Subgroups — C₃₁×#33251 #21490010

Order 7936.

1036 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#501	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#496	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#490	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#489	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2861	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2858	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1973	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1972	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1970	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1968	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#124	—	2	2	no	—
18	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
19	C₃₁×#501	—	2	4	no	—
20	C₃₁×#496	—	2	4	no	—
21	C₃₁×#490	—	2	4	no	—
22	C₃₁×#489	—	2	4	no	—
23	C₂×C₃₁×#107	C₂²	1	2	yes	—
24	C₂×C₃₁×#106	C₂²	1	2	yes	—
25	C₂×C₃₁×#103	C₂²	1	2	yes	—
26	C₂×C₃₁×#102	C₂²	1	2	yes	—
27	C₂×C₃₁×#102	C₂²	1	2	yes	—
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×#97	C₂²	1	2	yes	—
30	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
33	C₃₁×#391	C₂²	1	1	yes	characteristic
34	C₃₁×#391	C₂²	1	1	yes	characteristic
35	C₃₁×#390	—	2	2	no	—
36	C₃₁×#390	C₂²	1	2	yes	—
37	C₃₁×#390	C₂²	1	2	yes	—
38	C₃₁×#390	C₂²	1	1	yes	characteristic
39	C₃₁×#390	C₂²	1	1	yes	characteristic
40	C₃₁×#388	—	2	2	no	—
41	C₃₁×#388	—	2	2	no	—
42	C₄³×C₃₁	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₂²×D₄×C₃₁	—	2	1	no	—
47	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
49	C₂×SD₁₆×C₃₁	—	2	4	no	—
50	C₂×SD₁₆×C₃₁	—	2	4	no	—
51	C₂×SD₁₆×C₃₁	—	2	4	no	—
52	C₂×D₈×C₃₁	—	2	4	no	—
53	C₂×D₈×C₃₁	—	2	4	no	—
54	C₂×D₈×C₃₁	—	2	4	no	—
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
57	C₃₁×C₄⋊Q₈	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	—	2	2	no	—
59	C₃₁×#127	—	2	2	no	—
60	C₃₁×#127	—	2	2	no	—
61	C₃₁×#125	—	2	2	no	—
62	C₃₁×#125	—	2	2	no	—
63	C₃₁×#124	—	2	2	no	—
64	C₃₁×#124	—	2	2	no	—
65	C₃₁×#124	—	4	8	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₄²×C₃₁	D₄	1	2	yes	—
79	C₂×C₄²×C₃₁	D₄	1	2	yes	—
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×#107	—	2	4	no	—
84	C₃₁×#106	—	2	4	no	—
85	C₃₁×#103	—	2	4	no	—
86	C₃₁×#102	—	2	4	no	—
87	C₃₁×#102	—	2	4	no	—
88	C₃₁×#102	—	2	4	no	—
89	C₃₁×#97	—	2	4	no	—
90	C₃₁×#97	D₄	1	4	yes	—
91	C₃₁×#97	D₄	1	4	yes	—
92	C₄×C₈×C₃₁	—	2	4	no	—
93	C₃₁×#95	—	2	2	no	—
94	C₃₁×#95	—	2	2	no	—
95	C₂⁴×C₃₁	—	2	1	no	—
96	C₂⁴×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	—	2	1	no	—
98	C₂×Q₈×C₃₁	—	2	1	no	—
99	C₂×Q₈×C₃₁	—	2	2	no	—
100	C₂×Q₈×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	—	4	4	no	—
102	C₂×Q₈×C₃₁	—	4	2	no	—
103	C₂×D₄×C₃₁	—	2	1	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	1	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	4	4	no	—
114	C₂×D₄×C₃₁	—	4	4	no	—
115	C₂×D₄×C₃₁	—	4	2	no	—
116	C₂×D₄×C₃₁	—	4	2	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
126	SD₁₆×C₃₁	—	4	8	no	—
127	D₈×C₃₁	—	4	8	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
131	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
132	C₂×C₈×C₃₁	#41	1	4	yes	—
