Subgroups — C₃₁×#29256 #21486015

Order 7936.

884 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2092	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2025	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1982	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1982	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1947	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1864	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1862	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	—	2	1	no	—
15	C₂×C₄×D₄×C₃₁	—	2	1	no	—
16	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₃₁×#390	—	2	1	no	—
26	C₃₁×#390	—	2	1	no	—
27	C₃₁×#390	C₂²	1	2	yes	—
28	C₃₁×#389	C₂²	1	1	yes	characteristic
29	C₃₁×#389	C₂²	1	1	yes	characteristic
30	C₃₁×#387	—	2	1	no	—
31	C₃₁×#387	—	2	1	no	—
32	C₃₁×#384	C₂²	1	1	yes	characteristic
33	C₃₁×#382	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
35	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
36	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₄³×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×D₄×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
46	C₂³×C₄×C₃₁	—	2	2	no	—
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₃₁×#127	—	2	2	no	—
50	C₃₁×#127	Q₈	1	4	yes	—
51	C₃₁×#125	—	2	2	no	—
52	C₃₁×#125	—	2	2	no	—
53	C₄×D₄×C₃₁	—	4	2	no	—
54	C₄×D₄×C₃₁	—	4	2	no	—
55	C₄×D₄×C₃₁	—	4	4	no	—
56	C₄×D₄×C₃₁	—	4	4	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
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×C₄²×C₃₁	—	2	1	no	—
72	C₂×C₄²×C₃₁	—	2	1	no	—
73	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×#107	—	4	2	no	—
81	C₃₁×#106	—	4	2	no	—
82	C₃₁×C₄⋊C₈	—	2	2	no	—
83	C₃₁×C₄⋊C₈	—	2	2	no	—
84	C₃₁×#102	—	2	4	no	—
85	C₃₁×#102	—	2	4	no	—
86	C₃₁×#102	—	4	2	no	—
87	C₃₁×#102	—	4	2	no	—
88	C₃₁×#95	—	2	1	no	—
89	C₃₁×#95	—	2	1	no	—
90	C₃₁×#95	—	2	1	no	—
91	C₃₁×#95	—	2	1	no	—
92	C₃₁×#95	—	2	1	no	—
93	C₃₁×#95	—	2	1	no	—
94	C₂⁴×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	4	no	—
96	C₂×D₄×C₃₁	—	2	4	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	—	4	2	no	—
103	C₂×D₄×C₃₁	—	4	2	no	—
104	C₂×D₄×C₃₁	—	4	2	no	—
105	C₂×D₄×C₃₁	—	4	2	no	—
106	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	—
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	1	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	—	4	4	no	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	4	no	—
129	C₄⋊C₄×C₃₁	—	2	4	no	—
130	C₄⋊C₄×C₃₁	—	2	1	no	—
131	C₄⋊C₄×C₃₁	—	2	1	no	—
132	C₄⋊C₄×C₃₁	—	2	1	no	—
133	C₄⋊C₄×C₃₁	—	2	1	no	—
134	C₄⋊C₄×C₃₁	—	2	1	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	4	2	no	—
139	C₄⋊C₄×C₃₁	—	4	2	no	—
140	C₄⋊C₄×C₃₁	—	4	2	no	—
141	C₄⋊C₄×C₃₁	—	4	2	no	—
142	C₄⋊C₄×C₃₁	#41	1	2	yes	—
143	C₄⋊C₄×C₃₁	#41	1	2	yes	—
144	C₂²⋊C₄×C₃₁	—	4	4	no	—
145	C₂²⋊C₄×C₃₁	—	4	4	no	—
146	C₂²⋊C₄×C₃₁	—	4	4	no	—
147	C₄²×C₃₁	—	2	2	no	—
148	C₄²×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	—	2	2	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	C₄²×C₃₁	—	4	1	no	—
155	C₄²×C₃₁	—	4	2	no	—
156	C₄²×C₃₁	—	4	2	no	—
157	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
158	C₄²×C₃₁	#41	1	2	yes	—
