Subgroups — C₃₁×#26264 #21483023

Order 7936.

1038 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#442	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2445	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2054	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2041	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1972	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1972	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1972	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1941	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1879	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1771	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
14	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#127	C₂²	1	2	yes	—
16	C₂×C₃₁×#123	—	2	2	no	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#493	—	2	4	no	—
19	C₃₁×#442	—	2	4	no	—
20	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₃₁×#102	C₂²	1	2	yes	—
27	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#394	—	2	2	no	—
31	C₃₁×#394	—	2	2	no	—
32	C₃₁×#390	C₂²	1	2	yes	—
33	C₃₁×#390	C₂²	1	2	yes	—
34	C₃₁×#390	C₂²	1	2	yes	—
35	C₃₁×#389	C₂²	1	2	yes	—
36	C₃₁×#386	—	2	2	no	—
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	1	yes	characteristic
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₂²×D₄×C₃₁	—	2	2	no	—
42	C₂²×D₄×C₃₁	—	2	2	no	—
43	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
44	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
45	C₂³×C₄×C₃₁	—	2	2	no	—
46	C₂×D₈×C₃₁	—	2	4	no	—
47	C₂×D₈×C₃₁	—	2	2	no	—
48	C₂×D₈×C₃₁	—	2	2	no	—
49	C₂×D₈×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	—	2	4	no	—
51	C₂×D₈×C₃₁	—	2	2	no	—
52	C₂×D₈×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	4	2	no	—
54	C₂×D₈×C₃₁	D₄	1	4	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×#127	—	2	4	no	—
60	C₃₁×#127	—	4	2	no	—
61	C₃₁×#123	—	4	8	no	—
62	C₄×D₄×C₃₁	—	2	4	no	—
63	C₄×D₄×C₃₁	—	4	2	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₃₁×#106	—	2	2	no	—
79	C₃₁×#106	—	2	2	no	—
80	C₃₁×C₄⋊C₈	—	2	2	no	—
81	C₃₁×#102	—	2	4	no	—
82	C₃₁×#102	—	4	2	no	—
83	C₃₁×#102	—	4	2	no	—
84	C₃₁×#102	—	4	2	no	—
85	C₃₁×#97	—	2	2	no	—
86	C₃₁×#97	—	2	2	no	—
87	C₄×C₈×C₃₁	—	2	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₃₁×#95	D₄	1	2	yes	—
90	C₂⁴×C₃₁	—	2	2	no	—
91	C₂⁴×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	4	4	no	—
104	C₂×D₄×C₃₁	—	4	4	no	—
105	C₂×D₄×C₃₁	—	4	4	no	—
106	C₂×D₄×C₃₁	—	4	4	no	—
107	C₂×D₄×C₃₁	—	8	2	no	—
108	C₂×D₄×C₃₁	—	8	2	no	—
109	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	4	4	no	—
113	C₂²×C₄×C₃₁	—	4	4	no	—
114	C₂²×C₄×C₃₁	—	8	2	no	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
120	D₈×C₃₁	—	2	8	no	—
121	D₈×C₃₁	—	4	8	no	—
122	D₈×C₃₁	—	4	4	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	2	1	no	—
130	C₂×C₈×C₃₁	—	2	1	no	—
131	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
132	C₂×C₈×C₃₁	#41	1	2	yes	—
133	C₂×C₈×C₃₁	#41	1	2	yes	—
134	C₂×C₈×C₃₁	#41	1	2	yes	—
135	C₄⋊C₄×C₃₁	—	2	4	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	4	4	no	—
141	C₂²⋊C₄×C₃₁	—	4	4	no	—
142	C₂²⋊C₄×C₃₁	—	4	4	no	—
143	C₂²⋊C₄×C₃₁	—	4	4	no	—
144	C₂²⋊C₄×C₃₁	—	4	4	no	—
145	C₂²⋊C₄×C₃₁	—	8	2	no	—
