Subgroups — C₂×C₃₁×#2405 #21491415

Order 7936.

7822 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#123	C₂	1	2	yes	maximal, complemented
3	C₂³×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2405	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₃₁×#394	C₂	1	4	yes	maximal, complemented
6	C₂×C₃₁×#386	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#385	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#379	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#379	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁶×C₃₁	C₂²	1	1	yes	characteristic
11	C₂⁶×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₃₁	—	2	8	no	—
13	C₂³×D₄×C₃₁	—	2	16	no	—
14	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂³×D₄×C₃₁	C₂²	1	8	yes	—
16	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
17	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
19	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₃₁×#123	—	2	16	no	—
21	C₂×C₃₁×#123	C₂²	1	16	yes	complemented
22	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
23	C₂²×C₂²⋊C₄×C₃₁	—	2	8	no	—
24	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
27	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
28	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
29	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
30	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
31	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
32	C₃₁×#394	C₂²	1	32	yes	complemented
33	C₃₁×#386	C₂²	1	16	yes	complemented
34	C₃₁×#385	C₂²	1	16	yes	complemented
35	C₃₁×#379	C₂²	1	16	yes	complemented
36	C₃₁×#379	C₂²	1	8	yes	complemented
37	C₂×C₃₁×#95	C₂²	1	4	yes	complemented
38	C₂⁵×C₃₁	—	2	8	no	—
39	C₂⁵×C₃₁	—	2	8	no	—
40	C₂⁵×C₃₁	—	2	4	no	—
41	C₂⁵×C₃₁	—	2	8	no	—
42	C₂⁵×C₃₁	—	2	8	no	—
43	C₂⁵×C₃₁	—	4	4	no	—
44	C₂⁵×C₃₁	—	4	4	no	—
45	C₂⁵×C₃₁	D₄	1	8	yes	—
46	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
47	C₂⁵×C₃₁	C₂³	1	2	yes	—
48	C₂⁵×C₃₁	C₂³	1	4	yes	—
49	C₂⁵×C₃₁	C₂³	1	4	yes	—
50	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
51	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×D₄×C₃₁	—	2	128	no	—
53	C₂²×D₄×C₃₁	—	2	8	no	—
54	C₂²×D₄×C₃₁	—	2	8	no	—
55	C₂²×D₄×C₃₁	—	2	32	no	—
56	C₂²×D₄×C₃₁	—	2	8	no	—
57	C₂²×D₄×C₃₁	—	2	8	no	—
58	C₂²×D₄×C₃₁	—	2	64	no	—
59	C₂²×D₄×C₃₁	—	2	8	no	—
60	C₂²×D₄×C₃₁	D₄	1	16	yes	—
61	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
62	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
63	C₂³×C₄×C₃₁	—	2	4	no	—
64	C₂³×C₄×C₃₁	—	2	8	no	—
65	C₂³×C₄×C₃₁	—	2	4	no	—
66	C₂³×C₄×C₃₁	—	2	4	no	—
67	C₂³×C₄×C₃₁	—	2	8	no	—
68	C₂³×C₄×C₃₁	—	4	4	no	—
69	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
70	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
71	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
72	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
73	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
74	C₂³×C₄×C₃₁	C₂³	1	4	yes	complemented
75	C₂³×C₄×C₃₁	C₂³	1	4	yes	complemented
76	C₃₁×#123	—	2	64	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	complemented
79	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	32	no	—
