Subgroups — C₅₃×#14693 #27222205

Order 13568.

6494 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#385	C₂	1	6	yes	maximal, complemented
3	C₂×C₅₃×#379	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#379	C₂	1	1	yes	maximal, characteristic
5	C₅₃×C₂²⋊#95	C₂	1	6	yes	maximal, complemented
6	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄×C₅₃	—	2	4	no	—
8	C₂⁴×C₄×C₅₃	—	2	24	no	—
9	C₂⁴×C₄×C₅₃	C₂²	1	6	yes	—
10	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	complemented
11	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
12	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
13	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	6	yes	—
14	C₅₃×#385	—	2	24	no	—
15	C₅₃×#379	—	2	4	no	—
16	C₅₃×#379	C₄	1	8	yes	complemented
17	C₂×#95×C₅₃	C₂²	1	12	yes	complemented
18	C₂⁵×C₅₃	—	2	12	no	—
19	C₂⁵×C₅₃	—	2	12	no	—
20	C₂⁵×C₅₃	C₂×C₄	1	4	yes	—
21	C₂⁵×C₅₃	D₄	1	4	yes	—
22	C₂⁵×C₅₃	Q₈	1	4	yes	—
23	C₂⁵×C₅₃	C₂³	1	3	yes	—
24	C₂³×C₄×C₅₃	—	2	96	no	—
25	C₂³×C₄×C₅₃	—	2	128	no	—
26	C₂³×C₄×C₅₃	—	2	12	no	—
27	C₂³×C₄×C₅₃	—	2	12	no	—
28	C₂³×C₄×C₅₃	—	2	48	no	—
29	C₂³×C₄×C₅₃	D₄	1	24	yes	—
30	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
32	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
33	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
34	C₂×C₄⋊C₄×C₅₃	—	2	24	no	—
35	C₂×C₂²⋊C₄×C₅₃	—	2	12	no	—
36	C₂×C₂²⋊C₄×C₅₃	—	2	6	no	—
37	C₂×C₂²⋊C₄×C₅₃	—	2	6	no	—
38	C₂×C₂²⋊C₄×C₅₃	—	2	12	no	—
39	C₂×C₂²⋊C₄×C₅₃	—	2	12	no	—
40	C₂×C₂²⋊C₄×C₅₃	—	2	12	no	—
41	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	24	yes	complemented
42	#95×C₅₃	—	2	24	no	—
43	C₂⁴×C₅₃	—	2	32	no	—
44	C₂⁴×C₅₃	—	2	96	no	—
45	C₂⁴×C₅₃	—	2	48	no	—
46	C₂⁴×C₅₃	—	2	24	no	—
47	C₂⁴×C₅₃	—	2	48	no	—
48	C₂⁴×C₅₃	—	2	24	no	—
49	C₂⁴×C₅₃	—	2	12	no	—
50	C₂⁴×C₅₃	—	2	12	no	—
51	C₂⁴×C₅₃	C₄⋊C₄	1	16	yes	complemented
52	C₂⁴×C₅₃	C₂²×C₄	1	6	yes	—
53	C₂⁴×C₅₃	C₂×D₄	1	6	yes	—
54	C₂⁴×C₅₃	C₂×D₄	1	12	yes	—
55	C₂⁴×C₅₃	C₂×Q₈	1	6	yes	—
56	C₂⁴×C₅₃	#41	1	12	yes	—
57	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	—	2	256	no	—
59	C₂²×C₄×C₅₃	—	2	96	no	—
60	C₂²×C₄×C₅₃	—	2	96	no	—
61	C₂²×C₄×C₅₃	—	2	48	no	—
62	C₂²×C₄×C₅₃	—	2	4	no	—
63	C₂²×C₄×C₅₃	—	2	12	no	—
64	C₂²×C₄×C₅₃	—	2	96	no	—
65	C₂²×C₄×C₅₃	—	2	12	no	—
66	C₂²×C₄×C₅₃	—	2	12	no	—
67	C₂²×C₄×C₅₃	—	2	6	no	—
68	C₂²×C₄×C₅₃	—	2	1	no	—
69	C₂²×C₄×C₅₃	—	2	3	no	—
70	C₂²×C₄×C₅₃	—	2	6	no	—
71	C₂²×C₄×C₅₃	—	2	12	no	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	complemented
