Subgroups — C₃₁×#8563 #21465322

Order 7936.

708 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×D₄⋊C₈	C₂	1	2	yes	maximal
5	C₃₁×#1279	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₃₁	—	2	1	no	—
10	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
11	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
12	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
15	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
19	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
20	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
21	C₄³×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#336	—	2	1	no	—
23	C₃₁×D₄⋊C₈	—	2	4	no	—
24	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₃₁	—	2	1	no	—
26	C₂³×C₄×C₃₁	—	2	1	no	—
27	C₂³×C₄×C₃₁	—	2	1	no	—
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	—	2	2	no	—
30	C₄×D₄×C₃₁	—	2	2	no	—
31	C₄×D₄×C₃₁	—	2	2	no	—
32	C₄×D₄×C₃₁	—	2	2	no	—
33	C₄×D₄×C₃₁	—	2	1	no	—
34	C₄×D₄×C₃₁	—	2	1	no	—
35	C₄×D₄×C₃₁	C₈	1	4	yes	complemented
36	C₄×D₄×C₃₁	D₄	1	2	yes	—
37	C₄×D₄×C₃₁	Q₈	1	2	yes	—
38	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
42	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
44	C₂×C₄²×C₃₁	—	2	1	no	—
45	C₂×C₄²×C₃₁	—	2	1	no	—
46	C₂×C₄²×C₃₁	—	2	1	no	—
47	C₂×C₄²×C₃₁	—	4	2	no	—
48	C₂×C₄²×C₃₁	—	4	2	no	—
49	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×C₄⋊C₈	—	2	4	no	—
57	C₃₁×C₄⋊C₈	—	2	2	no	—
58	C₃₁×C₄⋊C₈	—	2	2	no	—
59	C₃₁×C₄⋊C₈	—	4	2	no	—
60	C₃₁×C₄⋊C₈	—	4	2	no	—
61	C₄×C₈×C₃₁	—	2	2	no	—
62	C₄×C₈×C₃₁	—	2	2	no	—
63	C₃₁×#95	—	2	1	no	—
64	C₃₁×#95	—	2	1	no	—
65	C₃₁×#95	—	2	1	no	—
66	C₂⁴×C₃₁	—	2	1	no	—
67	C₂×D₄×C₃₁	—	2	1	no	—
68	C₂×D₄×C₃₁	—	2	1	no	—
69	C₂×D₄×C₃₁	—	2	1	no	—
70	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
71	C₂×D₄×C₃₁	C₂×C₈	1	2	yes	—
72	C₂×D₄×C₃₁	M₁₆	1	2	yes	—
73	C₂²×C₄×C₃₁	—	2	4	no	—
74	C₂²×C₄×C₃₁	—	2	1	no	—
75	C₂²×C₄×C₃₁	—	2	1	no	—
76	C₂²×C₄×C₃₁	—	2	1	no	—
77	C₂²×C₄×C₃₁	—	2	4	no	—
78	C₂²×C₄×C₃₁	—	2	2	no	—
79	C₂²×C₄×C₃₁	—	2	2	no	—
80	C₂²×C₄×C₃₁	—	4	2	no	—
81	C₂²×C₄×C₃₁	—	4	2	no	—
82	C₂²×C₄×C₃₁	—	4	2	no	—
83	C₂²×C₄×C₃₁	—	4	2	no	—
84	C₂²×C₄×C₃₁	—	4	1	no	—
85	C₂²×C₄×C₃₁	—	4	1	no	—
86	C₂²×C₄×C₃₁	—	4	1	no	—
87	C₂²×C₄×C₃₁	—	4	1	no	—
88	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	—	2	4	no	—
96	C₂×C₈×C₃₁	—	4	2	no	—
97	C₂×C₈×C₃₁	—	4	2	no	—
98	C₂×C₈×C₃₁	—	4	2	no	—
99	C₂×C₈×C₃₁	—	4	2	no	—
100	C₂×C₈×C₃₁	—	4	2	no	—
101	C₄⋊C₄×C₃₁	—	2	1	no	—
102	C₄⋊C₄×C₃₁	—	2	1	no	—
103	C₄⋊C₄×C₃₁	—	2	1	no	—
104	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
105	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
106	C₄⋊C₄×C₃₁	M₁₆	1	2	yes	—
107	C₂²⋊C₄×C₃₁	—	2	4	no	—
108	C₂²⋊C₄×C₃₁	—	2	4	no	—
109	C₂²⋊C₄×C₃₁	—	2	2	no	—
110	C₂²⋊C₄×C₃₁	—	2	2	no	—
111	C₄²×C₃₁	—	2	2	no	—
112	C₄²×C₃₁	—	2	1	no	—
113	C₄²×C₃₁	—	2	1	no	—
114	C₄²×C₃₁	—	2	1	no	—
115	C₄²×C₃₁	—	2	1	no	—
116	C₄²×C₃₁	—	4	1	no	—
117	C₄²×C₃₁	—	4	2	no	—
118	C₄²×C₃₁	—	4	2	no	—
119	C₄²×C₃₁	—	4	1	no	—
120	C₄²×C₃₁	—	4	1	no	—
121	C₄²×C₃₁	C₂×C₈	1	2	yes	—
122	C₄²×C₃₁	M₁₆	1	2	yes	—
123	C₄²×C₃₁	D₈	1	2	yes	—
124	C₄²×C₃₁	SD₁₆	1	2	yes	—
125	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
127	C₄²×C₃₁	#41	1	1	yes	characteristic
128	C₄²×C₃₁	#41	1	1	yes	characteristic
129	#8563	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂³×C₃₁	—	2	2	no	—
131	C₂³×C₃₁	—	2	2	no	—
132	C₂³×C₃₁	—	2	2	no	—
133	C₂³×C₃₁	—	4	1	no	—
134	C₂³×C₃₁	—	4	1	no	—
