Subgroups — C₃₁×#24829 #21481588

Order 7936.

956 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#442	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2542	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2349	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2084	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2065	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1879	C₂	1	2	yes	maximal
10	C₃₁×#1804	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1802	C₂	1	2	yes	maximal, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#476	—	2	4	no	—
19	C₃₁×#442	—	2	4	no	—
20	C₂×C₃₁×#106	C₂²	1	2	yes	—
21	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#102	C₂²	1	2	yes	—
28	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
29	C₃₁×#399	—	2	1	no	—
30	C₃₁×#399	—	2	1	no	—
31	C₃₁×#390	C₂²	1	2	yes	—
32	C₃₁×#389	C₂²	1	2	yes	—
33	C₃₁×#385	—	2	2	no	—
34	C₃₁×#384	C₂²	1	2	yes	—
35	C₃₁×#382	C₂²	1	1	yes	characteristic
36	C₃₁×#380	—	2	1	no	—
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×#95	—	2	1	no	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂³×C₄×C₃₁	—	2	1	no	—
48	C₂³×C₄×C₃₁	—	2	2	no	—
49	C₂×D₈×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	—	2	2	no	—
51	C₂×D₈×C₃₁	Q₈	1	4	yes	complemented
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×C₄⋊Q₈	—	2	2	no	—
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₃₁×C₂²⋊Q₈	—	4	8	no	—
58	C₄×D₄×C₃₁	—	2	4	no	—
59	C₄×D₄×C₃₁	—	2	2	no	—
60	C₄×D₄×C₃₁	—	2	2	no	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	—	4	2	no	—
63	C₄×D₄×C₃₁	D₄	1	4	yes	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×#106	—	2	2	no	—
81	C₃₁×#106	—	2	4	no	—
82	C₃₁×C₄⋊C₈	—	2	2	no	—
83	C₃₁×C₄⋊C₈	—	4	2	no	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	—	2	4	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	—	4	2	no	—
88	C₃₁×#97	—	2	2	no	—
89	C₃₁×#95	—	2	1	no	—
90	C₃₁×#95	—	2	2	no	—
91	C₃₁×#95	—	2	1	no	—
92	C₃₁×#95	D₄	1	1	yes	characteristic
93	C₃₁×#95	D₄	1	1	yes	characteristic
94	C₂⁴×C₃₁	—	2	1	no	—
95	C₂⁴×C₃₁	—	2	1	no	—
96	C₂×Q₈×C₃₁	—	4	4	no	—
97	C₂×Q₈×C₃₁	—	4	2	no	—
98	C₂×D₄×C₃₁	—	2	1	no	—
99	C₂×D₄×C₃₁	—	2	1	no	—
100	C₂×D₄×C₃₁	—	2	1	no	—
101	C₂×D₄×C₃₁	—	2	1	no	—
102	C₂×D₄×C₃₁	—	2	1	no	—
103	C₂×D₄×C₃₁	—	2	1	no	—
104	C₂×D₄×C₃₁	—	2	1	no	—
105	C₂×D₄×C₃₁	—	2	1	no	—
106	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
107	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
108	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
109	C₂×D₄×C₃₁	#41	1	2	yes	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	4	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	4	4	no	—
116	C₂²×C₄×C₃₁	—	4	4	no	—
117	C₂²×C₄×C₃₁	—	4	2	no	—
118	C₂²×C₄×C₃₁	—	4	4	no	—
119	C₂²×C₄×C₃₁	—	8	2	no	—
120	C₂²×C₄×C₃₁	—	8	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	D₈×C₃₁	—	2	8	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	2	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
136	C₄⋊C₄×C₃₁	—	2	4	no	—
137	C₄⋊C₄×C₃₁	—	2	1	no	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₄⋊C₄×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₄⋊C₄×C₃₁	—	4	4	no	—
145	C₄⋊C₄×C₃₁	—	4	2	no	—
146	C₄⋊C₄×C₃₁	—	4	4	no	—
147	C₄⋊C₄×C₃₁	—	8	2	no	—
148	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
149	C₂²⋊C₄×C₃₁	—	2	4	no	—
150	C₂²⋊C₄×C₃₁	—	4	4	no	—
151	C₂²⋊C₄×C₃₁	—	4	4	no	—
152	C₂²⋊C₄×C₃₁	—	4	2	no	—
