Subgroups — C₃₁×#22289 #21479048

Order 7936.

754 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1984	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1940	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1917	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1916	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1851	C₂	1	2	yes	maximal
7	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1753	C₂	1	2	yes	maximal
9	C₃₁×#1728	C₂	1	1	yes	maximal, characteristic
10	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₂²⋊Q₈	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#391	C₄	1	1	yes	characteristic
23	C₃₁×#389	C₄	1	1	yes	characteristic
24	C₃₁×#387	C₄	1	2	yes	—
25	C₃₁×#385	C₄	1	2	yes	—
26	C₃₁×#380	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₃₁×#336	C₂²	1	2	yes	—
31	C₃₁×#336	C₂²	1	2	yes	—
32	C₃₁×#336	C₂²	1	2	yes	—
33	C₃₁×#336	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	2	yes	—
37	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₃₁	—	2	2	no	—
42	C₂×M₁₆×C₃₁	—	4	1	no	—
43	C₂×M₁₆×C₃₁	—	4	1	no	—
44	C₂×M₁₆×C₃₁	—	4	1	no	—
45	C₂×M₁₆×C₃₁	—	4	1	no	—
46	C₂×M₁₆×C₃₁	Q₈	1	4	yes	complemented
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₃₁×C₂²⋊Q₈	—	2	4	no	—
52	C₄×Q₈×C₃₁	—	4	1	no	—
53	C₄×Q₈×C₃₁	—	4	1	no	—
54	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₃₁	—	4	1	no	—
63	C₂×C₄²×C₃₁	—	4	1	no	—
64	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
68	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×C₄⋊C₈	—	4	2	no	—
71	C₃₁×C₄⋊C₈	—	4	1	no	—
72	C₃₁×C₄⋊C₈	—	4	1	no	—
73	C₃₁×C₄⋊C₈	—	4	2	no	—
74	C₃₁×C₂²⋊C₈	—	2	4	no	—
75	C₃₁×#97	—	4	2	no	—
76	C₄×C₈×C₃₁	—	4	2	no	—
77	C₃₁×#95	C₂×C₄	1	2	yes	—
78	C₃₁×#95	C₂×C₄	1	2	yes	—
79	C₃₁×#95	C₂×C₄	1	2	yes	—
80	C₃₁×#95	C₂×C₄	1	2	yes	—
81	C₃₁×#95	C₂×C₄	1	2	yes	—
82	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×Q₈×C₃₁	—	4	1	no	—
84	C₂×Q₈×C₃₁	—	4	1	no	—
85	C₂×Q₈×C₃₁	M₁₆	1	4	yes	complemented
86	C₂²×C₄×C₃₁	—	2	1	no	—
87	C₂²×C₄×C₃₁	—	2	1	no	—
88	C₂²×C₄×C₃₁	—	2	1	no	—
89	C₂²×C₄×C₃₁	—	2	1	no	—
90	C₂²×C₄×C₃₁	—	2	2	no	—
91	C₂²×C₄×C₃₁	—	2	2	no	—
92	C₂²×C₄×C₃₁	—	4	1	no	—
93	C₂²×C₄×C₃₁	—	4	1	no	—
94	C₂²×C₄×C₃₁	—	4	1	no	—
95	C₂²×C₄×C₃₁	—	4	1	no	—
96	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
106	C₂²×C₄×C₃₁	#41	1	2	yes	—
107	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
108	M₁₆×C₃₁	—	4	4	no	—
109	M₁₆×C₃₁	—	4	4	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	4	no	—
124	C₄⋊C₄×C₃₁	—	2	4	no	—
125	C₄⋊C₄×C₃₁	—	4	1	no	—
126	C₄⋊C₄×C₃₁	—	4	1	no	—
127	C₄⋊C₄×C₃₁	—	4	2	no	—
128	C₂²⋊C₄×C₃₁	—	2	4	no	—
129	C₂²⋊C₄×C₃₁	—	4	2	no	—
130	C₄²×C₃₁	—	2	2	no	—
131	C₄²×C₃₁	—	2	2	no	—
132	C₄²×C₃₁	—	2	1	no	—
133	C₄²×C₃₁	—	2	1	no	—
134	C₄²×C₃₁	—	2	2	no	—
135	C₄²×C₃₁	—	2	1	no	—
136	C₄²×C₃₁	—	2	1	no	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	—	4	1	no	—
139	C₄²×C₃₁	—	4	1	no	—
140	#22289	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	—	2	1	no	—
142	C₂³×C₃₁	—	2	1	no	—
143	C₂³×C₃₁	—	2	1	no	—
144	C₂³×C₃₁	—	2	1	no	—
145	C₂³×C₃₁	—	2	1	no	—
146	C₂³×C₃₁	—	2	1	no	—
147	C₂³×C₃₁	C₄×Q₈	1	2	yes	—
148	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
149	Q₈×C₃₁	—	4	4	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	1	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	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	4	4	no	—
178	C₂×C₄×C₃₁	—	4	4	no	—
179	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
180	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
181	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
182	C₂×C₄×C₃₁	#131	1	2	yes	—
183	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
185	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
186	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
188	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
189	C₈×C₃₁	—	2	4	no	—
190	C₈×C₃₁	—	4	2	no	—
191	C₈×C₃₁	—	4	2	no	—
192	C₈×C₃₁	—	4	2	no	—
193	C₄×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
194	#1984	C₂×C₃₁	1	2	yes	complemented
195	#1940	C₂×C₃₁	1	2	yes	complemented
196	#1917	C₂×C₃₁	1	2	yes	complemented
197	#1916	C₂×C₃₁	1	2	yes	complemented
198	#1851	C₂×C₃₁	1	2	yes	—
199	#1797	C₂×C₃₁	1	1	yes	characteristic
