Subgroups — C₃₁×#22811 #21479570

Order 7936.

846 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2355	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2354	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1973	C₂	1	2	yes	maximal
5	C₃₁×#1928	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1925	C₂	1	2	yes	maximal
7	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1776	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#117	—	2	1	no	—
16	C₄×SD₁₆×C₃₁	C₄	1	8	yes	complemented
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×#107	C₂²	1	2	yes	—
19	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#103	C₂²	1	2	yes	—
22	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#97	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#391	C₂²	1	2	yes	—
28	C₃₁×#390	C₂²	1	2	yes	—
29	C₃₁×#388	—	2	2	no	—
30	C₃₁×#387	—	2	2	no	—
31	C₃₁×#384	C₂²	1	1	yes	characteristic
32	C₃₁×#384	C₂²	1	1	yes	characteristic
33	C₃₁×#382	—	2	1	no	—
34	C₃₁×#380	—	2	1	no	—
35	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
36	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
38	C₄×C₄⋊C₄×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	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	—	2	1	no	—
45	C₂×SD₁₆×C₃₁	—	2	2	no	—
46	C₂×SD₁₆×C₃₁	—	2	2	no	—
47	C₂×SD₁₆×C₃₁	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₃₁	—	2	2	no	—
52	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
53	C₄×D₄×C₃₁	—	2	2	no	—
54	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
55	C₃₁×#117	—	4	4	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	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	D₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄²×C₃₁	—	2	1	no	—
71	C₂×C₄²×C₃₁	—	2	2	no	—
72	C₂×C₄²×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₃₁×#107	—	2	4	no	—
76	C₃₁×#107	—	4	2	no	—
77	C₃₁×#106	C₂×C₄	1	4	yes	—
78	C₃₁×C₄⋊C₈	—	2	2	no	—
79	C₃₁×#103	—	4	2	no	—
80	C₃₁×#103	C₂×C₄	1	4	yes	—
81	C₃₁×#102	—	4	2	no	—
82	C₃₁×#102	C₂×C₄	1	4	yes	—
83	C₃₁×#97	—	2	4	no	—
84	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
85	C₃₁×#95	—	2	2	no	—
86	C₃₁×#95	—	2	2	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₃₁×#95	D₄	1	2	yes	—
90	C₂⁴×C₃₁	—	2	1	no	—
91	C₂×Q₈×C₃₁	—	2	1	no	—
92	C₂×Q₈×C₃₁	—	2	1	no	—
93	C₂×Q₈×C₃₁	—	2	1	no	—
94	C₂×Q₈×C₃₁	—	2	1	no	—
95	C₂×Q₈×C₃₁	—	2	1	no	—
96	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂×D₄×C₃₁	—	2	1	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	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	2	1	no	—
106	C₂²×C₄×C₃₁	—	2	4	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	4	2	no	—
110	C₂²×C₄×C₃₁	—	4	2	no	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₃₁	C₂×D₄	1	2	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	C₂⁴	1	1	yes	characteristic
116	SD₁₆×C₃₁	—	2	8	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
124	C₂×C₈×C₃₁	#41	1	2	yes	—
125	C₂×C₈×C₃₁	#41	1	2	yes	—
126	C₄⋊C₄×C₃₁	—	2	4	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	1	no	—
130	C₄⋊C₄×C₃₁	—	2	1	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	—	4	2	no	—
136	C₄⋊C₄×C₃₁	—	4	2	no	—
137	C₄⋊C₄×C₃₁	—	8	1	no	—
138	C₄⋊C₄×C₃₁	—	8	1	no	—
139	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
140	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
141	C₂²⋊C₄×C₃₁	—	2	4	no	—
