Subgroups — C₂×C₅₃×#2815 #27244159

Order 13568.

3774 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#121×C₅₃	C₂	1	2	yes	maximal, complemented
2	C₅₃×#2815	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₅₃×#397	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#392	C₂	1	4	yes	maximal, complemented
5	C₂×C₅₃×#392	C₂	1	4	yes	maximal, complemented
6	C₂×C₅₃×#381	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₅₃	C₂²	1	4	yes	—
8	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
9	C₂³×D₄×C₅₃	C₂²	1	4	yes	—
10	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#121×C₅₃	—	2	16	no	—
12	C₂×#121×C₅₃	C₂²	1	16	yes	complemented
13	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
15	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
17	C₅₃×#397	C₂²	1	32	yes	complemented
18	C₅₃×#392	C₂²	1	32	yes	complemented
19	C₅₃×#392	C₂²	1	32	yes	complemented
20	C₅₃×#381	C₂²	1	8	yes	complemented
21	C₂×#95×C₅₃	C₂²	1	4	yes	—
22	C₂×#95×C₅₃	C₂²	1	4	yes	—
23	C₂⁵×C₅₃	C₂³	1	4	yes	—
24	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×D₄×C₅₃	—	2	16	no	—
26	C₂²×D₄×C₅₃	—	2	8	no	—
27	C₂²×D₄×C₅₃	—	4	8	no	—
28	C₂²×D₄×C₅₃	—	4	16	no	—
29	C₂²×D₄×C₅₃	—	4	8	no	—
30	C₂²×D₄×C₅₃	—	4	16	no	—
31	C₂²×D₄×C₅₃	C₂³	1	16	yes	—
32	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
33	C₂²×D₄×C₅₃	C₂³	1	16	yes	—
34	C₂³×C₄×C₅₃	—	4	4	no	—
35	C₂³×C₄×C₅₃	D₄	1	8	yes	complemented
36	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
37	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
38	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
39	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
40	#121×C₅₃	—	2	64	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
42	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
43	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
44	C₂×C₂²⋊C₄×C₅₃	—	4	4	no	—
45	C₂×C₂²⋊C₄×C₅₃	—	4	16	no	—
46	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
47	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
48	C₂×C₂²⋊C₄×C₅₃	C₂³	1	32	yes	—
49	#95×C₅₃	C₂³	1	16	yes	—
50	#95×C₅₃	C₂³	1	16	yes	—
51	C₂⁴×C₅₃	—	2	4	no	—
52	C₂⁴×C₅₃	—	2	8	no	—
53	C₂⁴×C₅₃	—	2	8	no	—
54	C₂⁴×C₅₃	—	2	8	no	—
55	C₂⁴×C₅₃	—	2	16	no	—
56	C₂⁴×C₅₃	—	2	16	no	—
57	C₂⁴×C₅₃	—	2	8	no	—
58	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
59	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
60	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
61	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
62	C₂×D₄×C₅₃	—	2	32	no	—
63	C₂×D₄×C₅₃	—	2	16	no	—
64	C₂×D₄×C₅₃	—	4	16	no	—
65	C₂×D₄×C₅₃	—	4	32	no	—
66	C₂×D₄×C₅₃	—	4	16	no	—
67	C₂×D₄×C₅₃	—	4	32	no	—
68	C₂×D₄×C₅₃	—	4	32	no	—
69	C₂×D₄×C₅₃	—	4	64	no	—
70	C₂×D₄×C₅₃	—	4	64	no	—
71	C₂²×C₄×C₅₃	—	2	8	no	—
72	C₂²×C₄×C₅₃	—	2	8	no	—
