Subgroups — C₃₁×#22640 #21479399

Order 7936.

700 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1924	C₂	1	2	yes	maximal
5	C₃₁×#1923	C₂	1	2	yes	maximal
6	C₃₁×#1840	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1768	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
24	C₃₁×#383	C₄	1	2	yes	—
25	C₃₁×#382	C₄	1	1	yes	characteristic
26	C₃₁×#382	C₄	1	1	yes	characteristic
27	C₃₁×#381	C₄	1	2	yes	—
28	C₃₁×#380	C₄	1	1	yes	characteristic
29	C₃₁×#380	C₄	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₃₁	C₂²	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×#376	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	1	yes	characteristic
36	C₃₁×#336	C₂²	1	2	yes	—
37	C₃₁×#336	C₂²	1	2	yes	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₃₁	—	2	2	no	—
57	C₂×C₄²×C₃₁	—	2	2	no	—
58	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₃₁×C₄⋊C₈	—	2	4	no	—
65	C₃₁×C₄⋊C₈	—	2	4	no	—
66	C₃₁×C₄⋊C₈	—	2	2	no	—
67	C₃₁×C₄⋊C₈	—	2	2	no	—
68	C₃₁×C₂²⋊C₈	—	2	2	no	—
69	C₃₁×C₂²⋊C₈	—	2	2	no	—
70	C₃₁×C₂²⋊C₈	—	4	2	no	—
71	C₃₁×#97	—	4	2	no	—
72	C₄×C₈×C₃₁	—	2	2	no	—
73	C₄×C₈×C₃₁	—	2	2	no	—
74	C₄×C₈×C₃₁	—	4	2	no	—
75	C₃₁×#95	C₂×C₄	1	2	yes	—
76	C₃₁×#95	C₂×C₄	1	2	yes	—
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	C₃₁×#95	C₂×C₄	1	2	yes	—
83	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	2	no	—
86	C₂²×C₄×C₃₁	—	2	2	no	—
87	C₂²×C₄×C₃₁	—	2	2	no	—
88	C₂²×C₄×C₃₁	—	2	2	no	—
89	C₂²×C₄×C₃₁	—	2	1	no	—
90	C₂²×C₄×C₃₁	—	2	1	no	—
91	C₂²×C₄×C₃₁	—	2	1	no	—
92	C₂²×C₄×C₃₁	—	2	1	no	—
93	C₂²×C₄×C₃₁	—	2	1	no	—
94	C₂²×C₄×C₃₁	—	2	1	no	—
95	C₂²×C₄×C₃₁	—	4	1	no	—
96	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
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	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	C₂⁴	1	1	yes	characteristic
106	C₂×C₈×C₃₁	—	2	1	no	—
107	C₂×C₈×C₃₁	—	2	1	no	—
108	C₂×C₈×C₃₁	—	2	1	no	—
109	C₂×C₈×C₃₁	—	2	1	no	—
110	C₂×C₈×C₃₁	—	2	1	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	1	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	1	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	#41	1	2	yes	—
123	C₂×C₈×C₃₁	#41	1	2	yes	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	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	2	no	—
133	C₄²×C₃₁	—	2	2	no	—
134	C₄²×C₃₁	—	2	1	no	—
135	C₄²×C₃₁	—	2	1	no	—
136	C₄²×C₃₁	—	2	2	no	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	#41	1	2	yes	—
139	C₄²×C₃₁	#41	1	2	yes	—
140	#22640	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	#131	1	2	yes	—
148	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	2	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	1	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	2	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	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	—	4	4	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	1	no	—
176	C₂×C₄×C₃₁	—	4	1	no	—
177	C₂×C₄×C₃₁	—	4	4	no	—
178	C₂×C₄×C₃₁	#117	1	2	yes	—
179	C₂×C₄×C₃₁	#117	1	2	yes	—
180	C₂×C₄×C₃₁	#131	1	2	yes	—
181	C₂×C₄×C₃₁	#131	1	2	yes	—
182	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
188	C₈×C₃₁	—	2	2	no	—
189	C₈×C₃₁	—	2	2	no	—
190	C₈×C₃₁	—	4	2	no	—
191	C₈×C₃₁	—	4	2	no	—
192	C₈×C₃₁	—	4	1	no	—
193	C₈×C₃₁	—	4	1	no	—
194	#2283	C₂×C₃₁	1	1	yes	characteristic
195	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1924	C₂×C₃₁	1	2	yes	—
198	#1923	C₂×C₃₁	1	2	yes	—
199	#1840	C₂×C₃₁	1	2	yes	complemented
200	#1797	C₂×C₃₁	1	1	yes	characteristic
201	#1770	C₂×C₃₁	1	1	yes	characteristic, complemented
202	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#1768	C₂×C₃₁	1	2	yes	complemented
204	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic
205	C₂²×C₃₁	—	2	2	no	—
206	C₂²×C₃₁	—	2	2	no	—
207	C₂²×C₃₁	—	2	2	no	—
