Subgroups — C₃₁×#20918 #21477677

Order 7936.

1714 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#575	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#568	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#104	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃₁×#104	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2879	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1876	C₂	1	8	yes	maximal, complemented
7	C₃₁×#1738	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1728	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#575	—	2	4	no	—
12	C₃₁×#575	—	2	4	no	—
13	C₂×C₃₁×#137	C₄	1	2	yes	complemented
14	C₂×C₃₁×#137	C₄	1	2	yes	complemented
15	C₂×C₃₁×AGL(1,ℤ/8)	C₄	1	2	yes	complemented
16	C₂×C₃₁×AGL(1,ℤ/8)	C₄	1	2	yes	complemented
17	C₂×C₃₁×#135	C₂²	1	4	yes	—
18	C₃₁×#568	—	2	4	no	—
19	C₃₁×#568	—	2	4	no	—
20	C₂×C₃₁×#131	—	2	2	no	—
21	C₂×C₃₁×#131	C₂²	1	4	yes	complemented
22	C₂²×M₁₆×C₃₁	—	2	2	no	—
23	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
25	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
26	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#104	—	2	4	no	—
29	C₂×C₃₁×#104	—	2	4	no	—
30	C₂×C₃₁×#104	C₂²	1	4	yes	complemented
31	C₂×C₃₁×#104	C₂²	1	4	yes	complemented
32	C₃₁×#408	—	2	2	no	—
33	C₃₁×#408	C₂²	1	4	yes	complemented
34	C₃₁×#407	—	2	2	no	—
35	C₃₁×#341	C₂²	1	4	yes	complemented
36	C₃₁×#336	C₂²	1	4	yes	complemented
37	C₂×#41×C₃₁	—	2	1	no	—
38	C₂×#41×C₃₁	—	2	2	no	—
39	C₂×#41×C₃₁	—	2	1	no	—
40	C₂×#41×C₃₁	—	2	1	no	—
41	C₂×#41×C₃₁	—	2	2	no	—
42	C₂×#41×C₃₁	—	2	1	no	—
43	C₂×#41×C₃₁	—	2	2	no	—
44	C₂×#41×C₃₁	—	2	2	no	—
45	C₂×#41×C₃₁	—	2	2	no	—
46	C₂×#41×C₃₁	—	2	2	no	—
47	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
48	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
49	C₂×#41×C₃₁	D₄	1	4	yes	—
50	C₂×#41×C₃₁	D₄	1	4	yes	—
51	C₂×#41×C₃₁	C₂³	1	2	yes	—
52	C₂×#41×C₃₁	C₂³	1	2	yes	—
53	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×D₄×C₃₁	—	2	1	no	—
56	C₂²×D₄×C₃₁	—	2	1	no	—
57	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂³×C₄×C₃₁	—	2	1	no	—
60	C₂³×C₄×C₃₁	—	2	1	no	—
61	C₂³×C₄×C₃₁	—	2	1	no	—
62	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₃₁×#137	—	2	8	no	—
64	C₃₁×#137	—	2	8	no	—
65	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
66	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
67	C₃₁×#135	—	2	8	no	—
68	C₃₁×#135	—	2	8	no	—
69	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
70	C₂×SD₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
71	C₂×SD₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
72	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	complemented
73	C₃₁×#131	—	2	16	no	—
74	C₃₁×#131	—	2	8	no	—
75	C₃₁×#131	—	2	8	no	—
76	C₂×M₁₆×C₃₁	—	2	16	no	—
77	C₂×M₁₆×C₃₁	—	2	1	no	—
78	C₂×M₁₆×C₃₁	—	2	2	no	—
79	C₂×M₁₆×C₃₁	—	2	2	no	—
80	C₂×M₁₆×C₃₁	—	2	2	no	—
81	C₂×M₁₆×C₃₁	—	2	2	no	—
82	C₂×M₁₆×C₃₁	—	2	2	no	—
83	C₂×M₁₆×C₃₁	—	2	1	no	—
84	C₂×M₁₆×C₃₁	—	2	4	no	—
