Subgroups — C₅₃×#23365 #27230877

Order 13568.

636 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1985	C₂	1	2	yes	maximal
3	C₅₃×#1984	C₂	1	2	yes	maximal
4	C₅₃×#1917	C₂	1	4	yes	maximal
5	C₅₃×#1916	C₂	1	4	yes	maximal
6	C₅₃×#1839	C₂	1	2	yes	maximal
7	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×#97×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#391	C₄	1	1	yes	characteristic
15	C₅₃×#391	C₄	1	1	yes	characteristic
16	C₅₃×#389	C₄	1	4	yes	—
17	C₅₃×#384	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
19	C₄³×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#336	C₂²	1	2	yes	—
21	C₅₃×#336	C₂²	1	4	yes	—
22	C₅₃×#336	C₂²	1	4	yes	—
23	C₅₃×#336	C₂²	1	4	yes	—
24	C₅₃×#336	C₂²	1	2	yes	—
25	C₅₃×#336	C₂²	1	4	yes	—
26	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
29	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
30	C₄⋊Q₈×C₅₃	—	2	4	no	—
31	C₄×Q₈×C₅₃	—	4	1	no	—
32	C₄×Q₈×C₅₃	—	4	1	no	—
33	C₄×Q₈×C₅₃	—	4	1	no	—
34	C₄×Q₈×C₅₃	—	4	1	no	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
47	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
48	C₄⋊C₈×C₅₃	—	4	2	no	—
49	C₄⋊C₈×C₅₃	—	4	2	no	—
50	#97×C₅₃	—	4	2	no	—
51	C₄×C₈×C₅₃	—	4	2	no	—
52	#95×C₅₃	C₂×C₄	1	4	yes	—
53	#95×C₅₃	C₂×C₄	1	4	yes	—
54	C₂×Q₈×C₅₃	—	2	2	no	—
55	C₂×Q₈×C₅₃	—	4	1	no	—
56	C₂×Q₈×C₅₃	—	4	1	no	—
57	C₂×Q₈×C₅₃	—	4	1	no	—
58	C₂×Q₈×C₅₃	—	4	1	no	—
59	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₅₃	—	2	4	no	—
70	C₂×C₈×C₅₃	—	2	4	no	—
71	C₂×C₈×C₅₃	—	2	8	no	—
72	C₂×C₈×C₅₃	—	2	8	no	—
73	C₄⋊C₄×C₅₃	—	2	8	no	—
74	C₄⋊C₄×C₅₃	—	2	8	no	—
75	C₄⋊C₄×C₅₃	—	4	1	no	—
76	C₄⋊C₄×C₅₃	—	4	1	no	—
77	C₄⋊C₄×C₅₃	—	4	2	no	—
78	C₄⋊C₄×C₅₃	—	4	2	no	—
79	C₄²×C₅₃	—	2	2	no	—
80	C₄²×C₅₃	—	2	4	no	—
81	C₄²×C₅₃	—	2	2	no	—
82	C₄²×C₅₃	—	2	2	no	—
83	C₄²×C₅₃	—	2	2	no	—
84	C₄²×C₅₃	—	2	2	no	—
85	C₄²×C₅₃	—	2	1	no	—
86	C₄²×C₅₃	—	2	1	no	—
87	C₄²×C₅₃	—	2	4	no	—
88	C₄²×C₅₃	—	4	1	no	—
89	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
90	Q₈×C₅₃	—	4	4	no	—
91	Q₈×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
112	C₂×C₄×C₅₃	#131	1	4	yes	—
113	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
116	C₈×C₅₃	—	4	4	no	—
117	C₈×C₅₃	—	4	4	no	—
118	#23365	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂²×C₅₃	#518	1	1	yes	characteristic
120	C₂²×C₅₃	#519	1	2	yes	—
121	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
122	C₂²×C₅₃	C₂×#131	1	2	yes	—
123	C₂²×C₅₃	#568	1	1	yes	characteristic
124	C₄×C₅₃	—	2	4	no	—
125	C₄×C₅₃	—	2	2	no	—
126	C₄×C₅₃	—	2	4	no	—
127	C₄×C₅₃	—	4	2	no	—
128	C₄×C₅₃	—	4	2	no	—
129	C₄×C₅₃	—	4	2	no	—
130	C₄×C₅₃	—	4	1	no	—
131	C₄×C₅₃	—	4	2	no	—
132	C₄×C₅₃	#563	1	2	yes	—
133	C₄×C₅₃	#563	1	2	yes	—
134	C₄×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
135	#1985	C₂×C₅₃	1	2	yes	—
136	#1984	C₂×C₅₃	1	2	yes	—
137	#1917	C₂×C₅₃	1	4	yes	—
138	#1916	C₂×C₅₃	1	4	yes	—
139	#1839	C₂×C₅₃	1	2	yes	—
140	C₂×C₅₃	#2381	1	1	yes	characteristic
141	C₂×C₅₃	#2973	1	1	yes	characteristic
142	C₂×C₅₃	#2979	1	1	yes	characteristic
143	C₂×C₅₃	#2984	1	2	yes	—
144	C₂×C₅₃	#2985	1	2	yes	—
145	C₂×C₄⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
146	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
147	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
148	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
149	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
150	C₂×#97	C₂²×C₅₃	1	2	yes	—
151	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
152	#391	C₄×C₅₃	1	1	yes	characteristic