81	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
82	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
83	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
84	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
85	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
86	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
87	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
88	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
89	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	complemented
90	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	complemented
91	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	complemented
92	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	complemented
93	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	complemented
94	C₃₁×#95	C₂³	1	16	yes	complemented
95	C₂⁴×C₃₁	—	2	4	no	—
96	C₂⁴×C₃₁	—	2	4	no	—
97	C₂⁴×C₃₁	—	2	4	no	—
98	C₂⁴×C₃₁	—	2	4	no	—
99	C₂⁴×C₃₁	—	2	32	no	—
100	C₂⁴×C₃₁	—	2	32	no	—
101	C₂⁴×C₃₁	—	2	32	no	—
102	C₂⁴×C₃₁	—	2	16	no	—
103	C₂⁴×C₃₁	—	2	16	no	—
104	C₂⁴×C₃₁	—	2	8	no	—
105	C₂⁴×C₃₁	—	2	4	no	—
106	C₂⁴×C₃₁	—	2	4	no	—
107	C₂⁴×C₃₁	—	2	4	no	—
108	C₂⁴×C₃₁	—	2	16	no	—
109	C₂⁴×C₃₁	—	2	4	no	—
110	C₂⁴×C₃₁	—	2	8	no	—
111	C₂⁴×C₃₁	—	2	4	no	—
112	C₂⁴×C₃₁	—	2	8	no	—
113	C₂⁴×C₃₁	—	2	16	no	—
114	C₂⁴×C₃₁	—	4	16	no	—
115	C₂⁴×C₃₁	—	4	8	no	—
116	C₂⁴×C₃₁	—	4	8	no	—
117	C₂⁴×C₃₁	—	4	64	no	—
118	C₂⁴×C₃₁	—	4	64	no	—
119	C₂⁴×C₃₁	—	4	16	no	—
120	C₂⁴×C₃₁	—	4	16	no	—
121	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
122	C₂⁴×C₃₁	C₂×D₄	1	8	yes	—
123	C₂⁴×C₃₁	C₂×D₄	1	16	yes	—
124	C₂⁴×C₃₁	C₂×D₄	1	8	yes	—
125	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
126	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
127	C₂⁴×C₃₁	#41	1	8	yes	—
128	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
129	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
130	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
131	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
132	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
133	C₂×D₄×C₃₁	—	2	256	no	—
134	C₂×D₄×C₃₁	—	2	64	no	—
135	C₂×D₄×C₃₁	—	2	64	no	—
136	C₂×D₄×C₃₁	—	2	16	no	—
137	C₂×D₄×C₃₁	—	2	16	no	—
138	C₂×D₄×C₃₁	—	2	16	no	—
139	C₂×D₄×C₃₁	—	2	16	no	—
140	C₂×D₄×C₃₁	—	2	16	no	—
141	C₂×D₄×C₃₁	C₂×D₄	1	32	yes	—
142	C₂²×C₄×C₃₁	—	2	32	no	—
143	C₂²×C₄×C₃₁	—	2	8	no	—
144	C₂²×C₄×C₃₁	—	2	4	no	—
145	C₂²×C₄×C₃₁	—	2	4	no	—
146	C₂²×C₄×C₃₁	—	2	16	no	—
147	C₂²×C₄×C₃₁	—	2	8	no	—
148	C₂²×C₄×C₃₁	—	2	4	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	8	no	—
151	C₂²×C₄×C₃₁	—	2	8	no	—
152	C₂²×C₄×C₃₁	—	2	8	no	—
153	C₂²×C₄×C₃₁	—	2	16	no	—
154	C₂²×C₄×C₃₁	—	2	8	no	—
155	C₂²×C₄×C₃₁	—	2	8	no	—
156	C₂²×C₄×C₃₁	—	2	8	no	—
157	C₂²×C₄×C₃₁	—	4	8	no	—
158	C₂²×C₄×C₃₁	—	4	16	no	—
159	C₂²×C₄×C₃₁	—	4	16	no	—
160	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
163	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
165	C₄⋊C₄×C₃₁	—	2	32	no	—
166	C₂²⋊C₄×C₃₁	—	2	16	no	—
167	C₂²⋊C₄×C₃₁	—	2	16	no	—
168	C₂²⋊C₄×C₃₁	—	2	16	no	—
169	C₂²⋊C₄×C₃₁	—	2	32	no	—
170	C₂²⋊C₄×C₃₁	—	2	16	no	—
171	C₂²⋊C₄×C₃₁	—	4	8	no	—
172	C₂²⋊C₄×C₃₁	—	4	16	no	—
