Subgroups — C₅₃×#22283 #27229795

Order 13568.

826 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3150	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2932	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1962	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1876	C₂	1	8	yes	maximal, complemented
5	C₅₃×#1838	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1745	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#135×C₅₃	C₂²	1	4	yes	—
8	C₂×#131×C₅₃	C₂²	1	4	yes	—
9	C₅₃×#532	C₂²	1	2	yes	—
10	C₅₃×#497	C₄	1	4	yes	complemented
11	C₅₃×#496	C₄	1	4	yes	complemented
12	C₅₃×#435	—	2	8	no	—
13	C₅₃×#423	C₂²	1	2	yes	—
14	C₂×#104×C₅₃	C₂²	1	8	yes	—
15	C₅₃×#408	C₂²	1	4	yes	—
16	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
18	C₄³×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#341	C₂²	1	4	yes	—
20	C₅₃×#336	C₂²	1	4	yes	—
21	C₂×#41×C₅₃	C₂³	1	4	yes	—
22	#135×C₅₃	—	4	8	no	—
23	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
24	C₂×SD₁₆×C₅₃	C₂×C₄	1	8	yes	—
25	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
26	#131×C₅₃	—	4	8	no	—
27	C₂×M₁₆×C₅₃	—	2	4	no	—
28	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
29	C₂²×C₈×C₅₃	—	2	4	no	—
30	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
31	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
32	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
33	#127×C₅₃	C₂×C₄	1	2	yes	—
34	#124×C₅₃	C₂×C₄	1	4	yes	—
35	#121×C₅₃	—	2	4	no	—
36	C₄×D₄×C₅₃	—	2	4	no	—
37	C₂×C₄²×C₅₃	—	2	2	no	—
38	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	4	no	—
41	#104×C₅₃	—	4	8	no	—
42	C₂²⋊C₈×C₅₃	—	2	8	no	—
43	#97×C₅₃	C₂×C₄	1	4	yes	—
44	#97×C₅₃	D₄	1	4	yes	—
45	#41×C₅₃	—	2	8	no	—
46	#41×C₅₃	—	2	8	no	—
47	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
48	C₂×D₄×C₅₃	—	2	4	no	—
49	C₂×D₄×C₅₃	—	2	4	no	—
50	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	#41	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
55	Q₁₆×C₅₃	—	4	4	no	—
56	SD₁₆×C₅₃	—	4	8	no	—
57	D₈×C₅₃	—	4	4	no	—
58	M₁₆×C₅₃	—	2	8	no	—
59	M₁₆×C₅₃	—	4	8	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	—	2	4	no	—
62	C₂×C₈×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	—	4	8	no	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
66	C₄⋊C₄×C₅₃	—	2	4	no	—
67	C₂²⋊C₄×C₅₃	—	2	8	no	—
68	C₄²×C₅₃	—	2	4	no	—
69	C₄²×C₅₃	—	4	4	no	—
70	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
71	C₄²×C₅₃	C₂×D₄	1	2	yes	—
72	C₂³×C₅₃	—	2	4	no	—
73	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
74	Q₈×C₅₃	—	2	8	no	—
75	D₄×C₅₃	—	2	8	no	—
76	D₄×C₅₃	—	4	8	no	—
77	D₄×C₅₃	—	8	4	no	—
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	—	4	2	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	—	4	4	no	—
84	C₂×C₄×C₅₃	—	4	4	no	—
85	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
86	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
89	C₈×C₅₃	—	2	4	no	—
90	C₈×C₅₃	—	2	4	no	—
91	C₈×C₅₃	—	4	4	no	—
92	#22283	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	—	4	4	no	—
95	C₂²×C₅₃	—	4	8	no	—
96	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
97	C₂²×C₅₃	#540	1	2	yes	—
98	C₄×C₅₃	—	2	4	no	—
99	C₄×C₅₃	—	4	4	no	—
100	C₄×C₅₃	—	4	4	no	—
101	C₄×C₅₃	#518	1	2	yes	—
102	C₄×C₅₃	#535	1	2	yes	—
103	#3150	C₂×C₅₃	1	1	yes	characteristic
104	#2932	C₂×C₅₃	1	2	yes	complemented
105	#1962	C₂×C₅₃	1	2	yes	complemented
106	#1876	C₂×C₅₃	1	8	yes	complemented
107	#1838	C₂×C₅₃	1	1	yes	characteristic, complemented
108	#1745	C₂×C₅₃	1	1	yes	characteristic, complemented
109	C₂×C₅₃	—	2	2	no	—
110	C₂×C₅₃	—	8	4	no	—
111	C₂×C₅₃	#2328	1	1	yes	characteristic
112	C₂×C₅₃	#2956	1	2	yes	—
113	C₂×#135	C₂²×C₅₃	1	4	yes	—
114	C₂×#131	C₂²×C₅₃	1	4	yes	—