208	C₂²×C₃₁	—	4	1	no	—
209	C₂²×C₃₁	—	4	1	no	—
210	C₂²×C₃₁	—	4	1	no	—
211	C₂²×C₃₁	—	4	1	no	—
212	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
213	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
214	C₂²×C₃₁	#520	1	1	yes	characteristic
215	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
216	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
217	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
218	C₂²×C₃₁	#568	1	1	yes	characteristic
219	C₄×C₃₁	—	2	2	no	—
220	C₄×C₃₁	—	4	1	no	—
221	C₄×C₃₁	—	4	1	no	—
222	C₄×C₃₁	—	4	1	no	—
223	C₄×C₃₁	—	4	2	no	—
224	C₄×C₃₁	—	4	2	no	—
225	C₄×C₃₁	—	4	2	no	—
226	C₄×C₃₁	—	4	2	no	—
227	C₄×C₃₁	#433	1	4	yes	—
228	C₄×C₃₁	#553	1	1	yes	characteristic
229	C₄×C₃₁	#553	1	1	yes	characteristic
230	C₄×C₃₁	#556	1	1	yes	characteristic
231	C₄×C₃₁	#556	1	1	yes	characteristic
232	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
236	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
237	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
238	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
240	C₂×#97	C₂²×C₃₁	1	2	yes	—
241	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
244	#383	C₄×C₃₁	1	2	yes	—
245	#382	C₄×C₃₁	1	1	yes	characteristic
246	#382	C₄×C₃₁	1	1	yes	characteristic
247	#381	C₄×C₃₁	1	2	yes	—
248	#380	C₄×C₃₁	1	1	yes	characteristic
249	#380	C₄×C₃₁	1	1	yes	characteristic
250	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
251	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
252	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
253	#376	C₂²×C₃₁	1	2	yes	—
254	#336	C₂²×C₃₁	1	2	yes	—
255	#336	C₂²×C₃₁	1	1	yes	characteristic
256	#336	C₂²×C₃₁	1	2	yes	—
257	#336	C₂²×C₃₁	1	2	yes	—
258	#336	C₂²×C₃₁	1	2	yes	—
259	#336	C₂²×C₃₁	1	2	yes	—
260	#336	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₃₁	—	4	2	no	—
262	C₂×C₃₁	#2347	1	1	yes	characteristic
263	C₂×C₃₁	C₂×#433	1	1	yes	characteristic
264	C₂×C₃₁	#2924	1	1	yes	characteristic
265	C₂×C₃₁	#2966	1	1	yes	characteristic
266	C₂×C₃₁	#2969	1	1	yes	characteristic
267	C₂×C₃₁	#2993	1	1	yes	characteristic
268	C₂×C₃₁	#2994	1	1	yes	characteristic
269	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
270	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
271	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
272	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
273	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
274	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
275	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
276	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
277	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
278	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
280	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
281	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
282	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
283	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
284	C₂×C₄²	—	2	2	no	—
285	C₂×C₄²	—	2	2	no	—
286	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
287	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
288	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
289	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
290	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
291	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
292	C₄⋊C₈	—	2	4	no	—
293	C₄⋊C₈	—	2	4	no	—
294	C₄⋊C₈	—	2	2	no	—
295	C₄⋊C₈	—	2	2	no	—
296	C₂²⋊C₈	—	2	2	no	—
297	C₂²⋊C₈	—	2	2	no	—
298	C₂²⋊C₈	—	4	2	no	—
299	#97	—	4	2	no	—
300	C₄×C₈	—	2	2	no	—
301	C₄×C₈	—	2	2	no	—
302	C₄×C₈	—	4	2	no	—
303	#95	C₂×C₄×C₃₁	1	2	yes	—
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	#95	C₂×C₄×C₃₁	1	2	yes	—
310	#95	C₂×C₄×C₃₁	1	2	yes	—
