Subgroups — C₅₃×#20825 #27228337

Order 13568.

842 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2283	C₂	1	2	yes	maximal
3	C₅₃×#1846	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1835	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1809	C₂	1	4	yes	maximal
6	C₅₃×#1776	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#107×C₅₃	C₂²	1	4	yes	—
10	C₂×#106×C₅₃	C₂²	1	4	yes	—
11	C₄×M₁₆×C₅₃	C₄	1	8	yes	complemented
12	C₂×#97×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#383	—	2	2	no	—
16	C₅₃×#382	C₂²	1	4	yes	—
17	C₅₃×#381	—	2	2	no	—
18	C₅₃×#380	C₂²	1	4	yes	—
19	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
20	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
21	C₅₃×#376	—	2	2	no	—
22	C₅₃×#340	C₂²	1	8	yes	—
23	C₂³×C₄×C₅₃	D₄	1	2	yes	—
24	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂×M₁₆×C₅₃	—	2	4	no	—
26	C₂×M₁₆×C₅₃	—	2	4	no	—
27	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
30	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
31	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
32	C₂×C₄²×C₅₃	—	2	4	no	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#107×C₅₃	—	2	8	no	—
39	#106×C₅₃	—	2	8	no	—
40	#97×C₅₃	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
43	#95×C₅₃	—	2	4	no	—
44	#95×C₅₃	—	2	4	no	—
45	#95×C₅₃	—	2	4	no	—
46	#95×C₅₃	—	2	4	no	—
47	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	—	2	8	no	—
51	C₂²×C₄×C₅₃	—	4	4	no	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
59	M₁₆×C₅₃	—	2	16	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₅₃	#41	1	4	yes	—
65	C₂×C₈×C₅₃	#41	1	4	yes	—
66	C₄⋊C₄×C₅₃	—	2	8	no	—
67	C₄⋊C₄×C₅₃	—	2	8	no	—
68	C₂²⋊C₄×C₅₃	—	4	8	no	—
69	C₄²×C₅₃	—	4	8	no	—
70	C₄²×C₅₃	C₄⋊C₄	1	8	yes	—
71	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
72	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
74	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
75	C₂³×C₅₃	—	2	2	no	—
76	C₂³×C₅₃	—	2	4	no	—
77	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	2	4	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	4	8	no	—
86	C₂×C₄×C₅₃	—	4	8	no	—
87	C₂×C₄×C₅₃	—	4	8	no	—
88	C₂×C₄×C₅₃	—	4	8	no	—
89	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
96	C₈×C₅₃	—	2	4	no	—
97	C₈×C₅₃	#117	1	8	yes	—
98	#20825	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₅₃	—	2	4	no	—
100	C₂²×C₅₃	—	2	2	no	—
101	C₂²×C₅₃	—	4	4	no	—
102	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₅₃	#532	1	2	yes	—
104	C₂²×C₅₃	#575	1	4	yes	—
105	C₄×C₅₃	—	2	4	no	—
106	C₄×C₅₃	—	2	4	no	—
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	—	8	8	no	—
109	C₄×C₅₃	C₂×#117	1	2	yes	—
110	C₄×C₅₃	#533	1	1	yes	characteristic
111	C₄×C₅₃	#533	1	1	yes	characteristic
112	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
113	#2283	C₂×C₅₃	1	2	yes	—
114	#1846	C₂×C₅₃	1	2	yes	complemented
115	#1835	C₂×C₅₃	1	2	yes	complemented
116	#1809	C₂×C₅₃	1	4	yes	—
117	#1776	C₂×C₅₃	1	4	yes	complemented
118	C₂×C₅₃	—	4	2	no	—
119	C₂×C₅₃	#2285	1	1	yes	characteristic
120	C₂×C₅₃	#2911	1	2	yes	—
121	C₂×C₅₃	#3183	1	4	yes	—
122	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
123	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
124	C₂×#107	C₂²×C₅₃	1	4	yes	—
125	C₂×#106	C₂²×C₅₃	1	4	yes	—
126	C₄×M₁₆	C₄×C₅₃	1	8	yes	complemented
127	C₂×#97	C₂²×C₅₃	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
130	#383	—	2	2	no	—
131	#382	C₂²×C₅₃	1	4	yes	—
132	#381	—	2	2	no	—
133	#380	C₂²×C₅₃	1	4	yes	—
134	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
135	C₄×C₂²⋊C₄	—	2	2	no	—
136	#376	—	2	2	no	—
137	#340	C₂²×C₅₃	1	8	yes	—
138	C₅₃	#20825	1	1	yes	characteristic, direct product factor, Sylow
