Subgroups — C₂×C₃₁×#2513 #21491546

Order 7936.

2334 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2513	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃₁×#397	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#397	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₁×#390	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#390	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×#390	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#388	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#388	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×#382	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
30	C₃₁×#397	C₂²	1	16	yes	complemented
31	C₃₁×#397	C₂²	1	16	yes	complemented
32	C₃₁×#390	C₂²	1	8	yes	—
33	C₃₁×#390	C₂²	1	8	yes	—
34	C₃₁×#390	C₂²	1	8	yes	—
35	C₃₁×#389	C₂²	1	8	yes	complemented
36	C₃₁×#388	C₂²	1	16	yes	complemented
37	C₃₁×#388	C₂²	1	16	yes	complemented
38	C₃₁×#382	C₂²	1	8	yes	complemented
39	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	complemented
40	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	complemented
41	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#95	C₂²	1	2	yes	—
43	C₂×C₃₁×#95	C₂²	1	2	yes	—
44	C₂×C₃₁×#95	C₂²	1	2	yes	—
45	C₂×C₃₁×#95	C₂²	1	2	yes	—
46	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
47	C₂⁵×C₃₁	C₂³	1	2	yes	—
48	C₂²×D₄×C₃₁	—	2	4	no	—
49	C₂²×D₄×C₃₁	—	2	4	no	—
50	C₂²×D₄×C₃₁	—	2	4	no	—
51	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
52	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
54	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
58	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
59	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
67	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
80	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
82	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
84	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
86	C₂×C₄²×C₃₁	—	2	4	no	—
87	C₂×C₄²×C₃₁	—	2	4	no	—
88	C₂×C₄²×C₃₁	—	2	4	no	—
89	C₂×C₄²×C₃₁	—	4	4	no	—
90	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
91	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
92	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
93	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
94	C₃₁×#95	C₂³	1	4	yes	—
95	C₃₁×#95	C₂³	1	8	yes	—
96	C₃₁×#95	C₂³	1	8	yes	—
97	C₃₁×#95	C₂³	1	8	yes	—
98	C₃₁×#95	C₂³	1	8	yes	—
99	C₃₁×#95	C₂³	1	4	yes	—
100	C₂⁴×C₃₁	—	2	4	no	—
101	C₂⁴×C₃₁	—	2	4	no	—
102	C₂⁴×C₃₁	—	2	4	no	—
103	C₂⁴×C₃₁	—	2	4	no	—
104	C₂⁴×C₃₁	—	2	4	no	—
105	C₂⁴×C₃₁	—	2	4	no	—
106	C₂⁴×C₃₁	—	2	4	no	—
107	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
108	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
109	C₂×D₄×C₃₁	—	2	8	no	—
110	C₂×D₄×C₃₁	—	2	8	no	—
111	C₂×D₄×C₃₁	—	2	8	no	—
112	C₂×D₄×C₃₁	—	4	16	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	2	2	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	4	no	—
122	C₂²×C₄×C₃₁	—	2	4	no	—
123	C₂²×C₄×C₃₁	—	2	4	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	2	4	no	—
128	C₂²×C₄×C₃₁	—	2	4	no	—
129	C₂²×C₄×C₃₁	—	2	4	no	—
130	C₂²×C₄×C₃₁	—	2	4	no	—
131	C₂²×C₄×C₃₁	—	2	2	no	—
132	C₂²×C₄×C₃₁	—	2	2	no	—
133	C₂²×C₄×C₃₁	—	2	2	no	—
134	C₂²×C₄×C₃₁	—	2	2	no	—
135	C₂²×C₄×C₃₁	—	2	4	no	—
136	C₂²×C₄×C₃₁	—	2	4	no	—
137	C₂²×C₄×C₃₁	—	2	2	no	—
138	C₂²×C₄×C₃₁	#41	1	4	yes	—
139	C₂²×C₄×C₃₁	#41	1	4	yes	—
140	C₂²×C₄×C₃₁	#41	1	4	yes	—
141	C₂²×C₄×C₃₁	#41	1	4	yes	—
142	C₂²×C₄×C₃₁	#41	1	4	yes	—