135	C₂³×C₃₁	—	4	1	no	—
136	C₂³×C₃₁	—	4	1	no	—
137	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	D₄×C₃₁	—	2	2	no	—
139	D₄×C₃₁	—	2	2	no	—
140	D₄×C₃₁	—	2	2	no	—
141	D₄×C₃₁	C₄⋊C₈	1	4	yes	complemented
142	C₂×C₄×C₃₁	—	2	1	no	—
143	C₂×C₄×C₃₁	—	2	1	no	—
144	C₂×C₄×C₃₁	—	2	1	no	—
145	C₂×C₄×C₃₁	—	2	1	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	2	2	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	4	8	no	—
155	C₂×C₄×C₃₁	—	4	1	no	—
156	C₂×C₄×C₃₁	—	4	1	no	—
157	C₂×C₄×C₃₁	—	4	1	no	—
158	C₂×C₄×C₃₁	—	4	1	no	—
159	C₂×C₄×C₃₁	—	4	1	no	—
160	C₂×C₄×C₃₁	—	4	1	no	—
161	C₂×C₄×C₃₁	—	4	8	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
167	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
168	C₂×C₄×C₃₁	#102	1	2	yes	—
169	C₂×C₄×C₃₁	#102	1	2	yes	—
170	C₂×C₄×C₃₁	#104	1	2	yes	—
171	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
173	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
174	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
176	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
182	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
183	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
185	C₈×C₃₁	—	4	4	no	—
186	C₈×C₃₁	—	8	2	no	—
187	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic
188	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
189	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
190	C₂×D₄⋊C₈	C₂×C₃₁	1	2	yes	—
191	#1279	C₂×C₃₁	1	2	yes	complemented
192	C₂²×C₃₁	—	2	4	no	—
193	C₂²×C₃₁	—	4	2	no	—
194	C₂²×C₃₁	—	4	2	no	—
195	C₂²×C₃₁	—	4	2	no	—
196	C₂²×C₃₁	—	4	2	no	—
197	C₂²×C₃₁	—	4	2	no	—
198	C₂²×C₃₁	—	4	2	no	—
199	C₂²×C₃₁	#380	1	1	yes	characteristic
200	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
201	C₂²×C₃₁	#407	1	1	yes	characteristic
202	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
203	C₂²×C₃₁	#417	1	1	yes	characteristic
204	C₂²×C₃₁	C₂×#104	1	1	yes	characteristic
205	C₂²×C₃₁	#421	1	1	yes	characteristic
206	C₄×C₃₁	—	2	1	no	—
207	C₄×C₃₁	—	2	1	no	—
208	C₄×C₃₁	—	2	2	no	—
209	C₄×C₃₁	—	2	2	no	—
210	C₄×C₃₁	—	4	2	no	—
211	C₄×C₃₁	—	4	2	no	—
212	C₄×C₃₁	—	4	2	no	—
213	C₄×C₃₁	—	4	1	no	—
214	C₄×C₃₁	D₄⋊C₈	1	4	yes	complemented
215	C₄×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
216	C₄×C₃₁	#423	1	1	yes	characteristic
217	C₄×C₃₁	#431	1	1	yes	characteristic
218	C₄×C₃₁	#432	1	1	yes	characteristic
219	C₄×C₃₁	#459	1	1	yes	characteristic
220	C₄×C₃₁	#461	1	1	yes	characteristic
221	C₄×C₃₁	#474	1	1	yes	characteristic
222	C₄×C₃₁	#476	1	1	yes	characteristic
223	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
224	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
225	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
226	C₂²×C₄²	—	2	1	no	—
227	C₂×C₄⋊C₈	—	2	1	no	—
228	C₂×C₄⋊C₈	—	2	1	no	—
229	C₂×C₄⋊C₈	—	2	1	no	—
230	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
231	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
232	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
233	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
234	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
235	C₄×C₂²⋊C₄	—	2	1	no	—
236	C₄×C₂²⋊C₄	—	2	1	no	—
237	C₄×C₂²⋊C₄	—	2	1	no	—
238	C₄³	C₂²×C₃₁	1	1	yes	characteristic
239	#336	—	2	1	no	—
240	D₄⋊C₈	—	2	4	no	—
241	C₂×C₃₁	—	4	4	no	—
242	C₂×C₃₁	C₂×D₄⋊C₈	1	1	yes	characteristic
243	C₂×C₃₁	#1477	1	1	yes	characteristic
244	C₂×C₃₁	#1486	1	1	yes	characteristic
