Subgroups — C₃₁×#22545 #21479304

Order 7936.

1366 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#550	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2355	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1982	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1982	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1961	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1804	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1762	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	—	2	1	no	—
13	C₃₁×#550	—	2	4	no	—
14	C₃₁×#550	C₄	1	8	yes	complemented
15	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#127	C₂²	1	2	yes	—
17	C₂×C₃₁×#121	—	2	1	no	—
18	C₂×C₃₁×#121	—	2	2	no	—
19	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#446	—	2	4	no	—
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₃₁×#102	C₂²	1	2	yes	—
31	C₂×C₃₁×#102	C₂²	1	2	yes	—
32	C₂×C₃₁×#97	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#391	—	2	1	no	—
35	C₃₁×#387	—	2	2	no	—
36	C₃₁×#384	C₂²	1	1	yes	characteristic
37	C₃₁×#382	C₂²	1	2	yes	—
38	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
39	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	1	yes	characteristic
43	C₃₁×#340	C₂²	1	1	yes	characteristic
44	C₂×#41×C₃₁	—	2	8	no	—
45	C₂×#41×C₃₁	—	4	4	no	—
46	C₂×#41×C₃₁	—	4	4	no	—
47	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
48	C₂²×D₄×C₃₁	—	2	2	no	—
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	—	2	2	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
53	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₃₁	—	2	2	no	—
55	C₂³×C₄×C₃₁	—	2	1	no	—
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×#127	—	2	2	no	—
60	C₃₁×#127	—	2	4	no	—
61	C₃₁×#127	C₂×C₄	1	4	yes	—
62	C₃₁×#127	C₂×C₄	1	8	yes	—
63	C₃₁×#121	—	2	16	no	—
64	C₃₁×#121	—	2	8	no	—
65	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
66	C₄×Q₈×C₃₁	D₄	1	4	yes	—
67	C₄×D₄×C₃₁	—	2	2	no	—
68	C₄×D₄×C₃₁	—	2	4	no	—
69	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
70	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
71	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₄²×C₃₁	—	2	1	no	—
82	C₂×C₄²×C₃₁	—	2	2	no	—
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
85	C₃₁×C₄⋊C₈	—	2	2	no	—
86	C₃₁×C₄⋊C₈	—	2	4	no	—
87	C₃₁×#102	—	2	4	no	—
88	C₃₁×#102	—	2	2	no	—
89	C₃₁×#102	—	2	2	no	—
90	C₃₁×#102	—	2	4	no	—
91	C₃₁×#102	—	4	2	no	—
92	C₃₁×#102	—	4	2	no	—
93	C₃₁×#102	—	4	2	no	—
94	C₃₁×#97	—	2	4	no	—
95	C₄×C₈×C₃₁	—	2	2	no	—
96	C₃₁×#95	—	2	2	no	—
97	C₃₁×#95	—	2	1	no	—
98	C₃₁×#95	—	2	2	no	—
99	C₃₁×#95	—	2	2	no	—
100	C₂⁴×C₃₁	—	2	2	no	—
101	C₂⁴×C₃₁	—	2	1	no	—
102	#41×C₃₁	—	4	32	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	C₂²⋊C₄	1	4	yes	—
106	C₂×D₄×C₃₁	—	2	8	no	—
107	C₂×D₄×C₃₁	—	2	4	no	—
108	C₂×D₄×C₃₁	—	2	4	no	—
109	C₂×D₄×C₃₁	—	2	8	no	—
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	—	2	2	no	—
118	C₂×D₄×C₃₁	—	2	2	no	—
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	—	4	4	no	—
121	C₂×D₄×C₃₁	—	4	4	no	—
122	C₂×D₄×C₃₁	—	4	2	no	—
123	C₂×D₄×C₃₁	—	4	2	no	—
124	C₂×D₄×C₃₁	—	8	2	no	—
125	C₂×D₄×C₃₁	—	8	1	no	—
126	C₂×D₄×C₃₁	—	8	2	no	—
127	C₂×D₄×C₃₁	—	8	1	no	—
128	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
129	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	2	8	no	—
133	C₂²×C₄×C₃₁	—	2	4	no	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	2	2	no	—
136	C₂²×C₄×C₃₁	—	4	4	no	—