143	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
144	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
145	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
146	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
149	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
152	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
153	C₄⋊C₄×C₃₁	—	2	8	no	—
154	C₄⋊C₄×C₃₁	—	2	8	no	—
155	C₄⋊C₄×C₃₁	—	2	8	no	—
156	C₄⋊C₄×C₃₁	—	2	8	no	—
157	C₄⋊C₄×C₃₁	—	2	8	no	—
158	C₄⋊C₄×C₃₁	—	4	8	no	—
159	C₂²⋊C₄×C₃₁	—	2	16	no	—
160	C₂²⋊C₄×C₃₁	—	2	16	no	—
161	C₂²⋊C₄×C₃₁	—	4	8	no	—
162	C₂²⋊C₄×C₃₁	—	4	8	no	—
163	C₂²⋊C₄×C₃₁	—	4	8	no	—
164	C₂²⋊C₄×C₃₁	—	4	8	no	—
165	C₄²×C₃₁	—	2	8	no	—
166	C₄²×C₃₁	—	2	8	no	—
167	C₄²×C₃₁	—	2	8	no	—
168	C₄²×C₃₁	—	4	8	no	—
169	C₂×#2513	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂³×C₃₁	—	2	4	no	—
171	C₂³×C₃₁	—	2	4	no	—
172	C₂³×C₃₁	—	2	4	no	—
173	C₂³×C₃₁	—	2	4	no	—
174	C₂³×C₃₁	—	2	4	no	—
175	C₂³×C₃₁	—	2	4	no	—
176	C₂³×C₃₁	—	2	4	no	—
177	C₂³×C₃₁	—	4	8	no	—
178	C₂³×C₃₁	—	4	8	no	—
179	C₂³×C₃₁	—	4	8	no	—
180	C₂³×C₃₁	—	4	8	no	—
181	C₂³×C₃₁	—	4	8	no	—
182	C₂³×C₃₁	—	4	8	no	—
183	C₂³×C₃₁	—	4	8	no	—
184	C₂³×C₃₁	C₂×#41	1	2	yes	—
185	C₂³×C₃₁	C₂×#41	1	2	yes	—
186	C₂³×C₃₁	C₂×#41	1	2	yes	—
187	C₂³×C₃₁	C₂×#41	1	2	yes	—
188	C₂³×C₃₁	C₂×#41	1	2	yes	—
189	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
190	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
191	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
192	D₄×C₃₁	—	4	16	no	—
193	C₂×C₄×C₃₁	—	2	8	no	—
194	C₂×C₄×C₃₁	—	2	8	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	4	no	—
204	C₂×C₄×C₃₁	—	2	4	no	—
205	C₂×C₄×C₃₁	—	2	4	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	4	no	—
212	C₂×C₄×C₃₁	—	2	4	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	4	no	—
218	C₂×C₄×C₃₁	—	2	4	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	4	4	no	—
221	C₂×C₄×C₃₁	—	4	8	no	—
222	C₂×C₄×C₃₁	—	4	4	no	—
223	C₂×C₄×C₃₁	—	4	4	no	—
224	C₂×C₄×C₃₁	—	4	4	no	—
225	C₂×C₄×C₃₁	—	4	8	no	—
226	C₂×C₄×C₃₁	—	4	8	no	—
227	C₂×C₄×C₃₁	#126	1	16	yes	—
228	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
229	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
230	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
231	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
232	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
233	#2513	C₂×C₃₁	1	16	yes	direct product factor
234	C₂×#397	C₂×C₃₁	1	2	yes	complemented
235	C₂×#397	C₂×C₃₁	1	2	yes	complemented
236	C₂×#390	C₂×C₃₁	1	1	yes	characteristic
237	C₂×#390	C₂×C₃₁	1	1	yes	characteristic
238	C₂×#390	C₂×C₃₁	1	1	yes	characteristic
239	C₂×#389	C₂×C₃₁	1	1	yes	characteristic, complemented
240	C₂×#388	C₂×C₃₁	1	2	yes	complemented
241	C₂×#388	C₂×C₃₁	1	2	yes	complemented
242	C₂×#382	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
244	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
245	C₂²×C₃₁	—	4	4	no	—
246	C₂²×C₃₁	—	4	4	no	—
247	C₂²×C₃₁	—	4	4	no	—
248	C₂²×C₃₁	—	4	4	no	—
249	C₂²×C₃₁	—	4	4	no	—
250	C₂²×C₃₁	—	4	4	no	—
251	C₂²×C₃₁	—	4	4	no	—
252	C₂²×C₃₁	—	8	16	no	—
253	C₂²×C₃₁	C₂×#126	1	4	yes	—
254	C₂²×C₃₁	#529	1	4	yes	—
255	C₂²×C₃₁	#540	1	4	yes	—
256	C₂²×C₃₁	#553	1	4	yes	—
257	C₂²×C₃₁	#555	1	4	yes	—
258	C₂²×C₃₁	#559	1	4	yes	—
259	C₂²×C₃₁	#559	1	4	yes	—
260	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
261	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
262	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