115	#532	C₂²×C₅₃	1	2	yes	—
116	#497	C₄×C₅₃	1	4	yes	complemented
117	#496	C₄×C₅₃	1	4	yes	complemented
118	#435	—	2	8	no	—
119	#423	C₂²×C₅₃	1	2	yes	—
120	C₂×#104	C₂²×C₅₃	1	8	yes	—
121	#408	C₂²×C₅₃	1	4	yes	—
122	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
123	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
124	C₄³	C₂²×C₅₃	1	1	yes	characteristic
125	#341	C₂²×C₅₃	1	4	yes	—
126	#336	C₂²×C₅₃	1	4	yes	—
127	C₅₃	#22283	1	1	yes	characteristic, direct product factor, Sylow
128	C₂×#41	C₂³×C₅₃	1	4	yes	—
129	#135	—	4	8	no	—
130	C₂×Q₁₆	C₂×C₄×C₅₃	1	4	yes	—
131	C₂×SD₁₆	C₂×C₄×C₅₃	1	8	yes	—
132	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	—
133	#131	—	4	8	no	—
134	C₂×M₁₆	—	2	4	no	—
135	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
136	C₂²×C₈	—	2	4	no	—
137	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
138	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
139	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
140	#127	C₂×C₄×C₅₃	1	2	yes	—
141	#124	C₂×C₄×C₅₃	1	4	yes	—
142	#121	—	2	4	no	—
143	C₄×D₄	—	2	4	no	—
144	C₂×C₄²	—	2	2	no	—
145	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
146	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
147	C₄⋊C₈	—	2	4	no	—
148	#104	—	4	8	no	—
149	C₂²⋊C₈	—	2	8	no	—
150	#97	C₂×C₄×C₅₃	1	4	yes	—
151	#97	D₄×C₅₃	1	4	yes	—
152	#41	—	2	8	no	—
153	#41	—	2	8	no	—
154	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
155	C₂×D₄	—	2	4	no	—
156	C₂×D₄	—	2	4	no	—
157	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	#41×C₅₃	1	2	yes	—
161	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
162	Q₁₆	—	4	4	no	—
163	SD₁₆	—	4	8	no	—
164	D₈	—	4	4	no	—
165	M₁₆	—	2	8	no	—
166	M₁₆	—	4	8	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	—	4	8	no	—
171	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
172	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
173	C₄⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	2	8	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	4	4	no	—
177	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
178	C₄²	C₂×D₄×C₅₃	1	2	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
181	Q₈	—	2	8	no	—
182	D₄	—	2	8	no	—
183	D₄	—	4	8	no	—
184	D₄	—	8	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
193	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
194	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
195	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
196	C₈	—	2	4	no	—
197	C₈	—	2	4	no	—
198	C₈	—	4	4	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	8	no	—
201	C₂²	—	4	4	no	—
202	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
203	C₂²	C₅₃×#540	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	4	4	no	—
206	C₄	—	4	4	no	—
207	C₄	C₅₃×#518	1	2	yes	—
208	C₄	C₅₃×#535	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	8	4	no	—
211	C₂	C₅₃×#2328	1	1	yes	characteristic
212	C₂	C₅₃×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#3150	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2932	C₂	semidirect
3. C₅₃×#1962	C₂	semidirect
4. C₅₃×#1876	C₂	semidirect
5. C₅₃×#1838	C₂	semidirect
6. C₅₃×#1745	C₂	semidirect
10. C₅₃×#497	C₄	semidirect
11. C₅₃×#496	C₄	semidirect
92. #22283	C₅₃	direct (N × H)
104. #2932	C₂×C₅₃	semidirect
105. #1962	C₂×C₅₃	semidirect
106. #1876	C₂×C₅₃	semidirect
107. #1838	C₂×C₅₃	semidirect
108. #1745	C₂×C₅₃	semidirect
116. #497	C₄×C₅₃	semidirect
117. #496	C₄×C₅₃	semidirect
127. C₅₃	#22283	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3150	2
C₅₃×#2932	2
C₅₃×#1962	2
C₅₃×#1876	2
C₅₃×#1838	2
C₅₃×#1745	2
#22283	53

Subgroups — C53×#22283 #27229795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22283 #27229795