Subgroups — C₅₃×#2955 #20271892

Order 6784.

456 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#575	C₂	1	1	yes	maximal, characteristic
2	C₂×#131×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₅₃×#444	C₂	1	4	yes	maximal, complemented
4	C₅₃×#443	C₂	1	4	yes	maximal, complemented
5	C₅₃×#430	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#421	C₂	1	2	yes	maximal, complemented
7	C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₅₃	C₂²	1	2	yes	—
9	#137×C₅₃	C₄	1	4	yes	complemented
10	AGL(1,ℤ/8)×C₅₃	C₄	1	4	yes	complemented
11	#135×C₅₃	C₂²	1	4	yes	—
12	#131×C₅₃	—	2	2	no	—
13	#131×C₅₃	—	2	2	no	—
14	#131×C₅₃	C₂²	1	4	yes	complemented
15	#131×C₅₃	C₂²	1	4	yes	complemented
16	C₂×M₁₆×C₅₃	—	2	2	no	—
17	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	—	2	2	no	—
20	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	#117×C₅₃	C₂²	1	1	yes	characteristic
22	#108×C₅₃	C₂²	1	4	yes	complemented
23	#104×C₅₃	C₂²	1	8	yes	complemented
24	#97×C₅₃	C₂²	1	2	yes	complemented
25	C₄×C₈×C₅₃	C₂²	1	2	yes	complemented
26	#41×C₅₃	—	2	2	no	—
27	#41×C₅₃	—	2	2	no	—
28	#41×C₅₃	C₂×C₄	1	4	yes	—
29	#41×C₅₃	C₂³	1	4	yes	—
30	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×D₄×C₅₃	—	2	2	no	—
32	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₅₃	—	2	2	no	—
34	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
36	SD₁₆×C₅₃	C₂×C₄	1	8	yes	—
37	D₈×C₅₃	C₂×C₄	1	4	yes	—
38	M₁₆×C₅₃	—	2	2	no	—
39	M₁₆×C₅₃	—	2	4	no	—
40	M₁₆×C₅₃	—	2	4	no	—
41	M₁₆×C₅₃	C₂×C₄	1	4	yes	—
42	M₁₆×C₅₃	C₂³	1	4	yes	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₂×C₈×C₅₃	—	2	4	no	—
46	C₂×C₈×C₅₃	D₄	1	2	yes	—
47	C₂×C₈×C₅₃	D₄	1	2	yes	—
48	C₂×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₅₃	—	2	1	no	—
52	C₂²⋊C₄×C₅₃	—	2	1	no	—
53	C₄²×C₅₃	C₂³	1	2	yes	—
54	C₂³×C₅₃	—	2	2	no	—
55	C₂³×C₅₃	#41	1	1	yes	characteristic
56	Q₈×C₅₃	—	2	2	no	—
57	Q₈×C₅₃	C₂²×C₄	1	4	yes	—
58	D₄×C₅₃	—	2	4	no	—
59	D₄×C₅₃	—	2	4	no	—
60	D₄×C₅₃	—	2	2	no	—
61	D₄×C₅₃	C₂²×C₄	1	4	yes	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	4	no	—
64	C₂×C₄×C₅₃	—	2	4	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
70	C₈×C₅₃	—	2	4	no	—
71	C₈×C₅₃	C₂²×C₄	1	4	yes	—
72	C₈×C₅₃	C₂×D₄	1	2	yes	—
73	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₅₃	—	2	4	no	—
76	C₂²×C₅₃	—	4	1	no	—
77	C₂²×C₅₃	—	4	4	no	—
78	C₂²×C₅₃	C₄×D₄	1	2	yes	—
79	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
80	C₄×C₅₃	—	2	4	no	—
81	C₄×C₅₃	—	4	2	no	—
82	C₄×C₅₃	C₄×D₄	1	2	yes	—
83	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
84	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
85	#2955	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	—	2	2	no	—
88	C₂×C₅₃	—	4	4	no	—
89	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
90	#575	C₂×C₅₃	1	1	yes	characteristic
91	C₂×#131	C₂×C₅₃	1	2	yes	complemented
92	#444	C₂×C₅₃	1	4	yes	complemented
93	#443	C₂×C₅₃	1	4	yes	complemented
94	#430	C₂×C₅₃	1	1	yes	characteristic, complemented
95	#421	C₂×C₅₃	1	2	yes	complemented
96	#405	C₂×C₅₃	1	1	yes	characteristic, complemented
97	C₅₃	#2955	1	1	yes	characteristic, direct product factor, Sylow
98	C₂×#41	C₂²×C₅₃	1	2	yes	—
99	#137	C₄×C₅₃	1	4	yes	complemented
100	AGL(1,ℤ/8)	C₄×C₅₃	1	4	yes	complemented
101	#135	C₂²×C₅₃	1	4	yes	—
102	#131	—	2	2	no	—
103	#131	—	2	2	no	—
104	#131	C₂²×C₅₃	1	4	yes	complemented
105	#131	C₂²×C₅₃	1	4	yes	complemented