263	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
264	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
265	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
266	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
267	C₄×C₃₁	—	2	4	no	—
268	C₄×C₃₁	—	2	4	no	—
269	C₄×C₃₁	—	4	2	no	—
270	C₄×C₃₁	—	4	4	no	—
271	C₄×C₃₁	—	4	2	no	—
272	C₄×C₃₁	—	4	2	no	—
273	C₄×C₃₁	—	4	2	no	—
274	C₄×C₃₁	—	4	4	no	—
275	C₄×C₃₁	—	4	4	no	—
276	C₄×C₃₁	C₂×#126	1	8	yes	—
277	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
278	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
279	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
280	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
281	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
282	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
283	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
284	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
285	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
286	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
287	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
288	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
289	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
290	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
291	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
292	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
293	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
294	#397	C₂²×C₃₁	1	16	yes	complemented
295	#397	C₂²×C₃₁	1	16	yes	complemented
296	#390	C₂²×C₃₁	1	8	yes	—
297	#390	C₂²×C₃₁	1	8	yes	—
298	#390	C₂²×C₃₁	1	8	yes	—
299	#389	C₂²×C₃₁	1	8	yes	complemented
300	#388	C₂²×C₃₁	1	16	yes	complemented
301	#388	C₂²×C₃₁	1	16	yes	complemented
302	#382	C₂²×C₃₁	1	8	yes	complemented
303	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	complemented
304	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	complemented
305	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
306	C₂×#95	C₂²×C₃₁	1	2	yes	—
307	C₂×#95	C₂²×C₃₁	1	2	yes	—
308	C₂×#95	C₂²×C₃₁	1	2	yes	—
309	C₂×#95	C₂²×C₃₁	1	2	yes	—
310	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
311	C₂×C₃₁	—	8	4	no	—
312	C₂×C₃₁	#2513	1	8	yes	direct product factor
313	C₂×C₃₁	C₂²×#126	1	1	yes	characteristic
314	C₂×C₃₁	C₂×#529	1	1	yes	characteristic
315	C₂×C₃₁	C₂×#540	1	1	yes	characteristic
316	C₂×C₃₁	C₂×#553	1	1	yes	characteristic
317	C₂×C₃₁	C₂×#555	1	1	yes	characteristic
318	C₂×C₃₁	C₂×#559	1	1	yes	characteristic
319	C₂×C₃₁	C₂×#559	1	1	yes	characteristic
320	C₂⁵	C₂³×C₃₁	1	2	yes	—
321	C₂²×D₄	—	2	4	no	—
322	C₂²×D₄	—	2	4	no	—
323	C₂²×D₄	—	2	4	no	—
324	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
325	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
326	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
327	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
328	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
329	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
330	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
331	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
332	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
333	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
334	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
335	C₂×C₄⋊C₄	—	2	4	no	—
336	C₂×C₄⋊C₄	—	2	4	no	—
337	C₂×C₄⋊C₄	—	2	4	no	—
338	C₂×C₄⋊C₄	—	2	4	no	—
339	C₂×C₄⋊C₄	—	2	4	no	—
340	C₂×C₄⋊C₄	—	4	4	no	—
341	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
