Subgroups — C₂×C₅₃×#1371 #27219518

Order 13568.

262 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#335	C₂	1	2	yes	maximal
3	C₅₃×#1371	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
9	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#335	—	2	2	no	—
11	C₅₃×#335	C₂²	1	4	yes	—
12	C₂²×C₈×C₅₃	—	4	2	no	—
13	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
15	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	—	2	2	no	—
17	C₂×C₁₆×C₅₃	C₈	1	4	yes	complemented
18	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
20	C₄⋊C₈×C₅₃	—	2	4	no	—
21	C₄⋊C₈×C₅₃	—	4	2	no	—
22	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	—	4	4	no	—
31	C₂×C₈×C₅₃	—	8	4	no	—
32	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
36	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₅₃	D₈	1	1	yes	characteristic
40	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
41	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₅₃	—	2	2	no	—
43	C₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
44	C₂³×C₅₃	#107	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
47	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#108	1	2	yes	—
51	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
52	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
56	C₈×C₅₃	—	2	1	no	—
57	C₈×C₅₃	—	2	1	no	—
58	C₈×C₅₃	—	8	4	no	—
59	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
60	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
61	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
62	C₂×#1371	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂²×C₅₃	#335	1	2	yes	—
64	C₂²×C₅₃	#366	1	2	yes	—
65	C₂²×C₅₃	#368	1	2	yes	—
66	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	#335	1	2	yes	—
70	C₄×C₅₃	#366	1	2	yes	—
71	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
72	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
73	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
74	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
75	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic
76	C₂×#335	C₂×C₅₃	1	2	yes	—
77	#1371	C₂×C₅₃	1	4	yes	direct product factor
78	C₂×C₅₃	#1371	1	4	yes	direct product factor
79	C₂×C₅₃	C₂×#335	1	1	yes	characteristic
80	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
81	C₂×C₅₃	C₂×#368	1	1	yes	characteristic
82	C₂²×C₁₆	C₄×C₅₃	1	1	yes	characteristic
83	C₂²×C₁₆	C₄×C₅₃	1	1	yes	characteristic
84	C₂×C₄⋊C₈	—	2	2	no	—
85	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
86	C₄×C₁₆	C₄×C₅₃	1	2	yes	—
87	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
88	#335	—	2	2	no	—
89	#335	C₂²×C₅₃	1	4	yes	—
90	C₅₃	C₂×#1371	1	1	yes	characteristic, direct product factor, Sylow
91	C₂²×C₈	—	4	2	no	—
92	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
93	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
94	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	C₈×C₅₃	1	4	yes	complemented
97	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
98	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
99	C₄⋊C₈	—	2	4	no	—
100	C₄⋊C₈	—	4	2	no	—
101	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
102	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
103	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
104	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
105	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
106	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
107	C₂²×C₄	Q₁₆×C₅₃	1	1	yes	characteristic
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	4	4	no	—
110	C₂×C₈	—	8	4	no	—
111	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
112	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
113	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
114	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
115	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
116	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
117	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
118	C₄²	D₈×C₅₃	1	1	yes	characteristic
119	C₄²	Q₁₆×C₅₃	1	1	yes	characteristic
120	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
121	C₁₆	—	2	2	no	—
122	C₁₆	C₂×C₈×C₅₃	1	4	yes	complemented
123	C₂³	#107×C₅₃	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
126	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
127	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
128	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
129	C₂×C₄	#108×C₅₃	1	2	yes	—
130	C₂×C₄	D₁₆×C₅₃	1	2	yes	—
131	C₂×C₄	Q₃₂×C₅₃	1	2	yes	—
132	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
133	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
134	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	8	4	no	—
138	C₈	C₄⋊C₈×C₅₃	1	2	yes	—
139	C₈	C₂²×C₈×C₅₃	1	1	yes	characteristic
140	C₈	C₂×M₁₆×C₅₃	1	1	yes	characteristic
141	C₂²	C₅₃×#335	1	2	yes	—
142	C₂²	C₅₃×#366	1	2	yes	—
143	C₂²	C₅₃×#368	1	2	yes	—
144	C₂²	C₂×#107×C₅₃	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	C₅₃×#335	1	2	yes	—
148	C₄	C₅₃×#366	1	2	yes	—
149	C₄	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
150	C₄	C₂×#108×C₅₃	1	1	yes	characteristic
151	C₄	C₂×D₁₆×C₅₃	1	1	yes	characteristic
152	C₄	C₂×Q₃₂×C₅₃	1	1	yes	characteristic
153	C₂	C₅₃×#1371	1	4	yes	direct product factor
154	C₂	C₂×C₅₃×#335	1	1	yes	characteristic
155	C₂	C₂×C₅₃×#366	1	1	yes	characteristic
156	C₂	C₂×C₅₃×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	D₈
commutator	C₈	C₂²×C₈×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
3. C₅₃×#1371	C₂	direct (N × H)
17. C₂×C₁₆×C₅₃	C₈	semidirect
43. C₁₆×C₅₃	C₂×C₈	semidirect
62. C₂×#1371	C₅₃	direct (N × H)
77. #1371	C₂×C₅₃	direct (N × H)
78. C₂×C₅₃	#1371	direct (N × H)
90. C₅₃	C₂×#1371	direct (N × H)
96. C₂×C₁₆	C₈×C₅₃	semidirect
122. C₁₆	C₂×C₈×C₅₃	semidirect
153. C₂	C₅₃×#1371	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₆×C₅₃	2
C₂×C₅₃×#335	2
C₅₃×#1371	2
C₂×#1371	53

Subgroups — C2×C53×#1371 #27219518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1371 #27219518