85	C₂×M₁₆×C₃₁	—	2	2	no	—
86	C₂×M₁₆×C₃₁	—	2	2	no	—
87	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
88	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
89	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
90	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
91	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
92	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
93	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
94	C₂²×C₈×C₃₁	—	2	1	no	—
95	C₂²×C₈×C₃₁	—	2	4	no	—
96	C₂²×C₈×C₃₁	—	2	1	no	—
97	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
98	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
99	C₂×C₄²×C₃₁	—	2	2	no	—
100	C₂×C₄²×C₃₁	—	2	2	no	—
101	C₂×C₄²×C₃₁	—	2	1	no	—
102	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
103	C₃₁×#104	—	2	8	no	—
104	C₃₁×#104	—	2	8	no	—
105	C₃₁×C₂²⋊C₈	—	2	8	no	—
106	C₂⁴×C₃₁	—	2	1	no	—
107	C₂⁴×C₃₁	—	2	1	no	—
108	C₂⁴×C₃₁	#41	1	1	yes	characteristic
109	#41×C₃₁	—	2	4	no	—
110	#41×C₃₁	—	2	4	no	—
111	#41×C₃₁	—	2	4	no	—
112	#41×C₃₁	—	2	4	no	—
113	#41×C₃₁	—	2	4	no	—
114	#41×C₃₁	—	2	2	no	—
115	#41×C₃₁	—	2	2	no	—
116	#41×C₃₁	—	2	4	no	—
117	#41×C₃₁	—	2	4	no	—
118	#41×C₃₁	—	2	2	no	—
119	#41×C₃₁	—	2	2	no	—
120	#41×C₃₁	—	2	2	no	—
121	#41×C₃₁	—	2	2	no	—
122	#41×C₃₁	—	4	2	no	—
123	#41×C₃₁	—	4	2	no	—
124	#41×C₃₁	—	4	2	no	—
125	#41×C₃₁	—	4	2	no	—
126	#41×C₃₁	C₂²⋊C₄	1	8	yes	—
127	#41×C₃₁	C₂×D₄	1	4	yes	—
128	#41×C₃₁	C₂×D₄	1	4	yes	—
129	C₂×Q₈×C₃₁	—	2	1	no	—
130	C₂×Q₈×C₃₁	—	2	2	no	—
131	C₂×Q₈×C₃₁	—	2	1	no	—
132	C₂×Q₈×C₃₁	—	2	2	no	—
133	C₂×Q₈×C₃₁	—	2	2	no	—
134	C₂×Q₈×C₃₁	C₂²⋊C₄	1	4	yes	—
135	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
136	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
137	C₂×D₄×C₃₁	—	2	2	no	—
138	C₂×D₄×C₃₁	—	2	4	no	—
139	C₂×D₄×C₃₁	—	2	4	no	—
140	C₂×D₄×C₃₁	—	2	2	no	—
141	C₂×D₄×C₃₁	—	2	2	no	—
142	C₂×D₄×C₃₁	—	2	2	no	—
143	C₂×D₄×C₃₁	—	2	2	no	—
144	C₂×D₄×C₃₁	—	2	2	no	—
145	C₂×D₄×C₃₁	—	2	1	no	—
146	C₂×D₄×C₃₁	—	2	2	no	—
147	C₂×D₄×C₃₁	—	2	1	no	—
148	C₂×D₄×C₃₁	—	4	2	no	—
149	C₂×D₄×C₃₁	—	4	2	no	—
150	C₂×D₄×C₃₁	C₂²⋊C₄	1	4	yes	—
151	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
152	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
153	C₂²×C₄×C₃₁	—	2	1	no	—
154	C₂²×C₄×C₃₁	—	2	2	no	—
155	C₂²×C₄×C₃₁	—	2	4	no	—
156	C₂²×C₄×C₃₁	—	2	4	no	—
157	C₂²×C₄×C₃₁	—	2	2	no	—
158	C₂²×C₄×C₃₁	—	2	1	no	—
159	C₂²×C₄×C₃₁	—	2	2	no	—
160	C₂²×C₄×C₃₁	—	2	2	no	—
161	C₂²×C₄×C₃₁	—	2	2	no	—
162	C₂²×C₄×C₃₁	—	4	4	no	—
163	C₂²×C₄×C₃₁	—	4	2	no	—
164	C₂²×C₄×C₃₁	—	4	2	no	—
165	C₂²×C₄×C₃₁	—	4	2	no	—
166	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	#41	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
176	Q₁₆×C₃₁	—	2	8	no	—
177	SD₁₆×C₃₁	—	2	8	no	—
178	SD₁₆×C₃₁	—	2	8	no	—
179	D₈×C₃₁	—	2	8	no	—
180	M₁₆×C₃₁	—	2	16	no	—
181	M₁₆×C₃₁	—	2	4	no	—
182	M₁₆×C₃₁	—	2	4	no	—
183	M₁₆×C₃₁	—	2	4	no	—
184	M₁₆×C₃₁	—	2	4	no	—
185	M₁₆×C₃₁	—	2	4	no	—
186	M₁₆×C₃₁	—	2	2	no	—
187	M₁₆×C₃₁	—	2	2	no	—
