Subgroups — C₃₁×#22169 #21478928

Order 7936.

954 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2392	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1923	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1916	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1870	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1869	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1850	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1798	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
12	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₈×D₄×C₃₁	—	2	8	no	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#398	C₄	1	1	yes	characteristic
26	C₃₁×#397	C₄	1	1	yes	characteristic
27	C₃₁×#396	C₄	1	2	yes	—
28	C₃₁×#394	C₄	1	1	yes	characteristic
29	C₃₁×#394	C₄	1	2	yes	—
30	C₃₁×#392	C₄	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊C₄×C₃₁	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	C₃₁×#336	C₂²	1	2	yes	—
38	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
39	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
41	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₃₁	—	2	2	no	—
43	C₂×M₁₆×C₃₁	—	4	1	no	—
44	C₂×M₁₆×C₃₁	—	4	1	no	—
45	C₂²×C₈×C₃₁	—	2	4	no	—
46	C₂²×C₈×C₃₁	—	4	1	no	—
47	C₂²×C₈×C₃₁	—	4	1	no	—
48	C₂²×C₈×C₃₁	—	4	2	no	—
49	C₂²×C₈×C₃₁	—	4	2	no	—
50	C₂²×C₈×C₃₁	D₄	1	4	yes	complemented
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₄×D₄×C₃₁	—	2	4	no	—
56	C₄×D₄×C₃₁	—	2	4	no	—
57	C₄×D₄×C₃₁	—	4	1	no	—
58	C₄×D₄×C₃₁	—	4	1	no	—
59	C₃₁×#117	—	4	1	no	—
60	C₃₁×#117	—	4	1	no	—
61	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
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×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×C₄⋊C₈	—	2	4	no	—
77	C₃₁×C₄⋊C₈	—	4	1	no	—
78	C₃₁×C₄⋊C₈	—	4	1	no	—
79	C₃₁×C₂²⋊C₈	—	2	4	no	—
80	C₃₁×C₂²⋊C₈	—	2	4	no	—
81	C₃₁×C₂²⋊C₈	—	4	2	no	—
82	C₃₁×C₂²⋊C₈	—	4	2	no	—
83	C₄×C₈×C₃₁	—	2	4	no	—
84	C₃₁×#95	C₂×C₄	1	2	yes	—
85	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
86	C₃₁×#95	C₂×C₄	1	2	yes	—
87	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
88	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
89	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
90	C₂×D₄×C₃₁	—	2	4	no	—
91	C₂×D₄×C₃₁	—	2	4	no	—
92	C₂×D₄×C₃₁	—	2	4	no	—
93	C₂×D₄×C₃₁	—	2	4	no	—
94	C₂×D₄×C₃₁	—	4	1	no	—
95	C₂²×C₄×C₃₁	—	2	1	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
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	2	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₃₁	#41	1	2	yes	—
115	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
116	M₁₆×C₃₁	—	4	4	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	—	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	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	4	4	no	—
129	C₂×C₈×C₃₁	—	4	8	no	—
130	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
131	C₄⋊C₄×C₃₁	—	2	4	no	—
132	C₄⋊C₄×C₃₁	—	2	4	no	—
133	C₄⋊C₄×C₃₁	—	4	1	no	—
134	C₄⋊C₄×C₃₁	—	4	1	no	—
135	C₄⋊C₄×C₃₁	—	4	1	no	—
136	C₂²⋊C₄×C₃₁	—	2	4	no	—
137	C₂²⋊C₄×C₃₁	—	2	4	no	—
138	C₂²⋊C₄×C₃₁	—	2	4	no	—
139	C₂²⋊C₄×C₃₁	—	2	4	no	—
140	C₂²⋊C₄×C₃₁	—	4	1	no	—
141	C₂²⋊C₄×C₃₁	—	4	1	no	—
142	C₂²⋊C₄×C₃₁	—	4	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	1	no	—