342	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
343	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
344	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
345	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
346	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
347	C₂×C₂²⋊C₄	—	2	8	no	—
348	C₂×C₂²⋊C₄	—	2	8	no	—
349	C₂×C₂²⋊C₄	—	4	4	no	—
350	C₂×C₂²⋊C₄	—	4	4	no	—
351	C₂×C₂²⋊C₄	—	4	4	no	—
352	C₂×C₂²⋊C₄	—	4	4	no	—
353	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
359	C₂×C₄²	—	2	4	no	—
360	C₂×C₄²	—	2	4	no	—
361	C₂×C₄²	—	2	4	no	—
362	C₂×C₄²	—	4	4	no	—
363	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
364	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
365	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
366	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
367	#95	C₂³×C₃₁	1	4	yes	—
368	#95	C₂³×C₃₁	1	4	yes	—
369	#95	C₂³×C₃₁	1	8	yes	—
370	#95	C₂³×C₃₁	1	8	yes	—
371	#95	C₂³×C₃₁	1	8	yes	—
372	#95	C₂³×C₃₁	1	8	yes	—
373	C₃₁	C₂×#2513	1	1	yes	characteristic, direct product factor, Sylow
374	C₂⁴	—	2	4	no	—
375	C₂⁴	—	2	4	no	—
376	C₂⁴	—	2	4	no	—
377	C₂⁴	—	2	4	no	—
378	C₂⁴	—	2	4	no	—
379	C₂⁴	—	2	4	no	—
380	C₂⁴	—	2	4	no	—
381	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
382	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
383	C₂×D₄	—	2	8	no	—
384	C₂×D₄	—	2	8	no	—
385	C₂×D₄	—	2	8	no	—
386	C₂×D₄	—	4	16	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	4	no	—
396	C₂²×C₄	—	2	4	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	4	no	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	#41×C₃₁	1	4	yes	—
413	C₂²×C₄	#41×C₃₁	1	4	yes	—
414	C₂²×C₄	#41×C₃₁	1	4	yes	—
415	C₂²×C₄	#41×C₃₁	1	4	yes	—
416	C₂²×C₄	#41×C₃₁	1	4	yes	—
417	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
418	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
419	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
420	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
421	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
422	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
423	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
424	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
425	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
426	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
427	C₄⋊C₄	—	2	8	no	—
428	C₄⋊C₄	—	2	8	no	—
429	C₄⋊C₄	—	2	8	no	—
430	C₄⋊C₄	—	2	8	no	—
431	C₄⋊C₄	—	2	8	no	—
432	C₄⋊C₄	—	4	8	no	—
433	C₂²⋊C₄	—	2	16	no	—
434	C₂²⋊C₄	—	2	16	no	—
435	C₂²⋊C₄	—	4	8	no	—
436	C₂²⋊C₄	—	4	8	no	—
437	C₂²⋊C₄	—	4	8	no	—
438	C₂²⋊C₄	—	4	8	no	—
439	C₄²	—	2	8	no	—
440	C₄²	—	2	8	no	—
441	C₄²	—	2	8	no	—
442	C₄²	—	4	8	no	—
443	C₂³	—	2	4	no	—
444	C₂³	—	2	4	no	—
445	C₂³	—	2	4	no	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	4	no	—
448	C₂³	—	2	4	no	—
449	C₂³	—	2	4	no	—
450	C₂³	—	4	8	no	—
451	C₂³	—	4	8	no	—
452	C₂³	—	4	8	no	—
453	C₂³	—	4	8	no	—
454	C₂³	—	4	8	no	—
455	C₂³	—	4	8	no	—
456	C₂³	—	4	8	no	—
457	C₂³	C₂×#41×C₃₁	1	2	yes	—
458	C₂³	C₂×#41×C₃₁	1	2	yes	—
459	C₂³	C₂×#41×C₃₁	1	2	yes	—
460	C₂³	C₂×#41×C₃₁	1	2	yes	—
461	C₂³	C₂×#41×C₃₁	1	2	yes	—
462	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
463	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
464	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
465	D₄	—	4	16	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	8	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	4	4	no	—
494	C₂×C₄	—	4	8	no	—
495	C₂×C₄	—	4	4	no	—