73	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
74	C₄⋊C₄×C₅₃	—	4	16	no	—
75	C₂²⋊C₄×C₅₃	—	2	48	no	—
76	C₂²⋊C₄×C₅₃	—	4	12	no	—
77	C₂²⋊C₄×C₅₃	—	4	12	no	—
78	C₂³×C₅₃	—	2	32	no	—
79	C₂³×C₅₃	—	2	128	no	—
80	C₂³×C₅₃	—	2	96	no	—
81	C₂³×C₅₃	—	2	128	no	—
82	C₂³×C₅₃	—	2	24	no	—
83	C₂³×C₅₃	—	2	24	no	—
84	C₂³×C₅₃	—	2	12	no	—
85	C₂³×C₅₃	—	2	24	no	—
86	C₂³×C₅₃	—	2	12	no	—
87	C₂³×C₅₃	—	2	24	no	—
88	C₂³×C₅₃	—	2	48	no	—
89	C₂³×C₅₃	—	2	96	no	—
90	C₂³×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
91	C₂³×C₅₃	C₄×D₄	1	24	yes	—
92	C₂³×C₅₃	#120	1	12	yes	—
93	C₂³×C₅₃	C₂²⋊Q₈	1	24	yes	—
94	C₂³×C₅₃	#123	1	12	yes	—
95	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
96	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
97	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
98	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
99	C₂³×C₅₃	C₂×#41	1	6	yes	—
100	C₂×C₄×C₅₃	—	2	192	no	—
101	C₂×C₄×C₅₃	—	2	24	no	—
102	C₂×C₄×C₅₃	—	2	24	no	—
103	C₂×C₄×C₅₃	—	2	24	no	—
104	C₂×C₄×C₅₃	—	4	4	no	—
105	C₂×C₄×C₅₃	—	4	12	no	—
106	C₂×C₄×C₅₃	—	4	3	no	—
107	C₂×C₄×C₅₃	—	4	3	no	—
108	C₂×C₄×C₅₃	—	4	4	no	—
109	C₂×C₄×C₅₃	—	4	6	no	—
110	C₂×C₄×C₅₃	—	4	12	no	—
111	C₂×C₄×C₅₃	#120	1	16	yes	—
112	#14693	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₅₃	—	2	128	no	—
114	C₂²×C₅₃	—	2	12	no	—
115	C₂²×C₅₃	—	2	48	no	—
116	C₂²×C₅₃	—	2	12	no	—
117	C₂²×C₅₃	—	2	24	no	—
118	C₂²×C₅₃	—	2	24	no	—
119	C₂²×C₅₃	—	2	48	no	—
120	C₂²×C₅₃	#385	1	24	yes	complemented
121	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₅₃	C₂×C₄×D₄	1	6	yes	—
123	C₂²×C₅₃	C₂×#120	1	4	yes	—
124	C₂²×C₅₃	C₂×#120	1	3	yes	—
125	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	6	yes	—
126	C₂²×C₅₃	C₂×#123	1	3	yes	—
127	C₂²×C₅₃	#525	1	12	yes	—
128	C₄×C₅₃	—	2	16	no	—
129	C₄×C₅₃	—	4	8	no	—
130	C₄×C₅₃	—	8	1	no	—
131	C₄×C₅₃	—	8	3	no	—
132	C₂⁵×C₄	C₂×C₅₃	1	1	yes	characteristic
133	C₂×#385	C₂×C₅₃	1	6	yes	complemented
134	C₂×#379	C₂×C₅₃	1	1	yes	characteristic
135	C₂×#379	C₂×C₅₃	1	1	yes	characteristic
136	C₂²⋊#95	C₂×C₅₃	1	6	yes	complemented
137	C₂×C₅₃	—	2	24	no	—
138	C₂×C₅₃	C₂×#385	1	3	yes	—
139	C₂×C₅₃	C₄×#120	1	4	yes	—
140	C₂×C₅₃	#2846	1	1	yes	characteristic
141	C₂×C₅₃	#2848	1	4	yes	—
142	C₂×C₅₃	#2849	1	3	yes	—
143	C₂⁶	C₂²×C₅₃	1	1	yes	characteristic
144	C₂⁴×C₄	—	2	4	no	—
145	C₂⁴×C₄	—	2	24	no	—
146	C₂⁴×C₄	C₂²×C₅₃	1	6	yes	—
147	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	complemented