137	C₂²×C₄×C₃₁	—	4	4	no	—
138	C₂²×C₄×C₃₁	—	4	2	no	—
139	C₂²×C₄×C₃₁	—	4	2	no	—
140	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	—	2	1	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	1	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₂×C₈×C₃₁	#41	1	2	yes	complemented
154	C₄⋊C₄×C₃₁	—	2	1	no	—
155	C₄⋊C₄×C₃₁	—	2	1	no	—
156	C₄⋊C₄×C₃₁	—	2	1	no	—
157	C₄⋊C₄×C₃₁	—	2	2	no	—
158	C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	—	4	2	no	—
161	C₄⋊C₄×C₃₁	—	8	1	no	—
162	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
163	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
164	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
165	C₄⋊C₄×C₃₁	#41	1	2	yes	complemented
166	C₂²⋊C₄×C₃₁	—	2	4	no	—
167	C₂²⋊C₄×C₃₁	—	2	8	no	—
168	C₄²×C₃₁	—	4	2	no	—
169	C₄²×C₃₁	—	8	2	no	—
170	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
171	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
172	C₄²×C₃₁	C₂×D₄	1	2	yes	—
173	C₄²×C₃₁	C₂×D₄	1	4	yes	—
174	#22545	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂³×C₃₁	—	2	4	no	—
176	C₂³×C₃₁	—	2	2	no	—
177	C₂³×C₃₁	—	4	2	no	—
178	C₂³×C₃₁	—	4	1	no	—
179	C₂³×C₃₁	—	4	2	no	—
180	C₂³×C₃₁	—	4	1	no	—
181	C₂³×C₃₁	—	4	2	no	—
182	C₂³×C₃₁	—	4	1	no	—
183	C₂³×C₃₁	—	4	2	no	—
184	C₂³×C₃₁	—	4	1	no	—
185	C₂³×C₃₁	—	4	1	no	—
186	C₂³×C₃₁	—	4	2	no	—
187	C₂³×C₃₁	—	4	2	no	—
188	C₂³×C₃₁	—	4	1	no	—
189	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
190	Q₈×C₃₁	—	2	8	no	—
191	D₄×C₃₁	—	2	4	no	—
192	D₄×C₃₁	—	2	2	no	—
193	D₄×C₃₁	—	2	2	no	—
194	D₄×C₃₁	—	2	4	no	—
195	D₄×C₃₁	—	4	16	no	—
196	D₄×C₃₁	—	4	1	no	—
197	D₄×C₃₁	—	4	2	no	—
198	D₄×C₃₁	—	4	1	no	—
199	D₄×C₃₁	—	4	2	no	—
200	D₄×C₃₁	—	4	8	no	—
201	D₄×C₃₁	—	8	8	no	—
202	D₄×C₃₁	—	8	4	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	1	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	4	16	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	1	no	—
215	C₂×C₄×C₃₁	—	4	1	no	—
216	C₂×C₄×C₃₁	—	4	8	no	—
217	C₂×C₄×C₃₁	—	4	1	no	—
218	C₂×C₄×C₃₁	—	4	2	no	—
219	C₂×C₄×C₃₁	—	4	1	no	—
220	C₂×C₄×C₃₁	—	4	2	no	—
221	C₂×C₄×C₃₁	—	4	2	no	—
222	C₂×C₄×C₃₁	—	4	1	no	—
223	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
224	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
225	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
226	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
227	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
228	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
229	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
230	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
232	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
233	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
234	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
235	C₈×C₃₁	—	2	2	no	—
236	C₈×C₃₁	—	4	1	no	—
237	C₈×C₃₁	—	4	2	no	—
238	C₂×#550	C₂×C₃₁	1	1	yes	characteristic
239	C₂×#446	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#2355	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#1982	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#1982	C₂×C₃₁	1	2	yes	complemented
243	#1961	C₂×C₃₁	1	2	yes	complemented
244	#1957	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#1804	C₂×C₃₁	1	2	yes	complemented
246	#1802	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#1762	C₂×C₃₁	1	2	yes	complemented
248	C₄×#102	C₂×C₃₁	1	1	yes	characteristic, complemented
249	C₂²×C₃₁	—	4	4	no	—
250	C₂²×C₃₁	—	4	2	no	—