173	C₂×#2405	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂³×C₃₁	—	2	8	no	—
175	C₂³×C₃₁	—	2	8	no	—
176	C₂³×C₃₁	—	2	8	no	—
177	C₂³×C₃₁	—	2	8	no	—
178	C₂³×C₃₁	—	2	32	no	—
179	C₂³×C₃₁	—	2	32	no	—
180	C₂³×C₃₁	—	2	4	no	—
181	C₂³×C₃₁	—	2	4	no	—
182	C₂³×C₃₁	—	2	4	no	—
183	C₂³×C₃₁	—	2	4	no	—
184	C₂³×C₃₁	—	2	32	no	—
185	C₂³×C₃₁	—	2	32	no	—
186	C₂³×C₃₁	—	2	8	no	—
187	C₂³×C₃₁	—	2	16	no	—
188	C₂³×C₃₁	—	2	8	no	—
189	C₂³×C₃₁	—	2	32	no	—
190	C₂³×C₃₁	—	2	8	no	—
191	C₂³×C₃₁	—	2	32	no	—
192	C₂³×C₃₁	—	2	4	no	—
193	C₂³×C₃₁	—	2	4	no	—
194	C₂³×C₃₁	—	2	4	no	—
195	C₂³×C₃₁	—	2	4	no	—
196	C₂³×C₃₁	—	2	8	no	—
197	C₂³×C₃₁	—	2	4	no	—
198	C₂³×C₃₁	—	2	8	no	—
199	C₂³×C₃₁	—	4	16	no	—
200	C₂³×C₃₁	—	4	128	no	—
201	C₂³×C₃₁	—	4	64	no	—
202	C₂³×C₃₁	—	4	64	no	—
203	C₂³×C₃₁	—	4	128	no	—
204	C₂³×C₃₁	—	4	16	no	—
205	C₂³×C₃₁	—	4	8	no	—
206	C₂³×C₃₁	—	4	16	no	—
207	C₂³×C₃₁	—	4	8	no	—
208	C₂³×C₃₁	—	4	16	no	—
209	C₂³×C₃₁	—	4	16	no	—
210	C₂³×C₃₁	#120	1	16	yes	—
211	C₂³×C₃₁	#123	1	16	yes	complemented
212	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
213	C₂³×C₃₁	C₂²×D₄	1	8	yes	—
214	C₂³×C₃₁	C₂²×D₄	1	8	yes	—
215	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
216	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
217	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
218	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
219	C₂³×C₃₁	C₂×#41	1	8	yes	—
220	C₂³×C₃₁	C₂×#41	1	4	yes	—
221	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
222	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
223	D₄×C₃₁	—	2	64	no	—
224	D₄×C₃₁	—	2	64	no	—
225	C₂×C₄×C₃₁	—	2	32	no	—
226	C₂×C₄×C₃₁	—	2	16	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	8	no	—
230	C₂×C₄×C₃₁	—	2	4	no	—
231	C₂×C₄×C₃₁	—	2	4	no	—
232	C₂×C₄×C₃₁	—	2	8	no	—
233	C₂×C₄×C₃₁	—	2	8	no	—
234	C₂×C₄×C₃₁	—	2	8	no	—
235	C₂×C₄×C₃₁	—	2	8	no	—
236	C₂×C₄×C₃₁	—	4	16	no	—
237	C₂×C₄×C₃₁	—	4	16	no	—
238	C₂×C₄×C₃₁	—	4	8	no	—
239	C₂×C₄×C₃₁	—	4	8	no	—
240	C₂×C₄×C₃₁	—	4	16	no	—
241	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
242	C₂⁴×D₄	C₂×C₃₁	1	1	yes	characteristic
243	C₂²×#123	C₂×C₃₁	1	2	yes	complemented
244	C₂³×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#2405	C₂×C₃₁	1	16	yes	direct product factor
246	C₂×#394	C₂×C₃₁	1	4	yes	complemented
247	C₂×#386	C₂×C₃₁	1	2	yes	complemented
248	C₂×#385	C₂×C₃₁	1	2	yes	complemented
249	C₂×#379	C₂×C₃₁	1	2	yes	complemented
250	C₂×#379	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₂²×C₃₁	—	2	32	no	—
252	C₂²×C₃₁	—	2	32	no	—
253	C₂²×C₃₁	—	2	4	no	—
254	C₂²×C₃₁	—	2	4	no	—
255	C₂²×C₃₁	—	2	4	no	—
256	C₂²×C₃₁	—	2	4	no	—
257	C₂²×C₃₁	—	2	16	no	—
258	C₂²×C₃₁	—	2	16	no	—
259	C₂²×C₃₁	—	2	4	no	—
260	C₂²×C₃₁	—	2	8	no	—
261	C₂²×C₃₁	—	2	4	no	—
262	C₂²×C₃₁	—	2	4	no	—
263	C₂²×C₃₁	—	2	16	no	—
264	C₂²×C₃₁	—	4	64	no	—
265	C₂²×C₃₁	—	4	64	no	—
266	C₂²×C₃₁	—	4	16	no	—
267	C₂²×C₃₁	—	4	32	no	—
268	C₂²×C₃₁	—	4	4	no	—
269	C₂²×C₃₁	—	4	4	no	—
270	C₂²×C₃₁	—	4	8	no	—
271	C₂²×C₃₁	—	4	8	no	—
272	C₂²×C₃₁	C₂×#120	1	8	yes	—
273	C₂²×C₃₁	C₂×#120	1	4	yes	—
274	C₂²×C₃₁	C₂×#123	1	4	yes	—
275	C₂²×C₃₁	C₂×#123	1	8	yes	complemented
276	C₂²×C₃₁	#534	1	4	yes	—
277	C₂²×C₃₁	#546	1	16	yes	—