148	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	3	yes	—
149	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	3	yes	—
150	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	6	yes	—
151	#385	—	2	24	no	—
152	#379	—	2	4	no	—
153	#379	C₄×C₅₃	1	8	yes	complemented
154	C₂×#95	C₂²×C₅₃	1	12	yes	complemented
155	C₅₃	#14693	1	1	yes	characteristic, direct product factor, Sylow
156	C₂⁵	—	2	12	no	—
157	C₂⁵	—	2	12	no	—
158	C₂⁵	C₂×C₄×C₅₃	1	4	yes	—
159	C₂⁵	D₄×C₅₃	1	4	yes	—
160	C₂⁵	Q₈×C₅₃	1	4	yes	—
161	C₂⁵	C₂³×C₅₃	1	3	yes	—
162	C₂³×C₄	—	2	96	no	—
163	C₂³×C₄	—	2	128	no	—
164	C₂³×C₄	—	2	12	no	—
165	C₂³×C₄	—	2	12	no	—
166	C₂³×C₄	—	2	48	no	—
167	C₂³×C₄	D₄×C₅₃	1	24	yes	—
168	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
169	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
170	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
171	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
172	C₂×C₄⋊C₄	—	2	24	no	—
173	C₂×C₂²⋊C₄	—	2	12	no	—
174	C₂×C₂²⋊C₄	—	2	12	no	—
175	C₂×C₂²⋊C₄	—	2	6	no	—
176	C₂×C₂²⋊C₄	—	2	6	no	—
177	C₂×C₂²⋊C₄	—	2	12	no	—
178	C₂×C₂²⋊C₄	—	2	12	no	—
179	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	24	yes	complemented
180	#95	—	2	24	no	—
181	C₂⁴	—	2	32	no	—
182	C₂⁴	—	2	96	no	—
183	C₂⁴	—	2	48	no	—
184	C₂⁴	—	2	24	no	—
185	C₂⁴	—	2	48	no	—
186	C₂⁴	—	2	24	no	—
187	C₂⁴	—	2	12	no	—
188	C₂⁴	—	2	12	no	—
189	C₂⁴	C₄⋊C₄×C₅₃	1	16	yes	complemented
190	C₂⁴	C₂²×C₄×C₅₃	1	6	yes	—
191	C₂⁴	C₂×D₄×C₅₃	1	6	yes	—
192	C₂⁴	C₂×D₄×C₅₃	1	12	yes	—
193	C₂⁴	C₂×Q₈×C₅₃	1	6	yes	—
194	C₂⁴	#41×C₅₃	1	12	yes	—
195	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
196	C₂²×C₄	—	2	256	no	—
197	C₂²×C₄	—	2	96	no	—
198	C₂²×C₄	—	2	96	no	—
199	C₂²×C₄	—	2	48	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	12	no	—
202	C₂²×C₄	—	2	96	no	—
203	C₂²×C₄	—	2	12	no	—
204	C₂²×C₄	—	2	12	no	—
205	C₂²×C₄	—	2	6	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	3	no	—
208	C₂²×C₄	—	2	6	no	—
209	C₂²×C₄	—	2	12	no	—
210	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	complemented
211	C₂²×C₄	C₂×D₄×C₅₃	1	24	yes	—
212	C₄⋊C₄	—	4	16	no	—
213	C₂²⋊C₄	—	2	48	no	—
214	C₂²⋊C₄	—	4	12	no	—
215	C₂²⋊C₄	—	4	12	no	—
216	C₂³	—	2	32	no	—
217	C₂³	—	2	128	no	—
218	C₂³	—	2	128	no	—
219	C₂³	—	2	96	no	—
220	C₂³	—	2	24	no	—
221	C₂³	—	2	12	no	—
222	C₂³	—	2	24	no	—
223	C₂³	—	2	24	no	—
224	C₂³	—	2	12	no	—
225	C₂³	—	2	24	no	—
226	C₂³	—	2	48	no	—
227	C₂³	—	2	96	no	—
228	C₂³	C₂×C₄⋊C₄×C₅₃	1	12	yes	—