159	#29256	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₃₁	—	2	4	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	C₂²×D₄	1	1	yes	characteristic
168	D₄×C₃₁	—	2	4	no	—
169	D₄×C₃₁	—	2	4	no	—
170	D₄×C₃₁	—	4	8	no	—
171	D₄×C₃₁	—	4	2	no	—
172	D₄×C₃₁	—	4	2	no	—
173	D₄×C₃₁	—	4	8	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	1	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	1	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	4	1	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	1	no	—
201	C₂×C₄×C₃₁	—	4	1	no	—
202	C₂×C₄×C₃₁	—	8	8	no	—
203	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
204	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
205	C₂×C₄×C₃₁	#135	1	2	yes	—
206	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
207	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
208	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
209	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
212	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
213	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
214	C₈×C₃₁	—	4	2	no	—
215	C₈×C₃₁	—	4	2	no	—
216	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
217	C₄×#125	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#2092	C₂×C₃₁	1	2	yes	complemented
219	#2025	C₂×C₃₁	1	2	yes	complemented
220	#1982	C₂×C₃₁	1	1	yes	characteristic
221	#1982	C₂×C₃₁	1	1	yes	characteristic
222	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#1947	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#1864	C₂×C₃₁	1	2	yes	complemented
225	#1862	C₂×C₃₁	1	2	yes	complemented
226	#1831	C₂×C₃₁	1	1	yes	characteristic, complemented
227	C₂²×C₃₁	—	4	4	no	—
228	C₂²×C₃₁	—	4	4	no	—
229	C₂²×C₃₁	—	4	4	no	—
230	C₂²×C₃₁	—	8	2	no	—
231	C₂²×C₃₁	—	8	2	no	—
232	C₂²×C₃₁	—	8	2	no	—
233	C₂²×C₃₁	—	8	2	no	—
234	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
235	C₂²×C₃₁	#525	1	1	yes	characteristic
236	C₂²×C₃₁	#535	1	1	yes	characteristic
237	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
238	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
239	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
240	C₂²×C₃₁	#575	1	1	yes	characteristic
241	C₄×C₃₁	—	2	2	no	—
242	C₄×C₃₁	—	2	2	no	—
243	C₄×C₃₁	—	4	1	no	—
244	C₄×C₃₁	—	4	2	no	—
245	C₄×C₃₁	—	4	1	no	—
246	C₄×C₃₁	—	4	2	no	—
247	C₄×C₃₁	—	4	1	no	—
248	C₄×C₃₁	—	8	1	no	—
249	C₄×C₃₁	—	8	1	no	—
250	C₄×C₃₁	—	8	2	no	—
251	C₄×C₃₁	#478	1	4	yes	—
252	C₄×C₃₁	#554	1	1	yes	characteristic
253	C₄×C₃₁	#554	1	1	yes	characteristic
254	C₄×C₃₁	#559	1	1	yes	characteristic
255	C₄×C₃₁	#559	1	1	yes	characteristic
256	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×C₄×D₄	—	2	1	no	—
259	C₂×C₄×D₄	—	2	1	no	—
260	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
261	C₂×#107	C₂²×C₃₁	1	2	yes	—
262	C₂×#106	C₂²×C₃₁	1	2	yes	—
263	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×#102	C₂²×C₃₁	1	2	yes	—
266	C₂×#102	C₂²×C₃₁	1	2	yes	—
267	C₂×#102	C₂²×C₃₁	1	2	yes	—
268	C₂×#102	C₂²×C₃₁	1	2	yes	—
269	#390	—	2	1	no	—
270	#390	—	2	1	no	—
271	#390	C₂²×C₃₁	1	2	yes	—
272	#389	C₂²×C₃₁	1	1	yes	characteristic
273	#389	C₂²×C₃₁	1	1	yes	characteristic
274	#387	—	2	1	no	—
275	#387	—	2	1	no	—
276	#384	C₂²×C₃₁	1	1	yes	characteristic
277	#382	C₂²×C₃₁	1	1	yes	characteristic