496	C₂×C₄	—	4	4	no	—
497	C₂×C₄	—	4	4	no	—
498	C₂×C₄	—	4	8	no	—
499	C₂×C₄	—	4	8	no	—
500	C₂×C₄	C₃₁×#126	1	16	yes	—
501	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
502	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
503	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
504	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
505	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
506	C₂²	—	4	4	no	—
507	C₂²	—	4	4	no	—
508	C₂²	—	4	4	no	—
509	C₂²	—	4	4	no	—
510	C₂²	—	4	4	no	—
511	C₂²	—	4	4	no	—
512	C₂²	—	4	4	no	—
513	C₂²	—	8	16	no	—
514	C₂²	C₂×C₃₁×#126	1	4	yes	—
515	C₂²	C₃₁×#529	1	4	yes	—
516	C₂²	C₃₁×#540	1	4	yes	—
517	C₂²	C₃₁×#553	1	4	yes	—
518	C₂²	C₃₁×#555	1	4	yes	—
519	C₂²	C₃₁×#559	1	4	yes	—
520	C₂²	C₃₁×#559	1	4	yes	—
521	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
522	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
523	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
524	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
525	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
526	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
527	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
528	C₄	—	2	4	no	—
529	C₄	—	2	4	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	4	no	—
532	C₄	—	4	2	no	—
533	C₄	—	4	2	no	—
534	C₄	—	4	2	no	—
535	C₄	—	4	4	no	—
536	C₄	—	4	4	no	—
537	C₄	C₂×C₃₁×#126	1	8	yes	—
538	C₂	—	8	4	no	—
539	C₂	C₃₁×#2513	1	8	yes	direct product factor
540	C₂	C₂²×C₃₁×#126	1	1	yes	characteristic
541	C₂	C₂×C₃₁×#529	1	1	yes	characteristic
542	C₂	C₂×C₃₁×#540	1	1	yes	characteristic
543	C₂	C₂×C₃₁×#553	1	1	yes	characteristic
544	C₂	C₂×C₃₁×#555	1	1	yes	characteristic
545	C₂	C₂×C₃₁×#559	1	1	yes	characteristic
546	C₂	C₂×C₃₁×#559	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2513	C₂	direct (N × H)
2. C₂×C₃₁×#397	C₂	semidirect
3. C₂×C₃₁×#397	C₂	semidirect
7. C₂×C₃₁×#389	C₂	semidirect
8. C₂×C₃₁×#388	C₂	semidirect
9. C₂×C₃₁×#388	C₂	semidirect
10. C₂×C₃₁×#382	C₂	semidirect
11. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
12. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
30. C₃₁×#397	C₂²	semidirect
31. C₃₁×#397	C₂²	semidirect
35. C₃₁×#389	C₂²	semidirect
36. C₃₁×#388	C₂²	semidirect
37. C₃₁×#388	C₂²	semidirect
38. C₃₁×#382	C₂²	semidirect
39. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
40. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
169. C₂×#2513	C₃₁	direct (N × H)
233. #2513	C₂×C₃₁	direct (N × H)
234. C₂×#397	C₂×C₃₁	semidirect
235. C₂×#397	C₂×C₃₁	semidirect
239. C₂×#389	C₂×C₃₁	semidirect
240. C₂×#388	C₂×C₃₁	semidirect
241. C₂×#388	C₂×C₃₁	semidirect
242. C₂×#382	C₂×C₃₁	semidirect
243. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
244. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
294. #397	C₂²×C₃₁	semidirect
295. #397	C₂²×C₃₁	semidirect
299. #389	C₂²×C₃₁	semidirect
300. #388	C₂²×C₃₁	semidirect
301. #388	C₂²×C₃₁	semidirect
302. #382	C₂²×C₃₁	semidirect
303. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
304. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
312. C₂×C₃₁	#2513	direct (N × H)
373. C₃₁	C₂×#2513	direct (N × H)
539. C₂	C₃₁×#2513	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×#2513	2
C₂×C₃₁×#397	2
C₂×C₃₁×#390	2
C₂×C₃₁×#389	2
C₂×C₃₁×#388	2
C₂×C₃₁×#382	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₂×#2513	31

Subgroups — C2×C31×#2513 #21491546

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2513 #21491546