Subgroups — C₃₁×#22701 #21479460

Order 7936.

916 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	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2293	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1851	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1850	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1798	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1729	C₂	1	2	yes	maximal, complemented
12	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#117	C₂²	1	2	yes	—
16	C₃₁×#436	—	2	4	no	—
17	C₈×D₄×C₃₁	—	2	4	no	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#390	C₄	1	1	yes	characteristic
29	C₃₁×#390	C₄	1	1	yes	characteristic
30	C₃₁×#389	C₄	1	1	yes	characteristic
31	C₃₁×#389	C₄	1	1	yes	characteristic
32	C₃₁×#388	C₄	1	2	yes	—
33	C₃₁×#388	C₄	1	2	yes	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₄³×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#336	C₂²	1	1	yes	characteristic
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
45	C₂×M₁₆×C₃₁	—	2	4	no	—
46	C₂×M₁₆×C₃₁	—	2	4	no	—
47	C₂×M₁₆×C₃₁	—	4	2	no	—
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	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₄×D₄×C₃₁	D₄	1	4	yes	—
58	C₄×D₄×C₃₁	Q₈	1	4	yes	—
59	C₃₁×#117	—	4	2	no	—
60	C₃₁×#117	—	4	2	no	—
61	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
69	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
70	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
71	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊C₈	—	2	2	no	—
80	C₃₁×C₄⋊C₈	—	2	2	no	—
81	C₃₁×C₄⋊C₈	—	2	2	no	—
82	C₃₁×C₄⋊C₈	—	2	2	no	—
83	C₃₁×C₄⋊C₈	—	4	2	no	—
84	C₃₁×C₄⋊C₈	—	4	2	no	—
85	C₃₁×C₂²⋊C₈	—	2	4	no	—
86	C₃₁×C₂²⋊C₈	—	2	4	no	—
87	C₄×C₈×C₃₁	—	2	2	no	—
88	C₄×C₈×C₃₁	—	2	2	no	—
89	C₃₁×#95	C₂×C₄	1	2	yes	—
90	C₃₁×#95	C₂×C₄	1	2	yes	—
91	C₃₁×#95	C₂×C₄	1	2	yes	—
92	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
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	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
110	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
111	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
112	M₁₆×C₃₁	—	4	8	no	—
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₂×C₈×C₃₁	—	2	1	no	—
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	1	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	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	4	8	no	—
130	C₂×C₈×C₃₁	#41	1	2	yes	—
131	C₂×C₈×C₃₁	#41	1	2	yes	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	4	2	no	—
135	C₄⋊C₄×C₃₁	—	4	2	no	—
136	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
137	C₂²⋊C₄×C₃₁	—	2	4	no	—
138	C₂²⋊C₄×C₃₁	—	4	2	no	—
139	C₂²⋊C₄×C₃₁	—	4	2	no	—
140	C₂²⋊C₄×C₃₁	—	4	2	no	—
141	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
142	C₄²×C₃₁	—	2	2	no	—
143	C₄²×C₃₁	—	2	2	no	—
144	C₄²×C₃₁	—	2	2	no	—
145	C₄²×C₃₁	—	2	1	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	—	2	2	no	—
148	C₄²×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	—	2	1	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	C₄²×C₃₁	—	4	1	no	—
152	C₄²×C₃₁	C₂×D₄	1	2	yes	—
153	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
154	#22701	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂³×C₃₁	—	2	2	no	—
156	C₂³×C₃₁	—	2	2	no	—
157	C₂³×C₃₁	—	2	2	no	—
158	C₂³×C₃₁	—	2	2	no	—
159	C₂³×C₃₁	—	2	2	no	—
160	C₂³×C₃₁	—	2	2	no	—
161	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
162	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
163	D₄×C₃₁	—	2	8	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	2	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	4	no	—
171	C₂×C₄×C₃₁	—	2	4	no	—
172	C₂×C₄×C₃₁	—	2	4	no	—
173	C₂×C₄×C₃₁	—	2	4	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	1	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	1	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
190	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
191	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
192	C₂×C₄×C₃₁	#131	1	2	yes	—
193	C₂×C₄×C₃₁	#131	1	2	yes	—
194	C₂×C₄×C₃₁	#131	1	2	yes	—
195	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂²×Q₈	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	H(5,𝔽₂)	1	1	yes	characteristic
200	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
201	C₈×C₃₁	—	2	2	no	—
202	C₈×C₃₁	—	2	2	no	—
203	C₈×C₃₁	—	4	1	no	—
204	C₈×C₃₁	—	4	1	no	—
205	C₈×C₃₁	—	4	2	no	—
206	C₈×C₃₁	—	4	2	no	—
207	C₂×#436	C₂×C₃₁	1	1	yes	characteristic
208	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