106	C₂×M₁₆	—	2	2	no	—
107	C₂×M₁₆	C₂²×C₅₃	1	2	yes	—
108	C₂×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
111	#117	C₂²×C₅₃	1	1	yes	characteristic
112	#108	C₂²×C₅₃	1	4	yes	complemented
113	#104	C₂²×C₅₃	1	8	yes	complemented
114	#97	C₂²×C₅₃	1	2	yes	complemented
115	C₄×C₈	C₂²×C₅₃	1	2	yes	complemented
116	#41	—	2	2	no	—
117	#41	—	2	2	no	—
118	#41	C₂×C₄×C₅₃	1	4	yes	—
119	#41	C₂³×C₅₃	1	4	yes	—
120	C₂×Q₈	C₂×C₄×C₅₃	1	2	yes	—
121	C₂×D₄	—	2	2	no	—
122	C₂×D₄	C₂×C₄×C₅₃	1	2	yes	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
125	Q₁₆	C₂×C₄×C₅₃	1	4	yes	—
126	SD₁₆	C₂×C₄×C₅₃	1	8	yes	—
127	D₈	C₂×C₄×C₅₃	1	4	yes	—
128	M₁₆	—	2	2	no	—
129	M₁₆	—	2	4	no	—
130	M₁₆	—	2	4	no	—
131	M₁₆	C₂×C₄×C₅₃	1	4	yes	—
132	M₁₆	C₂³×C₅₃	1	4	yes	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	D₄×C₅₃	1	2	yes	—
137	C₂×C₈	D₄×C₅₃	1	2	yes	—
138	C₂×C₈	C₂³×C₅₃	1	2	yes	—
139	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
140	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
141	C₄⋊C₄	—	2	1	no	—
142	C₂²⋊C₄	—	2	1	no	—
143	C₄²	C₂³×C₅₃	1	2	yes	—
144	C₂³	—	2	2	no	—
145	C₂³	#41×C₅₃	1	1	yes	characteristic
146	Q₈	—	2	2	no	—
147	Q₈	C₂²×C₄×C₅₃	1	4	yes	—
148	D₄	—	2	4	no	—
149	D₄	—	2	4	no	—
150	D₄	—	2	2	no	—
151	D₄	C₂²×C₄×C₅₃	1	4	yes	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
157	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
158	C₂×C₄	#41×C₅₃	1	1	yes	characteristic
159	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
160	C₈	—	2	4	no	—
161	C₈	C₂²×C₄×C₅₃	1	4	yes	—
162	C₈	C₂×D₄×C₅₃	1	2	yes	—
163	C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
164	C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
165	C₂²	—	2	4	no	—
166	C₂²	—	4	1	no	—
167	C₂²	—	4	4	no	—
168	C₂²	C₄×D₄×C₅₃	1	2	yes	—
169	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
170	C₄	—	2	4	no	—
171	C₄	—	4	2	no	—
172	C₄	C₄×D₄×C₅₃	1	2	yes	—
173	C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
174	C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	2	no	—
177	C₂	—	4	4	no	—
178	C₂	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₅₃	C₂×D₄
commutator	C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
socle	C₂×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#575	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#131×C₅₃	C₂	semidirect
3. C₅₃×#444	C₂	semidirect
4. C₅₃×#443	C₂	semidirect
5. C₅₃×#430	C₂	semidirect
6. C₅₃×#421	C₂	semidirect
7. C₅₃×#405	C₂	semidirect
9. #137×C₅₃	C₄	semidirect
10. AGL(1,ℤ/8)×C₅₃	C₄	semidirect
14. #131×C₅₃	C₂²	semidirect
15. #131×C₅₃	C₂²	semidirect
22. #108×C₅₃	C₂²	semidirect
23. #104×C₅₃	C₂²	semidirect
24. #97×C₅₃	C₂²	semidirect
25. C₄×C₈×C₅₃	C₂²	semidirect
85. #2955	C₅₃	direct (N × H)
91. C₂×#131	C₂×C₅₃	semidirect
92. #444	C₂×C₅₃	semidirect
93. #443	C₂×C₅₃	semidirect
94. #430	C₂×C₅₃	semidirect
95. #421	C₂×C₅₃	semidirect
96. #405	C₂×C₅₃	semidirect
97. C₅₃	#2955	direct (N × H)
99. #137	C₄×C₅₃	semidirect
100. AGL(1,ℤ/8)	C₄×C₅₃	semidirect
104. #131	C₂²×C₅₃	semidirect
105. #131	C₂²×C₅₃	semidirect
112. #108	C₂²×C₅₃	semidirect
113. #104	C₂²×C₅₃	semidirect
114. #97	C₂²×C₅₃	semidirect
115. C₄×C₈	C₂²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#575	2
C₂×#131×C₅₃	2
C₅₃×#444	2
C₅₃×#443	2
C₅₃×#430	2
C₅₃×#421	2
C₅₃×#405	2
#2955	53

Subgroups — C53×#2955 #20271892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2955 #20271892