Subgroups — C₅₃×#22242 #27229754

Order 13568.

722 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#446	C₂	1	2	yes	maximal
2	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1984	C₂	1	2	yes	maximal
4	C₅₃×#1917	C₂	1	8	yes	maximal
5	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#446	—	2	8	no	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×#97×C₅₃	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#391	C₄	1	2	yes	—
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	8	yes	—
21	C₅₃×#336	C₂²	1	4	yes	—
22	C₅₃×#336	C₂²	1	4	yes	—
23	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
26	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
27	C₄⋊Q₈×C₅₃	—	2	4	no	—
28	C₄×Q₈×C₅₃	—	2	4	no	—
29	C₄×Q₈×C₅₃	—	2	4	no	—
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	8	no	—
41	C₄⋊C₈×C₅₃	—	2	4	no	—
42	#97×C₅₃	—	2	8	no	—
43	C₄×C₈×C₅₃	—	2	2	no	—
44	C₄×C₈×C₅₃	D₄	1	4	yes	—
45	#95×C₅₃	C₂×C₄	1	8	yes	—
46	C₂×Q₈×C₅₃	—	2	4	no	—
47	C₂×Q₈×C₅₃	—	2	4	no	—
48	C₂×Q₈×C₅₃	—	2	4	no	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₂×C₈×C₅₃	—	2	4	no	—
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	8	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
62	C₄⋊C₄×C₅₃	—	2	8	no	—
63	C₄⋊C₄×C₅₃	—	2	4	no	—
64	C₄⋊C₄×C₅₃	—	2	8	no	—
65	C₄⋊C₄×C₅₃	—	4	4	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	—	2	4	no	—
68	C₄²×C₅₃	—	2	4	no	—
69	C₄²×C₅₃	—	2	4	no	—
70	C₄²×C₅₃	—	4	2	no	—
71	C₄²×C₅₃	M₁₆	1	4	yes	—
72	C₄²×C₅₃	C₂×D₄	1	2	yes	—
73	C₄²×C₅₃	#41	1	2	yes	—
74	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
75	Q₈×C₅₃	—	4	8	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	4	no	—
83	C₂×C₄×C₅₃	—	2	4	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	8	no	—
87	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
90	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
91	C₂×C₄×C₅₃	#131	1	2	yes	—
92	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
94	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
95	C₈×C₅₃	—	2	4	no	—
96	C₈×C₅₃	—	4	4	no	—
97	C₈×C₅₃	—	4	2	no	—
98	#22242	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
100	C₂²×C₅₃	#519	1	2	yes	—
101	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
102	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
103	C₂²×C₅₃	#568	1	2	yes	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	2	4	no	—
106	C₄×C₅₃	—	4	4	no	—
107	C₄×C₅₃	—	4	4	no	—
108	C₄×C₅₃	—	4	2	no	—
109	C₄×C₅₃	#436	1	4	yes	—
110	C₄×C₅₃	#536	1	2	yes	—
111	C₄×C₅₃	#541	1	2	yes	—
112	C₂×#446	C₂×C₅₃	1	2	yes	—
113	C₄×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
114	#1984	C₂×C₅₃	1	2	yes	—
115	#1917	C₂×C₅₃	1	8	yes	—
116	#1831	C₂×C₅₃	1	1	yes	characteristic
117	#1744	C₂×C₅₃	1	1	yes	characteristic
118	C₂×C₅₃	#2327	1	1	yes	characteristic
119	C₂×C₅₃	C₂×#436	1	1	yes	characteristic
120	C₂×C₅₃	#2932	1	1	yes	characteristic
121	C₂×C₅₃	#2983	1	2	yes	—
122	C₂×C₅₃	#2985	1	2	yes	—
123	C₂×C₄⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
124	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
125	#446	—	2	8	no	—
126	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
127	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
128	C₂×#97	C₂²×C₅₃	1	4	yes	—
129	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
131	#391	C₄×C₅₃	1	2	yes	—
132	#389	C₄×C₅₃	1	4	yes	—
133	#384	C₄×C₅₃	1	1	yes	characteristic
134	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
135	C₄³	C₂²×C₅₃	1	1	yes	characteristic
136	#336	C₂²×C₅₃	1	4	yes	—
137	#336	C₂²×C₅₃	1	8	yes	—
138	#336	C₂²×C₅₃	1	4	yes	—
139	C₅₃	#22242	1	1	yes	characteristic, direct product factor, Sylow
140	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
141	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
142	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
143	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
144	C₄⋊Q₈	—	2	4	no	—
145	C₄×Q₈	—	2	4	no	—
146	C₄×Q₈	—	2	4	no	—
147	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
148	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
149	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
150	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
151	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
152	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
153	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
154	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
155	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
156	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
157	C₄⋊C₈	—	2	8	no	—
158	C₄⋊C₈	—	2	4	no	—
159	#97	—	2	8	no	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	D₄×C₅₃	1	4	yes	—
162	#95	C₂×C₄×C₅₃	1	8	yes	—
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	2	4	no	—
166	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
167	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
168	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
169	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
170	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
171	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	8	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	8	no	—
181	C₄⋊C₄	—	2	8	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	4	no	—
185	C₄²	—	2	4	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	4	2	no	—
188	C₄²	M₁₆×C₅₃	1	4	yes	—
189	C₄²	C₂×D₄×C₅₃	1	2	yes	—
190	C₄²	#41×C₅₃	1	2	yes	—
191	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
192	Q₈	—	4	8	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
205	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
206	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
207	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
208	C₂×C₄	#131×C₅₃	1	2	yes	—
209	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
210	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
211	C₂×C₄	2₋¹⁺⁴×C₅₃	1	4	yes	—
212	C₈	—	2	4	no	—
213	C₈	—	4	4	no	—
214	C₈	—	4	2	no	—
215	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
216	C₂²	C₅₃×#519	1	2	yes	—
217	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
218	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
219	C₂²	C₅₃×#568	1	2	yes	—
220	C₄	—	2	4	no	—
221	C₄	—	2	2	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	4	no	—
224	C₄	—	4	4	no	—
225	C₄	C₅₃×#436	1	4	yes	—
226	C₄	C₅₃×#536	1	2	yes	—
227	C₄	C₅₃×#541	1	2	yes	—
228	C₂	C₅₃×#2327	1	1	yes	characteristic
229	C₂	C₂×C₅₃×#436	1	1	yes	characteristic
230	C₂	C₅₃×#2932	1	1	yes	characteristic
231	C₂	C₅₃×#2983	1	2	yes	—
232	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
98. #22242	C₅₃	direct (N × H)
139. C₅₃	#22242	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#446	2
C₄×C₄⋊Q₈×C₅₃	2
C₅₃×#1984	2
C₅₃×#1917	2
C₅₃×#1831	2
C₅₃×#1744	2
#22242	53

Subgroups — C53×#22242 #27229754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22242 #27229754