245	C₂×C₃₁	#1489	1	1	yes	characteristic
246	C₂×C₃₁	#1797	1	1	yes	characteristic
247	C₂×C₃₁	#1802	1	1	yes	characteristic
248	C₂×C₃₁	#1806	1	1	yes	characteristic
249	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
250	C₂³×C₄	—	2	1	no	—
251	C₂³×C₄	—	2	1	no	—
252	C₂³×C₄	—	2	1	no	—
253	C₂²×C₈	—	2	2	no	—
254	C₂²×C₈	—	2	2	no	—
255	C₄×D₄	—	2	2	no	—
256	C₄×D₄	—	2	2	no	—
257	C₄×D₄	—	2	2	no	—
258	C₄×D₄	—	2	1	no	—
259	C₄×D₄	—	2	1	no	—
260	C₄×D₄	C₈×C₃₁	1	4	yes	complemented
261	C₄×D₄	D₄×C₃₁	1	2	yes	—
262	C₄×D₄	Q₈×C₃₁	1	2	yes	—
263	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
264	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
266	C₂×C₂²⋊C₄	—	2	1	no	—
267	C₂×C₂²⋊C₄	—	2	1	no	—
268	C₂×C₂²⋊C₄	—	2	1	no	—
269	C₂×C₄²	—	2	1	no	—
270	C₂×C₄²	—	2	1	no	—
271	C₂×C₄²	—	2	1	no	—
272	C₂×C₄²	—	4	2	no	—
273	C₂×C₄²	—	4	2	no	—
274	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
275	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
276	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
277	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
278	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
280	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
281	C₄⋊C₈	—	2	4	no	—
282	C₄⋊C₈	—	2	2	no	—
283	C₄⋊C₈	—	2	2	no	—
284	C₄⋊C₈	—	4	2	no	—
285	C₄⋊C₈	—	4	2	no	—
286	C₄×C₈	—	2	2	no	—
287	C₄×C₈	—	2	2	no	—
288	#95	—	2	1	no	—
289	#95	—	2	1	no	—
290	#95	—	2	1	no	—
291	C₃₁	#8563	1	1	yes	characteristic, direct product factor, Sylow
292	C₂⁴	—	2	1	no	—
293	C₂×D₄	—	2	1	no	—
294	C₂×D₄	—	2	1	no	—
295	C₂×D₄	—	2	1	no	—
296	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
297	C₂×D₄	C₂×C₈×C₃₁	1	2	yes	—
298	C₂×D₄	M₁₆×C₃₁	1	2	yes	—
299	C₂²×C₄	—	2	4	no	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	4	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	4	2	no	—
307	C₂²×C₄	—	4	2	no	—
308	C₂²×C₄	—	4	2	no	—
309	C₂²×C₄	—	4	2	no	—
310	C₂²×C₄	—	4	1	no	—
311	C₂²×C₄	—	4	1	no	—
312	C₂²×C₄	—	4	1	no	—
313	C₂²×C₄	—	4	1	no	—
314	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
315	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
316	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
317	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
318	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₂×C₈	—	2	4	no	—
322	C₂×C₈	—	4	2	no	—
323	C₂×C₈	—	4	2	no	—
324	C₂×C₈	—	4	2	no	—
325	C₂×C₈	—	4	2	no	—
326	C₂×C₈	—	4	2	no	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	2	1	no	—
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
331	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
332	C₄⋊C₄	M₁₆×C₃₁	1	2	yes	—
333	C₂²⋊C₄	—	2	4	no	—
334	C₂²⋊C₄	—	2	4	no	—
335	C₂²⋊C₄	—	2	2	no	—
336	C₂²⋊C₄	—	2	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	2	1	no	—
339	C₄²	—	2	1	no	—
340	C₄²	—	2	1	no	—
341	C₄²	—	2	1	no	—
342	C₄²	—	4	1	no	—
343	C₄²	—	4	2	no	—
344	C₄²	—	4	2	no	—
345	C₄²	—	4	1	no	—
346	C₄²	—	4	1	no	—
347	C₄²	C₂×C₈×C₃₁	1	2	yes	—
348	C₄²	M₁₆×C₃₁	1	2	yes	—
349	C₄²	D₈×C₃₁	1	2	yes	—
350	C₄²	SD₁₆×C₃₁	1	2	yes	—
351	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
353	C₄²	#41×C₃₁	1	1	yes	characteristic
354	C₄²	#41×C₃₁	1	1	yes	characteristic
355	C₂³	—	2	2	no	—
356	C₂³	—	2	2	no	—
357	C₂³	—	2	2	no	—
358	C₂³	—	4	1	no	—
359	C₂³	—	4	1	no	—
360	C₂³	—	4	1	no	—
361	C₂³	—	4	1	no	—
362	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
363	D₄	—	2	2	no	—