139	C₂³×C₄	D₄×C₅₃	1	2	yes	—
140	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
141	C₂×M₁₆	—	2	4	no	—
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	C₂×C₄×C₅₃	1	8	yes	—
144	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
145	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
146	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
147	C₂×C₂²⋊C₄	—	2	4	no	—
148	C₂×C₄²	—	2	4	no	—
149	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
150	C₂×C₄²	D₄×C₅₃	1	2	yes	—
151	C₂×C₄²	Q₈×C₅₃	1	4	yes	—
152	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
153	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
154	#107	—	2	8	no	—
155	#106	—	2	8	no	—
156	#97	C₂×C₄×C₅₃	1	8	yes	—
157	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
158	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
159	#95	—	2	4	no	—
160	#95	—	2	4	no	—
161	#95	—	2	4	no	—
162	#95	—	2	4	no	—
163	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	4	4	no	—
168	C₂²×C₄	C₄⋊C₄×C₅₃	1	8	yes	—
169	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
170	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
171	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
172	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
173	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
174	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
175	M₁₆	—	2	16	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
179	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
180	C₂×C₈	#41×C₅₃	1	4	yes	—
181	C₂×C₈	#41×C₅₃	1	4	yes	—
182	C₄⋊C₄	—	2	8	no	—
183	C₄⋊C₄	—	2	8	no	—
184	C₂²⋊C₄	—	4	8	no	—
185	C₄²	—	4	8	no	—
186	C₄²	C₄⋊C₄×C₅₃	1	8	yes	—
187	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
188	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
189	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
190	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	4	no	—
193	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
194	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	8	no	—
202	C₂×C₄	—	4	8	no	—
203	C₂×C₄	—	4	8	no	—
204	C₂×C₄	—	4	8	no	—
205	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
206	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
207	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
208	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
209	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
210	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
211	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
212	C₈	—	2	4	no	—
213	C₈	#117×C₅₃	1	8	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	4	4	no	—
217	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
218	C₂²	C₅₃×#532	1	2	yes	—
219	C₂²	C₅₃×#575	1	4	yes	—
220	C₄	—	2	2	no	—
221	C₄	—	2	4	no	—
222	C₄	—	2	4	no	—
223	C₄	—	8	8	no	—
224	C₄	C₂×#117×C₅₃	1	2	yes	—
225	C₄	C₅₃×#533	1	1	yes	characteristic
226	C₄	C₅₃×#533	1	1	yes	characteristic
227	C₂	—	4	2	no	—
228	C₂	C₅₃×#2285	1	1	yes	characteristic
229	C₂	C₅₃×#2911	1	2	yes	—
230	C₂	C₅₃×#3183	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1846	C₂	semidirect
4. C₅₃×#1835	C₂	semidirect
6. C₅₃×#1776	C₂	semidirect
11. C₄×M₁₆×C₅₃	C₄	semidirect
98. #20825	C₅₃	direct (N × H)
114. #1846	C₂×C₅₃	semidirect
115. #1835	C₂×C₅₃	semidirect
117. #1776	C₂×C₅₃	semidirect
126. C₄×M₁₆	C₄×C₅₃	semidirect
138. C₅₃	#20825	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×C₄×M₁₆×C₅₃	2
C₅₃×#2283	2
C₅₃×#1846	2
C₅₃×#1835	2
C₅₃×#1809	2
C₅₃×#1776	2
#20825	53

Subgroups — C53×#20825 #27228337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#20825 #27228337