Subgroups — C₅₃×#12668 #27220180

Order 13568.

308 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×#2155	C₂	1	2	yes	maximal
2	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1772	C₂	1	2	yes	maximal
4	C₅₃×#1385	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1384	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#108×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#427	C₂²	1	2	yes	—
10	C₅₃×#405	—	2	2	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#365	C₄	1	8	yes	complemented
13	C₅₃×#346	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₅₃	C₄	1	2	yes	complemented
15	C₅₃×#339	—	2	2	no	—
16	C₂×M₁₆×C₅₃	—	2	2	no	—
17	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	#117×C₅₃	—	2	2	no	—
20	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
21	M₃₂×C₅₃	—	2	2	no	—
22	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
23	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
25	#108×C₅₃	—	2	2	no	—
26	#108×C₅₃	C₂×C₄	1	4	yes	—
27	#107×C₅₃	—	2	2	no	—
28	#106×C₅₃	C₂×C₄	1	4	yes	—
29	#97×C₅₃	—	4	2	no	—
30	C₄×C₈×C₅₃	—	4	2	no	—
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	—	2	1	no	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	M₁₆×C₅₃	—	2	4	no	—
38	M₁₆×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	—	4	2	no	—
40	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
43	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	#41	1	2	yes	—
48	C₄⋊C₄×C₅₃	—	2	4	no	—
49	C₂²⋊C₄×C₅₃	—	4	2	no	—
50	C₄²×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	—	4	2	no	—
52	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
54	C₁₆×C₅₃	—	2	2	no	—
55	C₁₆×C₅₃	C₄²	1	4	yes	—
56	C₂³×C₅₃	#135	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	4	2	no	—
60	C₂×C₄×C₅₃	—	4	4	no	—
61	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
64	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
65	C₈×C₅₃	—	4	4	no	—
66	C₈×C₅₃	#106	1	2	yes	—
67	C₈×C₅₃	#106	1	1	yes	characteristic
68	C₈×C₅₃	#106	1	1	yes	characteristic
69	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
70	C₈×C₅₃	#117	1	1	yes	characteristic
71	C₈×C₅₃	C₄×Q₈	1	2	yes	—
72	#12668	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂²×C₅₃	—	4	1	no	—
74	C₂²×C₅₃	#427	1	1	yes	characteristic
75	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	4	1	no	—
78	C₄×C₅₃	—	8	4	no	—
79	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
81	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
82	#2155	C₂×C₅₃	1	2	yes	—
83	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic
84	#1772	C₂×C₅₃	1	2	yes	—
85	#1385	C₂×C₅₃	1	1	yes	characteristic
86	#1384	C₂×C₅₃	1	1	yes	characteristic
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	—	2	2	no	—
89	C₂×C₅₃	C₄×#106	1	1	yes	characteristic
90	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
91	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
92	C₂×#108	C₂²×C₅₃	1	2	yes	—
93	#427	C₂²×C₅₃	1	2	yes	—
94	#405	—	2	2	no	—
95	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
96	#365	C₄×C₅₃	1	8	yes	complemented
97	#346	C₄×C₅₃	1	2	yes	complemented
98	C₄×C₁₆	C₄×C₅₃	1	2	yes	complemented
99	#339	—	2	2	no	—
100	C₅₃	#12668	1	1	yes	characteristic, direct product factor, Sylow
101	C₂×M₁₆	—	2	2	no	—
102	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
103	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
104	#117	—	2	2	no	—
105	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	C₂×C₄×C₅₃	1	4	yes	—
108	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
109	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
110	#108	—	2	2	no	—
111	#108	C₂×C₄×C₅₃	1	4	yes	—
112	#107	—	2	2	no	—
113	#106	C₂×C₄×C₅₃	1	4	yes	—
114	#97	—	4	2	no	—
115	C₄×C₈	—	4	2	no	—
116	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
117	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
118	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
119	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
122	M₁₆	—	2	4	no	—
123	M₁₆	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
126	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
127	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
128	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
129	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
130	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
131	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
132	C₂×C₈	#41×C₅₃	1	2	yes	—
133	C₄⋊C₄	—	2	4	no	—
134	C₂²⋊C₄	—	4	2	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	4	2	no	—
137	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
138	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
139	C₁₆	—	2	2	no	—
140	C₁₆	C₄²×C₅₃	1	4	yes	—
141	C₂³	#135×C₅₃	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
147	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
148	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
149	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
150	C₈	—	4	4	no	—
151	C₈	#106×C₅₃	1	2	yes	—
152	C₈	#106×C₅₃	1	1	yes	characteristic
153	C₈	#106×C₅₃	1	1	yes	characteristic
154	C₈	C₂×C₄²×C₅₃	1	1	yes	characteristic
155	C₈	#117×C₅₃	1	1	yes	characteristic
156	C₈	C₄×Q₈×C₅₃	1	2	yes	—
157	C₂²	—	4	1	no	—
158	C₂²	C₅₃×#427	1	1	yes	characteristic
159	C₂²	C₄×SD₁₆×C₅₃	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	1	no	—
162	C₄	—	8	4	no	—
163	C₄	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
164	C₄	C₂×#106×C₅₃	1	1	yes	characteristic
165	C₄	C₄×SD₁₆×C₅₃	1	2	yes	—
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	C₄×#106×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₅₃	#106
commutator	C₈	C₂×C₄²×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	C₄×#106
prime radical	C₂³×C₅₃	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₅₃×#365	C₄	semidirect
13. C₅₃×#346	C₄	semidirect
14. C₄×C₁₆×C₅₃	C₄	semidirect
72. #12668	C₅₃	direct (N × H)
96. #365	C₄×C₅₃	semidirect
97. #346	C₄×C₅₃	semidirect
98. C₄×C₁₆	C₄×C₅₃	semidirect
100. C₅₃	#12668	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×#2155	2
C₄×M₃₂×C₅₃	2
C₅₃×#1772	2
C₅₃×#1385	2
C₅₃×#1384	2
#12668	53

Subgroups — C53×#12668 #27220180

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12668 #27220180