364	D₄	—	2	2	no	—
365	D₄	—	2	2	no	—
366	D₄	C₃₁×C₄⋊C₈	1	4	yes	complemented
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	8	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	8	no	—
391	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
392	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
393	C₂×C₄	C₃₁×#102	1	2	yes	—
394	C₂×C₄	C₃₁×#102	1	2	yes	—
395	C₂×C₄	C₃₁×#104	1	2	yes	—
396	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
397	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
398	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
399	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
400	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
401	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
402	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
403	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
404	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
405	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
406	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
407	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
408	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
409	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
410	C₈	—	4	4	no	—
411	C₈	—	8	2	no	—
412	C₂²	—	2	4	no	—
413	C₂²	—	4	2	no	—
414	C₂²	—	4	2	no	—
415	C₂²	—	4	2	no	—
416	C₂²	—	4	2	no	—
417	C₂²	—	4	2	no	—
418	C₂²	—	4	2	no	—
419	C₂²	C₃₁×#380	1	1	yes	characteristic
420	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
421	C₂²	C₃₁×#407	1	1	yes	characteristic
422	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
423	C₂²	C₃₁×#417	1	1	yes	characteristic
424	C₂²	C₂×C₃₁×#104	1	1	yes	characteristic
425	C₂²	C₃₁×#421	1	1	yes	characteristic
426	C₄	—	2	1	no	—
427	C₄	—	2	1	no	—
428	C₄	—	2	2	no	—
429	C₄	—	2	2	no	—
430	C₄	—	4	2	no	—
431	C₄	—	4	2	no	—
432	C₄	—	4	2	no	—
433	C₄	—	4	1	no	—
434	C₄	C₃₁×D₄⋊C₈	1	4	yes	complemented
435	C₄	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
436	C₄	C₃₁×#423	1	1	yes	characteristic
437	C₄	C₃₁×#431	1	1	yes	characteristic
438	C₄	C₃₁×#432	1	1	yes	characteristic
439	C₄	C₃₁×#459	1	1	yes	characteristic
440	C₄	C₃₁×#461	1	1	yes	characteristic
441	C₄	C₃₁×#474	1	1	yes	characteristic
442	C₄	C₃₁×#476	1	1	yes	characteristic
443	C₂	—	4	4	no	—
444	C₂	C₂×C₃₁×D₄⋊C₈	1	1	yes	characteristic
445	C₂	C₃₁×#1477	1	1	yes	characteristic
446	C₂	C₃₁×#1486	1	1	yes	characteristic
447	C₂	C₃₁×#1489	1	1	yes	characteristic
448	C₂	C₃₁×#1797	1	1	yes	characteristic
449	C₂	C₃₁×#1802	1	1	yes	characteristic
450	C₂	C₃₁×#1806	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂³×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₃₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1842	C₂	semidirect
3. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
5. C₃₁×#1279	C₂	semidirect
35. C₄×D₄×C₃₁	C₈	semidirect
129. #8563	C₃₁	direct (N × H)
141. D₄×C₃₁	C₄⋊C₈	semidirect
188. #1842	C₂×C₃₁	semidirect
189. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
191. #1279	C₂×C₃₁	semidirect
214. C₄×C₃₁	D₄⋊C₈	semidirect
260. C₄×D₄	C₈×C₃₁	semidirect
291. C₃₁	#8563	direct (N × H)
366. D₄	C₃₁×C₄⋊C₈	semidirect
434. C₄	C₃₁×D₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₄×C₃₁	2
C₃₁×#1842	2
C₄×C₃₁×C₄⋊C₈	2
C₂×C₃₁×D₄⋊C₈	2
C₃₁×#1279	2
#8563	31

Subgroups — C31×#8563 #21465322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8563 #21465322