278	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
279	C₂²×C₃₁	C₂³×D₄	1	2	yes	—
280	C₂²×C₃₁	C₂²×#41	1	2	yes	—
281	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
282	C₄×C₃₁	—	2	8	no	—
283	C₄×C₃₁	—	4	4	no	—
284	C₄×C₃₁	—	4	4	no	—
285	C₄×C₃₁	—	4	8	no	—
286	C₂⁶	C₂²×C₃₁	1	1	yes	characteristic
287	C₂⁶	C₂²×C₃₁	1	1	yes	characteristic
288	C₂³×D₄	—	2	8	no	—
289	C₂³×D₄	—	2	16	no	—
290	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
291	C₂³×D₄	C₂²×C₃₁	1	8	yes	—
292	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
293	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
294	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
295	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
296	C₂×#123	—	2	16	no	—
297	C₂×#123	C₂²×C₃₁	1	16	yes	complemented
298	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
299	C₂²×C₂²⋊C₄	—	2	8	no	—
300	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
301	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
302	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
303	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
304	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
305	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
306	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
307	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
308	#394	C₂²×C₃₁	1	32	yes	complemented
309	#386	C₂²×C₃₁	1	16	yes	complemented
310	#385	C₂²×C₃₁	1	16	yes	complemented
311	#379	C₂²×C₃₁	1	16	yes	complemented
312	#379	C₂²×C₃₁	1	8	yes	complemented
313	C₂×#95	C₂²×C₃₁	1	4	yes	complemented
314	C₂×C₃₁	—	2	8	no	—
315	C₂×C₃₁	—	2	8	no	—
316	C₂×C₃₁	—	4	4	no	—
317	C₂×C₃₁	—	4	8	no	—
318	C₂×C₃₁	#2405	1	8	yes	direct product factor
319	C₂×C₃₁	C₂²×#120	1	1	yes	characteristic
320	C₂×C₃₁	C₂²×#123	1	1	yes	characteristic
321	C₂×C₃₁	C₂×#534	1	1	yes	characteristic
322	C₂×C₃₁	C₂×#546	1	4	yes	—
323	C₂⁵	—	2	8	no	—
324	C₂⁵	—	2	8	no	—
325	C₂⁵	—	2	4	no	—
326	C₂⁵	—	2	8	no	—
327	C₂⁵	—	2	8	no	—
328	C₂⁵	—	4	4	no	—
329	C₂⁵	—	4	4	no	—
330	C₂⁵	D₄×C₃₁	1	8	yes	—
331	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
332	C₂⁵	C₂³×C₃₁	1	2	yes	—
333	C₂⁵	C₂³×C₃₁	1	4	yes	—
334	C₂⁵	C₂³×C₃₁	1	4	yes	—
335	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
336	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
337	C₂²×D₄	—	2	128	no	—
338	C₂²×D₄	—	2	8	no	—
339	C₂²×D₄	—	2	8	no	—
340	C₂²×D₄	—	2	32	no	—
341	C₂²×D₄	—	2	8	no	—
342	C₂²×D₄	—	2	8	no	—
343	C₂²×D₄	—	2	64	no	—
344	C₂²×D₄	—	2	8	no	—
345	C₂²×D₄	D₄×C₃₁	1	16	yes	—
346	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
347	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
348	C₂³×C₄	—	2	4	no	—
349	C₂³×C₄	—	2	8	no	—
350	C₂³×C₄	—	2	4	no	—
351	C₂³×C₄	—	2	4	no	—
352	C₂³×C₄	—	2	8	no	—
353	C₂³×C₄	—	4	4	no	—
354	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
355	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
356	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
357	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
358	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
359	C₂³×C₄	C₂³×C₃₁	1	4	yes	complemented
360	C₂³×C₄	C₂³×C₃₁	1	4	yes	complemented