153	C₂²⋊C₄×C₃₁	—	4	4	no	—
154	C₄²×C₃₁	—	2	2	no	—
155	C₄²×C₃₁	—	2	2	no	—
156	C₄²×C₃₁	C₂×D₄	1	2	yes	—
157	C₄²×C₃₁	#41	1	2	yes	—
158	#24829	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂³×C₃₁	—	2	2	no	—
160	C₂³×C₃₁	—	2	2	no	—
161	C₂³×C₃₁	—	2	2	no	—
162	C₂³×C₃₁	—	2	2	no	—
163	C₂³×C₃₁	—	4	2	no	—
164	C₂³×C₃₁	—	4	2	no	—
165	C₂³×C₃₁	—	4	1	no	—
166	C₂³×C₃₁	—	4	1	no	—
167	C₂³×C₃₁	—	4	2	no	—
168	C₂³×C₃₁	—	4	2	no	—
169	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
170	Q₈×C₃₁	—	8	4	no	—
171	D₄×C₃₁	—	2	2	no	—
172	D₄×C₃₁	—	2	2	no	—
173	D₄×C₃₁	—	2	2	no	—
174	D₄×C₃₁	—	2	2	no	—
175	D₄×C₃₁	—	2	2	no	—
176	D₄×C₃₁	—	4	1	no	—
177	D₄×C₃₁	—	4	1	no	—
178	D₄×C₃₁	C₂²⋊Q₈	1	4	yes	complemented
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	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	1	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	—	4	2	no	—
196	C₂×C₄×C₃₁	—	8	4	no	—
197	C₂×C₄×C₃₁	—	8	8	no	—
198	C₂×C₄×C₃₁	—	8	4	no	—
199	C₂×C₄×C₃₁	#121	1	2	yes	—
200	C₂×C₄×C₃₁	#121	1	2	yes	—
201	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
202	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
204	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
207	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
208	C₈×C₃₁	—	2	2	no	—
209	C₈×C₃₁	—	4	1	no	—
210	C₈×C₃₁	—	4	2	no	—
211	C₂×#476	C₂×C₃₁	1	1	yes	characteristic, complemented
212	C₂×#442	C₂×C₃₁	1	1	yes	characteristic
213	#2542	C₂×C₃₁	1	1	yes	characteristic, complemented
214	#2349	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#2084	C₂×C₃₁	1	2	yes	complemented
216	#2065	C₂×C₃₁	1	2	yes	complemented
217	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
219	#1879	C₂×C₃₁	1	2	yes	—
220	#1804	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#1802	C₂×C₃₁	1	2	yes	complemented
222	C₂²×C₃₁	—	2	4	no	—
223	C₂²×C₃₁	—	4	4	no	—
224	C₂²×C₃₁	—	4	2	no	—
225	C₂²×C₃₁	—	4	2	no	—
226	C₂²×C₃₁	—	4	4	no	—
227	C₂²×C₃₁	—	4	4	no	—
228	C₂²×C₃₁	—	4	2	no	—
229	C₂²×C₃₁	—	8	1	no	—
230	C₂²×C₃₁	—	8	1	no	—
231	C₂²×C₃₁	—	8	2	no	—
232	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
233	C₂²×C₃₁	#546	1	1	yes	characteristic
234	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
235	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
236	C₂²×C₃₁	#577	1	2	yes	—
237	C₄×C₃₁	—	2	2	no	—
238	C₄×C₃₁	—	2	2	no	—
239	C₄×C₃₁	—	2	4	no	—
240	C₄×C₃₁	—	4	2	no	—
241	C₄×C₃₁	—	4	2	no	—
242	C₄×C₃₁	—	8	1	no	—
243	C₄×C₃₁	—	8	2	no	—
244	C₄×C₃₁	—	8	2	no	—
245	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
246	C₄×C₃₁	#537	1	1	yes	characteristic
247	C₄×C₃₁	#552	1	1	yes	characteristic
248	C₄×C₃₁	#554	1	1	yes	characteristic
249	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×C₂²⋊Q₈	—	2	2	no	—
252	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
254	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
255	#476	—	2	4	no	—
256	#442	—	2	4	no	—
257	C₂×#106	C₂²×C₃₁	1	2	yes	—
258	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
261	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×#102	C₂²×C₃₁	1	2	yes	—
263	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×#102	C₂²×C₃₁	1	2	yes	—
265	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
266	#399	—	2	1	no	—
267	#399	—	2	1	no	—
268	#390	C₂²×C₃₁	1	2	yes	—
269	#389	C₂²×C₃₁	1	2	yes	—
270	#385	—	2	2	no	—
271	#384	C₂²×C₃₁	1	2	yes	—
272	#382	C₂²×C₃₁	1	1	yes	characteristic