146	C₂²⋊C₄×C₃₁	—	8	2	no	—
147	C₂²⋊C₄×C₃₁	—	8	2	no	—
148	C₄²×C₃₁	—	2	1	no	—
149	C₄²×C₃₁	—	2	1	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	C₄²×C₃₁	—	2	1	no	—
152	C₄²×C₃₁	—	2	1	no	—
153	C₄²×C₃₁	—	2	1	no	—
154	#26264	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂³×C₃₁	—	2	4	no	—
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	—	4	2	no	—
158	C₂³×C₃₁	—	4	2	no	—
159	C₂³×C₃₁	—	4	2	no	—
160	C₂³×C₃₁	—	4	2	no	—
161	C₂³×C₃₁	—	4	2	no	—
162	C₂³×C₃₁	—	4	2	no	—
163	C₂³×C₃₁	—	4	2	no	—
164	C₂³×C₃₁	—	4	2	no	—
165	C₂³×C₃₁	—	4	2	no	—
166	C₂³×C₃₁	—	4	2	no	—
167	C₂³×C₃₁	—	4	2	no	—
168	C₂³×C₃₁	—	4	2	no	—
169	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
170	D₄×C₃₁	—	2	4	no	—
171	D₄×C₃₁	—	2	4	no	—
172	D₄×C₃₁	—	4	2	no	—
173	D₄×C₃₁	—	4	2	no	—
174	D₄×C₃₁	—	4	2	no	—
175	D₄×C₃₁	—	4	2	no	—
176	D₄×C₃₁	—	4	2	no	—
177	D₄×C₃₁	—	4	2	no	—
178	D₄×C₃₁	—	8	8	no	—
179	D₄×C₃₁	—	8	8	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	8	8	no	—
196	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
197	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
201	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
202	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
203	C₈×C₃₁	—	2	2	no	—
204	C₈×C₃₁	—	4	1	no	—
205	C₈×C₃₁	—	4	1	no	—
206	C₈×C₃₁	#123	1	4	yes	complemented
207	C₂×#493	C₂×C₃₁	1	1	yes	characteristic
208	C₂×#442	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#2445	C₂×C₃₁	1	2	yes	complemented
210	#2054	C₂×C₃₁	1	2	yes	complemented
211	#2041	C₂×C₃₁	1	2	yes	complemented
212	#1972	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#1972	C₂×C₃₁	1	1	yes	characteristic, complemented
214	#1972	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#1941	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#1879	C₂×C₃₁	1	2	yes	complemented
217	#1771	C₂×C₃₁	1	1	yes	characteristic, complemented
218	C₂²×C₃₁	—	4	4	no	—
219	C₂²×C₃₁	—	4	4	no	—
220	C₂²×C₃₁	—	4	4	no	—
221	C₂²×C₃₁	—	8	2	no	—
222	C₂²×C₃₁	—	8	2	no	—
223	C₂²×C₃₁	—	8	2	no	—
224	C₂²×C₃₁	—	8	2	no	—
225	C₂²×C₃₁	—	8	2	no	—
226	C₂²×C₃₁	—	8	2	no	—
227	C₂²×C₃₁	—	8	2	no	—
228	C₂²×C₃₁	—	8	2	no	—
229	C₂²×C₃₁	—	8	2	no	—
230	C₂²×C₃₁	—	8	2	no	—
231	C₂²×C₃₁	—	8	2	no	—
232	C₂²×C₃₁	#532	1	1	yes	characteristic
233	C₂²×C₃₁	D₄²	1	1	yes	characteristic
234	C₂²×C₃₁	#550	1	1	yes	characteristic
235	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
236	C₂²×C₃₁	#576	1	2	yes	—
237	C₄×C₃₁	—	2	4	no	—
238	C₄×C₃₁	—	4	1	no	—
239	C₄×C₃₁	—	4	1	no	—
240	C₄×C₃₁	—	4	2	no	—
241	C₄×C₃₁	—	8	2	no	—
242	C₄×C₃₁	—	8	2	no	—
243	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
244	C₄×C₃₁	#540	1	1	yes	characteristic
245	C₄×C₃₁	#546	1	1	yes	characteristic
246	C₄×C₃₁	#559	1	1	yes	characteristic
247	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
248	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
249	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×#127	C₂²×C₃₁	1	2	yes	—
251	C₂×#123	—	2	2	no	—
252	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
253	#493	—	2	4	no	—
254	#442	—	2	4	no	—
255	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×#102	C₂²×C₃₁	1	2	yes	—
259	C₂×#102	C₂²×C₃₁	1	2	yes	—
260	C₂×#102	C₂²×C₃₁	1	2	yes	—
261	C₂×#102	C₂²×C₃₁	1	2	yes	—