361	#123	—	2	64	no	—
362	C₂×C₄⋊C₄	—	2	16	no	—
363	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	complemented
364	C₂×C₂²⋊C₄	—	2	8	no	—
365	C₂×C₂²⋊C₄	—	2	32	no	—
366	C₂×C₂²⋊C₄	—	2	8	no	—
367	C₂×C₂²⋊C₄	—	2	8	no	—
368	C₂×C₂²⋊C₄	—	2	16	no	—
369	C₂×C₂²⋊C₄	—	2	8	no	—
370	C₂×C₂²⋊C₄	—	4	4	no	—
371	C₂×C₂²⋊C₄	—	4	8	no	—
372	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
373	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
374	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	complemented
375	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	complemented
376	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	complemented
377	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	complemented
378	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	complemented
379	#95	C₂³×C₃₁	1	16	yes	complemented
380	C₃₁	C₂×#2405	1	1	yes	characteristic, direct product factor, Sylow
381	C₂⁴	—	2	4	no	—
382	C₂⁴	—	2	4	no	—
383	C₂⁴	—	2	4	no	—
384	C₂⁴	—	2	4	no	—
385	C₂⁴	—	2	32	no	—
386	C₂⁴	—	2	32	no	—
387	C₂⁴	—	2	32	no	—
388	C₂⁴	—	2	16	no	—
389	C₂⁴	—	2	16	no	—
390	C₂⁴	—	2	8	no	—
391	C₂⁴	—	2	4	no	—
392	C₂⁴	—	2	4	no	—
393	C₂⁴	—	2	4	no	—
394	C₂⁴	—	2	16	no	—
395	C₂⁴	—	2	4	no	—
396	C₂⁴	—	2	8	no	—
397	C₂⁴	—	2	4	no	—
398	C₂⁴	—	2	8	no	—
399	C₂⁴	—	2	16	no	—
400	C₂⁴	—	4	16	no	—
401	C₂⁴	—	4	8	no	—
402	C₂⁴	—	4	8	no	—
403	C₂⁴	—	4	64	no	—
404	C₂⁴	—	4	64	no	—
405	C₂⁴	—	4	16	no	—
406	C₂⁴	—	4	16	no	—
407	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
408	C₂⁴	C₂×D₄×C₃₁	1	8	yes	—
409	C₂⁴	C₂×D₄×C₃₁	1	16	yes	—
410	C₂⁴	C₂×D₄×C₃₁	1	8	yes	—
411	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
412	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
413	C₂⁴	#41×C₃₁	1	8	yes	—
414	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
415	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
416	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
417	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
418	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
419	C₂×D₄	—	2	256	no	—
420	C₂×D₄	—	2	64	no	—
421	C₂×D₄	—	2	64	no	—
422	C₂×D₄	—	2	16	no	—
423	C₂×D₄	—	2	16	no	—
424	C₂×D₄	—	2	16	no	—
425	C₂×D₄	—	2	16	no	—
426	C₂×D₄	—	2	16	no	—
427	C₂×D₄	C₂×D₄×C₃₁	1	32	yes	—
428	C₂²×C₄	—	2	32	no	—
429	C₂²×C₄	—	2	8	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	2	16	no	—
433	C₂²×C₄	—	2	8	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	8	no	—
437	C₂²×C₄	—	2	8	no	—
438	C₂²×C₄	—	2	8	no	—
439	C₂²×C₄	—	2	16	no	—
440	C₂²×C₄	—	2	8	no	—
441	C₂²×C₄	—	2	8	no	—
442	C₂²×C₄	—	2	8	no	—
443	C₂²×C₄	—	4	8	no	—
444	C₂²×C₄	—	4	16	no	—
445	C₂²×C₄	—	4	16	no	—
446	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
447	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
448	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
449	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
450	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
451	C₄⋊C₄	—	2	32	no	—
452	C₂²⋊C₄	—	2	16	no	—
453	C₂²⋊C₄	—	2	16	no	—
454	C₂²⋊C₄	—	2	16	no	—
455	C₂²⋊C₄	—	2	32	no	—
456	C₂²⋊C₄	—	2	16	no	—
457	C₂²⋊C₄	—	4	8	no	—
458	C₂²⋊C₄	—	4	16	no	—
