Subgroups — C₃₁×#17383 #21474142

Order 7936.

1020 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#529	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1473	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1468	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1461	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1460	C₂	1	2	yes	maximal, complemented
8	C₃₁×#529	—	2	1	no	—
9	C₃₁×#529	—	2	1	no	—
10	C₃₁×#529	—	2	1	no	—
11	C₃₁×#529	—	2	1	no	—
12	C₃₁×#529	—	2	1	no	—
13	C₃₁×#529	—	2	1	no	—
14	C₃₁×#529	—	2	1	no	—
15	C₃₁×#529	—	2	1	no	—
16	C₂×C₃₁×#126	C₂²	1	2	yes	—
17	C₂×C₃₁×#125	C₄	1	1	yes	characteristic
18	C₂×C₃₁×#124	C₄	1	1	yes	characteristic
19	C₂×C₃₁×#123	C₂²	1	2	yes	—
20	C₂×C₃₁×C₂²⋊Q₈	C₄	1	1	yes	characteristic
21	C₂×C₃₁×#121	C₄	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₃₁	C₄	1	2	yes	—
24	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#117	C₄	1	2	yes	—
26	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#432	—	2	1	no	—
28	C₃₁×#432	C₂²	1	2	yes	—
29	C₃₁×#431	—	2	1	no	—
30	C₃₁×#431	—	2	1	no	—
31	C₃₁×#431	—	2	1	no	—
32	C₃₁×#431	C₂²	1	2	yes	—
33	C₃₁×#431	C₂²	1	2	yes	—
34	C₃₁×#431	C₂²	1	2	yes	—
35	C₃₁×#423	—	2	1	no	—
36	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
37	C₃₁×#407	C₂²	1	2	yes	—
38	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
39	C₄×M₁₆×C₃₁	—	2	1	no	—
40	C₂×C₄×C₈×C₃₁	—	2	1	no	—
41	C₃₁×#324	C₂²	1	8	yes	—
42	C₃₁×#323	C₂²	1	8	yes	—
43	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
48	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂²×C₈×C₃₁	—	2	2	no	—
52	C₃₁×#126	—	2	2	no	—
53	C₃₁×#126	—	2	2	no	—
54	C₃₁×#126	—	2	2	no	—
55	C₃₁×#126	—	2	2	no	—
56	C₃₁×#125	—	2	1	no	—
57	C₃₁×#125	—	2	1	no	—
58	C₃₁×#125	—	2	1	no	—
59	C₃₁×#125	—	2	1	no	—
60	C₃₁×#124	—	2	1	no	—
61	C₃₁×#124	—	2	1	no	—
62	C₃₁×#124	—	2	1	no	—
63	C₃₁×#124	—	2	1	no	—
64	C₃₁×#123	—	2	2	no	—
65	C₃₁×#123	—	2	2	no	—
66	C₃₁×#123	—	2	2	no	—
67	C₃₁×#123	—	2	2	no	—
68	C₃₁×C₂²⋊Q₈	—	2	1	no	—
69	C₃₁×C₂²⋊Q₈	—	2	1	no	—
70	C₃₁×C₂²⋊Q₈	—	2	1	no	—
71	C₃₁×C₂²⋊Q₈	—	2	1	no	—
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	C₄×Q₈×C₃₁	—	2	1	no	—
77	C₄×Q₈×C₃₁	—	2	1	no	—
78	C₄×Q₈×C₃₁	—	2	1	no	—
79	C₄×Q₈×C₃₁	—	2	1	no	—
80	C₄×D₄×C₃₁	—	2	2	no	—
81	C₄×D₄×C₃₁	—	2	2	no	—
82	C₄×D₄×C₃₁	—	2	2	no	—
83	C₄×D₄×C₃₁	—	2	2	no	—
84	C₄×D₄×C₃₁	—	2	1	no	—
85	C₄×D₄×C₃₁	—	2	1	no	—
86	C₄×D₄×C₃₁	—	2	1	no	—
87	C₄×D₄×C₃₁	—	2	1	no	—
88	C₃₁×#117	—	2	2	no	—
89	C₃₁×#117	—	2	2	no	—
90	C₃₁×#117	—	2	2	no	—
91	C₃₁×#117	—	2	2	no	—
92	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
93	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
94	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
95	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
98	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
99	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
100	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
101	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
102	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
104	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
105	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
106	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
107	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
108	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
109	C₂×C₄²×C₃₁	D₄	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	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
116	C₃₁×C₄⋊C₈	—	2	2	no	—
117	C₃₁×C₄⋊C₈	—	2	2	no	—
118	C₃₁×C₄⋊C₈	—	2	2	no	—
119	C₃₁×C₄⋊C₈	—	2	2	no	—
120	C₃₁×C₂²⋊C₈	—	2	2	no	—
121	C₃₁×C₂²⋊C₈	—	2	2	no	—
122	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