273	#380	—	2	1	no	—
274	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×#95	—	2	1	no	—
276	#340	C₂²×C₃₁	1	2	yes	—
277	#340	C₂²×C₃₁	1	2	yes	—
278	#340	C₂²×C₃₁	1	2	yes	—
279	#336	C₂²×C₃₁	1	2	yes	—
280	C₂×C₃₁	—	4	4	no	—
281	C₂×C₃₁	—	8	2	no	—
282	C₂×C₃₁	#2417	1	1	yes	characteristic
283	C₂×C₃₁	#3038	1	1	yes	characteristic
284	C₂×C₃₁	#3042	1	1	yes	characteristic
285	C₂×C₃₁	#3306	1	2	yes	—
286	C₂×C₃₁	#3384	1	2	yes	—
287	C₂²×Q₈	—	2	1	no	—
288	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
289	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
290	C₂³×C₄	—	2	1	no	—
291	C₂³×C₄	—	2	1	no	—
292	C₂³×C₄	—	2	2	no	—
293	C₂×D₈	—	2	2	no	—
294	C₂×D₈	—	2	2	no	—
295	C₂×D₈	Q₈×C₃₁	1	4	yes	complemented
296	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
297	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
299	C₄⋊Q₈	—	2	2	no	—
300	C₄⋊Q₈	—	2	2	no	—
301	C₂²⋊Q₈	—	4	8	no	—
302	C₄×D₄	—	2	4	no	—
303	C₄×D₄	—	2	2	no	—
304	C₄×D₄	—	2	2	no	—
305	C₄×D₄	—	2	2	no	—
306	C₄×D₄	—	4	2	no	—
307	C₄×D₄	D₄×C₃₁	1	4	yes	—
308	C₂×C₄⋊C₄	—	2	2	no	—
309	C₂×C₄⋊C₄	—	2	2	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
313	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
314	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
315	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
316	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
317	C₂×C₂²⋊C₄	—	2	1	no	—
318	C₂×C₂²⋊C₄	—	2	2	no	—
319	C₂×C₂²⋊C₄	—	2	1	no	—
320	C₂×C₂²⋊C₄	—	2	2	no	—
321	C₂×C₂²⋊C₄	—	2	2	no	—
322	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
323	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
324	#106	—	2	2	no	—
325	#106	—	2	4	no	—
326	C₄⋊C₈	—	2	2	no	—
327	C₄⋊C₈	—	4	2	no	—
328	#102	—	2	2	no	—
329	#102	—	2	4	no	—
330	#102	—	2	2	no	—
331	#102	—	4	2	no	—
332	#97	—	2	2	no	—
333	#95	—	2	1	no	—
334	#95	—	2	2	no	—
335	#95	—	2	1	no	—
336	#95	D₄×C₃₁	1	1	yes	characteristic
337	#95	D₄×C₃₁	1	1	yes	characteristic
338	C₃₁	#24829	1	1	yes	characteristic, direct product factor, Sylow
339	C₂⁴	—	2	1	no	—
340	C₂⁴	—	2	1	no	—
341	C₂×Q₈	—	4	4	no	—
342	C₂×Q₈	—	4	2	no	—
343	C₂×D₄	—	2	1	no	—
344	C₂×D₄	—	2	1	no	—
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	2	1	no	—
347	C₂×D₄	—	2	1	no	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	—	2	1	no	—
351	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
352	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
353	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
354	C₂×D₄	#41×C₃₁	1	2	yes	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	4	4	no	—
361	C₂²×C₄	—	4	4	no	—
362	C₂²×C₄	—	4	2	no	—
363	C₂²×C₄	—	4	4	no	—
364	C₂²×C₄	—	8	2	no	—
365	C₂²×C₄	—	8	2	no	—
366	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
367	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
369	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
370	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
371	D₈	—	2	8	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
381	C₄⋊C₄	—	2	4	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	4	4	no	—
390	C₄⋊C₄	—	4	2	no	—
391	C₄⋊C₄	—	4	4	no	—
392	C₄⋊C₄	—	8	2	no	—
393	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
394	C₂²⋊C₄	—	2	4	no	—
395	C₂²⋊C₄	—	4	4	no	—
396	C₂²⋊C₄	—	4	4	no	—
397	C₂²⋊C₄	—	4	2	no	—
398	C₂²⋊C₄	—	4	4	no	—
399	C₄²	—	2	2	no	—
400	C₄²	—	2	2	no	—
401	C₄²	C₂×D₄×C₃₁	1	2	yes	—