459	C₂³	—	2	8	no	—
460	C₂³	—	2	8	no	—
461	C₂³	—	2	8	no	—
462	C₂³	—	2	8	no	—
463	C₂³	—	2	32	no	—
464	C₂³	—	2	32	no	—
465	C₂³	—	2	4	no	—
466	C₂³	—	2	4	no	—
467	C₂³	—	2	4	no	—
468	C₂³	—	2	4	no	—
469	C₂³	—	2	32	no	—
470	C₂³	—	2	32	no	—
471	C₂³	—	2	8	no	—
472	C₂³	—	2	16	no	—
473	C₂³	—	2	8	no	—
474	C₂³	—	2	32	no	—
475	C₂³	—	2	8	no	—
476	C₂³	—	2	32	no	—
477	C₂³	—	2	4	no	—
478	C₂³	—	2	4	no	—
479	C₂³	—	2	4	no	—
480	C₂³	—	2	4	no	—
481	C₂³	—	2	8	no	—
482	C₂³	—	2	4	no	—
483	C₂³	—	2	8	no	—
484	C₂³	—	4	16	no	—
485	C₂³	—	4	128	no	—
486	C₂³	—	4	64	no	—
487	C₂³	—	4	64	no	—
488	C₂³	—	4	128	no	—
489	C₂³	—	4	16	no	—
490	C₂³	—	4	8	no	—
491	C₂³	—	4	16	no	—
492	C₂³	—	4	8	no	—
493	C₂³	—	4	16	no	—
494	C₂³	—	4	16	no	—
495	C₂³	C₃₁×#120	1	16	yes	—
496	C₂³	C₃₁×#123	1	16	yes	complemented
497	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
498	C₂³	C₂²×D₄×C₃₁	1	8	yes	—
499	C₂³	C₂²×D₄×C₃₁	1	8	yes	—
500	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
501	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
502	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
503	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
504	C₂³	C₂×#41×C₃₁	1	8	yes	—
505	C₂³	C₂×#41×C₃₁	1	4	yes	—
506	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
507	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
508	D₄	—	2	64	no	—
509	D₄	—	2	64	no	—
510	C₂×C₄	—	2	32	no	—
511	C₂×C₄	—	2	16	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	8	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	8	no	—
518	C₂×C₄	—	2	8	no	—
519	C₂×C₄	—	2	8	no	—
520	C₂×C₄	—	2	8	no	—
521	C₂×C₄	—	4	16	no	—
522	C₂×C₄	—	4	16	no	—
523	C₂×C₄	—	4	8	no	—
524	C₂×C₄	—	4	8	no	—
525	C₂×C₄	—	4	16	no	—
526	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
527	C₂²	—	2	32	no	—
528	C₂²	—	2	32	no	—
529	C₂²	—	2	4	no	—
530	C₂²	—	2	4	no	—
531	C₂²	—	2	4	no	—
532	C₂²	—	2	4	no	—
533	C₂²	—	2	16	no	—
534	C₂²	—	2	16	no	—
535	C₂²	—	2	4	no	—
536	C₂²	—	2	8	no	—
537	C₂²	—	2	4	no	—
538	C₂²	—	2	4	no	—
539	C₂²	—	2	16	no	—
540	C₂²	—	4	64	no	—
541	C₂²	—	4	64	no	—
542	C₂²	—	4	16	no	—
543	C₂²	—	4	32	no	—
544	C₂²	—	4	4	no	—
545	C₂²	—	4	4	no	—
546	C₂²	—	4	8	no	—
547	C₂²	—	4	8	no	—
548	C₂²	C₂×C₃₁×#120	1	8	yes	—
549	C₂²	C₂×C₃₁×#120	1	4	yes	—
550	C₂²	C₂×C₃₁×#123	1	4	yes	—
551	C₂²	C₂×C₃₁×#123	1	8	yes	complemented
552	C₂²	C₃₁×#534	1	4	yes	—
553	C₂²	C₃₁×#546	1	16	yes	—
554	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
555	C₂²	C₂³×D₄×C₃₁	1	2	yes	—
556	C₂²	C₂²×#41×C₃₁	1	2	yes	—
557	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
558	C₄	—	2	8	no	—
559	C₄	—	4	4	no	—
560	C₄	—	4	4	no	—
561	C₄	—	4	8	no	—
562	C₂	—	2	8	no	—
563	C₂	—	2	8	no	—
564	C₂	—	4	4	no	—
565	C₂	—	4	8	no	—
566	C₂	C₃₁×#2405	1	8	yes	direct product factor
567	C₂	C₂²×C₃₁×#120	1	1	yes	characteristic
568	C₂	C₂²×C₃₁×#123	1	1	yes	characteristic
569	C₂	C₂×C₃₁×#534	1	1	yes	characteristic
570	C₂	C₂×C₃₁×#546	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂⁴×D₄×C₃₁	C₂
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
2. C₂²×C₃₁×#123	C₂	semidirect
3. C₂³×C₂²⋊C₄×C₃₁	C₂	semidirect
4. C₃₁×#2405	C₂	direct (N × H)
5. C₂×C₃₁×#394	C₂	semidirect
6. C₂×C₃₁×#386	C₂	semidirect