251	C₂²×C₃₁	—	4	4	no	—
252	C₂²×C₃₁	—	4	2	no	—
253	C₂²×C₃₁	—	4	4	no	—
254	C₂²×C₃₁	—	4	2	no	—
255	C₂²×C₃₁	—	8	2	no	—
256	C₂²×C₃₁	—	8	1	no	—
257	C₂²×C₃₁	—	8	2	no	—
258	C₂²×C₃₁	—	8	1	no	—
259	C₂²×C₃₁	—	8	2	no	—
260	C₂²×C₃₁	—	8	1	no	—
261	C₂²×C₃₁	—	8	2	no	—
262	C₂²×C₃₁	—	8	1	no	—
263	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
264	C₂²×C₃₁	#548	1	1	yes	characteristic
265	C₂²×C₃₁	#550	1	1	yes	characteristic
266	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
267	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
268	C₂²×C₃₁	#576	1	1	yes	characteristic
269	C₂²×C₃₁	#577	1	1	yes	characteristic
270	C₄×C₃₁	—	2	4	no	—
271	C₄×C₃₁	—	2	4	no	—
272	C₄×C₃₁	—	2	2	no	—
273	C₄×C₃₁	—	4	2	no	—
274	C₄×C₃₁	—	4	2	no	—
275	C₄×C₃₁	—	8	1	no	—
276	C₄×C₃₁	—	8	2	no	—
277	C₄×C₃₁	#418	1	4	yes	—
278	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
279	C₄×C₃₁	#520	1	1	yes	characteristic
280	C₄×C₃₁	#548	1	1	yes	characteristic
281	C₄×C₃₁	#550	1	1	yes	characteristic
282	C₂²×#41	—	2	1	no	—
283	#550	—	2	4	no	—
284	#550	C₄×C₃₁	1	8	yes	complemented
285	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
286	C₂×#127	C₂²×C₃₁	1	2	yes	—
287	C₂×#121	—	2	1	no	—
288	C₂×#121	—	2	2	no	—
289	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
292	#446	—	2	4	no	—
293	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
294	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
295	C₂×#102	C₂²×C₃₁	1	2	yes	—
296	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
297	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×#102	C₂²×C₃₁	1	2	yes	—
299	C₂×#102	C₂²×C₃₁	1	2	yes	—
300	C₂×#102	C₂²×C₃₁	1	2	yes	—
301	C₂×#102	C₂²×C₃₁	1	2	yes	—
302	C₂×#97	C₂²×C₃₁	1	2	yes	—
303	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
304	#391	—	2	1	no	—
305	#387	—	2	2	no	—
306	#384	C₂²×C₃₁	1	1	yes	characteristic
307	#382	C₂²×C₃₁	1	2	yes	—
308	C₄×C₄⋊C₄	—	2	1	no	—
309	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
310	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
311	#340	C₂²×C₃₁	1	2	yes	—
312	#340	C₂²×C₃₁	1	1	yes	characteristic
313	#340	C₂²×C₃₁	1	1	yes	characteristic
314	C₂×C₃₁	—	8	4	no	—
315	C₂×C₃₁	—	8	2	no	—
316	C₂×C₃₁	#2341	1	1	yes	characteristic
317	C₂×C₃₁	C₂×#418	1	1	yes	characteristic
318	C₂×C₃₁	#2897	1	1	yes	characteristic
319	C₂×C₃₁	#3366	1	1	yes	characteristic
320	C₂×C₃₁	#3368	1	1	yes	characteristic
321	C₂×C₃₁	#3397	1	1	yes	characteristic
322	C₂×C₃₁	#3399	1	1	yes	characteristic
323	C₂×#41	—	2	8	no	—
324	C₂×#41	—	4	4	no	—
325	C₂×#41	—	4	4	no	—
326	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
327	C₂²×D₄	—	2	2	no	—
328	C₂²×D₄	—	2	1	no	—
329	C₂²×D₄	—	2	2	no	—
330	C₂²×D₄	—	2	1	no	—
331	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
332	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
333	C₂³×C₄	—	2	2	no	—
334	C₂³×C₄	—	2	1	no	—
335	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
336	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
337	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
338	#127	—	2	2	no	—
339	#127	—	2	4	no	—
340	#127	C₂×C₄×C₃₁	1	4	yes	—
341	#127	C₂×C₄×C₃₁	1	8	yes	—
342	#121	—	2	16	no	—
343	#121	—	2	8	no	—
344	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
345	C₄×Q₈	D₄×C₃₁	1	4	yes	—
346	C₄×D₄	—	2	2	no	—
347	C₄×D₄	—	2	4	no	—
348	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
349	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	—	2	1	no	—