142	C₂²⋊C₄×C₃₁	—	4	4	no	—
143	C₂²⋊C₄×C₃₁	—	8	2	no	—
144	C₄²×C₃₁	—	2	4	no	—
145	C₄²×C₃₁	—	2	2	no	—
146	C₄²×C₃₁	—	2	2	no	—
147	C₄²×C₃₁	—	4	4	no	—
148	C₄²×C₃₁	—	8	2	no	—
149	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
150	C₄²×C₃₁	#41	1	2	yes	—
151	#22811	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂³×C₃₁	—	2	2	no	—
153	C₂³×C₃₁	—	4	1	no	—
154	C₂³×C₃₁	—	4	1	no	—
155	C₂³×C₃₁	—	4	1	no	—
156	C₂³×C₃₁	—	4	1	no	—
157	C₂³×C₃₁	—	4	1	no	—
158	C₂³×C₃₁	—	4	1	no	—
159	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
160	Q₈×C₃₁	—	2	2	no	—
161	Q₈×C₃₁	—	2	2	no	—
162	Q₈×C₃₁	—	4	1	no	—
163	Q₈×C₃₁	—	4	1	no	—
164	D₄×C₃₁	—	2	2	no	—
165	D₄×C₃₁	—	2	2	no	—
166	D₄×C₃₁	—	4	1	no	—
167	D₄×C₃₁	—	4	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	4	1	no	—
180	C₂×C₄×C₃₁	—	4	1	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	8	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	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	C₄×D₄	1	2	yes	—
193	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
194	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
195	C₂×C₄×C₃₁	#137	1	2	yes	—
196	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
201	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
202	C₈×C₃₁	—	2	2	no	—
203	C₈×C₃₁	—	4	2	no	—
204	C₈×C₃₁	#117	1	4	yes	complemented
205	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
206	#2355	C₂×C₃₁	1	1	yes	characteristic, complemented
207	#2354	C₂×C₃₁	1	1	yes	characteristic
208	#1973	C₂×C₃₁	1	2	yes	—
209	#1928	C₂×C₃₁	1	2	yes	complemented
210	#1925	C₂×C₃₁	1	2	yes	—
211	#1845	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#1803	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#1802	C₂×C₃₁	1	1	yes	characteristic
214	#1776	C₂×C₃₁	1	2	yes	complemented
215	#1770	C₂×C₃₁	1	1	yes	characteristic, complemented
216	C₂²×C₃₁	—	4	2	no	—
217	C₂²×C₃₁	—	4	2	no	—
218	C₂²×C₃₁	—	4	2	no	—
219	C₂²×C₃₁	—	8	1	no	—
220	C₂²×C₃₁	—	8	1	no	—
221	C₂²×C₃₁	—	8	1	no	—
222	C₂²×C₃₁	—	8	1	no	—
223	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
224	C₂²×C₃₁	#547	1	1	yes	characteristic
225	C₂²×C₃₁	#550	1	1	yes	characteristic
226	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
227	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
228	C₂²×C₃₁	#576	1	1	yes	characteristic
229	C₂²×C₃₁	#578	1	1	yes	characteristic
230	C₄×C₃₁	—	2	2	no	—
231	C₄×C₃₁	—	4	2	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	—	4	2	no	—
234	C₄×C₃₁	—	4	2	no	—
235	C₄×C₃₁	—	4	2	no	—
236	C₄×C₃₁	—	8	2	no	—
237	C₄×C₃₁	—	8	2	no	—
238	C₄×C₃₁	#440	1	4	yes	—
239	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
240	C₄×C₃₁	#520	1	1	yes	characteristic
241	C₄×C₃₁	#555	1	1	yes	characteristic
242	C₄×C₃₁	#556	1	1	yes	characteristic
243	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×#117	—	2	1	no	—
247	C₄×SD₁₆	C₄×C₃₁	1	8	yes	complemented
248	C₂×#107	C₂²×C₃₁	1	2	yes	—
249	C₂×#107	C₂²×C₃₁	1	2	yes	—
250	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×#103	C₂²×C₃₁	1	2	yes	—
253	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
254	C₂×#102	C₂²×C₃₁	1	2	yes	—
255	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×#97	C₂²×C₃₁	1	2	yes	—
257	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
258	#391	C₂²×C₃₁	1	2	yes	—
259	#390	C₂²×C₃₁	1	2	yes	—
260	#388	—	2	2	no	—
261	#387	—	2	2	no	—