73	C₂²×C₄×C₅₃	—	2	16	no	—
74	C₂²×C₄×C₅₃	—	2	8	no	—
75	C₂²×C₄×C₅₃	—	2	16	no	—
76	C₂²×C₄×C₅₃	—	4	16	no	—
77	C₂²×C₄×C₅₃	—	4	8	no	—
78	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
79	C₂²×C₄×C₅₃	C₂×D₄	1	16	yes	complemented
80	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
81	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
82	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
83	C₄⋊C₄×C₅₃	—	2	16	no	—
84	C₂²⋊C₄×C₅₃	—	2	32	no	—
85	C₂²⋊C₄×C₅₃	—	4	8	no	—
86	C₂²⋊C₄×C₅₃	—	4	32	no	—
87	C₂³×C₅₃	—	2	8	no	—
88	C₂³×C₅₃	—	2	16	no	—
89	C₂³×C₅₃	—	2	4	no	—
90	C₂³×C₅₃	—	2	8	no	—
91	C₂³×C₅₃	—	2	8	no	—
92	C₂³×C₅₃	—	2	16	no	—
93	C₂³×C₅₃	—	2	16	no	—
94	C₂³×C₅₃	—	2	8	no	—
95	C₂³×C₅₃	—	2	8	no	—
96	C₂³×C₅₃	—	4	16	no	—
97	C₂³×C₅₃	—	4	16	no	—
98	C₂³×C₅₃	—	4	16	no	—
99	C₂³×C₅₃	—	4	16	no	—
100	C₂³×C₅₃	—	4	32	no	—
101	C₂³×C₅₃	—	4	32	no	—
102	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
103	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
104	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
105	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	—
106	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
107	D₄×C₅₃	—	4	32	no	—
108	D₄×C₅₃	—	4	64	no	—
109	D₄×C₅₃	—	4	64	no	—
110	C₂×C₄×C₅₃	—	2	16	no	—
111	C₂×C₄×C₅₃	—	2	8	no	—
112	C₂×C₄×C₅₃	—	2	8	no	—
113	C₂×C₄×C₅₃	—	2	16	no	—
114	C₂×C₄×C₅₃	—	2	8	no	—
115	C₂×C₄×C₅₃	—	2	16	no	—
116	C₂×C₄×C₅₃	—	4	16	no	—
117	C₂×C₄×C₅₃	—	4	16	no	—
118	C₂×C₄×C₅₃	—	4	16	no	—
119	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
120	C₂×#2815	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₅₃	—	2	4	no	—
122	C₂²×C₅₃	—	2	8	no	—
123	C₂²×C₅₃	—	4	8	no	—
124	C₂²×C₅₃	—	4	16	no	—
125	C₂²×C₅₃	—	4	8	no	—
126	C₂²×C₅₃	—	4	8	no	—
127	C₂²×C₅₃	—	4	8	no	—
128	C₂²×C₅₃	—	4	16	no	—
129	C₂²×C₅₃	—	4	16	no	—
130	C₂²×C₅₃	—	8	32	no	—
131	C₂²×C₅₃	#534	1	4	yes	—
132	C₂²×C₅₃	#535	1	8	yes	—
133	C₂²×C₅₃	#560	1	16	yes	—
134	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
135	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
136	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
137	C₄×C₅₃	—	2	8	no	—
138	C₄×C₅₃	—	4	8	no	—
139	C₄×C₅₃	—	4	8	no	—
140	C₂²×#121	C₂×C₅₃	1	2	yes	complemented
141	#2815	C₂×C₅₃	1	16	yes	direct product factor
142	C₂×#397	C₂×C₅₃	1	4	yes	complemented
143	C₂×#392	C₂×C₅₃	1	4	yes	complemented
144	C₂×#392	C₂×C₅₃	1	4	yes	complemented
145	C₂×#381	C₂×C₅₃	1	1	yes	characteristic, complemented
146	C₂×C₅₃	—	4	4	no	—
147	C₂×C₅₃	—	8	8	no	—
148	C₂×C₅₃	#2815	1	8	yes	direct product factor
149	C₂×C₅₃	C₂×#534	1	1	yes	characteristic
150	C₂×C₅₃	C₂×#535	1	2	yes	—
151	C₂×C₅₃	C₂×#560	1	4	yes	—
152	C₂³×D₄	C₂²×C₅₃	1	4	yes	—
153	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
154	C₂³×D₄	C₂²×C₅₃	1	4	yes	—
