Subgroups — C₄×C₃₁×#540 #21490852

Order 7936.

2518 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#540	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2335	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2333	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2330	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2328	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2324	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2319	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2314	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×#123	C₂	1	4	yes	maximal, complemented
12	C₄×C₃₁×#121	C₂	1	2	yes	maximal, complemented
13	C₄×C₃₁×#121	C₂	1	4	yes	maximal, complemented
14	C₄²×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
15	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#540	C₄	1	16	yes	direct product factor
17	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#123	C₂²	1	4	yes	—
20	C₂×C₃₁×#121	C₂²	1	2	yes	—
21	C₂×C₃₁×#121	C₂²	1	4	yes	—
22	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₃₁	C₂²	1	4	yes	—
31	C₃₁×#390	C₂²	1	2	yes	—
32	C₃₁×#390	C₂²	1	2	yes	—
33	C₃₁×#390	C₂²	1	4	yes	—
34	C₃₁×#390	C₂²	1	4	yes	—
35	C₃₁×#389	C₂²	1	2	yes	—
36	C₃₁×#389	C₂²	1	2	yes	—
37	C₃₁×#389	C₂²	1	4	yes	—
38	C₃₁×#388	C₂²	1	4	yes	—
39	C₃₁×#388	C₂²	1	4	yes	—
40	C₃₁×#388	C₂²	1	4	yes	—
41	C₃₁×#388	C₂²	1	4	yes	—
42	C₃₁×#388	C₂²	1	4	yes	—
43	C₃₁×#387	C₂²	1	4	yes	—
44	C₃₁×#387	C₂²	1	4	yes	—
45	C₃₁×#387	C₂²	1	4	yes	—
46	C₃₁×#386	C₂²	1	4	yes	—
47	C₃₁×#386	C₂²	1	4	yes	—
48	C₃₁×#386	C₂²	1	4	yes	—
49	C₃₁×#386	C₂²	1	4	yes	—
50	C₃₁×#385	C₂²	1	4	yes	—
51	C₃₁×#384	C₂²	1	1	yes	characteristic
52	C₃₁×#382	C₂²	1	1	yes	characteristic
53	C₃₁×#381	C₂²	1	4	yes	—
54	C₃₁×#380	C₂²	1	2	yes	—
55	C₃₁×#380	C₂²	1	4	yes	—
56	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
57	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
58	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
59	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
60	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
62	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
64	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
65	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
66	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
67	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
68	C₃₁×#376	C₂²	1	1	yes	characteristic
69	C₃₁×#376	C₂²	1	1	yes	characteristic
70	C₄³×C₃₁	C₂²	1	1	yes	characteristic
71	C₄³×C₃₁	C₂²	1	1	yes	characteristic
72	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
73	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
74	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
77	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
79	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
81	C₃₁×#127	C₂×C₄	1	8	yes	complemented
82	C₃₁×#125	C₂×C₄	1	8	yes	complemented
83	C₃₁×#123	C₂×C₄	1	32	yes	complemented
84	C₃₁×#121	C₂×C₄	1	32	yes	complemented
85	C₃₁×#121	C₂×C₄	1	16	yes	complemented
86	C₄×D₄×C₃₁	—	2	8	no	—
87	C₄×D₄×C₃₁	—	2	8	no	—
88	C₄×D₄×C₃₁	—	4	8	no	—
89	C₄×D₄×C₃₁	—	4	8	no	—
90	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
91	C₃₁×#117	—	2	4	no	—
92	C₃₁×#117	C₂×C₄	1	8	yes	complemented
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
108	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
109	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
113	C₂×C₄²×C₃₁	—	2	4	no	—
114	C₂×C₄²×C₃₁	—	4	8	no	—
115	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
119	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
122	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