262	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
265	#394	—	2	2	no	—
266	#394	—	2	2	no	—
267	#390	C₂²×C₃₁	1	2	yes	—
268	#390	C₂²×C₃₁	1	2	yes	—
269	#390	C₂²×C₃₁	1	2	yes	—
270	#389	C₂²×C₃₁	1	2	yes	—
271	#386	—	2	2	no	—
272	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
273	#340	C₂²×C₃₁	1	2	yes	—
274	#336	C₂²×C₃₁	1	1	yes	characteristic
275	#336	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×C₃₁	—	8	4	no	—
277	C₂×C₃₁	—	16	2	no	—
278	C₂×C₃₁	#2445	1	1	yes	characteristic
279	C₂×C₃₁	#3176	1	1	yes	characteristic
280	C₂×C₃₁	#3184	1	1	yes	characteristic
281	C₂×C₃₁	#3264	1	2	yes	—
282	C₂×C₃₁	#3399	1	2	yes	—
283	C₂²×D₄	—	2	2	no	—
284	C₂²×D₄	—	2	2	no	—
285	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
286	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
287	C₂³×C₄	—	2	2	no	—
288	C₂×D₈	—	2	4	no	—
289	C₂×D₈	—	2	2	no	—
290	C₂×D₈	—	2	2	no	—
291	C₂×D₈	—	2	2	no	—
292	C₂×D₈	—	2	4	no	—
293	C₂×D₈	—	2	2	no	—
294	C₂×D₈	—	2	2	no	—
295	C₂×D₈	—	4	2	no	—
296	C₂×D₈	D₄×C₃₁	1	4	yes	—
297	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
299	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
300	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
301	#127	—	2	4	no	—
302	#127	—	4	2	no	—
303	#123	—	4	8	no	—
304	C₄×D₄	—	2	4	no	—
305	C₄×D₄	—	4	2	no	—
306	C₂×C₄⋊C₄	—	2	2	no	—
307	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
308	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
309	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
310	C₂×C₂²⋊C₄	—	2	2	no	—
311	C₂×C₂²⋊C₄	—	2	2	no	—
312	C₂×C₂²⋊C₄	—	2	2	no	—
313	C₂×C₂²⋊C₄	—	2	2	no	—
314	C₂×C₂²⋊C₄	—	2	2	no	—
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₂²⋊C₄	—	2	2	no	—
317	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
318	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
319	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
320	#106	—	2	2	no	—
321	#106	—	2	2	no	—
322	C₄⋊C₈	—	2	2	no	—
323	#102	—	2	4	no	—
324	#102	—	4	2	no	—
325	#102	—	4	2	no	—
326	#102	—	4	2	no	—
327	#97	—	2	2	no	—
328	#97	—	2	2	no	—
329	C₄×C₈	—	2	2	no	—
330	#95	—	2	2	no	—
331	#95	D₄×C₃₁	1	2	yes	—
332	C₃₁	#26264	1	1	yes	characteristic, direct product factor, Sylow
333	C₂⁴	—	2	2	no	—
334	C₂⁴	—	2	2	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	4	4	no	—
347	C₂×D₄	—	4	4	no	—
348	C₂×D₄	—	4	4	no	—
349	C₂×D₄	—	4	4	no	—
350	C₂×D₄	—	8	2	no	—
351	C₂×D₄	—	8	2	no	—
352	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	4	4	no	—
356	C₂²×C₄	—	4	4	no	—
357	C₂²×C₄	—	8	2	no	—
358	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
360	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
361	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
362	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
363	D₈	—	2	8	no	—
364	D₈	—	4	8	no	—
365	D₈	—	4	4	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
375	C₂×C₈	#41×C₃₁	1	2	yes	—
376	C₂×C₈	#41×C₃₁	1	2	yes	—
377	C₂×C₈	#41×C₃₁	1	2	yes	—
378	C₄⋊C₄	—	2	4	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	4	4	no	—
384	C₂²⋊C₄	—	4	4	no	—
385	C₂²⋊C₄	—	4	4	no	—
386	C₂²⋊C₄	—	4	4	no	—
387	C₂²⋊C₄	—	4	4	no	—
388	C₂²⋊C₄	—	8	2	no	—
389	C₂²⋊C₄	—	8	2	no	—
390	C₂²⋊C₄	—	8	2	no	—
391	C₄²	—	2	1	no	—
392	C₄²	—	2	1	no	—