148	C₄²×C₃₁	—	2	1	no	—
149	#22169	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂³×C₃₁	—	2	2	no	—
151	C₂³×C₃₁	—	2	1	no	—
152	C₂³×C₃₁	—	2	2	no	—
153	C₂³×C₃₁	—	2	1	no	—
154	C₂³×C₃₁	—	2	2	no	—
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	C₄×D₄	1	2	yes	—
161	C₂³×C₃₁	#131	1	4	yes	—
162	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
163	D₄×C₃₁	—	2	16	no	—
164	D₄×C₃₁	—	4	4	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	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	1	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	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	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	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	C₄×D₄	1	2	yes	—
192	C₂×C₄×C₃₁	#131	1	4	yes	—
193	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
196	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
197	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
198	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
199	C₈×C₃₁	—	2	4	no	—
200	C₈×C₃₁	—	4	2	no	—
201	C₈×C₃₁	—	4	2	no	—
202	C₈×C₃₁	—	4	2	no	—
203	C₂×C₈×D₄	C₂×C₃₁	1	2	yes	complemented
204	#2392	C₂×C₃₁	1	1	yes	characteristic
205	#1923	C₂×C₃₁	1	2	yes	complemented
206	#1916	C₂×C₃₁	1	2	yes	complemented
207	#1870	C₂×C₃₁	1	2	yes	complemented
208	#1869	C₂×C₃₁	1	2	yes	complemented
209	#1850	C₂×C₃₁	1	2	yes	complemented
210	#1798	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1729	C₂×C₃₁	1	1	yes	characteristic, complemented
212	C₂²×C₃₁	—	2	2	no	—
213	C₂²×C₃₁	—	2	4	no	—
214	C₂²×C₃₁	—	2	4	no	—
215	C₂²×C₃₁	—	2	4	no	—
216	C₂²×C₃₁	—	2	4	no	—
217	C₂²×C₃₁	—	4	2	no	—
218	C₂²×C₃₁	—	4	2	no	—
219	C₂²×C₃₁	—	4	1	no	—
220	C₂²×C₃₁	—	4	1	no	—
221	C₂²×C₃₁	—	4	2	no	—
222	C₂²×C₃₁	—	4	2	no	—
223	C₂²×C₃₁	—	4	2	no	—
224	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
225	C₂²×C₃₁	#518	1	1	yes	characteristic
226	C₂²×C₃₁	#520	1	1	yes	characteristic
227	C₂²×C₃₁	C₂×#131	1	2	yes	—
228	C₂²×C₃₁	C₂×#131	1	2	yes	—
229	C₄×C₃₁	—	2	2	no	—
230	C₄×C₃₁	—	2	4	no	—
231	C₄×C₃₁	—	4	2	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	—	4	1	no	—
234	C₄×C₃₁	—	4	1	no	—
235	C₄×C₃₁	—	4	1	no	—
236	C₄×C₃₁	—	4	1	no	—
237	C₄×C₃₁	#534	1	1	yes	characteristic
238	C₄×C₃₁	#537	1	1	yes	characteristic
239	C₄×C₃₁	#538	1	1	yes	characteristic
240	C₄×C₃₁	#539	1	1	yes	characteristic
241	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
242	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
243	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
245	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
247	C₈×D₄	—	2	8	no	—
248	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
249	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
252	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
253	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
254	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
255	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
256	#398	C₄×C₃₁	1	1	yes	characteristic
257	#397	C₄×C₃₁	1	1	yes	characteristic
258	#396	C₄×C₃₁	1	2	yes	—
259	#394	C₄×C₃₁	1	2	yes	—
260	#394	C₄×C₃₁	1	1	yes	characteristic
261	#392	C₄×C₃₁	1	1	yes	characteristic
262	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
263	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