123	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
124	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
125	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
126	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
130	C₃₁×#95	C₂³	1	2	yes	—
131	C₃₁×#95	C₂³	1	2	yes	—
132	C₃₁×#95	C₂³	1	2	yes	—
133	C₃₁×#95	C₂³	1	2	yes	—
134	C₃₁×#95	C₂³	1	4	yes	—
135	C₃₁×#95	C₂³	1	4	yes	—
136	C₃₁×#95	C₂³	1	4	yes	—
137	C₃₁×#95	C₂³	1	4	yes	—
138	C₃₁×#95	C₂³	1	4	yes	—
139	C₃₁×#95	C₂³	1	4	yes	—
140	C₃₁×#95	C₂³	1	4	yes	—
141	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
142	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
143	C₂×D₄×C₃₁	—	2	4	no	—
144	C₂×D₄×C₃₁	—	2	4	no	—
145	C₂×D₄×C₃₁	—	4	4	no	—
146	C₂×D₄×C₃₁	—	4	4	no	—
147	C₂×D₄×C₃₁	—	4	4	no	—
148	C₂×D₄×C₃₁	—	4	4	no	—
149	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
150	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
151	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
152	C₂²×C₄×C₃₁	—	2	2	no	—
153	C₂²×C₄×C₃₁	—	2	2	no	—
154	C₂²×C₄×C₃₁	—	2	2	no	—
155	C₂²×C₄×C₃₁	—	2	2	no	—
156	C₂²×C₄×C₃₁	—	2	8	no	—
157	C₂²×C₄×C₃₁	—	2	8	no	—
158	C₂²×C₄×C₃₁	—	2	8	no	—
159	C₂²×C₄×C₃₁	—	2	4	no	—
160	C₂²×C₄×C₃₁	—	2	8	no	—
161	C₂²×C₄×C₃₁	—	4	4	no	—
162	C₂²×C₄×C₃₁	—	4	4	no	—
163	C₂²×C₄×C₃₁	—	4	2	no	—
164	C₂²×C₄×C₃₁	—	4	4	no	—
165	C₂²×C₄×C₃₁	—	4	4	no	—
166	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
167	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	—
168	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
181	C₄⋊C₄×C₃₁	—	2	4	no	—
182	C₄⋊C₄×C₃₁	—	2	4	no	—
183	C₄⋊C₄×C₃₁	—	2	4	no	—
184	C₄⋊C₄×C₃₁	—	4	4	no	—
185	C₄⋊C₄×C₃₁	—	4	4	no	—
186	C₄⋊C₄×C₃₁	—	4	4	no	—
187	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
188	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
189	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
190	C₂²⋊C₄×C₃₁	—	2	4	no	—
191	C₂²⋊C₄×C₃₁	—	2	8	no	—
192	C₂²⋊C₄×C₃₁	—	2	8	no	—
193	C₂²⋊C₄×C₃₁	—	4	4	no	—
194	C₂²⋊C₄×C₃₁	—	4	4	no	—
195	C₂²⋊C₄×C₃₁	—	4	4	no	—
196	C₂²⋊C₄×C₃₁	—	4	4	no	—
197	C₂²⋊C₄×C₃₁	—	4	4	no	—
198	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
199	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
200	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
201	C₄²×C₃₁	—	2	2	no	—
202	C₄²×C₃₁	—	2	2	no	—
203	C₄²×C₃₁	—	2	2	no	—
204	C₄²×C₃₁	—	2	2	no	—
205	C₄²×C₃₁	—	2	4	no	—
206	C₄²×C₃₁	—	2	8	no	—
207	C₄²×C₃₁	—	2	4	no	—
208	C₄²×C₃₁	—	4	4	no	—
209	C₄²×C₃₁	—	4	4	no	—
210	C₄²×C₃₁	—	4	2	no	—
211	C₄²×C₃₁	—	4	2	no	—
212	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
213	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
214	C₄²×C₃₁	#41	1	4	yes	—
215	C₄²×C₃₁	#41	1	4	yes	—
216	C₄×#540	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂³×C₃₁	—	2	1	no	—
218	C₂³×C₃₁	—	2	4	no	—
219	C₂³×C₃₁	—	2	4	no	—
220	C₂³×C₃₁	—	2	1	no	—
221	C₂³×C₃₁	—	2	4	no	—
222	C₂³×C₃₁	—	2	2	no	—
223	C₂³×C₃₁	—	2	4	no	—
224	C₂³×C₃₁	—	2	4	no	—
225	C₂³×C₃₁	—	2	2	no	—
226	C₂³×C₃₁	—	2	4	no	—
227	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
228	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
229	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
230	D₄×C₃₁	—	2	16	no	—
231	D₄×C₃₁	—	4	16	no	—
232	D₄×C₃₁	—	4	16	no	—
233	C₂×C₄×C₃₁	—	2	1	no	—
234	C₂×C₄×C₃₁	—	2	1	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	4	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	2	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	2	no	—
243	C₂×C₄×C₃₁	—	2	4	no	—
244	C₂×C₄×C₃₁	—	2	8	no	—
245	C₂×C₄×C₃₁	—	2	1	no	—
246	C₂×C₄×C₃₁	—	2	1	no	—
247	C₂×C₄×C₃₁	—	2	2	no	—
248	C₂×C₄×C₃₁	—	2	4	no	—
249	C₂×C₄×C₃₁	—	2	4	no	—
