Subgroups — C₅₃×#3396 #20272333

Order 6784.

384 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#559	C₂	1	2	yes	maximal, complemented
2	C₅₃×#495	C₂	1	2	yes	maximal, complemented
3	C₅₃×#493	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#463	C₂	1	4	yes	maximal, complemented
5	C₅₃×#460	C₂	1	2	yes	maximal, complemented
6	C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₈×C₅₃	C₂	1	2	yes	maximal, complemented
9	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×SD₁₆×C₅₃	C₂²	1	4	yes	—
11	C₂×D₈×C₅₃	C₂²	1	2	yes	—
12	C₂×D₈×C₅₃	C₂²	1	2	yes	—
13	#127×C₅₃	C₂²	1	2	yes	—
14	#124×C₅₃	—	2	2	no	—
15	#124×C₅₃	C₂²	1	4	yes	—
16	#123×C₅₃	—	2	4	no	—
17	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
18	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₅₃	—	2	2	no	—
20	C₄×D₄×C₅₃	C₂²	1	4	yes	—
21	#107×C₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
24	#103×C₅₃	C₂²	1	2	yes	—
25	#102×C₅₃	C₂²	1	4	yes	—
26	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂²	1	2	yes	—
28	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
30	C₂×D₄×C₅₃	—	2	2	no	—
31	C₂×D₄×C₅₃	—	2	2	no	—
32	C₂×D₄×C₅₃	C₂³	1	4	yes	—
33	C₂²×C₄×C₅₃	—	2	4	no	—
34	Q₁₆×C₅₃	—	2	2	no	—
35	SD₁₆×C₅₃	—	4	4	no	—
36	D₈×C₅₃	—	2	4	no	—
37	D₈×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	C₂³	1	2	yes	—
39	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
43	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
44	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
45	C₂²⋊C₄×C₅₃	—	2	4	no	—
46	C₂²⋊C₄×C₅₃	—	2	4	no	—
47	C₂²⋊C₄×C₅₃	—	2	4	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	C₂³	1	2	yes	—
50	C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	C₂³×C₅₃	—	2	4	no	—
52	Q₈×C₅₃	—	2	2	no	—
53	Q₈×C₅₃	—	2	2	no	—
54	Q₈×C₅₃	—	2	2	no	—
55	D₄×C₅₃	—	2	4	no	—
56	D₄×C₅₃	—	2	4	no	—
57	D₄×C₅₃	—	4	4	no	—
58	D₄×C₅₃	—	8	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	8	4	no	—
62	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
67	C₈×C₅₃	—	2	1	no	—
68	C₈×C₅₃	—	2	2	no	—
69	C₈×C₅₃	#41	1	2	yes	—
70	C₂²×C₅₃	—	4	4	no	—
71	C₂²×C₅₃	—	4	4	no	—
72	C₂²×C₅₃	—	4	4	no	—
73	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
74	C₄×C₅₃	—	2	2	no	—
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	—	4	2	no	—
78	C₄×C₅₃	—	4	2	no	—
79	C₄×C₅₃	#135	1	2	yes	—
80	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
81	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
82	#3396	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×C₅₃	—	8	4	no	—
84	C₂×C₅₃	#550	1	1	yes	characteristic
85	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
86	C₂×C₅₃	#576	1	1	yes	characteristic
87	#559	C₂×C₅₃	1	2	yes	complemented
88	#495	C₂×C₅₃	1	2	yes	complemented
89	#493	C₂×C₅₃	1	1	yes	characteristic
90	#463	C₂×C₅₃	1	4	yes	complemented
91	#460	C₂×C₅₃	1	2	yes	complemented
92	C₈×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
93	C₄×Q₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
94	C₄×D₈	C₂×C₅₃	1	2	yes	complemented