188	M₁₆×C₃₁	—	2	2	no	—
189	M₁₆×C₃₁	—	4	2	no	—
190	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
191	C₂×C₈×C₃₁	—	2	16	no	—
192	C₂×C₈×C₃₁	—	2	4	no	—
193	C₂×C₈×C₃₁	—	2	2	no	—
194	C₂×C₈×C₃₁	—	2	2	no	—
195	C₂×C₈×C₃₁	—	2	2	no	—
196	C₂×C₈×C₃₁	—	2	2	no	—
197	C₂×C₈×C₃₁	—	2	2	no	—
198	C₂×C₈×C₃₁	—	2	2	no	—
199	C₂×C₈×C₃₁	—	4	2	no	—
200	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
201	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
202	C₄²×C₃₁	—	2	2	no	—
203	C₄²×C₃₁	—	2	2	no	—
204	C₄²×C₃₁	—	4	2	no	—
205	#20918	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂³×C₃₁	—	2	2	no	—
207	C₂³×C₃₁	—	2	4	no	—
208	C₂³×C₃₁	—	2	2	no	—
209	C₂³×C₃₁	—	4	2	no	—
210	C₂³×C₃₁	—	4	2	no	—
211	C₂³×C₃₁	—	8	2	no	—
212	C₂³×C₃₁	—	8	2	no	—
213	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
214	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
215	C₂³×C₃₁	#121	1	2	yes	—
216	C₂³×C₃₁	#123	1	2	yes	—
217	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
218	Q₈×C₃₁	—	2	2	no	—
219	Q₈×C₃₁	—	2	4	no	—
220	Q₈×C₃₁	—	2	2	no	—
221	Q₈×C₃₁	—	2	2	no	—
222	Q₈×C₃₁	—	4	2	no	—
223	Q₈×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
224	D₄×C₃₁	—	2	4	no	—
225	D₄×C₃₁	—	2	4	no	—
226	D₄×C₃₁	—	2	8	no	—
227	D₄×C₃₁	—	2	2	no	—
228	D₄×C₃₁	—	2	2	no	—
229	D₄×C₃₁	—	2	2	no	—
230	D₄×C₃₁	—	2	4	no	—
231	D₄×C₃₁	—	4	2	no	—
232	D₄×C₃₁	—	4	2	no	—
233	D₄×C₃₁	—	4	4	no	—
234	D₄×C₃₁	—	4	2	no	—
235	D₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	4	no	—
238	C₂×C₄×C₃₁	—	2	4	no	—
239	C₂×C₄×C₃₁	—	2	4	no	—
240	C₂×C₄×C₃₁	—	2	8	no	—
241	C₂×C₄×C₃₁	—	2	4	no	—
242	C₂×C₄×C₃₁	—	2	2	no	—
243	C₂×C₄×C₃₁	—	4	2	no	—
244	C₂×C₄×C₃₁	—	4	8	no	—
245	C₂×C₄×C₃₁	—	4	2	no	—
246	C₂×C₄×C₃₁	—	4	2	no	—
247	C₂×C₄×C₃₁	—	4	2	no	—
248	C₂×C₄×C₃₁	—	4	2	no	—
249	C₂×C₄×C₃₁	—	4	4	no	—
250	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
251	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
252	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
253	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
254	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
255	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
256	C₂×C₄×C₃₁	#120	1	2	yes	—
257	C₂×C₄×C₃₁	#120	1	2	yes	—
258	C₂×C₄×C₃₁	#121	1	2	yes	—
259	C₂×C₄×C₃₁	#121	1	2	yes	—
260	C₂×C₄×C₃₁	#121	1	2	yes	—
261	C₂×C₄×C₃₁	#123	1	2	yes	—
262	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
263	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
268	C₈×C₃₁	—	2	4	no	—
269	C₈×C₃₁	—	2	4	no	—
270	C₈×C₃₁	—	2	4	no	—
271	C₈×C₃₁	—	4	2	no	—
272	C₂×#575	C₂×C₃₁	1	1	yes	characteristic
273	C₂×#568	C₂×C₃₁	1	1	yes	characteristic, complemented
274	C₂²×#104	C₂×C₃₁	1	1	yes	characteristic, complemented
275	C₂²×#104	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#2879	C₂×C₃₁	1	1	yes	characteristic, complemented
277	#1876	C₂×C₃₁	1	8	yes	complemented
278	#1738	C₂×C₃₁	1	1	yes	characteristic, complemented
279	#1728	C₂×C₃₁	1	1	yes	characteristic, complemented