123	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
124	C₃₁×#97	—	2	2	no	—
125	C₄×C₈×C₃₁	—	2	2	no	—
126	C₄×C₈×C₃₁	—	2	2	no	—
127	C₄×C₈×C₃₁	—	2	2	no	—
128	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
129	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
130	C₂×Q₈×C₃₁	—	2	1	no	—
131	C₂×Q₈×C₃₁	—	2	1	no	—
132	C₂×Q₈×C₃₁	—	2	4	no	—
133	C₂×D₄×C₃₁	—	2	2	no	—
134	C₂×D₄×C₃₁	—	2	2	no	—
135	C₂×D₄×C₃₁	—	2	1	no	—
136	C₂×D₄×C₃₁	—	2	1	no	—
137	C₂×D₄×C₃₁	—	2	4	no	—
138	C₂×D₄×C₃₁	—	4	4	no	—
139	C₂²×C₄×C₃₁	—	2	4	no	—
140	C₂²×C₄×C₃₁	—	2	2	no	—
141	C₂²×C₄×C₃₁	—	2	2	no	—
142	C₂²×C₄×C₃₁	—	4	2	no	—
143	C₂²×C₄×C₃₁	—	4	2	no	—
144	C₂²×C₄×C₃₁	—	4	4	no	—
145	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
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	2	yes	—
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	2	yes	—
157	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
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	C₂×D₄	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	C₂⁴	1	1	yes	characteristic
166	M₁₆×C₃₁	—	4	4	no	—
167	C₂×C₈×C₃₁	—	2	4	no	—
168	C₂×C₈×C₃₁	—	2	4	no	—
169	C₂×C₈×C₃₁	—	4	4	no	—
170	C₄⋊C₄×C₃₁	—	2	2	no	—
171	C₄⋊C₄×C₃₁	—	2	2	no	—
172	C₄⋊C₄×C₃₁	—	2	2	no	—
173	C₄⋊C₄×C₃₁	—	2	2	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	2	2	no	—
177	C₄⋊C₄×C₃₁	—	2	2	no	—
178	C₄⋊C₄×C₃₁	—	2	1	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	—	2	2	no	—
181	C₄⋊C₄×C₃₁	—	2	2	no	—
182	C₂²⋊C₄×C₃₁	—	2	2	no	—
183	C₂²⋊C₄×C₃₁	—	2	2	no	—
184	C₂²⋊C₄×C₃₁	—	2	2	no	—
185	C₂²⋊C₄×C₃₁	—	2	2	no	—
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	C₄²×C₃₁	—	2	2	no	—
193	C₄²×C₃₁	—	2	2	no	—
194	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
195	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
196	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
197	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
203	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
204	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
205	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
206	C₄²×C₃₁	C₂×D₄	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	#17383	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂³×C₃₁	—	2	2	no	—
212	C₂³×C₃₁	—	2	2	no	—
213	C₂³×C₃₁	—	2	2	no	—
214	C₂³×C₃₁	—	2	4	no	—
215	C₂³×C₃₁	—	4	1	no	—
216	C₂³×C₃₁	—	4	2	no	—
217	C₂³×C₃₁	—	4	2	no	—
218	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
219	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
220	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
221	Q₈×C₃₁	—	4	2	no	—
222	Q₈×C₃₁	—	4	2	no	—
223	D₄×C₃₁	—	4	4	no	—
224	D₄×C₃₁	—	4	4	no	—
225	D₄×C₃₁	—	4	2	no	—
226	D₄×C₃₁	—	4	2	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	2	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	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	2	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	4	1	no	—
242	C₂×C₄×C₃₁	—	4	1	no	—
243	C₂×C₄×C₃₁	—	4	1	no	—
244	C₂×C₄×C₃₁	—	4	1	no	—
245	C₂×C₄×C₃₁	—	4	1	no	—
246	C₂×C₄×C₃₁	—	4	1	no	—
247	C₂×C₄×C₃₁	—	4	1	no	—
248	C₂×C₄×C₃₁	—	4	4	no	—
249	C₂×C₄×C₃₁	—	4	4	no	—
250	C₂×C₄×C₃₁	—	4	2	no	—
251	C₂×C₄×C₃₁	—	4	2	no	—
252	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	#131	1	4	yes	—
263	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
265	C₈×C₃₁	—	4	4	no	—
266	C₈×C₃₁	—	4	4	no	—
267	C₂×#529	C₂×C₃₁	1	1	yes	characteristic
268	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
269	C₂×#431	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#1473	C₂×C₃₁	1	4	yes	complemented
271	#1468	C₂×C₃₁	1	4	yes	complemented
272	#1461	C₂×C₃₁	1	2	yes	complemented
273	#1460	C₂×C₃₁	1	2	yes	complemented
