirect
344. D₄⋊C₈	C₂²×C₃₁	semidirect
438. C₃₁	#17466	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
C₂²×C₃₁×C₄⋊C₈	2
C₃₁×#1483	2
C₃₁×#1481	2
C₃₁×#1480	2
C₃₁×#1477	2
#17466	31

Subgroups — C31×#17466 #21474225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17466 #21474225