402	C₄²	#41×C₃₁	1	2	yes	—
403	C₂³	—	2	2	no	—
404	C₂³	—	2	2	no	—
405	C₂³	—	2	2	no	—
406	C₂³	—	2	2	no	—
407	C₂³	—	4	2	no	—
408	C₂³	—	4	2	no	—
409	C₂³	—	4	1	no	—
410	C₂³	—	4	1	no	—
411	C₂³	—	4	2	no	—
412	C₂³	—	4	2	no	—
413	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
414	Q₈	—	8	4	no	—
415	D₄	—	2	2	no	—
416	D₄	—	2	2	no	—
417	D₄	—	2	2	no	—
418	D₄	—	2	2	no	—
419	D₄	—	2	2	no	—
420	D₄	—	4	1	no	—
421	D₄	—	4	1	no	—
422	D₄	C₃₁×C₂²⋊Q₈	1	4	yes	complemented
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	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	1	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	—	4	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	8	4	no	—
441	C₂×C₄	—	8	8	no	—
442	C₂×C₄	—	8	4	no	—
443	C₂×C₄	C₃₁×#121	1	2	yes	—
444	C₂×C₄	C₃₁×#121	1	2	yes	—
445	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
446	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
447	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
448	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
449	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
450	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
451	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
452	C₈	—	2	2	no	—
453	C₈	—	4	1	no	—
454	C₈	—	4	2	no	—
455	C₂²	—	2	4	no	—
456	C₂²	—	4	4	no	—
457	C₂²	—	4	2	no	—
458	C₂²	—	4	2	no	—
459	C₂²	—	4	4	no	—
460	C₂²	—	4	4	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	8	1	no	—
463	C₂²	—	8	1	no	—
464	C₂²	—	8	2	no	—
465	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
466	C₂²	C₃₁×#546	1	1	yes	characteristic
467	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
468	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
469	C₂²	C₃₁×#577	1	2	yes	—
470	C₄	—	2	2	no	—
471	C₄	—	2	2	no	—
472	C₄	—	2	4	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	2	no	—
475	C₄	—	8	1	no	—
476	C₄	—	8	2	no	—
477	C₄	—	8	2	no	—
478	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
479	C₄	C₃₁×#537	1	1	yes	characteristic
480	C₄	C₃₁×#552	1	1	yes	characteristic
481	C₄	C₃₁×#554	1	1	yes	characteristic
482	C₂	—	4	4	no	—
483	C₂	—	8	2	no	—
484	C₂	C₃₁×#2417	1	1	yes	characteristic
485	C₂	C₃₁×#3038	1	1	yes	characteristic
486	C₂	C₃₁×#3042	1	1	yes	characteristic
487	C₂	C₃₁×#3306	1	2	yes	—
488	C₂	C₃₁×#3384	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
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
1. C₂×C₃₁×#476	C₂	semidirect
3. C₃₁×#2542	C₂	semidirect
4. C₃₁×#2349	C₂	semidirect
5. C₃₁×#2084	C₂	semidirect
6. C₃₁×#2065	C₂	semidirect
7. C₃₁×#1993	C₂	semidirect
8. C₃₁×#1940	C₂	semidirect
10. C₃₁×#1804	C₂	semidirect
11. C₃₁×#1802	C₂	semidirect
51. C₂×D₈×C₃₁	Q₈	semidirect
158. #24829	C₃₁	direct (N × H)
178. D₄×C₃₁	C₂²⋊Q₈	semidirect
211. C₂×#476	C₂×C₃₁	semidirect
213. #2542	C₂×C₃₁	semidirect
214. #2349	C₂×C₃₁	semidirect
215. #2084	C₂×C₃₁	semidirect
216. #2065	C₂×C₃₁	semidirect
217. #1993	C₂×C₃₁	semidirect
218. #1940	C₂×C₃₁	semidirect
220. #1804	C₂×C₃₁	semidirect
221. #1802	C₂×C₃₁	semidirect
295. C₂×D₈	Q₈×C₃₁	semidirect
338. C₃₁	#24829	direct (N × H)
422. D₄	C₃₁×C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#476	2
C₂×C₃₁×#442	2
C₃₁×#2542	2
C₃₁×#2349	2
C₃₁×#2084	2
C₃₁×#2065	2
C₃₁×#1993	2
C₃₁×#1940	2
C₃₁×#1879	2
C₃₁×#1804	2
C₃₁×#1802	2
#24829	31

Subgroups — C31×#24829 #21481588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24829 #21481588