Subgroups — C₃₁×#29160 #21485919

Order 7936.

828 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2857	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2100	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2095	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2057	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2024	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1921	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1892	C₂	1	2	yes	maximal
9	C₃₁×#1863	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1808	C₂	1	2	yes	maximal
11	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
15	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#107	C₂²	1	2	yes	—
19	C₂×C₃₁×#107	C₂²	1	2	yes	—
20	C₂×C₃₁×#103	C₂²	1	2	yes	—
21	C₂×C₃₁×#103	C₂²	1	2	yes	—
22	C₃₁×#391	—	2	1	no	—
23	C₃₁×#391	C₂²	1	1	yes	characteristic
24	C₃₁×#389	—	2	1	no	—
25	C₃₁×#389	C₂²	1	1	yes	characteristic
26	C₃₁×#389	C₂²	1	2	yes	—
27	C₃₁×#387	—	2	1	no	—
28	C₃₁×#385	—	2	1	no	—
29	C₃₁×#384	C₂²	1	1	yes	characteristic
30	C₃₁×#380	—	2	1	no	—
31	C₃₁×#380	C₂²	1	1	yes	characteristic
32	C₃₁×#380	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₃₁×#336	C₂²	1	1	yes	characteristic
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	—	2	4	no	—
49	C₂²×C₈×C₃₁	—	2	4	no	—
50	C₂²×C₈×C₃₁	—	4	2	no	—
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₃₁×C₄⋊Q₈	—	4	2	no	—
54	C₃₁×C₂²⋊Q₈	—	2	2	no	—
55	C₃₁×C₂²⋊Q₈	—	4	4	no	—
56	C₃₁×C₂²⋊Q₈	Q₈	1	4	yes	complemented
57	C₄×Q₈×C₃₁	—	4	2	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	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₄²×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	—	4	2	no	—
76	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	D₄	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	—	2	4	no	—
81	C₃₁×#107	—	4	2	no	—
82	C₃₁×#103	—	2	4	no	—
83	C₃₁×#103	—	4	2	no	—
84	C₃₁×#95	—	2	1	no	—
85	C₃₁×#95	—	2	1	no	—
86	C₃₁×#95	—	2	1	no	—
87	C₃₁×#95	—	2	1	no	—
88	C₃₁×#95	—	2	1	no	—
89	C₃₁×#95	—	2	2	no	—
90	C₃₁×#95	—	2	2	no	—
91	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×Q₈×C₃₁	—	2	1	no	—
93	C₂×Q₈×C₃₁	—	2	2	no	—
94	C₂×Q₈×C₃₁	—	2	2	no	—
95	C₂×Q₈×C₃₁	—	4	2	no	—
96	C₂×Q₈×C₃₁	—	4	2	no	—
97	C₂×Q₈×C₃₁	—	4	2	no	—
98	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	1	no	—
106	C₂²×C₄×C₃₁	—	2	1	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	4	1	no	—
110	C₂²×C₄×C₃₁	—	4	1	no	—
111	C₂²×C₄×C₃₁	—	4	1	no	—
112	C₂²×C₄×C₃₁	Q₁₆	1	4	yes	—
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
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×Q₈	1	2	yes	—
119	C₂²×C₄×C₃₁	#41	1	2	yes	—
120	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
121	C₂×C₈×C₃₁	—	2	2	no	—
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	—	4	8	no	—
127	C₂×C₈×C₃₁	#41	1	4	yes	—
128	C₄⋊C₄×C₃₁	—	2	4	no	—
129	C₄⋊C₄×C₃₁	—	2	1	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	4	2	no	—
137	C₄⋊C₄×C₃₁	—	4	2	no	—
138	C₄⋊C₄×C₃₁	—	4	2	no	—
139	C₄⋊C₄×C₃₁	—	4	4	no	—
140	C₄⋊C₄×C₃₁	—	4	2	no	—
141	C₄⋊C₄×C₃₁	—	4	2	no	—
142	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
143	C₂²⋊C₄×C₃₁	—	2	4	no	—
144	C₂²⋊C₄×C₃₁	—	4	2	no	—
145	C₂²⋊C₄×C₃₁	—	4	4	no	—