133	C₂×C₈×C₃₁	#41	1	4	yes	—
134	C₄⋊C₄×C₃₁	—	2	1	no	—
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	1	no	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	—	4	2	no	—
141	C₄⋊C₄×C₃₁	—	4	2	no	—
142	C₄⋊C₄×C₃₁	—	4	2	no	—
143	C₂²⋊C₄×C₃₁	—	4	4	no	—
144	C₂²⋊C₄×C₃₁	—	4	8	no	—
145	C₂²⋊C₄×C₃₁	—	4	2	no	—
146	C₂²⋊C₄×C₃₁	—	4	2	no	—
147	C₂²⋊C₄×C₃₁	—	4	4	no	—
148	C₂²⋊C₄×C₃₁	—	4	8	no	—
149	C₄²×C₃₁	—	2	4	no	—
150	C₄²×C₃₁	—	2	2	no	—
151	C₄²×C₃₁	—	2	2	no	—
152	C₄²×C₃₁	—	4	2	no	—
153	C₄²×C₃₁	C₂×D₄	1	2	yes	—
154	C₄²×C₃₁	C₂×D₄	1	2	yes	—
155	#33251	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂³×C₃₁	—	4	1	no	—
157	C₂³×C₃₁	—	4	2	no	—
158	C₂³×C₃₁	—	4	1	no	—
159	C₂³×C₃₁	—	4	2	no	—
160	C₂³×C₃₁	—	4	1	no	—
161	C₂³×C₃₁	—	4	2	no	—
162	C₂³×C₃₁	—	4	4	no	—
163	C₂³×C₃₁	—	4	2	no	—
164	C₂³×C₃₁	—	4	2	no	—
165	C₂³×C₃₁	—	4	4	no	—
166	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
167	Q₈×C₃₁	—	4	2	no	—
168	Q₈×C₃₁	—	4	2	no	—
169	Q₈×C₃₁	—	8	4	no	—
170	D₄×C₃₁	—	4	2	no	—
171	D₄×C₃₁	—	4	2	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	—	8	4	no	—
177	D₄×C₃₁	—	8	4	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	1	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	1	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	4	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	#127	1	2	yes	—
195	C₂×C₄×C₃₁	#127	1	2	yes	—
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	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
200	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
201	C₈×C₃₁	—	2	4	no	—
202	C₈×C₃₁	#124	1	8	yes	complemented
203	C₂×#501	C₂×C₃₁	1	1	yes	characteristic, complemented
204	C₂×#496	C₂×C₃₁	1	1	yes	characteristic
205	C₂×#490	C₂×C₃₁	1	1	yes	characteristic, complemented
206	C₂×#489	C₂×C₃₁	1	1	yes	characteristic, complemented
207	#2861	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#2858	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#1973	C₂×C₃₁	1	2	yes	complemented
210	#1972	C₂×C₃₁	1	2	yes	complemented
211	#1970	C₂×C₃₁	1	2	yes	complemented
212	#1968	C₂×C₃₁	1	2	yes	complemented
213	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
214	C₂²×C₃₁	—	8	1	no	—
215	C₂²×C₃₁	—	8	2	no	—
216	C₂²×C₃₁	—	8	2	no	—
217	C₂²×C₃₁	—	8	1	no	—
218	C₂²×C₃₁	—	8	2	no	—
219	C₂²×C₃₁	—	8	1	no	—
220	C₂²×C₃₁	—	8	4	no	—
221	C₂²×C₃₁	—	8	2	no	—
222	C₂²×C₃₁	—	8	4	no	—
223	C₂²×C₃₁	—	8	2	no	—
224	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
225	C₂²×C₃₁	#532	1	1	yes	characteristic
226	C₂²×C₃₁	#532	1	1	yes	characteristic
227	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
228	C₂²×C₃₁	#575	1	2	yes	—
229	C₄×C₃₁	—	2	2	no	—
230	C₄×C₃₁	—	2	2	no	—
231	C₄×C₃₁	—	2	4	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	—	8	1	no	—
234	C₄×C₃₁	—	8	1	no	—
235	C₄×C₃₁	—	8	2	no	—
236	C₄×C₃₁	—	8	1	no	—
237	C₄×C₃₁	C₂×#124	1	2	yes	—
238	C₄×C₃₁	#532	1	2	yes	—
239	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
240	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
241	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×#124	—	2	2	no	—
245	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
246	#501	—	2	4	no	—
247	#496	—	2	4	no	—
248	#490	—	2	4	no	—