393	C₄²	—	2	1	no	—
394	C₄²	—	2	1	no	—
395	C₄²	—	2	1	no	—
396	C₄²	—	2	1	no	—
397	C₂³	—	2	4	no	—
398	C₂³	—	4	2	no	—
399	C₂³	—	4	2	no	—
400	C₂³	—	4	2	no	—
401	C₂³	—	4	2	no	—
402	C₂³	—	4	2	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	4	2	no	—
405	C₂³	—	4	2	no	—
406	C₂³	—	4	2	no	—
407	C₂³	—	4	2	no	—
408	C₂³	—	4	2	no	—
409	C₂³	—	4	2	no	—
410	C₂³	—	4	2	no	—
411	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
412	D₄	—	2	4	no	—
413	D₄	—	2	4	no	—
414	D₄	—	4	2	no	—
415	D₄	—	4	2	no	—
416	D₄	—	4	2	no	—
417	D₄	—	4	2	no	—
418	D₄	—	4	2	no	—
419	D₄	—	4	2	no	—
420	D₄	—	8	8	no	—
421	D₄	—	8	8	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	8	8	no	—
438	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
439	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
440	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
441	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
442	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
443	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
444	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
445	C₈	—	2	2	no	—
446	C₈	—	4	1	no	—
447	C₈	—	4	1	no	—
448	C₈	C₃₁×#123	1	4	yes	complemented
449	C₂²	—	4	4	no	—
450	C₂²	—	4	4	no	—
451	C₂²	—	4	4	no	—
452	C₂²	—	8	2	no	—
453	C₂²	—	8	2	no	—
454	C₂²	—	8	2	no	—
455	C₂²	—	8	2	no	—
456	C₂²	—	8	2	no	—
457	C₂²	—	8	2	no	—
458	C₂²	—	8	2	no	—
459	C₂²	—	8	2	no	—
460	C₂²	—	8	2	no	—
461	C₂²	—	8	2	no	—
462	C₂²	—	8	2	no	—
463	C₂²	C₃₁×#532	1	1	yes	characteristic
464	C₂²	D₄²×C₃₁	1	1	yes	characteristic
465	C₂²	C₃₁×#550	1	1	yes	characteristic
466	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
467	C₂²	C₃₁×#576	1	2	yes	—
468	C₄	—	2	4	no	—
469	C₄	—	4	1	no	—
470	C₄	—	4	1	no	—
471	C₄	—	4	2	no	—
472	C₄	—	8	2	no	—
473	C₄	—	8	2	no	—
474	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
475	C₄	C₃₁×#540	1	1	yes	characteristic
476	C₄	C₃₁×#546	1	1	yes	characteristic
477	C₄	C₃₁×#559	1	1	yes	characteristic
478	C₂	—	8	4	no	—
479	C₂	—	16	2	no	—
480	C₂	C₃₁×#2445	1	1	yes	characteristic
481	C₂	C₃₁×#3176	1	1	yes	characteristic
482	C₂	C₃₁×#3184	1	1	yes	characteristic
483	C₂	C₃₁×#3264	1	2	yes	—
484	C₂	C₃₁×#3399	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#493	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#442	C₂	semidirect
3. C₃₁×#2445	C₂	semidirect
4. C₃₁×#2054	C₂	semidirect
5. C₃₁×#2041	C₂	semidirect
6. C₃₁×#1972	C₂	semidirect
7. C₃₁×#1972	C₂	semidirect
8. C₃₁×#1972	C₂	semidirect
9. C₃₁×#1941	C₂	semidirect
10. C₃₁×#1879	C₂	semidirect
11. C₃₁×#1771	C₂	semidirect
154. #26264	C₃₁	direct (N × H)
206. C₈×C₃₁	#123	semidirect
208. C₂×#442	C₂×C₃₁	semidirect
209. #2445	C₂×C₃₁	semidirect
210. #2054	C₂×C₃₁	semidirect
211. #2041	C₂×C₃₁	semidirect
212. #1972	C₂×C₃₁	semidirect
213. #1972	C₂×C₃₁	semidirect
214. #1972	C₂×C₃₁	semidirect
215. #1941	C₂×C₃₁	semidirect
216. #1879	C₂×C₃₁	semidirect
217. #1771	C₂×C₃₁	semidirect
332. C₃₁	#26264	direct (N × H)
448. C₈	C₃₁×#123	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#493	2
C₂×C₃₁×#442	2
C₃₁×#2445	2
C₃₁×#2054	2
C₃₁×#2041	2
C₃₁×#1972	2
C₃₁×#1941	2
C₃₁×#1879	2
C₃₁×#1771	2
#26264	31

Subgroups — C31×#26264 #21483023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26264 #21483023