250	C₂×C₄×C₃₁	—	2	4	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	2	4	no	—
253	C₂×C₄×C₃₁	—	2	2	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	2	no	—
257	C₂×C₄×C₃₁	—	2	4	no	—
258	C₂×C₄×C₃₁	—	4	16	no	—
259	C₂×C₄×C₃₁	—	4	4	no	—
260	C₂×C₄×C₃₁	—	4	16	no	—
261	C₂×C₄×C₃₁	—	4	4	no	—
262	C₂×C₄×C₃₁	—	4	16	no	—
263	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
264	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
265	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
266	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
267	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
268	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
269	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
270	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
271	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
272	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
273	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
274	C₂×#540	C₂×C₃₁	1	1	yes	characteristic
275	#2335	C₂×C₃₁	1	4	yes	complemented
276	#2333	C₂×C₃₁	1	4	yes	complemented
277	#2330	C₂×C₃₁	1	1	yes	characteristic, complemented
278	#2328	C₂×C₃₁	1	1	yes	characteristic, complemented
279	#2324	C₂×C₃₁	1	2	yes	complemented
280	#2319	C₂×C₃₁	1	2	yes	complemented
281	#2314	C₂×C₃₁	1	1	yes	characteristic, complemented
282	C₄×#127	C₂×C₃₁	1	1	yes	characteristic, complemented
283	C₄×#125	C₂×C₃₁	1	1	yes	characteristic, complemented
284	C₄×#123	C₂×C₃₁	1	4	yes	complemented
285	C₄×#121	C₂×C₃₁	1	2	yes	complemented
286	C₄×#121	C₂×C₃₁	1	4	yes	complemented
287	C₄²×D₄	C₂×C₃₁	1	2	yes	complemented
288	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
289	C₂²×C₃₁	—	2	2	no	—
290	C₂²×C₃₁	—	2	8	no	—
291	C₂²×C₃₁	—	2	8	no	—
292	C₂²×C₃₁	—	2	4	no	—
293	C₂²×C₃₁	—	2	8	no	—
294	C₂²×C₃₁	—	4	4	no	—
295	C₂²×C₃₁	—	4	1	no	—
296	C₂²×C₃₁	—	4	1	no	—
297	C₂²×C₃₁	—	4	4	no	—
298	C₂²×C₃₁	—	4	4	no	—
299	C₂²×C₃₁	—	4	2	no	—
300	C₂²×C₃₁	—	4	4	no	—
301	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
302	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
303	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
304	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
305	C₂²×C₃₁	#585	1	2	yes	—
306	C₄×C₃₁	—	2	2	no	—
307	C₄×C₃₁	—	2	4	no	—
308	C₄×C₃₁	—	2	8	no	—
309	C₄×C₃₁	—	2	4	no	—
310	C₄×C₃₁	—	2	2	no	—
311	C₄×C₃₁	—	2	4	no	—
312	C₄×C₃₁	—	4	1	no	—
313	C₄×C₃₁	—	4	4	no	—
314	C₄×C₃₁	—	4	1	no	—
315	C₄×C₃₁	—	4	2	no	—
316	C₄×C₃₁	—	4	4	no	—
317	C₄×C₃₁	C₄×#41	1	4	yes	—
318	C₄×C₃₁	#540	1	4	yes	direct product factor
319	#540	C₄×C₃₁	1	16	yes	direct product factor
320	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
321	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
322	C₂×#123	C₂²×C₃₁	1	4	yes	—
323	C₂×#121	C₂²×C₃₁	1	2	yes	—
324	C₂×#121	C₂²×C₃₁	1	4	yes	—
325	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
326	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
327	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
328	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
329	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
330	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
332	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
333	C₂²×C₄²	C₂²×C₃₁	1	4	yes	—
334	#390	C₂²×C₃₁	1	2	yes	—
335	#390	C₂²×C₃₁	1	4	yes	—
336	#390	C₂²×C₃₁	1	2	yes	—
337	#390	C₂²×C₃₁	1	4	yes	—
338	#389	C₂²×C₃₁	1	2	yes	—
339	#389	C₂²×C₃₁	1	2	yes	—
340	#389	C₂²×C₃₁	1	4	yes	—
341	#388	C₂²×C₃₁	1	4	yes	—
342	#388	C₂²×C₃₁	1	4	yes	—
343	#388	C₂²×C₃₁	1	4	yes	—
344	#388	C₂²×C₃₁	1	4	yes	—
345	#388	C₂²×C₃₁	1	4	yes	—
346	#387	C₂²×C₃₁	1	4	yes	—