262	#384	C₂²×C₃₁	1	1	yes	characteristic
263	#384	C₂²×C₃₁	1	1	yes	characteristic
264	#382	—	2	1	no	—
265	#380	—	2	1	no	—
266	C₄×C₄⋊C₄	—	2	1	no	—
267	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
268	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
269	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
270	#340	C₂²×C₃₁	1	2	yes	—
271	#340	C₂²×C₃₁	1	2	yes	—
272	#336	C₂²×C₃₁	1	2	yes	—
273	C₂×C₃₁	—	8	2	no	—
274	C₂×C₃₁	#2354	1	1	yes	characteristic
275	C₂×C₃₁	C₂×#440	1	1	yes	characteristic
276	C₂×C₃₁	#2951	1	1	yes	characteristic
277	C₂×C₃₁	#3353	1	1	yes	characteristic
278	C₂×C₃₁	#3357	1	1	yes	characteristic
279	C₂×C₃₁	#3398	1	1	yes	characteristic
280	C₂×C₃₁	#3399	1	1	yes	characteristic
281	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
282	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
283	C₂³×C₄	—	2	1	no	—
284	C₂×SD₁₆	—	2	2	no	—
285	C₂×SD₁₆	—	2	2	no	—
286	C₂×SD₁₆	C₂×C₄×C₃₁	1	4	yes	—
287	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
288	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
289	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
290	C₄×Q₈	—	2	2	no	—
291	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
292	C₄×D₄	—	2	2	no	—
293	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
294	#117	—	4	4	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	—	2	1	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
301	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
303	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
304	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
306	C₂×C₂²⋊C₄	—	2	1	no	—
307	C₂×C₂²⋊C₄	—	2	2	no	—
308	C₂×C₂²⋊C₄	—	2	2	no	—
309	C₂×C₄²	—	2	1	no	—
310	C₂×C₄²	—	2	2	no	—
311	C₂×C₄²	—	2	2	no	—
312	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
313	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
314	#107	—	2	4	no	—
315	#107	—	4	2	no	—
316	#106	C₂×C₄×C₃₁	1	4	yes	—
317	C₄⋊C₈	—	2	2	no	—
318	#103	—	4	2	no	—
319	#103	C₂×C₄×C₃₁	1	4	yes	—
320	#102	—	4	2	no	—
321	#102	C₂×C₄×C₃₁	1	4	yes	—
322	#97	—	2	4	no	—
323	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
324	#95	—	2	2	no	—
325	#95	—	2	2	no	—
326	#95	—	2	2	no	—
327	#95	—	2	2	no	—
328	#95	D₄×C₃₁	1	2	yes	—
329	C₃₁	#22811	1	1	yes	characteristic, direct product factor, Sylow
330	C₂⁴	—	2	1	no	—
331	C₂×Q₈	—	2	1	no	—
332	C₂×Q₈	—	2	1	no	—
333	C₂×Q₈	—	2	1	no	—
334	C₂×Q₈	—	2	1	no	—
335	C₂×Q₈	—	2	1	no	—
336	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
337	C₂×D₄	—	2	1	no	—
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	4	2	no	—
350	C₂²×C₄	—	4	2	no	—
351	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
352	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
353	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
356	SD₁₆	—	2	8	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
363	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
364	C₂×C₈	#41×C₃₁	1	2	yes	—
365	C₂×C₈	#41×C₃₁	1	2	yes	—
366	C₄⋊C₄	—	2	4	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	4	2	no	—
376	C₄⋊C₄	—	4	2	no	—
377	C₄⋊C₄	—	8	1	no	—
378	C₄⋊C₄	—	8	1	no	—
379	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
380	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
381	C₂²⋊C₄	—	2	4	no	—
382	C₂²⋊C₄	—	4	4	no	—
383	C₂²⋊C₄	—	8	2	no	—
384	C₄²	—	2	4	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	4	4	no	—
