Subgroups — C₃₁×#10789 #21467548

Order 7936.

1370 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#103	C₂	1	2	yes	maximal
3	C₂×C₃₁×#340	C₂	1	2	yes	maximal
4	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic
5	C₃₁×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₃₁	—	2	4	no	—
9	C₂×C₄×Q₈×C₃₁	—	2	4	no	—
10	C₂×C₄×Q₈×C₃₁	—	2	2	no	—
11	C₂×C₄×Q₈×C₃₁	C₄	1	4	yes	complemented
12	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
13	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₃₁	—	2	1	no	—
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#103	—	2	4	no	—
18	C₂×C₃₁×#103	—	2	4	no	—
19	C₂×C₃₁×#103	C₄	1	8	yes	complemented
20	C₂×C₃₁×#95	—	2	1	no	—
21	C₂×C₃₁×#95	—	2	1	no	—
22	C₃₁×#340	—	2	4	no	—
23	C₃₁×#336	—	2	2	no	—
24	C₂²×Q₈×C₃₁	—	2	1	no	—
25	C₂²×Q₈×C₃₁	—	2	2	no	—
26	C₂²×Q₈×C₃₁	—	2	2	no	—
27	C₂²×Q₈×C₃₁	—	2	1	no	—
28	C₂²×Q₈×C₃₁	—	2	1	no	—
29	C₂²×Q₈×C₃₁	C₂×C₄	1	4	yes	—
30	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
31	C₂²×Q₈×C₃₁	D₄	1	2	yes	—
32	C₂²×Q₈×C₃₁	D₄	1	4	yes	—
33	C₂²×Q₈×C₃₁	Q₈	1	2	yes	—
34	C₂³×C₄×C₃₁	—	2	1	no	—
35	C₂³×C₄×C₃₁	—	2	1	no	—
36	C₂³×C₄×C₃₁	—	2	2	no	—
37	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	—	2	2	no	—
41	C₂²×C₈×C₃₁	—	2	2	no	—
42	C₂²×C₈×C₃₁	—	2	2	no	—
43	C₂²×C₈×C₃₁	—	2	2	no	—
44	C₂²×C₈×C₃₁	—	2	2	no	—
45	C₂²×C₈×C₃₁	—	2	2	no	—
46	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
47	C₄×Q₈×C₃₁	—	2	16	no	—
48	C₄×Q₈×C₃₁	—	2	8	no	—
49	C₄×Q₈×C₃₁	—	2	8	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	D₄	1	4	yes	—
62	C₂×C₄²×C₃₁	—	2	8	no	—
63	C₂×C₄²×C₃₁	—	2	4	no	—
64	C₂×C₄²×C₃₁	—	2	2	no	—
65	C₂×C₄²×C₃₁	—	2	2	no	—
66	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₃₁	D₄	1	2	yes	—
68	C₃₁×#103	—	2	16	no	—
69	C₃₁×#95	—	4	2	no	—
70	C₃₁×#95	—	4	2	no	—
71	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂×Q₈×C₃₁	—	2	2	no	—
73	C₂×Q₈×C₃₁	—	2	2	no	—
74	C₂×Q₈×C₃₁	—	2	2	no	—
75	C₂×Q₈×C₃₁	—	2	4	no	—
76	C₂×Q₈×C₃₁	—	2	4	no	—
77	C₂×Q₈×C₃₁	—	2	4	no	—
78	C₂×Q₈×C₃₁	—	2	4	no	—
79	C₂×Q₈×C₃₁	—	2	2	no	—
80	C₂×Q₈×C₃₁	—	2	4	no	—
81	C₂×Q₈×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	—	2	4	no	—
83	C₂×Q₈×C₃₁	—	2	4	no	—
84	C₂×Q₈×C₃₁	—	2	4	no	—
85	C₂×Q₈×C₃₁	C₄²	1	4	yes	—
86	C₂×Q₈×C₃₁	C₂²⋊C₄	1	8	yes	—
87	C₂×Q₈×C₃₁	C₂²⋊C₄	1	4	yes	—
88	C₂×Q₈×C₃₁	C₄⋊C₄	1	4	yes	—
89	C₂×Q₈×C₃₁	C₄⋊C₄	1	8	yes	—
90	C₂²×C₄×C₃₁	—	2	2	no	—
91	C₂²×C₄×C₃₁	—	2	4	no	—
92	C₂²×C₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	2	4	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	2	4	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	4	no	—
104	C₂²×C₄×C₃₁	—	4	2	no	—
105	C₂²×C₄×C₃₁	—	4	1	no	—
106	C₂²×C₄×C₃₁	—	4	1	no	—
107	C₂²×C₄×C₃₁	—	4	2	no	—
108	C₂²×C₄×C₃₁	—	4	2	no	—
109	C₂²×C₄×C₃₁	—	4	2	no	—
110	C₂²×C₄×C₃₁	—	4	2	no	—
111	C₂²×C₄×C₃₁	—	4	2	no	—
112	C₂²×C₄×C₃₁	—	4	2	no	—
113	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
114	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
115	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
116	C₂²×C₄×C₃₁	Q₁₆	1	2	yes	—
117	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	#41	1	2	yes	—