280	C₂²×C₃₁	—	2	2	no	—
281	C₂²×C₃₁	—	2	2	no	—
282	C₂²×C₃₁	—	2	2	no	—
283	C₂²×C₃₁	—	2	4	no	—
284	C₂²×C₃₁	—	4	2	no	—
285	C₂²×C₃₁	—	4	2	no	—
286	C₂²×C₃₁	—	4	4	no	—
287	C₂²×C₃₁	—	4	4	no	—
288	C₂²×C₃₁	—	4	2	no	—
289	C₂²×C₃₁	—	8	4	no	—
290	C₂²×C₃₁	—	8	4	no	—
291	C₂²×C₃₁	#386	1	2	yes	—
292	C₂²×C₃₁	#386	1	2	yes	—
293	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
294	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
295	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
296	C₄×C₃₁	—	2	4	no	—
297	C₄×C₃₁	—	4	4	no	—
298	C₄×C₃₁	—	4	2	no	—
299	C₄×C₃₁	#386	1	2	yes	—
300	C₄×C₃₁	#386	1	2	yes	—
301	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
302	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
303	C₄×C₃₁	C₂×#120	1	1	yes	characteristic
304	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
305	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
306	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
307	#575	—	2	4	no	—
308	#575	—	2	4	no	—
309	C₂×#137	C₄×C₃₁	1	2	yes	complemented
310	C₂×#137	C₄×C₃₁	1	2	yes	complemented
311	C₂×AGL(1,ℤ/8)	C₄×C₃₁	1	2	yes	complemented
312	C₂×AGL(1,ℤ/8)	C₄×C₃₁	1	2	yes	complemented
313	C₂×#135	C₂²×C₃₁	1	4	yes	—
314	#568	—	2	4	no	—
315	#568	—	2	4	no	—
316	C₂×#131	—	2	2	no	—
317	C₂×#131	C₂²×C₃₁	1	4	yes	complemented
318	C₂²×M₁₆	—	2	2	no	—
319	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
320	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
321	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
322	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
323	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
324	C₂×#104	—	2	4	no	—
325	C₂×#104	—	2	4	no	—
326	C₂×#104	C₂²×C₃₁	1	4	yes	complemented
327	C₂×#104	C₂²×C₃₁	1	4	yes	complemented
328	#408	—	2	2	no	—
329	#408	C₂²×C₃₁	1	4	yes	complemented
330	#407	—	2	2	no	—
331	#341	C₂²×C₃₁	1	4	yes	complemented
332	#336	C₂²×C₃₁	1	4	yes	complemented
333	C₂×C₃₁	—	2	1	no	—
334	C₂×C₃₁	—	2	2	no	—
335	C₂×C₃₁	—	2	2	no	—
336	C₂×C₃₁	—	2	1	no	—
337	C₂×C₃₁	—	4	4	no	—
338	C₂×C₃₁	—	8	2	no	—
339	C₂×C₃₁	C₂×#386	1	1	yes	characteristic
340	C₂×C₃₁	#2956	1	2	yes	—
341	C₂×#41	—	2	1	no	—
342	C₂×#41	—	2	2	no	—
343	C₂×#41	—	2	1	no	—
344	C₂×#41	—	2	1	no	—
345	C₂×#41	—	2	2	no	—
346	C₂×#41	—	2	1	no	—
347	C₂×#41	—	2	2	no	—
348	C₂×#41	—	2	2	no	—
349	C₂×#41	—	2	2	no	—
350	C₂×#41	—	2	2	no	—
351	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
352	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
353	C₂×#41	D₄×C₃₁	1	4	yes	—
354	C₂×#41	D₄×C₃₁	1	4	yes	—
355	C₂×#41	C₂³×C₃₁	1	2	yes	—
356	C₂×#41	C₂³×C₃₁	1	2	yes	—
357	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
358	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
359	C₂²×D₄	—	2	1	no	—
360	C₂²×D₄	—	2	1	no	—
361	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
362	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
363	C₂³×C₄	—	2	1	no	—
364	C₂³×C₄	—	2	1	no	—
365	C₂³×C₄	—	2	1	no	—