274	C₂²×C₃₁	—	2	1	no	—
275	C₂²×C₃₁	—	2	1	no	—
276	C₂²×C₃₁	—	2	1	no	—
277	C₂²×C₃₁	—	2	1	no	—
278	C₂²×C₃₁	—	2	1	no	—
279	C₂²×C₃₁	—	2	1	no	—
280	C₂²×C₃₁	—	2	1	no	—
281	C₂²×C₃₁	—	4	4	no	—
282	C₂²×C₃₁	—	4	2	no	—
283	C₂²×C₃₁	—	4	2	no	—
284	C₂²×C₃₁	—	4	2	no	—
285	C₂²×C₃₁	—	8	4	no	—
286	C₂²×C₃₁	#410	1	2	yes	—
287	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
288	C₂²×C₃₁	C₂×#131	1	2	yes	—
289	C₄×C₃₁	—	2	1	no	—
290	C₄×C₃₁	—	2	1	no	—
291	C₄×C₃₁	—	2	1	no	—
292	C₄×C₃₁	—	2	1	no	—
293	C₄×C₃₁	—	2	1	no	—
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	—	4	2	no	—
299	C₄×C₃₁	—	4	2	no	—
300	C₄×C₃₁	—	4	2	no	—
301	C₄×C₃₁	#408	1	2	yes	—
302	C₄×C₃₁	#408	1	2	yes	—
303	C₄×C₃₁	#413	1	2	yes	—
304	#529	—	2	1	no	—
305	#529	—	2	1	no	—
306	#529	—	2	1	no	—
307	#529	—	2	1	no	—
308	#529	—	2	1	no	—
309	#529	—	2	1	no	—
310	#529	—	2	1	no	—
311	#529	—	2	1	no	—
312	C₂×#126	C₂²×C₃₁	1	2	yes	—
313	C₂×#125	C₄×C₃₁	1	1	yes	characteristic
314	C₂×#124	C₄×C₃₁	1	1	yes	characteristic
315	C₂×#123	C₂²×C₃₁	1	2	yes	—
316	C₂×C₂²⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
317	C₂×#121	C₄×C₃₁	1	1	yes	characteristic
318	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×C₄×D₄	C₄×C₃₁	1	2	yes	—
320	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
321	C₂×#117	C₄×C₃₁	1	2	yes	—
322	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
323	#432	—	2	1	no	—
324	#432	C₂²×C₃₁	1	2	yes	—
325	#431	—	2	1	no	—
326	#431	—	2	1	no	—
327	#431	—	2	1	no	—
328	#431	C₂²×C₃₁	1	2	yes	—
329	#431	C₂²×C₃₁	1	2	yes	—
330	#431	C₂²×C₃₁	1	2	yes	—
331	#423	—	2	1	no	—
332	C₂×C₄⋊C₈	—	2	1	no	—
333	#407	C₂²×C₃₁	1	2	yes	—
334	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
335	C₄×M₁₆	—	2	1	no	—
336	C₂×C₄×C₈	—	2	1	no	—
337	#324	C₂²×C₃₁	1	8	yes	—
338	#323	C₂²×C₃₁	1	8	yes	—
339	C₂×C₃₁	—	2	2	no	—
340	C₂×C₃₁	—	2	2	no	—
341	C₂×C₃₁	—	4	1	no	—
342	C₂×C₃₁	—	8	2	no	—
343	C₂×C₃₁	C₂×#408	1	1	yes	characteristic
344	C₂×C₃₁	C₂×#410	1	1	yes	characteristic
345	C₂×C₃₁	C₂×#413	1	1	yes	characteristic
346	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
347	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
348	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
349	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
350	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
351	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
352	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
353	C₂×M₁₆	—	2	2	no	—
354	C₂²×C₈	—	2	2	no	—
355	#126	—	2	2	no	—
356	#126	—	2	2	no	—
357	#126	—	2	2	no	—
358	#126	—	2	2	no	—
359	#125	—	2	1	no	—
360	#125	—	2	1	no	—
361	#125	—	2	1	no	—
362	#125	—	2	1	no	—
363	#124	—	2	1	no	—
364	#124	—	2	1	no	—
365	#124	—	2	1	no	—
366	#124	—	2	1	no	—
367	#123	—	2	2	no	—
368	#123	—	2	2	no	—
369	#123	—	2	2	no	—
370	#123	—	2	2	no	—
371	C₂²⋊Q₈	—	2	1	no	—
372	C₂²⋊Q₈	—	2	1	no	—
373	C₂²⋊Q₈	—	2	1	no	—
374	C₂²⋊Q₈	—	2	1	no	—
375	#121	—	2	1	no	—
376	#121	—	2	1	no	—
377	#121	—	2	1	no	—
378	#121	—	2	1	no	—
379	C₄×Q₈	—	2	1	no	—
380	C₄×Q₈	—	2	1	no	—
381	C₄×Q₈	—	2	1	no	—
382	C₄×Q₈	—	2	1	no	—
383	C₄×D₄	—	2	2	no	—
384	C₄×D₄	—	2	2	no	—
385	C₄×D₄	—	2	2	no	—
386	C₄×D₄	—	2	2	no	—
387	C₄×D₄	—	2	1	no	—
388	C₄×D₄	—	2	1	no	—
389	C₄×D₄	—	2	1	no	—
390	C₄×D₄	—	2	1	no	—
391	#117	—	2	2	no	—
392	#117	—	2	2	no	—
393	#117	—	2	2	no	—
394	#117	—	2	2	no	—
395	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
396	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
397	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
398	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
399	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
400	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