127	C₂×C₈×C₃₁	—	2	4	no	—
128	C₂×C₈×C₃₁	—	2	4	no	—
129	C₂×C₈×C₃₁	—	2	4	no	—
130	C₂×C₈×C₃₁	—	4	2	no	—
131	C₂×C₈×C₃₁	—	4	2	no	—
132	C₂×C₈×C₃₁	—	4	2	no	—
133	C₂×C₈×C₃₁	—	4	2	no	—
134	C₄⋊C₄×C₃₁	—	2	16	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	2	4	no	—
137	C₄⋊C₄×C₃₁	—	2	4	no	—
138	C₄⋊C₄×C₃₁	—	2	4	no	—
139	C₄⋊C₄×C₃₁	—	4	2	no	—
140	C₄⋊C₄×C₃₁	—	4	2	no	—
141	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
142	C₄²×C₃₁	—	2	16	no	—
143	C₄²×C₃₁	—	2	2	no	—
144	C₄²×C₃₁	—	2	2	no	—
145	C₄²×C₃₁	—	2	4	no	—
146	#10789	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂³×C₃₁	C₄×D₄	1	2	yes	—
149	C₂³×C₃₁	#120	1	2	yes	—
150	C₂³×C₃₁	#121	1	1	yes	characteristic
151	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
152	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
153	C₂³×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
154	C₂³×C₃₁	#135	1	2	yes	—
155	C₂³×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂³×C₃₁	#137	1	1	yes	characteristic
157	C₂³×C₃₁	#137	1	2	yes	—
158	Q₈×C₃₁	—	2	4	no	—
159	Q₈×C₃₁	—	2	4	no	—
160	Q₈×C₃₁	—	2	4	no	—
161	Q₈×C₃₁	—	2	8	no	—
162	Q₈×C₃₁	—	2	8	no	—
163	Q₈×C₃₁	#95	1	8	yes	complemented
164	C₂×C₄×C₃₁	—	2	4	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	4	no	—
167	C₂×C₄×C₃₁	—	2	4	no	—
168	C₂×C₄×C₃₁	—	2	8	no	—
169	C₂×C₄×C₃₁	—	2	8	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	4	no	—
172	C₂×C₄×C₃₁	—	2	4	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	4	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	4	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	4	no	—
183	C₂×C₄×C₃₁	—	4	4	no	—
184	C₂×C₄×C₃₁	—	4	4	no	—
185	C₂×C₄×C₃₁	—	4	4	no	—
186	C₂×C₄×C₃₁	—	8	2	no	—
187	C₂×C₄×C₃₁	—	8	2	no	—
188	C₂×C₄×C₃₁	—	8	2	no	—
189	C₂×C₄×C₃₁	—	8	2	no	—
190	C₂×C₄×C₃₁	#103	1	4	yes	complemented
191	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
194	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
197	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
198	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
200	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
201	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
202	C₂×C₄×C₃₁	#120	1	2	yes	—
203	C₂×C₄×C₃₁	#121	1	2	yes	—
204	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
206	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
207	C₂×C₄×C₃₁	#123	1	2	yes	—
208	C₂×C₄×C₃₁	#123	1	2	yes	—
209	C₈×C₃₁	—	4	4	no	—
210	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
211	C₂²×#103	C₂×C₃₁	1	2	yes	—
212	C₂×#340	C₂×C₃₁	1	2	yes	—
213	C₂×#336	C₂×C₃₁	1	1	yes	characteristic
214	C₄⋊#95	C₂×C₃₁	1	1	yes	characteristic
215	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
216	C₂²×C₃₁	#379	1	1	yes	characteristic
217	C₂²×C₃₁	#380	1	1	yes	characteristic
218	C₂²×C₃₁	#385	1	2	yes	—
219	C₂²×C₃₁	#386	1	2	yes	—
220	C₂²×C₃₁	C₂×#103	1	1	yes	characteristic
221	C₂²×C₃₁	#416	1	1	yes	characteristic
222	C₂²×C₃₁	#417	1	1	yes	characteristic
223	C₂²×C₃₁	#419	1	1	yes	characteristic
224	C₂²×C₃₁	C₄×SD₁₆	1	2	yes	—
225	C₂²×C₃₁	C₄×Q₁₆	1	2	yes	—
226	C₂²×C₃₁	#440	1	2	yes	—