352	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
353	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
354	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
355	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
356	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
357	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
358	C₂×C₂²⋊C₄	—	2	2	no	—
359	C₂×C₂²⋊C₄	—	2	1	no	—
360	C₂×C₄²	—	2	1	no	—
361	C₂×C₄²	—	2	2	no	—
362	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
363	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
364	C₄⋊C₈	—	2	2	no	—
365	C₄⋊C₈	—	2	4	no	—
366	#102	—	2	4	no	—
367	#102	—	2	2	no	—
368	#102	—	2	2	no	—
369	#102	—	2	4	no	—
370	#102	—	4	2	no	—
371	#102	—	4	2	no	—
372	#102	—	4	2	no	—
373	#97	—	2	4	no	—
374	C₄×C₈	—	2	2	no	—
375	#95	—	2	2	no	—
376	#95	—	2	1	no	—
377	#95	—	2	2	no	—
378	#95	—	2	2	no	—
379	C₃₁	#22545	1	1	yes	characteristic, direct product factor, Sylow
380	C₂⁴	—	2	2	no	—
381	C₂⁴	—	2	1	no	—
382	#41	—	4	32	no	—
383	C₂×Q₈	—	2	2	no	—
384	C₂×Q₈	—	2	2	no	—
385	C₂×Q₈	C₂²⋊C₄×C₃₁	1	4	yes	—
386	C₂×D₄	—	2	8	no	—
387	C₂×D₄	—	2	4	no	—
388	C₂×D₄	—	2	4	no	—
389	C₂×D₄	—	2	8	no	—
390	C₂×D₄	—	2	1	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	—	2	1	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	—	2	1	no	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	4	4	no	—
401	C₂×D₄	—	4	4	no	—
402	C₂×D₄	—	4	2	no	—
403	C₂×D₄	—	4	2	no	—
404	C₂×D₄	—	8	2	no	—
405	C₂×D₄	—	8	1	no	—
406	C₂×D₄	—	8	2	no	—
407	C₂×D₄	—	8	1	no	—
408	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
409	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	2	8	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	4	4	no	—
417	C₂²×C₄	—	4	4	no	—
418	C₂²×C₄	—	4	2	no	—
419	C₂²×C₄	—	4	2	no	—
420	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
421	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
422	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
423	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	—	2	1	no	—
427	C₂×C₈	—	2	1	no	—
428	C₂×C₈	—	2	2	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	2	2	no	—
433	C₂×C₈	#41×C₃₁	1	2	yes	complemented
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	4	2	no	—
441	C₄⋊C₄	—	8	1	no	—
442	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
443	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
444	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
445	C₄⋊C₄	#41×C₃₁	1	2	yes	complemented
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	2	8	no	—
448	C₄²	—	4	2	no	—
449	C₄²	—	8	2	no	—
450	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
451	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
452	C₄²	C₂×D₄×C₃₁	1	2	yes	—
453	C₄²	C₂×D₄×C₃₁	1	4	yes	—
454	C₂³	—	2	4	no	—
455	C₂³	—	2	2	no	—
456	C₂³	—	4	2	no	—
457	C₂³	—	4	1	no	—
458	C₂³	—	4	2	no	—
459	C₂³	—	4	1	no	—
460	C₂³	—	4	2	no	—
461	C₂³	—	4	1	no	—
462	C₂³	—	4	2	no	—
463	C₂³	—	4	1	no	—
464	C₂³	—	4	1	no	—
465	C₂³	—	4	2	no	—
466	C₂³	—	4	2	no	—
467	C₂³	—	4	1	no	—
468	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
469	Q₈	—	2	8	no	—
470	D₄	—	2	4	no	—
471	D₄	—	2	2	no	—
472	D₄	—	2	2	no	—
473	D₄	—	2	4	no	—
474	D₄	—	4	16	no	—
475	D₄	—	4	8	no	—
476	D₄	—	4	1	no	—
477	D₄	—	4	2	no	—
478	D₄	—	4	1	no	—
479	D₄	—	4	2	no	—
480	D₄	—	8	8	no	—
481	D₄	—	8	4	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	4	16	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	4	1	no	—