146	C₄²×C₃₁	—	2	2	no	—
147	C₄²×C₃₁	—	2	2	no	—
148	C₄²×C₃₁	—	4	1	no	—
149	C₄²×C₃₁	—	4	1	no	—
150	C₄²×C₃₁	—	4	2	no	—
151	C₄²×C₃₁	—	4	2	no	—
152	#29160	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂³×C₃₁	—	2	1	no	—
154	C₂³×C₃₁	—	2	1	no	—
155	C₂³×C₃₁	—	2	1	no	—
156	C₂³×C₃₁	—	2	1	no	—
157	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
158	C₂³×C₃₁	C₂×Q₁₆	1	2	yes	—
159	C₂³×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
160	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
161	Q₈×C₃₁	—	2	4	no	—
162	Q₈×C₃₁	—	4	2	no	—
163	Q₈×C₃₁	—	8	4	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	4	no	—
173	C₂×C₄×C₃₁	—	2	4	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	2	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	—	4	4	no	—
180	C₂×C₄×C₃₁	—	4	4	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	—	4	1	no	—
183	C₂×C₄×C₃₁	—	4	1	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	2	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	2	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	2	no	—
197	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
198	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
199	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
200	C₂×C₄×C₃₁	#135	1	2	yes	—
201	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
204	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
205	C₈×C₃₁	—	2	4	no	—
206	C₈×C₃₁	—	4	2	no	—
207	#2857	C₂×C₃₁	1	1	yes	characteristic
208	C₄×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
209	#2100	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#2095	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#2057	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#2024	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#1921	C₂×C₃₁	1	2	yes	complemented
214	#1892	C₂×C₃₁	1	2	yes	—
215	#1863	C₂×C₃₁	1	2	yes	complemented
216	#1808	C₂×C₃₁	1	2	yes	—
217	C₂×#336	C₂×C₃₁	1	1	yes	characteristic
218	C₂²×C₃₁	—	2	2	no	—
219	C₂²×C₃₁	—	2	2	no	—
220	C₂²×C₃₁	—	2	2	no	—
221	C₂²×C₃₁	—	2	2	no	—
222	C₂²×C₃₁	—	2	2	no	—
223	C₂²×C₃₁	—	2	2	no	—
224	C₂²×C₃₁	#477	1	4	yes	—
225	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
226	C₂²×C₃₁	#532	1	1	yes	characteristic
227	C₂²×C₃₁	#534	1	1	yes	characteristic
228	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
229	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
230	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
231	C₂²×C₃₁	#575	1	1	yes	characteristic
232	C₄×C₃₁	—	2	2	no	—
233	C₄×C₃₁	—	2	2	no	—
234	C₄×C₃₁	—	2	2	no	—
235	C₄×C₃₁	—	2	2	no	—
236	C₄×C₃₁	—	4	2	no	—
237	C₄×C₃₁	—	4	2	no	—
238	C₄×C₃₁	—	4	1	no	—
239	C₄×C₃₁	—	8	2	no	—
240	C₄×C₃₁	—	8	1	no	—
241	C₄×C₃₁	—	8	1	no	—
242	C₄×C₃₁	—	8	2	no	—
243	C₄×C₃₁	#551	1	2	yes	—
244	C₄×C₃₁	#554	1	2	yes	—
245	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
246	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×C₂²⋊Q₈	—	2	1	no	—
248	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
250	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×#107	C₂²×C₃₁	1	2	yes	—
252	C₂×#107	C₂²×C₃₁	1	2	yes	—
253	C₂×#103	C₂²×C₃₁	1	2	yes	—
254	C₂×#103	C₂²×C₃₁	1	2	yes	—
255	#391	—	2	1	no	—
256	#391	C₂²×C₃₁	1	1	yes	characteristic