264	#336	C₂²×C₃₁	1	2	yes	—
265	#336	C₂²×C₃₁	1	2	yes	—
266	#336	C₂²×C₃₁	1	2	yes	—
267	#336	C₂²×C₃₁	1	2	yes	—
268	#336	C₂²×C₃₁	1	2	yes	—
269	C₂×C₃₁	—	4	2	no	—
270	C₂×C₃₁	—	4	4	no	—
271	C₂×C₃₁	#2325	1	1	yes	characteristic
272	C₂×C₃₁	#2929	1	2	yes	—
273	C₂×C₃₁	#2977	1	1	yes	characteristic
274	C₂×C₃₁	#2979	1	1	yes	characteristic
275	C₂×C₃₁	#2994	1	2	yes	—
276	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
277	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
278	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
279	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
280	C₂×M₁₆	—	2	2	no	—
281	C₂×M₁₆	—	4	1	no	—
282	C₂×M₁₆	—	4	1	no	—
283	C₂²×C₈	—	2	4	no	—
284	C₂²×C₈	—	4	1	no	—
285	C₂²×C₈	—	4	1	no	—
286	C₂²×C₈	—	4	2	no	—
287	C₂²×C₈	—	4	2	no	—
288	C₂²×C₈	D₄×C₃₁	1	4	yes	complemented
289	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
290	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
291	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
292	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
293	C₄×D₄	—	2	4	no	—
294	C₄×D₄	—	2	4	no	—
295	C₄×D₄	—	4	1	no	—
296	C₄×D₄	—	4	1	no	—
297	#117	—	4	1	no	—
298	#117	—	4	1	no	—
299	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
300	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
301	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
304	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
305	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
306	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
307	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
308	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
309	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
310	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
311	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
312	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
314	C₄⋊C₈	—	2	4	no	—
315	C₄⋊C₈	—	4	1	no	—
316	C₄⋊C₈	—	4	1	no	—
317	C₂²⋊C₈	—	2	4	no	—
318	C₂²⋊C₈	—	2	4	no	—
319	C₂²⋊C₈	—	4	2	no	—
320	C₂²⋊C₈	—	4	2	no	—
321	C₄×C₈	—	2	4	no	—
322	#95	C₂×C₄×C₃₁	1	2	yes	—
323	#95	C₂×C₄×C₃₁	1	2	yes	—
324	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
325	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
326	C₃₁	#22169	1	1	yes	characteristic, direct product factor, Sylow
327	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
328	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
329	C₂×D₄	—	2	4	no	—
330	C₂×D₄	—	2	4	no	—
331	C₂×D₄	—	2	4	no	—
332	C₂×D₄	—	2	4	no	—
333	C₂×D₄	—	4	1	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
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	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
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	C₂²×C₄×C₃₁	1	2	yes	—
352	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
353	C₂²×C₄	#41×C₃₁	1	2	yes	—
354	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
355	M₁₆	—	4	4	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
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	—	4	4	no	—
368	C₂×C₈	—	4	8	no	—
369	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
370	C₄⋊C₄	—	2	4	no	—
371	C₄⋊C₄	—	2	4	no	—
372	C₄⋊C₄	—	4	1	no	—
373	C₄⋊C₄	—	4	1	no	—