153	#391	C₄×C₅₃	1	1	yes	characteristic
154	#389	C₄×C₅₃	1	4	yes	—
155	#384	C₄×C₅₃	1	1	yes	characteristic
156	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
157	C₄³	C₂²×C₅₃	1	1	yes	characteristic
158	#336	C₂²×C₅₃	1	2	yes	—
159	#336	C₂²×C₅₃	1	4	yes	—
160	#336	C₂²×C₅₃	1	2	yes	—
161	#336	C₂²×C₅₃	1	4	yes	—
162	#336	C₂²×C₅₃	1	4	yes	—
163	#336	C₂²×C₅₃	1	4	yes	—
164	C₅₃	#23365	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
166	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
168	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
169	C₄⋊Q₈	—	2	4	no	—
170	C₄×Q₈	—	4	1	no	—
171	C₄×Q₈	—	4	1	no	—
172	C₄×Q₈	—	4	1	no	—
173	C₄×Q₈	—	4	1	no	—
174	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
175	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
176	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
178	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
179	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
180	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
181	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
182	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
185	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
186	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
187	C₄⋊C₈	—	4	2	no	—
188	C₄⋊C₈	—	4	2	no	—
189	#97	—	4	2	no	—
190	C₄×C₈	—	4	2	no	—
191	#95	C₂×C₄×C₅₃	1	4	yes	—
192	#95	C₂×C₄×C₅₃	1	4	yes	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	—	4	1	no	—
195	C₂×Q₈	—	4	1	no	—
196	C₂×Q₈	—	4	1	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂×Q₈	M₁₆×C₅₃	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
200	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
201	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
202	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
203	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
205	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
207	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	2	8	no	—
212	C₄⋊C₄	—	2	8	no	—
213	C₄⋊C₄	—	2	8	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	4	no	—
227	C₄²	—	4	1	no	—
228	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
229	Q₈	—	4	4	no	—
230	Q₈	—	4	4	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
250	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
251	C₂×C₄	#131×C₅₃	1	4	yes	—
252	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
253	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
254	C₂×C₄	2₋¹⁺⁴×C₅₃	1	4	yes	—
255	C₈	—	4	4	no	—
256	C₈	—	4	4	no	—
257	C₂²	C₅₃×#518	1	1	yes	characteristic
258	C₂²	C₅₃×#519	1	2	yes	—
259	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
260	C₂²	C₂×#131×C₅₃	1	2	yes	—
261	C₂²	C₅₃×#568	1	1	yes	characteristic
262	C₄	—	2	4	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	4	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	2	no	—
270	C₄	C₅₃×#563	1	2	yes	—
271	C₄	C₅₃×#563	1	2	yes	—
272	C₂	C₅₃×#2381	1	1	yes	characteristic
273	C₂	C₅₃×#2973	1	1	yes	characteristic
274	C₂	C₅₃×#2979	1	1	yes	characteristic
275	C₂	C₅₃×#2984	1	2	yes	—
276	C₂	C₅₃×#2985	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle, prime radical	C₂³×C₅₃	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
118. #23365	C₅₃	direct (N × H)
164. C₅₃	#23365	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₅₃	2
C₅₃×#1985	2
C₅₃×#1984	2
C₅₃×#1917	2
C₅₃×#1916	2
C₅₃×#1839	2
#23365	53

Subgroups — C53×#23365 #27230877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23365 #27230877