249	#489	—	2	4	no	—
250	C₂×#107	C₂²×C₃₁	1	2	yes	—
251	C₂×#106	C₂²×C₃₁	1	2	yes	—
252	C₂×#103	C₂²×C₃₁	1	2	yes	—
253	C₂×#102	C₂²×C₃₁	1	2	yes	—
254	C₂×#102	C₂²×C₃₁	1	2	yes	—
255	C₂×#102	C₂²×C₃₁	1	2	yes	—
256	C₂×#97	C₂²×C₃₁	1	2	yes	—
257	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
260	#391	C₂²×C₃₁	1	1	yes	characteristic
261	#391	C₂²×C₃₁	1	1	yes	characteristic
262	#390	—	2	2	no	—
263	#390	C₂²×C₃₁	1	2	yes	—
264	#390	C₂²×C₃₁	1	2	yes	—
265	#390	C₂²×C₃₁	1	1	yes	characteristic
266	#390	C₂²×C₃₁	1	1	yes	characteristic
267	#388	—	2	2	no	—
268	#388	—	2	2	no	—
269	C₄³	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₃₁	—	16	2	no	—
271	C₂×C₃₁	—	16	1	no	—
272	C₂×C₃₁	#2858	1	1	yes	characteristic
273	C₂×C₃₁	#3151	1	1	yes	characteristic
274	C₂×C₃₁	#3153	1	1	yes	characteristic
275	C₂×C₃₁	#3171	1	2	yes	—
276	C₂×C₃₁	#3180	1	2	yes	—
277	C₂²×Q₈	—	2	1	no	—
278	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
279	C₂²×D₄	—	2	1	no	—
280	C₂²×D₄	—	2	1	no	—
281	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
282	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
283	C₂×SD₁₆	—	2	4	no	—
284	C₂×SD₁₆	—	2	4	no	—
285	C₂×SD₁₆	—	2	4	no	—
286	C₂×D₈	—	2	4	no	—
287	C₂×D₈	—	2	4	no	—
288	C₂×D₈	—	2	4	no	—
289	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
290	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
291	C₄⋊Q₈	—	2	2	no	—
292	C₄⋊Q₈	—	2	2	no	—
293	#127	—	2	2	no	—
294	#127	—	2	2	no	—
295	#125	—	2	2	no	—
296	#125	—	2	2	no	—
297	#124	—	2	2	no	—
298	#124	—	2	2	no	—
299	#124	—	4	8	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
306	C₂×C₂²⋊C₄	—	2	4	no	—
307	C₂×C₂²⋊C₄	—	2	4	no	—
308	C₂×C₂²⋊C₄	—	2	1	no	—
309	C₂×C₂²⋊C₄	—	2	1	no	—
310	C₂×C₂²⋊C₄	—	2	2	no	—
311	C₂×C₂²⋊C₄	—	2	2	no	—
312	C₂×C₄²	D₄×C₃₁	1	2	yes	—
313	C₂×C₄²	D₄×C₃₁	1	2	yes	—
314	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
315	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
316	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
317	#107	—	2	4	no	—
318	#106	—	2	4	no	—
319	#103	—	2	4	no	—
320	#102	—	2	4	no	—
321	#102	—	2	4	no	—
322	#102	—	2	4	no	—
323	#97	—	2	4	no	—
324	#97	D₄×C₃₁	1	4	yes	—
325	#97	D₄×C₃₁	1	4	yes	—
326	C₄×C₈	—	2	4	no	—
327	#95	—	2	2	no	—
328	#95	—	2	2	no	—
329	C₃₁	#33251	1	1	yes	characteristic, direct product factor, Sylow
330	C₂⁴	—	2	1	no	—
331	C₂⁴	—	2	2	no	—
332	C₂×Q₈	—	2	1	no	—
333	C₂×Q₈	—	2	1	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	4	4	no	—
337	C₂×Q₈	—	4	2	no	—
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	1	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	—	2	2	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	4	4	no	—
349	C₂×D₄	—	4	4	no	—
350	C₂×D₄	—	4	2	no	—
351	C₂×D₄	—	4	2	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
357	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
358	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
360	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
361	SD₁₆	—	4	8	no	—
362	D₈	—	4	8	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
366	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
367	C₂×C₈	#41×C₃₁	1	4	yes	—
368	C₂×C₈	#41×C₃₁	1	4	yes	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	1	no	—