95	C₅₃	#3396	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
97	C₂×SD₁₆	C₂²×C₅₃	1	4	yes	—
98	C₂×D₈	C₂²×C₅₃	1	2	yes	—
99	C₂×D₈	C₂²×C₅₃	1	2	yes	—
100	#127	C₂²×C₅₃	1	2	yes	—
101	#124	—	2	2	no	—
102	#124	C₂²×C₅₃	1	4	yes	—
103	#123	—	2	4	no	—
104	C₄×Q₈	C₂²×C₅₃	1	2	yes	—
105	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
106	C₄×D₄	—	2	2	no	—
107	C₄×D₄	C₂²×C₅₃	1	4	yes	—
108	#107	C₂²×C₅₃	1	1	yes	characteristic
109	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
110	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
111	#103	C₂²×C₅₃	1	2	yes	—
112	#102	C₂²×C₅₃	1	4	yes	—
113	C₄×C₈	C₂²×C₅₃	1	2	yes	—
114	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
115	C₂×Q₈	D₄×C₅₃	1	1	yes	characteristic
116	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	C₂³×C₅₃	1	4	yes	—
120	C₂²×C₄	—	2	4	no	—
121	Q₁₆	—	2	2	no	—
122	SD₁₆	—	4	4	no	—
123	D₈	—	2	4	no	—
124	D₈	—	4	2	no	—
125	C₂×C₈	C₂³×C₅₃	1	2	yes	—
126	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
127	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
130	C₄⋊C₄	D₄×C₅₃	1	2	yes	—
131	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₂²⋊C₄	—	2	4	no	—
134	C₂²⋊C₄	—	2	4	no	—
135	C₄²	—	2	2	no	—
136	C₄²	C₂³×C₅₃	1	2	yes	—
137	C₄²	C₂³×C₅₃	1	1	yes	characteristic
138	C₂³	—	2	4	no	—
139	Q₈	—	2	2	no	—
140	Q₈	—	2	2	no	—
141	Q₈	—	2	2	no	—
142	D₄	—	2	4	no	—
143	D₄	—	2	4	no	—
144	D₄	—	4	4	no	—
145	D₄	—	8	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	8	4	no	—
149	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
150	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
151	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
152	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
153	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	2	2	no	—
156	C₈	#41×C₅₃	1	2	yes	—
157	C₂²	—	4	4	no	—
158	C₂²	—	4	4	no	—
159	C₂²	—	4	4	no	—
160	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	2	no	—
163	C₄	—	2	1	no	—
164	C₄	—	4	2	no	—
165	C₄	—	4	2	no	—
166	C₄	#135×C₅₃	1	2	yes	—
167	C₄	C₂×#41×C₅₃	1	1	yes	characteristic
168	C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
169	C₂	—	8	4	no	—
170	C₂	C₅₃×#550	1	1	yes	characteristic
171	C₂	C₂×#135×C₅₃	1	1	yes	characteristic
172	C₂	C₅₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
prime radical	C₅₃×#493	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#559	C₂	semidirect
2. C₅₃×#495	C₂	semidirect
4. C₅₃×#463	C₂	semidirect
5. C₅₃×#460	C₂	semidirect
6. C₈×Q₈×C₅₃	C₂	semidirect
7. C₄×Q₁₆×C₅₃	C₂	semidirect
8. C₄×D₈×C₅₃	C₂	semidirect
82. #3396	C₅₃	direct (N × H)
87. #559	C₂×C₅₃	semidirect
88. #495	C₂×C₅₃	semidirect
90. #463	C₂×C₅₃	semidirect
91. #460	C₂×C₅₃	semidirect
92. C₈×Q₈	C₂×C₅₃	semidirect
93. C₄×Q₁₆	C₂×C₅₃	semidirect
94. C₄×D₈	C₂×C₅₃	semidirect
95. C₅₃	#3396	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×#559	2
C₅₃×#495	2
C₅₃×#493	2
C₅₃×#463	2
C₅₃×#460	2
C₈×Q₈×C₅₃	2
C₄×Q₁₆×C₅₃	2
C₄×D₈×C₅₃	2
#3396	53

Subgroups — C53×#3396 #20272333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3396 #20272333