401	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
402	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
403	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
404	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
406	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
407	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
408	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
409	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
410	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
411	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
412	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
413	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
414	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
415	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
416	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
417	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
418	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
419	C₄⋊C₈	—	2	2	no	—
420	C₄⋊C₈	—	2	2	no	—
421	C₄⋊C₈	—	2	2	no	—
422	C₄⋊C₈	—	2	2	no	—
423	C₂²⋊C₈	—	2	2	no	—
424	C₂²⋊C₈	—	2	2	no	—
425	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
426	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
427	#97	—	2	2	no	—
428	C₄×C₈	—	2	2	no	—
429	C₄×C₈	—	2	2	no	—
430	C₄×C₈	—	2	2	no	—
431	C₃₁	#17383	1	1	yes	characteristic, direct product factor, Sylow
432	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
433	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	—	2	1	no	—
436	C₂×Q₈	—	2	4	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	2	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	2	1	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	4	4	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	2	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	4	2	no	—
447	C₂²×C₄	—	4	2	no	—
448	C₂²×C₄	—	4	4	no	—
449	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
450	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
451	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
452	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
453	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
454	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
455	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
456	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
457	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
458	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
459	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
460	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
461	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
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	C₂×D₄×C₃₁	1	1	yes	characteristic
466	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
467	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
468	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
469	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
470	M₁₆	—	4	4	no	—
471	C₂×C₈	—	2	4	no	—
472	C₂×C₈	—	2	4	no	—
473	C₂×C₈	—	4	4	no	—
474	C₄⋊C₄	—	2	2	no	—
475	C₄⋊C₄	—	2	2	no	—
476	C₄⋊C₄	—	2	2	no	—
477	C₄⋊C₄	—	2	2	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₂²⋊C₄	—	2	2	no	—
487	C₂²⋊C₄	—	2	2	no	—
488	C₂²⋊C₄	—	2	2	no	—
489	C₂²⋊C₄	—	2	2	no	—
490	C₂²⋊C₄	—	2	2	no	—
491	C₂²⋊C₄	—	2	2	no	—
492	C₂²⋊C₄	—	2	2	no	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	2	2	no	—
496	C₄²	—	2	2	no	—
497	C₄²	—	2	2	no	—
498	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
499	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
500	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
501	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