278	C₄×C₄⋊C₄	—	2	1	no	—
279	C₄×C₄⋊C₄	—	2	1	no	—
280	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
281	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
282	C₄³	C₂²×C₃₁	1	1	yes	characteristic
283	#340	C₂²×C₃₁	1	2	yes	—
284	#340	C₂²×C₃₁	1	2	yes	—
285	#336	C₂²×C₃₁	1	2	yes	—
286	#336	C₂²×C₃₁	1	2	yes	—
287	C₂×C₃₁	—	8	4	no	—
288	C₂×C₃₁	#2580	1	1	yes	characteristic
289	C₂×C₃₁	C₂×#478	1	1	yes	characteristic
290	C₂×C₃₁	#3077	1	1	yes	characteristic
291	C₂×C₃₁	#3106	1	1	yes	characteristic
292	C₂×C₃₁	#3115	1	1	yes	characteristic
293	C₂×C₃₁	#3212	1	1	yes	characteristic
294	C₂×C₃₁	#3217	1	1	yes	characteristic
295	C₂²×D₄	—	2	1	no	—
296	C₂²×D₄	—	2	1	no	—
297	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
298	C₂³×C₄	—	2	2	no	—
299	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
300	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
301	#127	—	2	2	no	—
302	#127	Q₈×C₃₁	1	4	yes	—
303	#125	—	2	2	no	—
304	#125	—	2	2	no	—
305	C₄×D₄	—	4	2	no	—
306	C₄×D₄	—	4	2	no	—
307	C₄×D₄	—	4	4	no	—
308	C₄×D₄	—	4	4	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄⋊C₄	—	2	1	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
316	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
317	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
318	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
319	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
320	C₂×C₂²⋊C₄	—	2	2	no	—
321	C₂×C₂²⋊C₄	—	2	2	no	—
322	C₂×C₂²⋊C₄	—	2	2	no	—
323	C₂×C₄²	—	2	1	no	—
324	C₂×C₄²	—	2	1	no	—
325	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
326	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
327	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
328	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
329	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
330	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
331	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
332	#107	—	4	2	no	—
333	#106	—	4	2	no	—
334	C₄⋊C₈	—	2	2	no	—
335	C₄⋊C₈	—	2	2	no	—
336	#102	—	2	4	no	—
337	#102	—	2	4	no	—
338	#102	—	4	2	no	—
339	#102	—	4	2	no	—
340	#95	—	2	1	no	—
341	#95	—	2	1	no	—
342	#95	—	2	1	no	—
343	#95	—	2	1	no	—
344	#95	—	2	1	no	—
345	#95	—	2	1	no	—
346	C₃₁	#29256	1	1	yes	characteristic, direct product factor, Sylow
347	C₂⁴	—	2	2	no	—
348	C₂×D₄	—	2	4	no	—
349	C₂×D₄	—	2	4	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	—	4	2	no	—
356	C₂×D₄	—	4	2	no	—
357	C₂×D₄	—	4	2	no	—
358	C₂×D₄	—	4	2	no	—
359	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	4	4	no	—
366	C₂²×C₄	—	4	4	no	—
367	C₂²×C₄	—	4	4	no	—
368	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
369	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
373	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₄⋊C₄	—	2	4	no	—
382	C₄⋊C₄	—	2	4	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	4	2	no	—
392	C₄⋊C₄	—	4	2	no	—
393	C₄⋊C₄	—	4	2	no	—
394	C₄⋊C₄	—	4	2	no	—
395	C₄⋊C₄	#41×C₃₁	1	2	yes	—
396	C₄⋊C₄	#41×C₃₁	1	2	yes	—
397	C₂²⋊C₄	—	4	4	no	—
398	C₂²⋊C₄	—	4	4	no	—
399	C₂²⋊C₄	—	4	4	no	—
400	C₄²	—	2	2	no	—
401	C₄²	—	2	2	no	—
402	C₄²	—	2	2	no	—
403	C₄²	—	2	1	no	—
404	C₄²	—	2	1	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	2	1	no	—
407	C₄²	—	4	1	no	—