347	#387	C₂²×C₃₁	1	4	yes	—
348	#387	C₂²×C₃₁	1	4	yes	—
349	#386	C₂²×C₃₁	1	4	yes	—
350	#386	C₂²×C₃₁	1	4	yes	—
351	#386	C₂²×C₃₁	1	4	yes	—
352	#386	C₂²×C₃₁	1	4	yes	—
353	#385	C₂²×C₃₁	1	4	yes	—
354	#384	C₂²×C₃₁	1	1	yes	characteristic
355	#382	C₂²×C₃₁	1	1	yes	characteristic
356	#381	C₂²×C₃₁	1	4	yes	—
357	#380	C₂²×C₃₁	1	2	yes	—
358	#380	C₂²×C₃₁	1	4	yes	—
359	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
360	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
361	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
362	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
363	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
364	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
365	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
366	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
367	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
368	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
369	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
370	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
371	#376	C₂²×C₃₁	1	1	yes	characteristic
372	#376	C₂²×C₃₁	1	1	yes	characteristic
373	C₄³	C₂²×C₃₁	1	1	yes	characteristic
374	C₄³	C₂²×C₃₁	1	1	yes	characteristic
375	C₂×C₃₁	—	4	2	no	—
376	C₂×C₃₁	—	4	8	no	—
377	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
378	C₂×C₃₁	C₄×H(5,𝔽₂)	1	2	yes	—
379	C₂×C₃₁	C₂×#540	1	1	yes	characteristic
380	C₂×C₃₁	#3455	1	1	yes	characteristic
381	C₂×C₃₁	#3460	1	2	yes	—
382	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
383	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
384	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
385	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
386	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
387	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
388	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
389	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
390	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
391	#127	C₂×C₄×C₃₁	1	8	yes	complemented
392	#125	C₂×C₄×C₃₁	1	8	yes	complemented
393	#123	C₂×C₄×C₃₁	1	32	yes	complemented
394	#121	C₂×C₄×C₃₁	1	32	yes	complemented
395	#121	C₂×C₄×C₃₁	1	16	yes	complemented
396	C₄×D₄	—	2	8	no	—
397	C₄×D₄	—	2	8	no	—
398	C₄×D₄	—	4	8	no	—
399	C₄×D₄	—	4	8	no	—
400	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
401	#117	—	2	4	no	—
402	#117	C₂×C₄×C₃₁	1	8	yes	complemented
403	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
404	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
405	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
407	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
408	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
409	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
410	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
411	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
412	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
423	C₂×C₄²	—	2	4	no	—
424	C₂×C₄²	—	4	8	no	—
425	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
426	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
427	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
428	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
429	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
430	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
431	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
433	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
434	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
435	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
436	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