366	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
367	#137	—	2	8	no	—
368	#137	—	2	8	no	—
369	AGL(1,ℤ/8)	—	2	8	no	—
370	AGL(1,ℤ/8)	—	2	8	no	—
371	#135	—	2	8	no	—
372	#135	—	2	8	no	—
373	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
374	C₂×SD₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
375	C₂×SD₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
376	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	complemented
377	#131	—	2	16	no	—
378	#131	—	2	8	no	—
379	#131	—	2	8	no	—
380	C₂×M₁₆	—	2	16	no	—
381	C₂×M₁₆	—	2	1	no	—
382	C₂×M₁₆	—	2	2	no	—
383	C₂×M₁₆	—	2	2	no	—
384	C₂×M₁₆	—	2	2	no	—
385	C₂×M₁₆	—	2	2	no	—
386	C₂×M₁₆	—	2	2	no	—
387	C₂×M₁₆	—	2	1	no	—
388	C₂×M₁₆	—	2	4	no	—
389	C₂×M₁₆	—	2	2	no	—
390	C₂×M₁₆	—	2	2	no	—
391	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
392	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
393	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
394	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
395	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
396	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
397	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
398	C₂²×C₈	—	2	1	no	—
399	C₂²×C₈	—	2	4	no	—
400	C₂²×C₈	—	2	1	no	—
401	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
402	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
403	C₂×C₄²	—	2	2	no	—
404	C₂×C₄²	—	2	2	no	—
405	C₂×C₄²	—	2	1	no	—
406	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
407	#104	—	2	8	no	—
408	#104	—	2	8	no	—
409	C₂²⋊C₈	—	2	8	no	—
410	C₃₁	#20918	1	1	yes	characteristic, direct product factor, Sylow
411	C₂⁴	—	2	1	no	—
412	C₂⁴	—	2	1	no	—
413	C₂⁴	#41×C₃₁	1	1	yes	characteristic
414	#41	—	2	4	no	—
415	#41	—	2	4	no	—
416	#41	—	2	4	no	—
417	#41	—	2	4	no	—
418	#41	—	2	4	no	—
419	#41	—	2	2	no	—
420	#41	—	2	2	no	—
421	#41	—	2	4	no	—
422	#41	—	2	4	no	—
423	#41	—	2	2	no	—
424	#41	—	2	2	no	—
425	#41	—	2	2	no	—
426	#41	—	2	2	no	—
427	#41	—	4	2	no	—
428	#41	—	4	2	no	—
429	#41	—	4	2	no	—
430	#41	—	4	2	no	—
431	#41	C₂²⋊C₄×C₃₁	1	8	yes	—
432	#41	C₂×D₄×C₃₁	1	4	yes	—
433	#41	C₂×D₄×C₃₁	1	4	yes	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	—	2	2	no	—
436	C₂×Q₈	—	2	1	no	—
437	C₂×Q₈	—	2	2	no	—
438	C₂×Q₈	—	2	2	no	—
439	C₂×Q₈	C₂²⋊C₄×C₃₁	1	4	yes	—
440	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
441	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	4	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	—	2	2	no	—
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	2	no	—
452	C₂×D₄	—	2	1	no	—
453	C₂×D₄	—	4	2	no	—
454	C₂×D₄	—	4	2	no	—
455	C₂×D₄	C₂²⋊C₄×C₃₁	1	4	yes	—
456	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
457	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	4	no	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	4	4	no	—
468	C₂²×C₄	—	4	2	no	—
469	C₂²×C₄	—	4	2	no	—
470	C₂²×C₄	—	4	2	no	—
471	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