200	#1753	C₂×C₃₁	1	2	yes	—
201	#1728	C₂×C₃₁	1	1	yes	characteristic
202	C₂²×C₃₁	—	2	2	no	—
203	C₂²×C₃₁	—	2	2	no	—
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	—	4	1	no	—
206	C₂²×C₃₁	—	4	1	no	—
207	C₂²×C₃₁	—	4	1	no	—
208	C₂²×C₃₁	—	4	1	no	—
209	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
210	C₂²×C₃₁	#518	1	1	yes	characteristic
211	C₂²×C₃₁	#520	1	1	yes	characteristic
212	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
213	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
214	C₂²×C₃₁	#568	1	1	yes	characteristic
215	C₂²×C₃₁	#568	1	1	yes	characteristic
216	C₄×C₃₁	—	2	2	no	—
217	C₄×C₃₁	—	2	2	no	—
218	C₄×C₃₁	—	2	4	no	—
219	C₄×C₃₁	—	2	2	no	—
220	C₄×C₃₁	—	4	1	no	—
221	C₄×C₃₁	—	4	2	no	—
222	C₄×C₃₁	—	4	1	no	—
223	C₄×C₃₁	—	4	2	no	—
224	C₄×C₃₁	—	4	2	no	—
225	C₄×C₃₁	#538	1	1	yes	characteristic
226	C₄×C₃₁	#539	1	1	yes	characteristic
227	C₄×C₃₁	#543	1	1	yes	characteristic
228	C₄×C₃₁	#544	1	1	yes	characteristic
229	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
230	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
231	C₂×C₂²⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
232	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
233	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
237	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
238	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
239	C₂×#97	C₂²×C₃₁	1	2	yes	—
240	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
241	#391	C₄×C₃₁	1	1	yes	characteristic
242	#389	C₄×C₃₁	1	1	yes	characteristic
243	#387	C₄×C₃₁	1	2	yes	—
244	#385	C₄×C₃₁	1	2	yes	—
245	#380	C₄×C₃₁	1	1	yes	characteristic
246	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
247	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
248	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
249	#336	C₂²×C₃₁	1	2	yes	—
250	#336	C₂²×C₃₁	1	2	yes	—
251	#336	C₂²×C₃₁	1	2	yes	—
252	#336	C₂²×C₃₁	1	2	yes	—
253	#336	C₂²×C₃₁	1	2	yes	—
254	#336	C₂²×C₃₁	1	2	yes	—
255	#336	C₂²×C₃₁	1	2	yes	—
256	C₂×C₃₁	—	4	2	no	—
257	C₂×C₃₁	#2329	1	1	yes	characteristic
258	C₂×C₃₁	#2962	1	1	yes	characteristic
259	C₂×C₃₁	Q₈×M₁₆	1	1	yes	characteristic
260	C₂×C₃₁	#2973	1	1	yes	characteristic
261	C₂×C₃₁	#2976	1	1	yes	characteristic
262	C₂×C₃₁	#2991	1	1	yes	characteristic
263	C₂×C₃₁	#2995	1	1	yes	characteristic
264	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
265	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
266	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
267	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
268	C₂×M₁₆	—	2	2	no	—
269	C₂×M₁₆	—	4	1	no	—
270	C₂×M₁₆	—	4	1	no	—
271	C₂×M₁₆	—	4	1	no	—
272	C₂×M₁₆	—	4	1	no	—
273	C₂×M₁₆	Q₈×C₃₁	1	4	yes	complemented
274	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
275	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
276	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
277	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
278	C₂²⋊Q₈	—	2	4	no	—
279	C₄×Q₈	—	4	1	no	—
280	C₄×Q₈	—	4	1	no	—
281	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
282	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
283	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
284	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
285	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
286	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
287	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
288	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
289	C₂×C₄²	—	4	1	no	—
290	C₂×C₄²	—	4	1	no	—
291	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
292	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
293	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
294	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
295	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
296	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
297	C₄⋊C₈	—	4	2	no	—
298	C₄⋊C₈	—	4	1	no	—
299	C₄⋊C₈	—	4	1	no	—
300	C₄⋊C₈	—	4	2	no	—
301	C₂²⋊C₈	—	2	4	no	—
302	#97	—	4	2	no	—
303	C₄×C₈	—	4	2	no	—
304	#95	C₂×C₄×C₃₁	1	2	yes	—
305	#95	C₂×C₄×C₃₁	1	2	yes	—
306	#95	C₂×C₄×C₃₁	1	2	yes	—
307	#95	C₂×C₄×C₃₁	1	2	yes	—