374	C₄⋊C₄	—	4	1	no	—
375	C₂²⋊C₄	—	2	4	no	—
376	C₂²⋊C₄	—	2	4	no	—
377	C₂²⋊C₄	—	2	4	no	—
378	C₂²⋊C₄	—	2	4	no	—
379	C₂²⋊C₄	—	4	1	no	—
380	C₂²⋊C₄	—	4	1	no	—
381	C₂²⋊C₄	—	4	2	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	2	1	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	1	no	—
387	C₄²	—	2	1	no	—
388	C₂³	—	2	2	no	—
389	C₂³	—	2	1	no	—
390	C₂³	—	2	2	no	—
391	C₂³	—	2	1	no	—
392	C₂³	—	2	2	no	—
393	C₂³	—	2	2	no	—
394	C₂³	—	2	2	no	—
395	C₂³	—	2	2	no	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	2	no	—
398	C₂³	C₄×D₄×C₃₁	1	2	yes	—
399	C₂³	C₃₁×#131	1	4	yes	—
400	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
401	D₄	—	2	16	no	—
402	D₄	—	4	4	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	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	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	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	C₄×D₄×C₃₁	1	2	yes	—
430	C₂×C₄	C₃₁×#131	1	4	yes	—
431	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
433	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
434	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
435	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
436	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
437	C₈	—	2	4	no	—
438	C₈	—	4	2	no	—
439	C₈	—	4	2	no	—
440	C₈	—	4	2	no	—
441	C₂²	—	2	2	no	—
442	C₂²	—	2	4	no	—
443	C₂²	—	2	4	no	—
444	C₂²	—	2	4	no	—
445	C₂²	—	2	4	no	—
446	C₂²	—	4	2	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	4	1	no	—
449	C₂²	—	4	1	no	—
450	C₂²	—	4	2	no	—
451	C₂²	—	4	2	no	—
452	C₂²	—	4	2	no	—
453	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
454	C₂²	C₃₁×#518	1	1	yes	characteristic
455	C₂²	C₃₁×#520	1	1	yes	characteristic
456	C₂²	C₂×C₃₁×#131	1	2	yes	—
457	C₂²	C₂×C₃₁×#131	1	2	yes	—
458	C₄	—	2	2	no	—
459	C₄	—	2	4	no	—
460	C₄	—	4	2	no	—
461	C₄	—	4	2	no	—
462	C₄	—	4	1	no	—
463	C₄	—	4	1	no	—
464	C₄	—	4	1	no	—
465	C₄	—	4	1	no	—
466	C₄	C₃₁×#534	1	1	yes	characteristic
467	C₄	C₃₁×#537	1	1	yes	characteristic
468	C₄	C₃₁×#538	1	1	yes	characteristic
469	C₄	C₃₁×#539	1	1	yes	characteristic
470	C₂	—	4	2	no	—
471	C₂	—	4	4	no	—
472	C₂	C₃₁×#2325	1	1	yes	characteristic
473	C₂	C₃₁×#2929	1	2	yes	—
474	C₂	C₃₁×#2977	1	1	yes	characteristic
475	C₂	C₃₁×#2979	1	1	yes	characteristic
476	C₂	C₃₁×#2994	1	2	yes	—

	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	C₃₁×#392	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
1. C₂×C₈×D₄×C₃₁	C₂	semidirect
3. C₃₁×#1923	C₂	semidirect
4. C₃₁×#1916	C₂	semidirect
5. C₃₁×#1870	C₂	semidirect
6. C₃₁×#1869	C₂	semidirect
7. C₃₁×#1850	C₂	semidirect
8. C₃₁×#1798	C₂	semidirect
9. C₃₁×#1729	C₂	semidirect
50. C₂²×C₈×C₃₁	D₄	semidirect
149. #22169	C₃₁	direct (N × H)
203. C₂×C₈×D₄	C₂×C₃₁	semidirect
205. #1923	C₂×C₃₁	semidirect
206. #1916	C₂×C₃₁	semidirect
207. #1870	C₂×C₃₁	semidirect
208. #1869	C₂×C₃₁	semidirect
209. #1850	C₂×C₃₁	semidirect
210. #1798	C₂×C₃₁	semidirect
211. #1729	C₂×C₃₁	semidirect
288. C₂²×C₈	D₄×C₃₁	semidirect
326. C₃₁	#22169	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×C₈×D₄×C₃₁	2
C₃₁×#2392	2
C₃₁×#1923	2
C₃₁×#1916	2
C₃₁×#1870	2
C₃₁×#1869	2
C₃₁×#1850	2
C₃₁×#1798	2
C₃₁×#1729	2
#22169	31

Subgroups — C31×#22169 #21478928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22169 #21478928