209	#2293	C₂×C₃₁	1	1	yes	characteristic
210	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#1851	C₂×C₃₁	1	2	yes	complemented
213	#1850	C₂×C₃₁	1	2	yes	complemented
214	#1831	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#1798	C₂×C₃₁	1	2	yes	complemented
216	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#1729	C₂×C₃₁	1	2	yes	complemented
218	C₂²×C₃₁	—	2	4	no	—
219	C₂²×C₃₁	—	2	4	no	—
220	C₂²×C₃₁	—	2	4	no	—
221	C₂²×C₃₁	—	4	2	no	—
222	C₂²×C₃₁	—	4	2	no	—
223	C₂²×C₃₁	—	4	2	no	—
224	C₂²×C₃₁	—	4	2	no	—
225	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
226	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
227	C₂²×C₃₁	#518	1	1	yes	characteristic
228	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
229	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
230	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
231	C₂²×C₃₁	#568	1	1	yes	characteristic
232	C₄×C₃₁	—	2	4	no	—
233	C₄×C₃₁	—	2	2	no	—
234	C₄×C₃₁	—	2	2	no	—
235	C₄×C₃₁	—	4	1	no	—
236	C₄×C₃₁	—	4	1	no	—
237	C₄×C₃₁	—	4	2	no	—
238	C₄×C₃₁	—	4	1	no	—
239	C₄×C₃₁	—	4	2	no	—
240	C₄×C₃₁	#424	1	4	yes	—
241	C₄×C₃₁	#546	1	1	yes	characteristic
242	C₄×C₃₁	#546	1	1	yes	characteristic
243	C₄×C₃₁	#554	1	1	yes	characteristic
244	C₄×C₃₁	#554	1	1	yes	characteristic
245	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
246	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
247	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×#117	C₂²×C₃₁	1	2	yes	—
249	#436	—	2	4	no	—
250	C₈×D₄	—	2	4	no	—
251	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
254	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
256	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
257	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
258	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
259	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
261	#390	C₄×C₃₁	1	1	yes	characteristic
262	#390	C₄×C₃₁	1	1	yes	characteristic
263	#389	C₄×C₃₁	1	1	yes	characteristic
264	#389	C₄×C₃₁	1	1	yes	characteristic
265	#388	C₄×C₃₁	1	2	yes	—
266	#388	C₄×C₃₁	1	2	yes	—
267	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
268	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
269	C₄³	C₂²×C₃₁	1	1	yes	characteristic
270	#336	C₂²×C₃₁	1	1	yes	characteristic
271	#336	C₂²×C₃₁	1	2	yes	—
272	#336	C₂²×C₃₁	1	2	yes	—
273	#336	C₂²×C₃₁	1	2	yes	—
274	#336	C₂²×C₃₁	1	2	yes	—
275	#336	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×C₃₁	—	4	4	no	—
277	C₂×C₃₁	#2349	1	1	yes	characteristic
278	C₂×C₃₁	C₂×#424	1	1	yes	characteristic
279	C₂×C₃₁	#2906	1	1	yes	characteristic
280	C₂×C₃₁	#2966	1	1	yes	characteristic
281	C₂×C₃₁	#2969	1	1	yes	characteristic
282	C₂×C₃₁	#2979	1	1	yes	characteristic
283	C₂×C₃₁	#2981	1	1	yes	characteristic
284	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
285	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
286	C₂×M₁₆	—	2	4	no	—
287	C₂×M₁₆	—	2	4	no	—
288	C₂×M₁₆	—	4	2	no	—
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	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
295	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
297	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₄×D₄	D₄×C₃₁	1	4	yes	—
299	C₄×D₄	Q₈×C₃₁	1	4	yes	—
300	#117	—	4	2	no	—
301	#117	—	4	2	no	—
302	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
305	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
306	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
307	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
308	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
309	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
310	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
311	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
312	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
313	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
314	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
315	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
316	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
317	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
318	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
319	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
320	C₄⋊C₈	—	2	2	no	—
321	C₄⋊C₈	—	2	2	no	—
322	C₄⋊C₈	—	2	2	no	—
323	C₄⋊C₈	—	2	2	no	—
324	C₄⋊C₈	—	4	2	no	—
325	C₄⋊C₈	—	4	2	no	—
326	C₂²⋊C₈	—	2	4	no	—
327	C₂²⋊C₈	—	2	4	no	—
328	C₄×C₈	—	2	2	no	—
329	C₄×C₈	—	2	2	no	—
330	#95	C₂×C₄×C₃₁	1	2	yes	—
331	#95	C₂×C₄×C₃₁	1	2	yes	—
332	#95	C₂×C₄×C₃₁	1	2	yes	—