155	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
156	C₂×#121	—	2	16	no	—
157	C₂×#121	C₂²×C₅₃	1	16	yes	complemented
158	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
159	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
160	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
161	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
162	#397	C₂²×C₅₃	1	32	yes	complemented
163	#392	C₂²×C₅₃	1	32	yes	complemented
164	#392	C₂²×C₅₃	1	32	yes	complemented
165	#381	C₂²×C₅₃	1	8	yes	complemented
166	C₂×#95	C₂²×C₅₃	1	4	yes	—
167	C₂×#95	C₂²×C₅₃	1	4	yes	—
168	C₅₃	C₂×#2815	1	1	yes	characteristic, direct product factor, Sylow
169	C₂⁵	C₂³×C₅₃	1	4	yes	—
170	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
171	C₂²×D₄	—	2	16	no	—
172	C₂²×D₄	—	2	8	no	—
173	C₂²×D₄	—	4	8	no	—
174	C₂²×D₄	—	4	16	no	—
175	C₂²×D₄	—	4	8	no	—
176	C₂²×D₄	—	4	16	no	—
177	C₂²×D₄	C₂³×C₅₃	1	16	yes	—
178	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
179	C₂²×D₄	C₂³×C₅₃	1	16	yes	—
180	C₂³×C₄	—	4	4	no	—
181	C₂³×C₄	D₄×C₅₃	1	8	yes	complemented
182	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
183	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
184	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
185	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
186	#121	—	2	64	no	—
187	C₂×C₄⋊C₄	—	2	8	no	—
188	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
189	C₂×C₂²⋊C₄	—	2	16	no	—
190	C₂×C₂²⋊C₄	—	4	4	no	—
191	C₂×C₂²⋊C₄	—	4	16	no	—
192	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
193	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
194	C₂×C₂²⋊C₄	C₂³×C₅₃	1	32	yes	—
195	#95	C₂³×C₅₃	1	16	yes	—
196	#95	C₂³×C₅₃	1	16	yes	—
197	C₂⁴	—	2	4	no	—
198	C₂⁴	—	2	8	no	—
199	C₂⁴	—	2	8	no	—
200	C₂⁴	—	2	8	no	—
201	C₂⁴	—	2	16	no	—
202	C₂⁴	—	2	16	no	—
203	C₂⁴	—	2	8	no	—
204	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
205	C₂⁴	C₂⁴×C₅₃	1	8	yes	—
206	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
207	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
208	C₂×D₄	—	2	32	no	—
209	C₂×D₄	—	2	16	no	—
210	C₂×D₄	—	4	16	no	—
211	C₂×D₄	—	4	32	no	—
212	C₂×D₄	—	4	16	no	—
213	C₂×D₄	—	4	32	no	—
214	C₂×D₄	—	4	32	no	—
215	C₂×D₄	—	4	64	no	—
216	C₂×D₄	—	4	64	no	—
217	C₂²×C₄	—	2	8	no	—
218	C₂²×C₄	—	2	8	no	—
219	C₂²×C₄	—	2	16	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	16	no	—
222	C₂²×C₄	—	4	16	no	—
223	C₂²×C₄	—	4	8	no	—
224	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
225	C₂²×C₄	C₂×D₄×C₅₃	1	16	yes	complemented
226	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
227	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
228	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
229	C₄⋊C₄	—	2	16	no	—
230	C₂²⋊C₄	—	2	32	no	—
231	C₂²⋊C₄	—	4	8	no	—