257	#389	—	2	1	no	—
258	#389	C₂²×C₃₁	1	1	yes	characteristic
259	#389	C₂²×C₃₁	1	2	yes	—
260	#387	—	2	1	no	—
261	#385	—	2	1	no	—
262	#384	C₂²×C₃₁	1	1	yes	characteristic
263	#380	—	2	1	no	—
264	#380	C₂²×C₃₁	1	1	yes	characteristic
265	#380	C₂²×C₃₁	1	2	yes	—
266	C₄×C₄⋊C₄	—	2	1	no	—
267	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
268	C₄×C₂²⋊C₄	—	2	1	no	—
269	#340	C₂²×C₃₁	1	2	yes	—
270	#340	C₂²×C₃₁	1	2	yes	—
271	#340	C₂²×C₃₁	1	2	yes	—
272	#340	C₂²×C₃₁	1	2	yes	—
273	#336	C₂²×C₃₁	1	1	yes	characteristic
274	#336	C₂²×C₃₁	1	1	yes	characteristic
275	#336	C₂²×C₃₁	1	2	yes	—
276	C₂×C₃₁	—	2	4	no	—
277	C₂×C₃₁	#2564	1	1	yes	characteristic
278	C₂×C₃₁	C₂×#477	1	1	yes	characteristic
279	C₂×C₃₁	#3076	1	1	yes	characteristic
280	C₂×C₃₁	#3170	1	1	yes	characteristic
281	C₂×C₃₁	#3180	1	1	yes	characteristic
282	C₂×C₃₁	#3188	1	1	yes	characteristic
283	C₂×C₃₁	#3189	1	1	yes	characteristic
284	C₂²×Q₈	—	2	1	no	—
285	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
286	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
287	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
288	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
289	C₂²×C₈	—	2	4	no	—
290	C₂²×C₈	—	2	4	no	—
291	C₂²×C₈	—	4	2	no	—
292	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
293	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
294	C₄⋊Q₈	—	4	2	no	—
295	C₂²⋊Q₈	—	2	2	no	—
296	C₂²⋊Q₈	—	4	4	no	—
297	C₂²⋊Q₈	Q₈×C₃₁	1	4	yes	complemented
298	C₄×Q₈	—	4	2	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	2	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
308	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
309	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
310	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
311	C₂×C₂²⋊C₄	—	2	1	no	—
312	C₂×C₂²⋊C₄	—	2	2	no	—
313	C₂×C₂²⋊C₄	—	2	1	no	—
314	C₂×C₄²	—	2	1	no	—
315	C₂×C₄²	—	2	1	no	—
316	C₂×C₄²	—	4	2	no	—
317	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
318	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
319	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
320	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
321	#107	—	2	4	no	—
322	#107	—	4	2	no	—
323	#103	—	2	4	no	—
324	#103	—	4	2	no	—
325	#95	—	2	1	no	—
326	#95	—	2	1	no	—
327	#95	—	2	1	no	—
328	#95	—	2	1	no	—
329	#95	—	2	1	no	—
330	#95	—	2	2	no	—
331	#95	—	2	2	no	—
332	C₃₁	#29160	1	1	yes	characteristic, direct product factor, Sylow
333	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₂×Q₈	—	2	1	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	4	2	no	—
338	C₂×Q₈	—	4	2	no	—
339	C₂×Q₈	—	4	2	no	—
340	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	4	1	no	—
352	C₂²×C₄	—	4	1	no	—
353	C₂²×C₄	—	4	1	no	—
354	C₂²×C₄	Q₁₆×C₃₁	1	4	yes	—
355	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
356	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
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	1	yes	characteristic
360	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
361	C₂²×C₄	#41×C₃₁	1	2	yes	—
362	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	4	8	no	—
369	C₂×C₈	#41×C₃₁	1	4	yes	—
370	C₄⋊C₄	—	2	4	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	4	2	no	—