408	C₄²	—	4	2	no	—
409	C₄²	—	4	2	no	—
410	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
411	C₄²	#41×C₃₁	1	2	yes	—
412	C₂³	—	2	4	no	—
413	C₂³	—	4	2	no	—
414	C₂³	—	4	2	no	—
415	C₂³	—	4	2	no	—
416	C₂³	—	4	2	no	—
417	C₂³	—	4	2	no	—
418	C₂³	—	4	2	no	—
419	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
420	D₄	—	2	4	no	—
421	D₄	—	2	4	no	—
422	D₄	—	4	8	no	—
423	D₄	—	4	8	no	—
424	D₄	—	4	2	no	—
425	D₄	—	4	2	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	8	8	no	—
455	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
456	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
457	C₂×C₄	C₃₁×#135	1	2	yes	—
458	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
459	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
460	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
461	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
462	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
463	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
464	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
465	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
466	C₈	—	4	2	no	—
467	C₈	—	4	2	no	—
468	C₂²	—	4	4	no	—
469	C₂²	—	4	4	no	—
470	C₂²	—	4	4	no	—
471	C₂²	—	8	2	no	—
472	C₂²	—	8	2	no	—
473	C₂²	—	8	2	no	—
474	C₂²	—	8	2	no	—
475	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
476	C₂²	C₃₁×#525	1	1	yes	characteristic
477	C₂²	C₃₁×#535	1	1	yes	characteristic
478	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
479	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
480	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
481	C₂²	C₃₁×#575	1	1	yes	characteristic
482	C₄	—	2	2	no	—
483	C₄	—	2	2	no	—
484	C₄	—	4	1	no	—
485	C₄	—	4	2	no	—
486	C₄	—	4	1	no	—
487	C₄	—	4	2	no	—
488	C₄	—	4	1	no	—
489	C₄	—	8	1	no	—
490	C₄	—	8	2	no	—
491	C₄	—	8	1	no	—
492	C₄	C₃₁×#478	1	4	yes	—
493	C₄	C₃₁×#554	1	1	yes	characteristic
494	C₄	C₃₁×#554	1	1	yes	characteristic
495	C₄	C₃₁×#559	1	1	yes	characteristic
496	C₄	C₃₁×#559	1	1	yes	characteristic
497	C₂	—	8	4	no	—
498	C₂	C₃₁×#2580	1	1	yes	characteristic
499	C₂	C₂×C₃₁×#478	1	1	yes	characteristic
500	C₂	C₃₁×#3077	1	1	yes	characteristic
501	C₂	C₃₁×#3106	1	1	yes	characteristic
502	C₂	C₃₁×#3115	1	1	yes	characteristic
503	C₂	C₃₁×#3212	1	1	yes	characteristic
504	C₂	C₃₁×#3217	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
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
2. C₄×C₃₁×#125	C₂	semidirect
3. C₃₁×#2092	C₂	semidirect
4. C₃₁×#2025	C₂	semidirect
7. C₃₁×#1948	C₂	semidirect
8. C₃₁×#1947	C₂	semidirect
9. C₃₁×#1864	C₂	semidirect
10. C₃₁×#1862	C₂	semidirect
11. C₃₁×#1831	C₂	semidirect
159. #29256	C₃₁	direct (N × H)
217. C₄×#125	C₂×C₃₁	semidirect
218. #2092	C₂×C₃₁	semidirect
219. #2025	C₂×C₃₁	semidirect
222. #1948	C₂×C₃₁	semidirect
223. #1947	C₂×C₃₁	semidirect
224. #1864	C₂×C₃₁	semidirect
225. #1862	C₂×C₃₁	semidirect
226. #1831	C₂×C₃₁	semidirect
346. C₃₁	#29256	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#127	2
C₄×C₃₁×#125	2
C₃₁×#2092	2
C₃₁×#2025	2
C₃₁×#1982	2
C₃₁×#1948	2
C₃₁×#1947	2
C₃₁×#1864	2
C₃₁×#1862	2
C₃₁×#1831	2
#29256	31

Subgroups — C31×#29256 #21486015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29256 #21486015