232	C₂²⋊C₄	—	4	32	no	—
233	C₂³	—	2	8	no	—
234	C₂³	—	2	16	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	8	no	—
237	C₂³	—	2	8	no	—
238	C₂³	—	2	8	no	—
239	C₂³	—	2	8	no	—
240	C₂³	—	2	16	no	—
241	C₂³	—	2	16	no	—
242	C₂³	—	4	16	no	—
243	C₂³	—	4	16	no	—
244	C₂³	—	4	16	no	—
245	C₂³	—	4	16	no	—
246	C₂³	—	4	32	no	—
247	C₂³	—	4	32	no	—
248	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
249	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
250	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
251	C₂³	H(5,𝔽₂)×C₅₃	1	8	yes	—
252	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
253	D₄	—	4	32	no	—
254	D₄	—	4	64	no	—
255	D₄	—	4	64	no	—
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	16	no	—
261	C₂×C₄	—	2	16	no	—
262	C₂×C₄	—	4	16	no	—
263	C₂×C₄	—	4	16	no	—
264	C₂×C₄	—	4	16	no	—
265	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	8	no	—
268	C₂²	—	4	8	no	—
269	C₂²	—	4	16	no	—
270	C₂²	—	4	16	no	—
271	C₂²	—	4	16	no	—
272	C₂²	—	4	8	no	—
273	C₂²	—	4	8	no	—
274	C₂²	—	4	8	no	—
275	C₂²	—	8	32	no	—
276	C₂²	C₅₃×#534	1	4	yes	—
277	C₂²	C₅₃×#535	1	8	yes	—
278	C₂²	C₅₃×#560	1	16	yes	—
279	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
280	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
281	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	4	yes	—
282	C₄	—	2	8	no	—
283	C₄	—	4	8	no	—
284	C₄	—	4	8	no	—
285	C₂	—	4	4	no	—
286	C₂	—	8	8	no	—
287	C₂	C₅₃×#2815	1	8	yes	direct product factor
288	C₂	C₂×C₅₃×#534	1	1	yes	characteristic
289	C₂	C₂×C₅₃×#535	1	2	yes	—
290	C₂	C₂×C₅₃×#560	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#121×C₅₃	C₂	semidirect
2. C₅₃×#2815	C₂	direct (N × H)
3. C₂×C₅₃×#397	C₂	semidirect
4. C₂×C₅₃×#392	C₂	semidirect
5. C₂×C₅₃×#392	C₂	semidirect
6. C₂×C₅₃×#381	C₂	semidirect
12. C₂×#121×C₅₃	C₂²	semidirect
17. C₅₃×#397	C₂²	semidirect
18. C₅₃×#392	C₂²	semidirect
19. C₅₃×#392	C₂²	semidirect
20. C₅₃×#381	C₂²	semidirect
35. C₂³×C₄×C₅₃	D₄	semidirect
79. C₂²×C₄×C₅₃	C₂×D₄	semidirect
120. C₂×#2815	C₅₃	direct (N × H)
140. C₂²×#121	C₂×C₅₃	semidirect
141. #2815	C₂×C₅₃	direct (N × H)
142. C₂×#397	C₂×C₅₃	semidirect
143. C₂×#392	C₂×C₅₃	semidirect
144. C₂×#392	C₂×C₅₃	semidirect
145. C₂×#381	C₂×C₅₃	semidirect
148. C₂×C₅₃	#2815	direct (N × H)
157. C₂×#121	C₂²×C₅₃	semidirect
162. #397	C₂²×C₅₃	semidirect
163. #392	C₂²×C₅₃	semidirect
164. #392	C₂²×C₅₃	semidirect
165. #381	C₂²×C₅₃	semidirect
168. C₅₃	C₂×#2815	direct (N × H)
181. C₂³×C₄	D₄×C₅₃	semidirect
225. C₂²×C₄	C₂×D₄×C₅₃	semidirect
287. C₂	C₅₃×#2815	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#121×C₅₃	2
C₅₃×#2815	2
C₂×C₅₃×#397	2
C₂×C₅₃×#392	2
C₂×C₅₃×#381	2
C₂×#2815	53

Subgroups — C2×C53×#2815 #27244159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2815 #27244159