493	C₂×C₄	—	4	1	no	—
494	C₂×C₄	—	4	1	no	—
495	C₂×C₄	—	4	8	no	—
496	C₂×C₄	—	4	1	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	4	1	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	4	2	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
503	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
504	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
505	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
506	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
507	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
508	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
509	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
510	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
511	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
512	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
513	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
514	C₈	—	2	2	no	—
515	C₈	—	4	1	no	—
516	C₈	—	4	2	no	—
517	C₂²	—	4	4	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	4	no	—
520	C₂²	—	4	2	no	—
521	C₂²	—	4	4	no	—
522	C₂²	—	4	2	no	—
523	C₂²	—	8	2	no	—
524	C₂²	—	8	1	no	—
525	C₂²	—	8	2	no	—
526	C₂²	—	8	1	no	—
527	C₂²	—	8	2	no	—
528	C₂²	—	8	1	no	—
529	C₂²	—	8	2	no	—
530	C₂²	—	8	1	no	—
531	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
532	C₂²	C₃₁×#548	1	1	yes	characteristic
533	C₂²	C₃₁×#550	1	1	yes	characteristic
534	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
535	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
536	C₂²	C₃₁×#576	1	1	yes	characteristic
537	C₂²	C₃₁×#577	1	1	yes	characteristic
538	C₄	—	2	4	no	—
539	C₄	—	2	4	no	—
540	C₄	—	2	2	no	—
541	C₄	—	4	2	no	—
542	C₄	—	4	2	no	—
543	C₄	—	8	1	no	—
544	C₄	—	8	2	no	—
545	C₄	C₃₁×#418	1	4	yes	—
546	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
547	C₄	C₃₁×#520	1	1	yes	characteristic
548	C₄	C₃₁×#548	1	1	yes	characteristic
549	C₄	C₃₁×#550	1	1	yes	characteristic
550	C₂	—	8	4	no	—
551	C₂	—	8	2	no	—
552	C₂	C₃₁×#2341	1	1	yes	characteristic
553	C₂	C₂×C₃₁×#418	1	1	yes	characteristic
554	C₂	C₃₁×#2897	1	1	yes	characteristic
555	C₂	C₃₁×#3366	1	1	yes	characteristic
556	C₂	C₃₁×#3368	1	1	yes	characteristic
557	C₂	C₃₁×#3397	1	1	yes	characteristic
558	C₂	C₃₁×#3399	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#550	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#446	C₂	semidirect
3. C₃₁×#2355	C₂	semidirect
4. C₃₁×#1982	C₂	semidirect
5. C₃₁×#1982	C₂	semidirect
6. C₃₁×#1961	C₂	semidirect
7. C₃₁×#1957	C₂	semidirect
8. C₃₁×#1804	C₂	semidirect
9. C₃₁×#1802	C₂	semidirect
10. C₃₁×#1762	C₂	semidirect
11. C₄×C₃₁×#102	C₂	semidirect
14. C₃₁×#550	C₄	semidirect
153. C₂×C₈×C₃₁	#41	semidirect
165. C₄⋊C₄×C₃₁	#41	semidirect
174. #22545	C₃₁	direct (N × H)
239. C₂×#446	C₂×C₃₁	semidirect
240. #2355	C₂×C₃₁	semidirect
241. #1982	C₂×C₃₁	semidirect
242. #1982	C₂×C₃₁	semidirect
243. #1961	C₂×C₃₁	semidirect
244. #1957	C₂×C₃₁	semidirect
245. #1804	C₂×C₃₁	semidirect
246. #1802	C₂×C₃₁	semidirect
247. #1762	C₂×C₃₁	semidirect
248. C₄×#102	C₂×C₃₁	semidirect
284. #550	C₄×C₃₁	semidirect
379. C₃₁	#22545	direct (N × H)
433. C₂×C₈	#41×C₃₁	semidirect
445. C₄⋊C₄	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#550	2
C₂×C₃₁×#446	2
C₃₁×#2355	2
C₃₁×#1982	2
C₃₁×#1961	2
C₃₁×#1957	2
C₃₁×#1804	2
C₃₁×#1802	2
C₃₁×#1762	2
C₄×C₃₁×#102	2
#22545	31

Subgroups — C31×#22545 #21479304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22545 #21479304