502	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
503	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
504	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
505	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
506	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
507	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
508	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
509	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
510	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
511	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
512	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
513	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
514	C₂³	—	2	2	no	—
515	C₂³	—	2	2	no	—
516	C₂³	—	2	2	no	—
517	C₂³	—	2	4	no	—
518	C₂³	—	4	1	no	—
519	C₂³	—	4	2	no	—
520	C₂³	—	4	2	no	—
521	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
522	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
523	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
524	Q₈	—	4	2	no	—
525	Q₈	—	4	2	no	—
526	D₄	—	4	4	no	—
527	D₄	—	4	4	no	—
528	D₄	—	4	2	no	—
529	D₄	—	4	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	4	no	—
552	C₂×C₄	—	4	4	no	—
553	C₂×C₄	—	4	2	no	—
554	C₂×C₄	—	4	2	no	—
555	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
556	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
557	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
558	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
559	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
560	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
561	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
562	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
563	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
564	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
565	C₂×C₄	C₃₁×#131	1	4	yes	—
566	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
567	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
568	C₈	—	4	4	no	—
569	C₈	—	4	4	no	—
570	C₂²	—	2	1	no	—
571	C₂²	—	2	1	no	—
572	C₂²	—	2	1	no	—
573	C₂²	—	2	1	no	—
574	C₂²	—	2	1	no	—
575	C₂²	—	2	1	no	—
576	C₂²	—	2	1	no	—
577	C₂²	—	4	4	no	—
578	C₂²	—	4	2	no	—
579	C₂²	—	4	2	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	8	4	no	—
582	C₂²	C₃₁×#410	1	2	yes	—
583	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
584	C₂²	C₂×C₃₁×#131	1	2	yes	—
585	C₄	—	2	1	no	—
586	C₄	—	2	1	no	—
587	C₄	—	2	1	no	—
588	C₄	—	2	1	no	—
589	C₄	—	2	1	no	—
590	C₄	—	2	1	no	—
591	C₄	—	2	1	no	—
592	C₄	—	2	1	no	—
593	C₄	—	2	1	no	—
594	C₄	—	4	2	no	—
595	C₄	—	4	2	no	—
596	C₄	—	4	2	no	—
597	C₄	C₃₁×#408	1	2	yes	—
598	C₄	C₃₁×#408	1	2	yes	—
599	C₄	C₃₁×#413	1	2	yes	—
600	C₂	—	2	2	no	—
601	C₂	—	2	2	no	—
602	C₂	—	4	1	no	—
603	C₂	—	8	2	no	—
604	C₂	C₂×C₃₁×#408	1	1	yes	characteristic
605	C₂	C₂×C₃₁×#410	1	1	yes	characteristic
606	C₂	C₂×C₃₁×#413	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₂×C₃₁×#121	C₄
LCS	G ▷ C₂³ ▷ 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. C₂×C₃₁×#431	C₂	semidirect
4. C₃₁×#1473	C₂	semidirect
5. C₃₁×#1468	C₂	semidirect
6. C₃₁×#1461	C₂	semidirect
7. C₃₁×#1460	C₂	semidirect
210. #17383	C₃₁	direct (N × H)
268. #2920	C₂×C₃₁	semidirect
269. C₂×#431	C₂×C₃₁	semidirect
270. #1473	C₂×C₃₁	semidirect
271. #1468	C₂×C₃₁	semidirect
272. #1461	C₂×C₃₁	semidirect
273. #1460	C₂×C₃₁	semidirect
431. C₃₁	#17383	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×#529	2
C₃₁×#2920	2
C₂×C₃₁×#431	2
C₃₁×#1473	2
C₃₁×#1468	2
C₃₁×#1461	2
C₃₁×#1460	2
#17383	31

Subgroups — C31×#17383 #21474142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17383 #21474142