472	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
474	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
475	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
476	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
478	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
479	C₂²×C₄	#41×C₃₁	1	2	yes	—
480	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
481	Q₁₆	—	2	8	no	—
482	SD₁₆	—	2	8	no	—
483	SD₁₆	—	2	8	no	—
484	D₈	—	2	8	no	—
485	M₁₆	—	2	16	no	—
486	M₁₆	—	2	4	no	—
487	M₁₆	—	2	4	no	—
488	M₁₆	—	2	4	no	—
489	M₁₆	—	2	4	no	—
490	M₁₆	—	2	4	no	—
491	M₁₆	—	2	2	no	—
492	M₁₆	—	2	2	no	—
493	M₁₆	—	2	2	no	—
494	M₁₆	—	4	2	no	—
495	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
496	C₂×C₈	—	2	16	no	—
497	C₂×C₈	—	2	4	no	—
498	C₂×C₈	—	2	2	no	—
499	C₂×C₈	—	2	2	no	—
500	C₂×C₈	—	2	2	no	—
501	C₂×C₈	—	2	2	no	—
502	C₂×C₈	—	2	2	no	—
503	C₂×C₈	—	2	2	no	—
504	C₂×C₈	—	4	2	no	—
505	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
506	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
507	C₄²	—	2	2	no	—
508	C₄²	—	2	2	no	—
509	C₄²	—	4	2	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	2	4	no	—
512	C₂³	—	2	2	no	—
513	C₂³	—	4	2	no	—
514	C₂³	—	4	2	no	—
515	C₂³	—	8	2	no	—
516	C₂³	—	8	2	no	—
517	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
518	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
519	C₂³	C₃₁×#121	1	2	yes	—
520	C₂³	C₃₁×#123	1	2	yes	—
521	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
522	Q₈	—	2	2	no	—
523	Q₈	—	2	4	no	—
524	Q₈	—	2	2	no	—
525	Q₈	—	2	2	no	—
526	Q₈	—	4	2	no	—
527	Q₈	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
528	D₄	—	2	4	no	—
529	D₄	—	2	4	no	—
530	D₄	—	2	8	no	—
531	D₄	—	2	2	no	—
532	D₄	—	2	2	no	—
533	D₄	—	2	2	no	—
534	D₄	—	2	4	no	—
535	D₄	—	4	2	no	—
536	D₄	—	4	2	no	—
537	D₄	—	4	4	no	—
538	D₄	—	4	2	no	—
539	D₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	8	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	4	2	no	—
548	C₂×C₄	—	4	8	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	4	2	no	—
552	C₂×C₄	—	4	2	no	—
553	C₂×C₄	—	4	4	no	—
554	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
555	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
556	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
557	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
558	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
559	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
560	C₂×C₄	C₃₁×#120	1	2	yes	—
561	C₂×C₄	C₃₁×#120	1	2	yes	—
562	C₂×C₄	C₃₁×#121	1	2	yes	—
563	C₂×C₄	C₃₁×#121	1	2	yes	—
564	C₂×C₄	C₃₁×#121	1	2	yes	—
565	C₂×C₄	C₃₁×#123	1	2	yes	—
566	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
567	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
568	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
569	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
570	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
571	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