227	C₂²×C₃₁	#441	1	2	yes	—
228	C₂²×C₃₁	#448	1	2	yes	—
229	C₂²×C₃₁	#449	1	2	yes	—
230	C₂²×C₃₁	#451	1	2	yes	—
231	C₂²×C₃₁	#452	1	2	yes	—
232	C₂²×C₃₁	#460	1	1	yes	characteristic
233	C₂²×C₃₁	#462	1	1	yes	characteristic
234	C₂²×C₃₁	#464	1	2	yes	—
235	C₂²×C₃₁	#475	1	1	yes	characteristic
236	C₂²×C₃₁	#477	1	1	yes	characteristic
237	C₂²×C₃₁	#479	1	2	yes	—
238	C₄×C₃₁	—	2	8	no	—
239	C₄×C₃₁	—	2	4	no	—
240	C₄×C₃₁	—	4	2	no	—
241	C₄×C₃₁	—	4	4	no	—
242	C₄×C₃₁	—	8	4	no	—
243	C₄×C₃₁	C₂×#95	1	1	yes	characteristic
244	C₄×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
245	C₄×C₃₁	#380	1	1	yes	characteristic
246	C₄×C₃₁	#381	1	1	yes	characteristic
247	C₄×C₃₁	#385	1	2	yes	—
248	C₄×C₃₁	#386	1	2	yes	—
249	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
250	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
251	C₂×C₄×Q₈	—	2	4	no	—
252	C₂×C₄×Q₈	—	2	4	no	—
253	C₂×C₄×Q₈	—	2	2	no	—
254	C₂×C₄×Q₈	C₄×C₃₁	1	4	yes	complemented
255	C₂²×C₄⋊C₄	—	2	1	no	—
256	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
257	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
258	C₂²×C₄²	—	2	1	no	—
259	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×#103	—	2	4	no	—
261	C₂×#103	—	2	4	no	—
262	C₂×#103	C₄×C₃₁	1	8	yes	complemented
263	C₂×#95	—	2	1	no	—
264	C₂×#95	—	2	1	no	—
265	#340	—	2	4	no	—
266	#336	—	2	2	no	—
267	C₂×C₃₁	C₂²⋊#95	1	1	yes	characteristic
268	C₂×C₃₁	C₄×#103	1	1	yes	characteristic
269	C₂×C₃₁	#1763	1	1	yes	characteristic
270	C₂×C₃₁	#1787	1	1	yes	characteristic
271	C₂×C₃₁	#1788	1	1	yes	characteristic
272	C₂×C₃₁	#1803	1	1	yes	characteristic
273	C₂×C₃₁	#1805	1	1	yes	characteristic
274	C₂×C₃₁	#1863	1	2	yes	—
275	C₂×C₃₁	#1865	1	2	yes	—
276	C₂×C₃₁	#1878	1	2	yes	—
277	C₂×C₃₁	#1880	1	2	yes	—
278	C₂²×Q₈	—	2	1	no	—
279	C₂²×Q₈	—	2	2	no	—
280	C₂²×Q₈	—	2	2	no	—
281	C₂²×Q₈	—	2	1	no	—
282	C₂²×Q₈	—	2	1	no	—
283	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
284	C₂²×Q₈	C₂×C₄×C₃₁	1	4	yes	—
285	C₂²×Q₈	D₄×C₃₁	1	4	yes	—
286	C₂²×Q₈	D₄×C₃₁	1	2	yes	—
287	C₂²×Q₈	Q₈×C₃₁	1	2	yes	—
288	C₂³×C₄	—	2	1	no	—
289	C₂³×C₄	—	2	1	no	—
290	C₂³×C₄	—	2	2	no	—
291	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
292	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
293	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
294	C₂²×C₈	—	2	2	no	—
295	C₂²×C₈	—	2	2	no	—
296	C₂²×C₈	—	2	2	no	—
297	C₂²×C₈	—	2	2	no	—
298	C₂²×C₈	—	2	2	no	—
299	C₂²×C₈	—	2	2	no	—
300	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
301	C₄×Q₈	—	2	16	no	—
302	C₄×Q₈	—	2	8	no	—
303	C₄×Q₈	—	2	8	no	—
304	C₂×C₄⋊C₄	—	2	8	no	—
305	C₂×C₄⋊C₄	—	2	4	no	—
306	C₂×C₄⋊C₄	—	2	2	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	—	2	2	no	—
309	C₂×C₄⋊C₄	—	2	2	no	—
310	C₂×C₄⋊C₄	—	2	2	no	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
314	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
315	C₂×C₄⋊C₄	D₄×C₃₁	1	4	yes	—
316	C₂×C₄²	—	2	8	no	—
317	C₂×C₄²	—	2	4	no	—
318	C₂×C₄²	—	2	2	no	—
319	C₂×C₄²	—	2	2	no	—
320	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
321	C₂×C₄²	D₄×C₃₁	1	2	yes	—
322	#103	—	2	16	no	—
