Subgroups — C₅₃×#12032 #27219544

Order 13568.

222 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2162	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1371	C₂	1	4	yes	maximal, complemented
5	C₅₃×#423	—	2	1	no	—
6	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×C₄⋊C₁₆	C₄	1	2	yes	—
9	C₅₃×C₂²⋊C₁₆	C₄	1	2	yes	—
10	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#335	—	2	1	no	—
12	C₅₃×#335	—	2	1	no	—
13	C₅₃×#335	C₂²	1	2	yes	—
14	C₅₃×#335	C₂²	1	2	yes	—
15	C₂×M₁₆×C₅₃	—	4	1	no	—
16	C₂²×C₈×C₅₃	—	4	1	no	—
17	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
21	C₄⋊C₈×C₅₃	—	2	2	no	—
22	C₄⋊C₈×C₅₃	—	2	2	no	—
23	C₄⋊C₈×C₅₃	—	4	1	no	—
24	C₄⋊C₈×C₅₃	—	4	1	no	—
25	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
32	M₁₆×C₅₃	—	8	2	no	—
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	—	4	1	no	—
35	C₂×C₈×C₅₃	—	4	2	no	—
36	C₂×C₈×C₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	—	8	2	no	—
38	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₅₃	D₈	1	1	yes	characteristic
44	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
45	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₁₆×C₅₃	—	2	4	no	—
47	C₂³×C₅₃	#107	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
54	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
55	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₅₃	—	2	1	no	—
60	C₈×C₅₃	—	2	1	no	—
61	C₈×C₅₃	—	2	1	no	—
62	C₈×C₅₃	—	8	2	no	—
63	C₈×C₅₃	—	8	2	no	—
64	C₈×C₅₃	#131	1	2	yes	—
65	#12032	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂²×C₅₃	—	2	1	no	—
67	C₂²×C₅₃	—	2	1	no	—
68	C₂²×C₅₃	#366	1	2	yes	—
69	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	2	1	no	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	#366	1	2	yes	—
74	C₄×C₅₃	#424	1	1	yes	characteristic
75	C₄×C₅₃	#430	1	1	yes	characteristic
76	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
77	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
78	#2162	C₂×C₅₃	1	1	yes	characteristic
79	#1567	C₂×C₅₃	1	1	yes	characteristic
80	C₂×#335	C₂×C₅₃	1	1	yes	characteristic
81	#1371	C₂×C₅₃	1	4	yes	complemented
82	C₂×C₅₃	—	2	2	no	—
83	C₂×C₅₃	#1571	1	1	yes	characteristic
84	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
85	C₂×C₅₃	#2161	1	1	yes	characteristic
86	#423	—	2	1	no	—
87	C₂×C₄⋊C₈	—	2	1	no	—
88	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
89	C₄⋊C₁₆	C₄×C₅₃	1	2	yes	—
90	C₂²⋊C₁₆	C₄×C₅₃	1	2	yes	—
91	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
92	#335	—	2	1	no	—
93	#335	—	2	1	no	—
94	#335	C₂²×C₅₃	1	2	yes	—
95	#335	C₂²×C₅₃	1	2	yes	—
96	C₅₃	#12032	1	1	yes	characteristic, direct product factor, Sylow
97	C₂×M₁₆	—	4	1	no	—
98	C₂²×C₈	—	4	1	no	—
99	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
100	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
101	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
102	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	4	1	no	—
106	C₄⋊C₈	—	4	1	no	—
107	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
108	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
109	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
110	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
111	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
112	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
113	C₂²×C₄	Q₁₆×C₅₃	1	1	yes	characteristic
114	M₁₆	—	8	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	8	2	no	—
120	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
121	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
122	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
123	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
124	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
125	C₄²	D₈×C₅₃	1	1	yes	characteristic
126	C₄²	Q₁₆×C₅₃	1	1	yes	characteristic
127	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
128	C₁₆	—	2	4	no	—
129	C₂³	#107×C₅₃	1	1	yes	characteristic
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
134	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
135	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
136	C₂×C₄	D₁₆×C₅₃	1	2	yes	—
137	C₂×C₄	Q₃₂×C₅₃	1	2	yes	—
138	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
139	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
140	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	8	2	no	—
145	C₈	—	8	2	no	—
146	C₈	#131×C₅₃	1	2	yes	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	C₅₃×#366	1	2	yes	—
150	C₂²	C₂×#107×C₅₃	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	C₅₃×#366	1	2	yes	—
155	C₄	C₅₃×#424	1	1	yes	characteristic
156	C₄	C₅₃×#430	1	1	yes	characteristic
157	C₄	C₂×D₁₆×C₅₃	1	1	yes	characteristic
158	C₄	C₂×Q₃₂×C₅₃	1	1	yes	characteristic
159	C₂	—	2	2	no	—
160	C₂	C₅₃×#1571	1	1	yes	characteristic
161	C₂	C₂×C₅₃×#366	1	1	yes	characteristic
162	C₂	C₅₃×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1371	C₂	semidirect
65. #12032	C₅₃	direct (N × H)
81. #1371	C₂×C₅₃	semidirect
96. C₅₃	#12032	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×#2162	2
C₅₃×#1567	2
C₂×C₅₃×#335	2
C₅₃×#1371	2
#12032	53

Subgroups — C53×#12032 #27219544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12032 #27219544