308	#95	C₂×C₄×C₃₁	1	2	yes	—
309	C₃₁	#22289	1	1	yes	characteristic, direct product factor, Sylow
310	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
311	C₂×Q₈	—	4	1	no	—
312	C₂×Q₈	—	4	1	no	—
313	C₂×Q₈	M₁₆×C₃₁	1	4	yes	complemented
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	4	1	no	—
321	C₂²×C₄	—	4	1	no	—
322	C₂²×C₄	—	4	1	no	—
323	C₂²×C₄	—	4	1	no	—
324	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
325	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
326	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
327	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
328	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
329	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
330	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
331	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
332	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
334	C₂²×C₄	#41×C₃₁	1	2	yes	—
335	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
336	M₁₆	—	4	4	no	—
337	M₁₆	—	4	4	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	4	no	—
352	C₄⋊C₄	—	2	4	no	—
353	C₄⋊C₄	—	4	1	no	—
354	C₄⋊C₄	—	4	1	no	—
355	C₄⋊C₄	—	4	2	no	—
356	C₂²⋊C₄	—	2	4	no	—
357	C₂²⋊C₄	—	4	2	no	—
358	C₄²	—	2	2	no	—
359	C₄²	—	2	2	no	—
360	C₄²	—	2	1	no	—
361	C₄²	—	2	1	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	2	1	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	4	1	no	—
367	C₄²	—	4	1	no	—
368	C₂³	—	2	1	no	—
369	C₂³	—	2	1	no	—
370	C₂³	—	2	1	no	—
371	C₂³	—	2	1	no	—
372	C₂³	—	2	1	no	—
373	C₂³	—	2	1	no	—
374	C₂³	C₄×Q₈×C₃₁	1	2	yes	—
375	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
376	Q₈	—	4	4	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	4	4	no	—
405	C₂×C₄	—	4	4	no	—
406	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
407	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
408	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
409	C₂×C₄	C₃₁×#131	1	2	yes	—
410	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
411	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
412	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
413	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
414	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
415	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
416	C₈	—	2	4	no	—
417	C₈	—	4	2	no	—
418	C₈	—	4	2	no	—
419	C₈	—	4	2	no	—
420	C₂²	—	2	2	no	—
421	C₂²	—	2	2	no	—
422	C₂²	—	2	2	no	—
423	C₂²	—	4	1	no	—
424	C₂²	—	4	1	no	—
425	C₂²	—	4	1	no	—
426	C₂²	—	4	1	no	—
427	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
428	C₂²	C₃₁×#518	1	1	yes	characteristic
429	C₂²	C₃₁×#520	1	1	yes	characteristic
430	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
431	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
432	C₂²	C₃₁×#568	1	1	yes	characteristic
433	C₂²	C₃₁×#568	1	1	yes	characteristic
434	C₄	—	2	2	no	—
435	C₄	—	2	2	no	—
436	C₄	—	2	4	no	—
437	C₄	—	2	2	no	—
438	C₄	—	4	1	no	—
439	C₄	—	4	1	no	—
440	C₄	—	4	2	no	—
441	C₄	—	4	2	no	—
442	C₄	—	4	2	no	—
443	C₄	C₃₁×#538	1	1	yes	characteristic
444	C₄	C₃₁×#539	1	1	yes	characteristic
445	C₄	C₃₁×#543	1	1	yes	characteristic
446	C₄	C₃₁×#544	1	1	yes	characteristic
447	C₂	—	4	2	no	—
448	C₂	C₃₁×#2329	1	1	yes	characteristic
449	C₂	C₃₁×#2962	1	1	yes	characteristic
450	C₂	Q₈×M₁₆×C₃₁	1	1	yes	characteristic
451	C₂	C₃₁×#2973	1	1	yes	characteristic
452	C₂	C₃₁×#2976	1	1	yes	characteristic
453	C₂	C₃₁×#2991	1	1	yes	characteristic
454	C₂	C₃₁×#2995	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1984	C₂	semidirect
3. C₃₁×#1940	C₂	semidirect
4. C₃₁×#1917	C₂	semidirect
5. C₃₁×#1916	C₂	semidirect
46. C₂×M₁₆×C₃₁	Q₈	semidirect
85. C₂×Q₈×C₃₁	M₁₆	semidirect
140. #22289	C₃₁	direct (N × H)
194. #1984	C₂×C₃₁	semidirect
195. #1940	C₂×C₃₁	semidirect
196. #1917	C₂×C₃₁	semidirect
197. #1916	C₂×C₃₁	semidirect
273. C₂×M₁₆	Q₈×C₃₁	semidirect
309. C₃₁	#22289	direct (N × H)
313. C₂×Q₈	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃₁×C₂²⋊Q₈	2
C₃₁×#1984	2
C₃₁×#1940	2
C₃₁×#1917	2
C₃₁×#1916	2
C₃₁×#1851	2
C₃₁×#1797	2
C₃₁×#1753	2
C₃₁×#1728	2
#22289	31

Subgroups — C31×#22289 #21479048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22289 #21479048