323	#95	—	4	2	no	—
324	#95	—	4	2	no	—
325	C₃₁	#10789	1	1	yes	characteristic, direct product factor, Sylow
326	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	—	2	2	no	—
330	C₂×Q₈	—	2	4	no	—
331	C₂×Q₈	—	2	4	no	—
332	C₂×Q₈	—	2	4	no	—
333	C₂×Q₈	—	2	4	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	4	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	2	4	no	—
338	C₂×Q₈	—	2	4	no	—
339	C₂×Q₈	—	2	4	no	—
340	C₂×Q₈	C₄²×C₃₁	1	4	yes	—
341	C₂×Q₈	C₂²⋊C₄×C₃₁	1	8	yes	—
342	C₂×Q₈	C₂²⋊C₄×C₃₁	1	4	yes	—
343	C₂×Q₈	C₄⋊C₄×C₃₁	1	4	yes	—
344	C₂×Q₈	C₄⋊C₄×C₃₁	1	8	yes	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	4	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	4	no	—
359	C₂²×C₄	—	4	2	no	—
360	C₂²×C₄	—	4	1	no	—
361	C₂²×C₄	—	4	1	no	—
362	C₂²×C₄	—	4	2	no	—
363	C₂²×C₄	—	4	2	no	—
364	C₂²×C₄	—	4	2	no	—
365	C₂²×C₄	—	4	2	no	—
366	C₂²×C₄	—	4	2	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
369	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
370	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
371	C₂²×C₄	Q₁₆×C₃₁	1	2	yes	—
372	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
373	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
374	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
375	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
377	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
379	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
380	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
381	C₂²×C₄	#41×C₃₁	1	2	yes	—
382	C₂×C₈	—	2	4	no	—
383	C₂×C₈	—	2	4	no	—
384	C₂×C₈	—	2	4	no	—
385	C₂×C₈	—	4	2	no	—
386	C₂×C₈	—	4	2	no	—
387	C₂×C₈	—	4	2	no	—
388	C₂×C₈	—	4	2	no	—
389	C₄⋊C₄	—	2	16	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	4	no	—
392	C₄⋊C₄	—	2	4	no	—
393	C₄⋊C₄	—	2	4	no	—
394	C₄⋊C₄	—	4	2	no	—
395	C₄⋊C₄	—	4	2	no	—
396	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
397	C₄²	—	2	16	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	2	2	no	—
400	C₄²	—	2	4	no	—
401	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
402	C₂³	C₄×D₄×C₃₁	1	2	yes	—
403	C₂³	C₃₁×#120	1	2	yes	—
404	C₂³	C₃₁×#121	1	1	yes	characteristic
405	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
406	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
407	C₂³	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
408	C₂³	C₃₁×#135	1	2	yes	—
409	C₂³	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
410	C₂³	C₃₁×#137	1	1	yes	characteristic
411	C₂³	C₃₁×#137	1	2	yes	—
412	Q₈	—	2	4	no	—
413	Q₈	—	2	4	no	—
414	Q₈	—	2	4	no	—
415	Q₈	—	2	8	no	—
416	Q₈	—	2	8	no	—
417	Q₈	C₃₁×#95	1	8	yes	complemented
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	4	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	8	no	—
426	C₂×C₄	—	2	8	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	4	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	4	no	—
437	C₂×C₄	—	4	4	no	—
438	C₂×C₄	—	4	4	no	—
439	C₂×C₄	—	4	4	no	—
440	C₂×C₄	—	8	2	no	—
441	C₂×C₄	—	8	2	no	—
442	C₂×C₄	—	8	2	no	—
443	C₂×C₄	—	8	2	no	—
444	C₂×C₄	C₃₁×#103	1	4	yes	complemented