333	C₃₁	#22701	1	1	yes	characteristic, direct product factor, Sylow
334	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
343	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
344	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
345	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
346	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
347	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
348	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
349	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
350	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
352	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
353	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
354	M₁₆	—	4	8	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	4	8	no	—
372	C₂×C₈	#41×C₃₁	1	2	yes	—
373	C₂×C₈	#41×C₃₁	1	2	yes	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	4	2	no	—
377	C₄⋊C₄	—	4	2	no	—
378	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
379	C₂²⋊C₄	—	2	4	no	—
380	C₂²⋊C₄	—	4	2	no	—
381	C₂²⋊C₄	—	4	2	no	—
382	C₂²⋊C₄	—	4	2	no	—
383	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
384	C₄²	—	2	2	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	1	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	2	2	no	—
391	C₄²	—	2	1	no	—
392	C₄²	—	2	1	no	—
393	C₄²	—	4	1	no	—
394	C₄²	C₂×D₄×C₃₁	1	2	yes	—
395	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	2	no	—
398	C₂³	—	2	2	no	—
399	C₂³	—	2	2	no	—
400	C₂³	—	2	2	no	—
401	C₂³	—	2	2	no	—
402	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
403	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
404	D₄	—	2	8	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
431	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
432	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
433	C₂×C₄	C₃₁×#131	1	2	yes	—
434	C₂×C₄	C₃₁×#131	1	2	yes	—
435	C₂×C₄	C₃₁×#131	1	2	yes	—
436	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
437	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
438	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
439	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
440	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
441	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
442	C₈	—	2	2	no	—
443	C₈	—	2	2	no	—
444	C₈	—	4	1	no	—
445	C₈	—	4	1	no	—
446	C₈	—	4	2	no	—
447	C₈	—	4	2	no	—
448	C₂²	—	2	4	no	—
449	C₂²	—	2	4	no	—
450	C₂²	—	2	4	no	—
451	C₂²	—	4	2	no	—
452	C₂²	—	4	2	no	—
453	C₂²	—	4	2	no	—
454	C₂²	—	4	2	no	—
455	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
456	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
457	C₂²	C₃₁×#518	1	1	yes	characteristic
458	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
459	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
460	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
461	C₂²	C₃₁×#568	1	1	yes	characteristic
462	C₄	—	2	4	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	2	no	—
465	C₄	—	4	1	no	—
466	C₄	—	4	1	no	—
467	C₄	—	4	2	no	—
468	C₄	—	4	1	no	—
469	C₄	—	4	2	no	—
470	C₄	C₃₁×#424	1	4	yes	—
471	C₄	C₃₁×#546	1	1	yes	characteristic
472	C₄	C₃₁×#546	1	1	yes	characteristic
473	C₄	C₃₁×#554	1	1	yes	characteristic
474	C₄	C₃₁×#554	1	1	yes	characteristic
475	C₂	—	4	4	no	—
476	C₂	C₃₁×#2349	1	1	yes	characteristic
477	C₂	C₂×C₃₁×#424	1	1	yes	characteristic
478	C₂	C₃₁×#2906	1	1	yes	characteristic
479	C₂	C₃₁×#2966	1	1	yes	characteristic
480	C₂	C₃₁×#2969	1	1	yes	characteristic
481	C₂	C₃₁×#2979	1	1	yes	characteristic
482	C₂	C₃₁×#2981	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	C₂²×D₄×C₃₁	C₂×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
4. C₃₁×#1939	C₂	semidirect
5. C₃₁×#1939	C₂	semidirect
6. C₃₁×#1851	C₂	semidirect
7. C₃₁×#1850	C₂	semidirect
8. C₃₁×#1831	C₂	semidirect
9. C₃₁×#1798	C₂	semidirect
10. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
11. C₃₁×#1729	C₂	semidirect
154. #22701	C₃₁	direct (N × H)
210. #1939	C₂×C₃₁	semidirect
211. #1939	C₂×C₃₁	semidirect
212. #1851	C₂×C₃₁	semidirect
213. #1850	C₂×C₃₁	semidirect
214. #1831	C₂×C₃₁	semidirect
215. #1798	C₂×C₃₁	semidirect
216. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
217. #1729	C₂×C₃₁	semidirect
333. C₃₁	#22701	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#436	2
C₂×C₈×D₄×C₃₁	2
C₃₁×#2293	2
C₃₁×#1939	2
C₃₁×#1851	2
C₃₁×#1850	2
C₃₁×#1831	2
C₃₁×#1798	2
C₄×C₃₁×C₄⋊C₈	2
C₃₁×#1729	2
#22701	31

Subgroups — C31×#22701 #21479460

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22701 #21479460