379	C₄⋊C₄	—	4	2	no	—
380	C₄⋊C₄	—	4	2	no	—
381	C₄⋊C₄	—	4	4	no	—
382	C₄⋊C₄	—	4	2	no	—
383	C₄⋊C₄	—	4	2	no	—
384	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
385	C₂²⋊C₄	—	2	4	no	—
386	C₂²⋊C₄	—	4	2	no	—
387	C₂²⋊C₄	—	4	4	no	—
388	C₄²	—	2	2	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	4	1	no	—
391	C₄²	—	4	1	no	—
392	C₄²	—	4	2	no	—
393	C₄²	—	4	2	no	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	1	no	—
396	C₂³	—	2	1	no	—
397	C₂³	—	2	1	no	—
398	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
399	C₂³	C₂×Q₁₆×C₃₁	1	2	yes	—
400	C₂³	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
401	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
402	Q₈	—	2	4	no	—
403	Q₈	—	4	2	no	—
404	Q₈	—	8	4	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	4	4	no	—
421	C₂×C₄	—	4	4	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
439	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
440	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
441	C₂×C₄	C₃₁×#135	1	2	yes	—
442	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
443	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
444	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
445	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
446	C₈	—	2	4	no	—
447	C₈	—	4	2	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	2	2	no	—
451	C₂²	—	2	2	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	2	no	—
454	C₂²	C₃₁×#477	1	4	yes	—
455	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
456	C₂²	C₃₁×#532	1	1	yes	characteristic
457	C₂²	C₃₁×#534	1	1	yes	characteristic
458	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
459	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
460	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
461	C₂²	C₃₁×#575	1	1	yes	characteristic
462	C₄	—	2	2	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	2	no	—
465	C₄	—	2	2	no	—
466	C₄	—	4	2	no	—
467	C₄	—	4	2	no	—
468	C₄	—	4	1	no	—
469	C₄	—	8	2	no	—
470	C₄	—	8	1	no	—
471	C₄	—	8	2	no	—
472	C₄	—	8	1	no	—
473	C₄	C₃₁×#551	1	2	yes	—
474	C₄	C₃₁×#554	1	2	yes	—
475	C₂	—	2	4	no	—
476	C₂	C₃₁×#2564	1	1	yes	characteristic
477	C₂	C₂×C₃₁×#477	1	1	yes	characteristic
478	C₂	C₃₁×#3076	1	1	yes	characteristic
479	C₂	C₃₁×#3170	1	1	yes	characteristic
480	C₂	C₃₁×#3180	1	1	yes	characteristic
481	C₂	C₃₁×#3188	1	1	yes	characteristic
482	C₂	C₃₁×#3189	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2100	C₂	semidirect
4. C₃₁×#2095	C₂	semidirect
5. C₃₁×#2057	C₂	semidirect
6. C₃₁×#2024	C₂	semidirect
7. C₃₁×#1921	C₂	semidirect
9. C₃₁×#1863	C₂	semidirect
56. C₃₁×C₂²⋊Q₈	Q₈	semidirect
152. #29160	C₃₁	direct (N × H)
209. #2100	C₂×C₃₁	semidirect
210. #2095	C₂×C₃₁	semidirect
211. #2057	C₂×C₃₁	semidirect
212. #2024	C₂×C₃₁	semidirect
213. #1921	C₂×C₃₁	semidirect
215. #1863	C₂×C₃₁	semidirect
297. C₂²⋊Q₈	Q₈×C₃₁	semidirect
332. C₃₁	#29160	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2857	2
C₄×C₃₁×C₂²⋊Q₈	2
C₃₁×#2100	2
C₃₁×#2095	2
C₃₁×#2057	2
C₃₁×#2024	2
C₃₁×#1921	2
C₃₁×#1892	2
C₃₁×#1863	2
C₃₁×#1808	2
C₂×C₃₁×#336	2
#29160	31

Subgroups — C31×#29160 #21485919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29160 #21485919