445	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
446	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
447	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
448	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
449	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
450	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
451	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
452	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
453	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
454	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
455	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
456	C₂×C₄	C₃₁×#120	1	2	yes	—
457	C₂×C₄	C₃₁×#121	1	2	yes	—
458	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
459	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
460	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
461	C₂×C₄	C₃₁×#123	1	2	yes	—
462	C₂×C₄	C₃₁×#123	1	2	yes	—
463	C₈	—	4	4	no	—
464	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
465	C₂²	C₃₁×#379	1	1	yes	characteristic
466	C₂²	C₃₁×#380	1	1	yes	characteristic
467	C₂²	C₃₁×#385	1	2	yes	—
468	C₂²	C₃₁×#386	1	2	yes	—
469	C₂²	C₂×C₃₁×#103	1	1	yes	characteristic
470	C₂²	C₃₁×#416	1	1	yes	characteristic
471	C₂²	C₃₁×#417	1	1	yes	characteristic
472	C₂²	C₃₁×#419	1	1	yes	characteristic
473	C₂²	C₄×SD₁₆×C₃₁	1	2	yes	—
474	C₂²	C₄×Q₁₆×C₃₁	1	2	yes	—
475	C₂²	C₃₁×#440	1	2	yes	—
476	C₂²	C₃₁×#441	1	2	yes	—
477	C₂²	C₃₁×#448	1	2	yes	—
478	C₂²	C₃₁×#449	1	2	yes	—
479	C₂²	C₃₁×#451	1	2	yes	—
480	C₂²	C₃₁×#452	1	2	yes	—
481	C₂²	C₃₁×#460	1	1	yes	characteristic
482	C₂²	C₃₁×#462	1	1	yes	characteristic
483	C₂²	C₃₁×#464	1	2	yes	—
484	C₂²	C₃₁×#475	1	1	yes	characteristic
485	C₂²	C₃₁×#477	1	1	yes	characteristic
486	C₂²	C₃₁×#479	1	2	yes	—
487	C₄	—	2	8	no	—
488	C₄	—	2	4	no	—
489	C₄	—	4	2	no	—
490	C₄	—	4	4	no	—
491	C₄	—	8	4	no	—
492	C₄	C₂×C₃₁×#95	1	1	yes	characteristic
493	C₄	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
494	C₄	C₃₁×#380	1	1	yes	characteristic
495	C₄	C₃₁×#381	1	1	yes	characteristic
496	C₄	C₃₁×#385	1	2	yes	—
497	C₄	C₃₁×#386	1	2	yes	—
498	C₂	C₃₁×C₂²⋊#95	1	1	yes	characteristic
499	C₂	C₄×C₃₁×#103	1	1	yes	characteristic
500	C₂	C₃₁×#1763	1	1	yes	characteristic
501	C₂	C₃₁×#1787	1	1	yes	characteristic
502	C₂	C₃₁×#1788	1	1	yes	characteristic
503	C₂	C₃₁×#1803	1	1	yes	characteristic
504	C₂	C₃₁×#1805	1	1	yes	characteristic
505	C₂	C₃₁×#1863	1	2	yes	—
506	C₂	C₃₁×#1865	1	2	yes	—
507	C₂	C₃₁×#1878	1	2	yes	—
508	C₂	C₃₁×#1880	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂×C₄×Q₈×C₃₁	C₄	semidirect
19. C₂×C₃₁×#103	C₄	semidirect
146. #10789	C₃₁	direct (N × H)
163. Q₈×C₃₁	#95	semidirect
190. C₂×C₄×C₃₁	#103	semidirect
254. C₂×C₄×Q₈	C₄×C₃₁	semidirect
262. C₂×#103	C₄×C₃₁	semidirect
325. C₃₁	#10789	direct (N × H)
417. Q₈	C₃₁×#95	semidirect
444. C₂×C₄	C₃₁×#103	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄×Q₈×C₃₁	2
C₂²×C₃₁×#103	2
C₂×C₃₁×#340	2
C₂×C₃₁×#336	2
C₃₁×C₄⋊#95	2
#10789	31

Subgroups — C31×#10789 #21467548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10789 #21467548