7. C₂×C₃₁×#385	C₂	semidirect
8. C₂×C₃₁×#379	C₂	semidirect
9. C₂×C₃₁×#379	C₂	semidirect
18. C₂⁴×C₄×C₃₁	C₂²	semidirect
19. C₂⁴×C₄×C₃₁	C₂²	semidirect
21. C₂×C₃₁×#123	C₂²	semidirect
22. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
26. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
27. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
28. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
29. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
30. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
31. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
32. C₃₁×#394	C₂²	semidirect
33. C₃₁×#386	C₂²	semidirect
34. C₃₁×#385	C₂²	semidirect
35. C₃₁×#379	C₂²	semidirect
36. C₃₁×#379	C₂²	semidirect
37. C₂×C₃₁×#95	C₂²	semidirect
74. C₂³×C₄×C₃₁	C₂³	semidirect
75. C₂³×C₄×C₃₁	C₂³	semidirect
78. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
89. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
90. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
91. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
92. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
93. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
94. C₃₁×#95	C₂³	semidirect
173. C₂×#2405	C₃₁	direct (N × H)
211. C₂³×C₃₁	#123	semidirect
243. C₂²×#123	C₂×C₃₁	semidirect
244. C₂³×C₂²⋊C₄	C₂×C₃₁	semidirect
245. #2405	C₂×C₃₁	direct (N × H)
246. C₂×#394	C₂×C₃₁	semidirect
247. C₂×#386	C₂×C₃₁	semidirect
248. C₂×#385	C₂×C₃₁	semidirect
249. C₂×#379	C₂×C₃₁	semidirect
250. C₂×#379	C₂×C₃₁	semidirect
275. C₂²×C₃₁	C₂×#123	semidirect
294. C₂⁴×C₄	C₂²×C₃₁	semidirect
295. C₂⁴×C₄	C₂²×C₃₁	semidirect
297. C₂×#123	C₂²×C₃₁	semidirect
298. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
302. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
303. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
304. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
305. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
306. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
307. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
308. #394	C₂²×C₃₁	semidirect
309. #386	C₂²×C₃₁	semidirect
310. #385	C₂²×C₃₁	semidirect
311. #379	C₂²×C₃₁	semidirect
312. #379	C₂²×C₃₁	semidirect
313. C₂×#95	C₂²×C₃₁	semidirect
318. C₂×C₃₁	#2405	direct (N × H)
359. C₂³×C₄	C₂³×C₃₁	semidirect
360. C₂³×C₄	C₂³×C₃₁	semidirect
363. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
374. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
375. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
376. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
377. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
378. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
379. #95	C₂³×C₃₁	semidirect
380. C₃₁	C₂×#2405	direct (N × H)
496. C₂³	C₃₁×#123	semidirect
551. C₂²	C₂×C₃₁×#123	semidirect
566. C₂	C₃₁×#2405	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₃₁	2
C₂²×C₃₁×#123	2
C₂³×C₂²⋊C₄×C₃₁	2
C₃₁×#2405	2
C₂×C₃₁×#394	2
C₂×C₃₁×#386	2
C₂×C₃₁×#385	2
C₂×C₃₁×#379	2
C₂×#2405	31

Subgroups — C2×C31×#2405 #21491415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2405 #21491415