437	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
438	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
439	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
440	#95	C₂³×C₃₁	1	2	yes	—
441	#95	C₂³×C₃₁	1	2	yes	—
442	#95	C₂³×C₃₁	1	2	yes	—
443	#95	C₂³×C₃₁	1	2	yes	—
444	#95	C₂³×C₃₁	1	4	yes	—
445	#95	C₂³×C₃₁	1	4	yes	—
446	#95	C₂³×C₃₁	1	4	yes	—
447	#95	C₂³×C₃₁	1	4	yes	—
448	#95	C₂³×C₃₁	1	4	yes	—
449	#95	C₂³×C₃₁	1	4	yes	—
450	#95	C₂³×C₃₁	1	4	yes	—
451	C₃₁	C₄×#540	1	1	yes	characteristic, direct product factor, Sylow
452	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
453	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
454	C₂×D₄	—	2	4	no	—
455	C₂×D₄	—	2	4	no	—
456	C₂×D₄	—	4	4	no	—
457	C₂×D₄	—	4	4	no	—
458	C₂×D₄	—	4	4	no	—
459	C₂×D₄	—	4	4	no	—
460	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
461	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
462	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	8	no	—
468	C₂²×C₄	—	2	8	no	—
469	C₂²×C₄	—	2	8	no	—
470	C₂²×C₄	—	2	4	no	—
471	C₂²×C₄	—	2	8	no	—
472	C₂²×C₄	—	4	4	no	—
473	C₂²×C₄	—	4	4	no	—
474	C₂²×C₄	—	4	2	no	—
475	C₂²×C₄	—	4	4	no	—
476	C₂²×C₄	—	4	4	no	—
477	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
478	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	—
479	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
481	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
482	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
483	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
484	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
485	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
486	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
487	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
488	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
489	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
490	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
491	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	4	4	no	—
496	C₄⋊C₄	—	4	4	no	—
497	C₄⋊C₄	—	4	4	no	—
498	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
499	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
500	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	8	no	—
503	C₂²⋊C₄	—	2	8	no	—
504	C₂²⋊C₄	—	4	4	no	—
505	C₂²⋊C₄	—	4	4	no	—
506	C₂²⋊C₄	—	4	4	no	—
507	C₂²⋊C₄	—	4	4	no	—
508	C₂²⋊C₄	—	4	4	no	—
509	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
510	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
511	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
512	C₄²	—	2	2	no	—
513	C₄²	—	2	2	no	—
514	C₄²	—	2	2	no	—
515	C₄²	—	2	2	no	—
516	C₄²	—	2	4	no	—
517	C₄²	—	2	8	no	—
518	C₄²	—	2	4	no	—
519	C₄²	—	4	4	no	—
520	C₄²	—	4	4	no	—
521	C₄²	—	4	2	no	—
522	C₄²	—	4	2	no	—
523	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
524	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
525	C₄²	#41×C₃₁	1	4	yes	—
526	C₄²	#41×C₃₁	1	4	yes	—
527	C₂³	—	2	1	no	—
528	C₂³	—	2	4	no	—
529	C₂³	—	2	4	no	—
530	C₂³	—	2	1	no	—
531	C₂³	—	2	4	no	—
532	C₂³	—	2	2	no	—
533	C₂³	—	2	4	no	—
534	C₂³	—	2	4	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	4	no	—
537	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
538	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
539	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
540	D₄	—	2	16	no	—
541	D₄	—	4	16	no	—
542	D₄	—	4	16	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	8	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	4	16	no	—
569	C₂×C₄	—	4	4	no	—
570	C₂×C₄	—	4	16	no	—
571	C₂×C₄	—	4	4	no	—
572	C₂×C₄	—	4	16	no	—