375	C₄⋊C₄	—	4	2	no	—
376	C₄⋊C₄	—	4	2	no	—
377	C₄⋊C₄	—	4	2	no	—
378	C₂²⋊C₄	—	4	4	no	—
379	C₂²⋊C₄	—	4	8	no	—
380	C₂²⋊C₄	—	4	2	no	—
381	C₂²⋊C₄	—	4	2	no	—
382	C₂²⋊C₄	—	4	4	no	—
383	C₂²⋊C₄	—	4	8	no	—
384	C₄²	—	2	4	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	4	2	no	—
388	C₄²	C₂×D₄×C₃₁	1	2	yes	—
389	C₄²	C₂×D₄×C₃₁	1	2	yes	—
390	C₂³	—	4	1	no	—
391	C₂³	—	4	2	no	—
392	C₂³	—	4	1	no	—
393	C₂³	—	4	2	no	—
394	C₂³	—	4	1	no	—
395	C₂³	—	4	2	no	—
396	C₂³	—	4	4	no	—
397	C₂³	—	4	2	no	—
398	C₂³	—	4	2	no	—
399	C₂³	—	4	4	no	—
400	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
401	Q₈	—	4	2	no	—
402	Q₈	—	4	2	no	—
403	Q₈	—	8	4	no	—
404	D₄	—	4	2	no	—
405	D₄	—	4	2	no	—
406	D₄	—	4	2	no	—
407	D₄	—	4	2	no	—
408	D₄	—	4	2	no	—
409	D₄	—	4	2	no	—
410	D₄	—	8	4	no	—
411	D₄	—	8	4	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	C₃₁×#127	1	2	yes	—
429	C₂×C₄	C₃₁×#127	1	2	yes	—
430	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
431	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
433	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
434	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
435	C₈	—	2	4	no	—
436	C₈	C₃₁×#124	1	8	yes	complemented
437	C₂²	—	8	1	no	—
438	C₂²	—	8	2	no	—
439	C₂²	—	8	2	no	—
440	C₂²	—	8	1	no	—
441	C₂²	—	8	2	no	—
442	C₂²	—	8	1	no	—
443	C₂²	—	8	4	no	—
444	C₂²	—	8	2	no	—
445	C₂²	—	8	4	no	—
446	C₂²	—	8	2	no	—
447	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
448	C₂²	C₃₁×#532	1	1	yes	characteristic
449	C₂²	C₃₁×#532	1	1	yes	characteristic
450	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
451	C₂²	C₃₁×#575	1	2	yes	—
452	C₄	—	2	2	no	—
453	C₄	—	2	2	no	—
454	C₄	—	2	4	no	—
455	C₄	—	4	2	no	—
456	C₄	—	8	1	no	—
457	C₄	—	8	2	no	—
458	C₄	—	8	1	no	—
459	C₄	—	8	1	no	—
460	C₄	C₂×C₃₁×#124	1	2	yes	—
461	C₄	C₃₁×#532	1	2	yes	—
462	C₂	—	16	2	no	—
463	C₂	—	16	1	no	—
464	C₂	C₃₁×#2858	1	1	yes	characteristic
465	C₂	C₃₁×#3151	1	1	yes	characteristic
466	C₂	C₃₁×#3153	1	1	yes	characteristic
467	C₂	C₃₁×#3171	1	2	yes	—
468	C₂	C₃₁×#3180	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#496	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#501	C₂	semidirect
3. C₂×C₃₁×#490	C₂	semidirect
4. C₂×C₃₁×#489	C₂	semidirect
5. C₃₁×#2861	C₂	semidirect
6. C₃₁×#2858	C₂	semidirect
7. C₃₁×#1973	C₂	semidirect
8. C₃₁×#1972	C₂	semidirect
9. C₃₁×#1970	C₂	semidirect
10. C₃₁×#1968	C₂	semidirect
11. C₄×C₃₁×#97	C₂	semidirect
155. #33251	C₃₁	direct (N × H)
202. C₈×C₃₁	#124	semidirect
203. C₂×#501	C₂×C₃₁	semidirect
205. C₂×#490	C₂×C₃₁	semidirect
206. C₂×#489	C₂×C₃₁	semidirect
207. #2861	C₂×C₃₁	semidirect
208. #2858	C₂×C₃₁	semidirect
209. #1973	C₂×C₃₁	semidirect
210. #1972	C₂×C₃₁	semidirect
211. #1970	C₂×C₃₁	semidirect
212. #1968	C₂×C₃₁	semidirect
213. C₄×#97	C₂×C₃₁	semidirect
329. C₃₁	#33251	direct (N × H)
436. C₈	C₃₁×#124	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#501	2
C₂×C₃₁×#496	2
C₂×C₃₁×#490	2
C₂×C₃₁×#489	2
C₃₁×#2861	2
C₃₁×#2858	2
C₃₁×#1973	2
C₃₁×#1972	2
C₃₁×#1970	2
C₃₁×#1968	2
C₄×C₃₁×#97	2
#33251	31

Subgroups — C31×#33251 #21490010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33251 #21490010