311	C₃₁	#22640	1	1	yes	characteristic, direct product factor, Sylow
312	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	—	2	1	no	—
320	C₂²×C₄	—	2	1	no	—
321	C₂²×C₄	—	2	1	no	—
322	C₂²×C₄	—	2	1	no	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	4	1	no	—
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	1	yes	characteristic
328	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
329	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
330	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
331	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
332	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
334	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	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	—	2	1	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	#41×C₃₁	1	2	yes	—
352	C₂×C₈	#41×C₃₁	1	2	yes	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₂²⋊C₄	—	2	4	no	—
358	C₂²⋊C₄	—	4	2	no	—
359	C₄²	—	2	2	no	—
360	C₄²	—	2	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	2	1	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	2	2	no	—
367	C₄²	#41×C₃₁	1	2	yes	—
368	C₄²	#41×C₃₁	1	2	yes	—
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	—	2	1	no	—
375	C₂³	C₃₁×#131	1	2	yes	—
376	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	4	4	no	—
401	C₂×C₄	—	4	4	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	C₃₁×#117	1	2	yes	—
407	C₂×C₄	C₃₁×#117	1	2	yes	—
408	C₂×C₄	C₃₁×#131	1	2	yes	—
409	C₂×C₄	C₃₁×#131	1	2	yes	—
410	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
411	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
412	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
413	C₂×C₄	C₂×#41×C₃₁	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	2	no	—
417	C₈	—	2	2	no	—
418	C₈	—	4	2	no	—
419	C₈	—	4	2	no	—
420	C₈	—	4	1	no	—
421	C₈	—	4	1	no	—
422	C₂²	—	2	2	no	—
423	C₂²	—	2	2	no	—
424	C₂²	—	2	2	no	—
425	C₂²	—	4	1	no	—
426	C₂²	—	4	1	no	—
427	C₂²	—	4	1	no	—
428	C₂²	—	4	1	no	—
429	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
430	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
431	C₂²	C₃₁×#520	1	1	yes	characteristic
432	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
433	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
434	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
435	C₂²	C₃₁×#568	1	1	yes	characteristic
436	C₄	—	2	2	no	—
437	C₄	—	4	1	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	—	4	2	no	—
444	C₄	C₃₁×#433	1	4	yes	—
445	C₄	C₃₁×#553	1	1	yes	characteristic
446	C₄	C₃₁×#553	1	1	yes	characteristic
447	C₄	C₃₁×#556	1	1	yes	characteristic
448	C₄	C₃₁×#556	1	1	yes	characteristic
449	C₂	—	4	2	no	—
450	C₂	C₃₁×#2347	1	1	yes	characteristic
451	C₂	C₂×C₃₁×#433	1	1	yes	characteristic
452	C₂	C₃₁×#2924	1	1	yes	characteristic
453	C₂	C₃₁×#2966	1	1	yes	characteristic
454	C₂	C₃₁×#2969	1	1	yes	characteristic
455	C₂	C₃₁×#2993	1	1	yes	characteristic
456	C₂	C₃₁×#2994	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×#1939	C₂	semidirect
3. C₃₁×#1939	C₂	semidirect
6. C₃₁×#1840	C₂	semidirect
8. C₃₁×#1770	C₂	semidirect
9. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
10. C₃₁×#1768	C₂	semidirect
140. #22640	C₃₁	direct (N × H)
195. #1939	C₂×C₃₁	semidirect
196. #1939	C₂×C₃₁	semidirect
199. #1840	C₂×C₃₁	semidirect
201. #1770	C₂×C₃₁	semidirect
202. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
203. #1768	C₂×C₃₁	semidirect
311. C₃₁	#22640	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×#2283	2
C₃₁×#1939	2
C₃₁×#1924	2
C₃₁×#1923	2
C₃₁×#1840	2
C₃₁×#1797	2
C₃₁×#1770	2
C₈×C₄⋊C₄×C₃₁	2
C₃₁×#1768	2
C₄×C₃₁×C₂²⋊C₈	2
#22640	31

Subgroups — C31×#22640 #21479399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22640 #21479399