573	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
574	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
575	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
576	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
577	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
578	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
579	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
580	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
581	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
582	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
583	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
584	C₂²	—	2	2	no	—
585	C₂²	—	2	8	no	—
586	C₂²	—	2	8	no	—
587	C₂²	—	2	4	no	—
588	C₂²	—	2	8	no	—
589	C₂²	—	4	4	no	—
590	C₂²	—	4	1	no	—
591	C₂²	—	4	1	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	4	4	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	4	4	no	—
596	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
597	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
598	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
599	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
600	C₂²	C₃₁×#585	1	2	yes	—
601	C₄	—	2	2	no	—
602	C₄	—	2	4	no	—
603	C₄	—	2	8	no	—
604	C₄	—	2	4	no	—
605	C₄	—	2	2	no	—
606	C₄	—	2	4	no	—
607	C₄	—	4	1	no	—
608	C₄	—	4	4	no	—
609	C₄	—	4	1	no	—
610	C₄	—	4	4	no	—
611	C₄	—	4	2	no	—
612	C₄	C₄×#41×C₃₁	1	4	yes	—
613	C₄	C₃₁×#540	1	4	yes	direct product factor
614	C₂	—	4	2	no	—
615	C₂	—	4	8	no	—
616	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
617	C₂	C₄×C₃₁×H(5,𝔽₂)	1	2	yes	—
618	C₂	C₂×C₃₁×#540	1	1	yes	characteristic
619	C₂	C₃₁×#3455	1	1	yes	characteristic
620	C₂	C₃₁×#3460	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
prime radical	C₂×C₃₁×#540	C₂
LCS	G ▷ C₂² ▷ 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×#2335	C₂	semidirect
3. C₃₁×#2333	C₂	semidirect
4. C₃₁×#2330	C₂	semidirect
5. C₃₁×#2328	C₂	semidirect
6. C₃₁×#2324	C₂	semidirect
7. C₃₁×#2319	C₂	semidirect
8. C₃₁×#2314	C₂	semidirect
9. C₄×C₃₁×#127	C₂	semidirect
10. C₄×C₃₁×#125	C₂	semidirect
11. C₄×C₃₁×#123	C₂	semidirect
12. C₄×C₃₁×#121	C₂	semidirect
13. C₄×C₃₁×#121	C₂	semidirect
14. C₄²×D₄×C₃₁	C₂	semidirect
15. C₄×C₃₁×#117	C₂	semidirect
16. C₃₁×#540	C₄	direct (N × H)
81. C₃₁×#127	C₂×C₄	semidirect
82. C₃₁×#125	C₂×C₄	semidirect
83. C₃₁×#123	C₂×C₄	semidirect
84. C₃₁×#121	C₂×C₄	semidirect
85. C₃₁×#121	C₂×C₄	semidirect
90. C₄×D₄×C₃₁	C₂×C₄	semidirect
92. C₃₁×#117	C₂×C₄	semidirect
216. C₄×#540	C₃₁	direct (N × H)
275. #2335	C₂×C₃₁	semidirect
276. #2333	C₂×C₃₁	semidirect
277. #2330	C₂×C₃₁	semidirect
278. #2328	C₂×C₃₁	semidirect
279. #2324	C₂×C₃₁	semidirect
280. #2319	C₂×C₃₁	semidirect
281. #2314	C₂×C₃₁	semidirect
282. C₄×#127	C₂×C₃₁	semidirect
283. C₄×#125	C₂×C₃₁	semidirect
284. C₄×#123	C₂×C₃₁	semidirect
285. C₄×#121	C₂×C₃₁	semidirect
286. C₄×#121	C₂×C₃₁	semidirect
287. C₄²×D₄	C₂×C₃₁	semidirect
288. C₄×#117	C₂×C₃₁	semidirect
318. C₄×C₃₁	#540	direct (N × H)
319. #540	C₄×C₃₁	direct (N × H)
391. #127	C₂×C₄×C₃₁	semidirect
392. #125	C₂×C₄×C₃₁	semidirect
393. #123	C₂×C₄×C₃₁	semidirect
394. #121	C₂×C₄×C₃₁	semidirect
395. #121	C₂×C₄×C₃₁	semidirect
400. C₄×D₄	C₂×C₄×C₃₁	semidirect
402. #117	C₂×C₄×C₃₁	semidirect
451. C₃₁	C₄×#540	direct (N × H)
613. C₄	C₃₁×#540	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×#540	2
C₃₁×#2335	2
C₃₁×#2333	2
C₃₁×#2330	2
C₃₁×#2328	2
C₃₁×#2324	2
C₃₁×#2319	2
C₃₁×#2314	2
C₄×C₃₁×#127	2
C₄×C₃₁×#125	2
C₄×C₃₁×#123	2
C₄×C₃₁×#121	2
C₄²×D₄×C₃₁	2
C₄×C₃₁×#117	2
C₄×#540	31

Subgroups — C4×C31×#540 #21490852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₃₁×#540 #21490852