229	C₂³	C₄×D₄×C₅₃	1	24	yes	—
230	C₂³	#120×C₅₃	1	12	yes	—
231	C₂³	C₂²⋊Q₈×C₅₃	1	24	yes	—
232	C₂³	#123×C₅₃	1	12	yes	—
233	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
234	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
235	C₂³	C₂²×D₄×C₅₃	1	6	yes	—
236	C₂³	C₂²×Q₈×C₅₃	1	1	yes	characteristic
237	C₂³	C₂×#41×C₅₃	1	6	yes	—
238	C₂×C₄	—	2	192	no	—
239	C₂×C₄	—	2	24	no	—
240	C₂×C₄	—	2	24	no	—
241	C₂×C₄	—	2	24	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	12	no	—
244	C₂×C₄	—	4	3	no	—
245	C₂×C₄	—	4	3	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	6	no	—
248	C₂×C₄	—	4	12	no	—
249	C₂×C₄	#120×C₅₃	1	16	yes	—
250	C₂²	—	2	128	no	—
251	C₂²	—	2	24	no	—
252	C₂²	—	2	48	no	—
253	C₂²	—	2	12	no	—
254	C₂²	—	2	48	no	—
255	C₂²	—	2	12	no	—
256	C₂²	—	2	24	no	—
257	C₂²	C₅₃×#385	1	24	yes	complemented
258	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
259	C₂²	C₂×C₄×D₄×C₅₃	1	6	yes	—
260	C₂²	C₂×#120×C₅₃	1	3	yes	—
261	C₂²	C₂×#120×C₅₃	1	4	yes	—
262	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	6	yes	—
263	C₂²	C₂×#123×C₅₃	1	3	yes	—
264	C₂²	C₅₃×#525	1	12	yes	—
265	C₄	—	2	16	no	—
266	C₄	—	4	8	no	—
267	C₄	—	8	1	no	—
268	C₄	—	8	3	no	—
269	C₂	—	2	24	no	—
270	C₂	C₂×C₅₃×#385	1	3	yes	—
271	C₂	C₄×#120×C₅₃	1	4	yes	—
272	C₂	C₅₃×#2846	1	1	yes	characteristic
273	C₂	C₅₃×#2848	1	4	yes	—
274	C₂	C₅₃×#2849	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂⁴	C₂⁴×C₅₃
prime radical	C₂⁶×C₅₃	C₂²
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
2. C₂×C₅₃×#385	C₂	semidirect
5. C₅₃×C₂²⋊#95	C₂	semidirect
10. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
16. C₅₃×#379	C₄	semidirect
17. C₂×#95×C₅₃	C₂²	semidirect
41. C₂×C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
51. C₂⁴×C₅₃	C₄⋊C₄	semidirect
72. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
112. #14693	C₅₃	direct (N × H)
120. C₂²×C₅₃	#385	semidirect
133. C₂×#385	C₂×C₅₃	semidirect
136. C₂²⋊#95	C₂×C₅₃	semidirect
147. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
153. #379	C₄×C₅₃	semidirect
154. C₂×#95	C₂²×C₅₃	semidirect
155. C₅₃	#14693	direct (N × H)
179. C₂×C₂²⋊C₄	C₂×C₄×C₅₃	semidirect
189. C₂⁴	C₄⋊C₄×C₅₃	semidirect
210. C₂²×C₄	C₂²×C₄×C₅₃	semidirect
257. C₂²	C₅₃×#385	semidirect

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₅₃	2
C₂×C₅₃×#385	2
C₂×C₅₃×#379	2
C₅₃×C₂²⋊#95	2
#14693	53

Subgroups — C53×#14693 #27222205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#14693 #27222205