Subgroups — C₅₃×#3176 #20272113

Order 6784.

386 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#540	C₂	1	2	yes	maximal, complemented
2	C₅₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#490	C₂	1	2	yes	maximal, complemented
5	C₅₃×#469	C₂	1	2	yes	maximal, complemented
6	C₅₃×#466	C₂	1	2	yes	maximal, complemented
7	C₅₃×#442	C₂	1	2	yes	maximal, complemented
8	C₄×D₈×C₅₃	C₂	1	2	yes	maximal, complemented
9	C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₈×C₅₃	C₂²	1	2	yes	—
11	C₂×D₈×C₅₃	C₂²	1	2	yes	—
12	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	#127×C₅₃	C₂²	1	2	yes	—
15	#125×C₅₃	C₂²	1	2	yes	—
16	#123×C₅₃	—	2	4	no	—
17	#121×C₅₃	—	2	4	no	—
18	#121×C₅₃	C₂²	1	4	yes	—
19	C₄×D₄×C₅₃	C₂²	1	4	yes	—
20	#117×C₅₃	C₂²	1	1	yes	characteristic
21	#107×C₅₃	C₂²	1	2	yes	—
22	#106×C₅₃	C₂²	1	2	yes	—
23	#102×C₅₃	C₂²	1	4	yes	—
24	#102×C₅₃	C₂²	1	4	yes	—
25	#97×C₅₃	C₂²	1	1	yes	characteristic
26	#97×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×D₄×C₅₃	—	2	4	no	—
30	C₂×D₄×C₅₃	—	2	4	no	—
31	C₂×D₄×C₅₃	C₂³	1	4	yes	—
32	C₂²×C₄×C₅₃	—	2	4	no	—
33	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	D₈×C₅₃	—	2	4	no	—
35	D₈×C₅₃	—	2	4	no	—
36	M₁₆×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₅₃	—	2	4	no	—
43	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
44	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
45	C₂²⋊C₄×C₅₃	—	2	4	no	—
46	C₂²⋊C₄×C₅₃	—	2	4	no	—
47	C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
48	C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	C₂³×C₅₃	—	2	4	no	—
51	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
52	D₄×C₅₃	—	2	4	no	—
53	D₄×C₅₃	—	2	4	no	—
54	D₄×C₅₃	—	8	4	no	—
55	D₄×C₅₃	—	8	4	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	4	no	—
59	C₂×C₄×C₅₃	—	8	4	no	—
60	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₈×C₅₃	—	2	1	no	—
65	C₈×C₅₃	—	2	1	no	—
66	C₈×C₅₃	#41	1	2	yes	—
67	C₈×C₅₃	#41	1	2	yes	—
68	C₂²×C₅₃	—	2	1	no	—
69	C₂²×C₅₃	—	2	2	no	—
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	1	no	—
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	4	4	no	—
78	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
79	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
80	#3176	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂×C₅₃	—	4	1	no	—
82	C₂×C₅₃	—	8	4	no	—
83	C₂×C₅₃	#532	1	1	yes	characteristic
84	C₂×C₅₃	#576	1	2	yes	—
85	#540	C₂×C₅₃	1	2	yes	complemented
86	#499	C₂×C₅₃	1	1	yes	characteristic, complemented
87	#493	C₂×C₅₃	1	1	yes	characteristic, complemented
88	#490	C₂×C₅₃	1	2	yes	complemented
89	#469	C₂×C₅₃	1	2	yes	complemented
90	#466	C₂×C₅₃	1	2	yes	complemented
91	#442	C₂×C₅₃	1	2	yes	complemented
92	C₄×D₈	C₂×C₅₃	1	2	yes	complemented
93	#405	C₂×C₅₃	1	1	yes	characteristic, complemented
94	C₅₃	#3176	1	1	yes	characteristic, direct product factor, Sylow
95	C₂×D₈	C₂²×C₅₃	1	2	yes	—
96	C₂×D₈	C₂²×C₅₃	1	2	yes	—
97	C₂×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
98	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
99	#127	C₂²×C₅₃	1	2	yes	—
100	#125	C₂²×C₅₃	1	2	yes	—
101	#123	—	2	4	no	—
102	#121	—	2	4	no	—
103	#121	C₂²×C₅₃	1	4	yes	—
104	C₄×D₄	C₂²×C₅₃	1	4	yes	—
105	#117	C₂²×C₅₃	1	1	yes	characteristic
106	#107	C₂²×C₅₃	1	2	yes	—
107	#106	C₂²×C₅₃	1	2	yes	—
108	#102	C₂²×C₅₃	1	4	yes	—
109	#102	C₂²×C₅₃	1	4	yes	—
110	#97	C₂²×C₅₃	1	1	yes	characteristic
111	#97	C₂²×C₅₃	1	1	yes	characteristic
112	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
113	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
114	C₂×D₄	—	2	4	no	—
115	C₂×D₄	—	2	4	no	—
116	C₂×D₄	C₂³×C₅₃	1	4	yes	—
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
119	D₈	—	2	4	no	—
120	D₈	—	2	4	no	—
121	M₁₆	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
124	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
125	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
126	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	D₄×C₅₃	1	2	yes	—
129	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
130	C₂²⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
133	C₄²	C₂³×C₅₃	1	1	yes	characteristic
134	C₄²	C₂³×C₅₃	1	1	yes	characteristic
135	C₂³	—	2	4	no	—
136	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
137	D₄	—	2	4	no	—
138	D₄	—	2	4	no	—
139	D₄	—	8	4	no	—
140	D₄	—	8	4	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	8	4	no	—
145	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
146	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
147	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
148	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	#41×C₅₃	1	2	yes	—
152	C₈	#41×C₅₃	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	—	4	4	no	—
158	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	4	4	no	—
163	C₄	C₂×#41×C₅₃	1	1	yes	characteristic
164	C₄	C₂×#41×C₅₃	1	1	yes	characteristic
165	C₂	—	4	1	no	—
166	C₂	—	8	4	no	—
167	C₂	C₅₃×#532	1	1	yes	characteristic
168	C₂	C₅₃×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ #117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#540	C₂	semidirect
2. C₅₃×#499	C₂	semidirect
3. C₅₃×#493	C₂	semidirect
4. C₅₃×#490	C₂	semidirect
5. C₅₃×#469	C₂	semidirect
6. C₅₃×#466	C₂	semidirect
7. C₅₃×#442	C₂	semidirect
8. C₄×D₈×C₅₃	C₂	semidirect
9. C₅₃×#405	C₂	semidirect
80. #3176	C₅₃	direct (N × H)
85. #540	C₂×C₅₃	semidirect
86. #499	C₂×C₅₃	semidirect
87. #493	C₂×C₅₃	semidirect
88. #490	C₂×C₅₃	semidirect
89. #469	C₂×C₅₃	semidirect
90. #466	C₂×C₅₃	semidirect
91. #442	C₂×C₅₃	semidirect
92. C₄×D₈	C₂×C₅₃	semidirect
93. #405	C₂×C₅₃	semidirect
94. C₅₃	#3176	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×#540	2
C₅₃×#499	2
C₅₃×#493	2
C₅₃×#490	2
C₅₃×#469	2
C₅₃×#466	2
C₅₃×#442	2
C₄×D₈×C₅₃	2
C₅₃×#405	2
#3176	53

Subgroups — C53×#3176 #20272113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3176 #20272113