388	C₄²	—	8	2	no	—
389	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
390	C₄²	#41×C₃₁	1	2	yes	—
391	C₂³	—	2	2	no	—
392	C₂³	—	4	1	no	—
393	C₂³	—	4	1	no	—
394	C₂³	—	4	1	no	—
395	C₂³	—	4	1	no	—
396	C₂³	—	4	1	no	—
397	C₂³	—	4	1	no	—
398	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
399	Q₈	—	2	2	no	—
400	Q₈	—	2	2	no	—
401	Q₈	—	4	1	no	—
402	Q₈	—	4	1	no	—
403	D₄	—	2	2	no	—
404	D₄	—	2	2	no	—
405	D₄	—	4	1	no	—
406	D₄	—	4	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	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	—	4	1	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	8	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
432	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
433	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
434	C₂×C₄	C₃₁×#137	1	2	yes	—
435	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
436	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
437	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
438	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
439	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
440	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
441	C₈	—	2	2	no	—
442	C₈	—	4	2	no	—
443	C₈	C₃₁×#117	1	4	yes	complemented
444	C₂²	—	4	2	no	—
445	C₂²	—	4	2	no	—
446	C₂²	—	4	2	no	—
447	C₂²	—	8	1	no	—
448	C₂²	—	8	1	no	—
449	C₂²	—	8	1	no	—
450	C₂²	—	8	1	no	—
451	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
452	C₂²	C₃₁×#547	1	1	yes	characteristic
453	C₂²	C₃₁×#550	1	1	yes	characteristic
454	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
455	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
456	C₂²	C₃₁×#576	1	1	yes	characteristic
457	C₂²	C₃₁×#578	1	1	yes	characteristic
458	C₄	—	2	2	no	—
459	C₄	—	4	2	no	—
460	C₄	—	4	2	no	—
461	C₄	—	4	2	no	—
462	C₄	—	4	2	no	—
463	C₄	—	4	2	no	—
464	C₄	—	8	2	no	—
465	C₄	—	8	2	no	—
466	C₄	C₃₁×#440	1	4	yes	—
467	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
468	C₄	C₃₁×#520	1	1	yes	characteristic
469	C₄	C₃₁×#555	1	1	yes	characteristic
470	C₄	C₃₁×#556	1	1	yes	characteristic
471	C₂	—	8	2	no	—
472	C₂	C₃₁×#2354	1	1	yes	characteristic
473	C₂	C₂×C₃₁×#440	1	1	yes	characteristic
474	C₂	C₃₁×#2951	1	1	yes	characteristic
475	C₂	C₃₁×#3353	1	1	yes	characteristic
476	C₂	C₃₁×#3357	1	1	yes	characteristic
477	C₂	C₃₁×#3398	1	1	yes	characteristic
478	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	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂×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
2. C₃₁×#2355	C₂	semidirect
5. C₃₁×#1928	C₂	semidirect
7. C₃₁×#1845	C₂	semidirect
8. C₃₁×#1803	C₂	semidirect
10. C₃₁×#1776	C₂	semidirect
11. C₃₁×#1770	C₂	semidirect
16. C₄×SD₁₆×C₃₁	C₄	semidirect
151. #22811	C₃₁	direct (N × H)
204. C₈×C₃₁	#117	semidirect
206. #2355	C₂×C₃₁	semidirect
209. #1928	C₂×C₃₁	semidirect
211. #1845	C₂×C₃₁	semidirect
212. #1803	C₂×C₃₁	semidirect
214. #1776	C₂×C₃₁	semidirect
215. #1770	C₂×C₃₁	semidirect
247. C₄×SD₁₆	C₄×C₃₁	semidirect
329. C₃₁	#22811	direct (N × H)
443. C₈	C₃₁×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×SD₁₆×C₃₁	2
C₃₁×#2355	2
C₃₁×#2354	2
C₃₁×#1973	2
C₃₁×#1928	2
C₃₁×#1925	2
C₃₁×#1845	2
C₃₁×#1803	2
C₃₁×#1802	2
C₃₁×#1776	2
C₃₁×#1770	2
#22811	31

Subgroups — C31×#22811 #21479570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22811 #21479570