572	C₈	—	2	4	no	—
573	C₈	—	2	4	no	—
574	C₈	—	2	4	no	—
575	C₈	—	4	2	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	2	2	no	—
578	C₂²	—	2	2	no	—
579	C₂²	—	2	4	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	4	4	no	—
583	C₂²	—	4	4	no	—
584	C₂²	—	4	2	no	—
585	C₂²	—	8	4	no	—
586	C₂²	—	8	4	no	—
587	C₂²	C₃₁×#386	1	2	yes	—
588	C₂²	C₃₁×#386	1	2	yes	—
589	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
590	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
591	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
592	C₄	—	2	4	no	—
593	C₄	—	4	4	no	—
594	C₄	—	4	2	no	—
595	C₄	C₃₁×#386	1	2	yes	—
596	C₄	C₃₁×#386	1	2	yes	—
597	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
598	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
599	C₄	C₂×C₃₁×#120	1	1	yes	characteristic
600	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
601	C₂	—	2	1	no	—
602	C₂	—	2	2	no	—
603	C₂	—	2	2	no	—
604	C₂	—	2	1	no	—
605	C₂	—	4	4	no	—
606	C₂	—	8	2	no	—
607	C₂	C₂×C₃₁×#386	1	1	yes	characteristic
608	C₂	C₃₁×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂×C₃₁×#575	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#568	C₂	semidirect
3. C₂²×C₃₁×#104	C₂	semidirect
4. C₂²×C₃₁×#104	C₂	semidirect
5. C₃₁×#2879	C₂	semidirect
6. C₃₁×#1876	C₂	semidirect
7. C₃₁×#1738	C₂	semidirect
8. C₃₁×#1728	C₂	semidirect
13. C₂×C₃₁×#137	C₄	semidirect
14. C₂×C₃₁×#137	C₄	semidirect
15. C₂×C₃₁×AGL(1,ℤ/8)	C₄	semidirect
16. C₂×C₃₁×AGL(1,ℤ/8)	C₄	semidirect
21. C₂×C₃₁×#131	C₂²	semidirect
30. C₂×C₃₁×#104	C₂²	semidirect
31. C₂×C₃₁×#104	C₂²	semidirect
33. C₃₁×#408	C₂²	semidirect
35. C₃₁×#341	C₂²	semidirect
36. C₃₁×#336	C₂²	semidirect
69. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
70. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
71. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
72. C₂×D₈×C₃₁	C₂×C₄	semidirect
205. #20918	C₃₁	direct (N × H)
273. C₂×#568	C₂×C₃₁	semidirect
274. C₂²×#104	C₂×C₃₁	semidirect
275. C₂²×#104	C₂×C₃₁	semidirect
276. #2879	C₂×C₃₁	semidirect
277. #1876	C₂×C₃₁	semidirect
278. #1738	C₂×C₃₁	semidirect
279. #1728	C₂×C₃₁	semidirect
309. C₂×#137	C₄×C₃₁	semidirect
310. C₂×#137	C₄×C₃₁	semidirect
311. C₂×AGL(1,ℤ/8)	C₄×C₃₁	semidirect
312. C₂×AGL(1,ℤ/8)	C₄×C₃₁	semidirect
317. C₂×#131	C₂²×C₃₁	semidirect
326. C₂×#104	C₂²×C₃₁	semidirect
327. C₂×#104	C₂²×C₃₁	semidirect
329. #408	C₂²×C₃₁	semidirect
331. #341	C₂²×C₃₁	semidirect
332. #336	C₂²×C₃₁	semidirect
373. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
374. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
375. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
376. C₂×D₈	C₂×C₄×C₃₁	semidirect
410. C₃₁	#20918	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×#575	2
C₂×C₃₁×#568	2
C₂²×C₃₁×#104	2
C₃₁×#2879	2
C₃₁×#1876	2
C₃₁×#1738	2
C₃₁×#1728	2
#20918	31

Subgroups — C31×#20918 #21477677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20918 #21477677