Subgroups — #27273469

Order 13568.

582 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133127	C₂	1	2	yes	maximal
2	C₄×#13133127	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810345	C₂	1	2	yes	maximal
4	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133127	C₄	1	8	yes	—
7	C₂×#13133127	C₄	1	4	yes	—
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
11	#16810345	C₄	1	8	yes	complemented
12	C₈×#88322	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#322	C₄	1	4	yes	complemented
16	C₂²×#88322	C₂×C₄	1	2	yes	—
17	C₂²×#88322	C₂×C₄	1	4	yes	—
18	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
20	#13133127	—	2	8	no	—
21	#13133127	—	2	16	no	—
22	C₄×#88322	C₂×C₄	1	4	yes	—
23	C₄×#88322	C₂×C₄	1	4	yes	—
24	C₄×#88322	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₅₃	C₄²	1	4	yes	—
35	C₂×C₈×C₅₃	—	2	8	no	—
36	C₂×C₈×C₅₃	C₄²	1	8	yes	—
37	C₂×C₈×C₅₃	C₄²	1	4	yes	—
38	C₂×C₈×C₅₃	M₁₆	1	8	yes	—
39	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
41	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₄×Dic₅₃	—	2	8	no	—
43	C₄×Dic₅₃	—	2	4	no	—
44	C₂×#88322	—	2	4	no	—
45	C₂×#88322	C₄²	1	4	yes	—
46	C₂×#88322	C₄²	1	8	yes	—
47	C₂×#88322	M₁₆	1	16	yes	—
48	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×M₁₆	1	8	yes	—
54	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
55	C₂×Dic₅₃	—	2	4	no	—
56	C₂×Dic₅₃	—	2	8	no	—
57	C₈×C₅₃	—	2	8	no	—
58	C₈×C₅₃	#97	1	8	yes	complemented
59	#88322	—	2	4	no	—
60	#88322	#97	1	16	yes	complemented
61	#7846	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	C₄³	1	1	yes	characteristic
63	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
64	C₂²×C₅₃	C₄×M₁₆	1	4	yes	—
65	C₄×C₅₃	C₂×#97	1	4	yes	—
66	C₄×C₅₃	C₂×#97	1	2	yes	—
67	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
68	C₄×C₅₃	C₄×M₁₆	1	4	yes	—
69	Dic₅₃	—	4	4	no	—
70	C₄×#97	—	53	2	no	—
71	C₄×#97	—	53	1	no	—
72	C₂×#322	—	53	2	no	—
73	C₂×#322	D₅₃	1	1	yes	characteristic
74	C₂×C₈²	—	53	1	no	—
75	C₂×C₅₃	#1455	1	4	yes	—
76	C₂×C₅₃	C₄×#97	1	2	yes	—
77	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
78	C₂×#97	—	53	8	no	—
79	C₂×#97	—	53	4	no	—
80	C₂×C₄×C₈	—	53	2	no	—
81	C₂×C₄×C₈	—	53	1	no	—
82	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
84	C₄³	—	53	1	no	—
85	#322	—	53	8	no	—
86	#322	Dic₅₃	1	4	yes	complemented
87	C₈²	—	53	4	no	—
88	C₅₃	#7846	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	—	53	4	no	—
90	C₂²×C₈	—	53	2	no	—
91	C₂²×C₈	C₄×D₅₃	1	4	yes	—
92	C₂²×C₈	C₄×D₅₃	1	2	yes	—
93	C₂×C₄²	—	53	4	no	—
94	C₂×C₄²	—	53	2	no	—
95	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
96	#97	—	106	16	no	—
97	#97	—	106	8	no	—
98	C₄×C₈	—	53	4	no	—
99	C₄×C₈	—	53	4	no	—
100	C₄×C₈	—	53	4	no	—
101	C₄×C₈	C₄×D₅₃	1	2	yes	—
102	C₄×C₈	C₄×D₅₃	1	4	yes	—
103	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
104	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
105	C₂²×C₄	—	53	4	no	—
106	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
108	C₂×C₈	—	2	8	no	—
109	C₂×C₈	—	53	8	no	—
110	C₂×C₈	—	53	16	no	—
111	C₂×C₈	—	53	4	no	—
112	C₂×C₈	—	106	4	no	—
113	C₂×C₈	#11719912	1	8	yes	—
114	C₂×C₈	C₄×Dic₅₃	1	8	yes	—
115	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
116	C₄²	—	106	8	no	—
117	C₄²	—	106	4	no	—
118	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
119	C₄²	C₂×C₄×D₅₃	1	2	yes	—
120	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
121	C₂³	C₄²×D₅₃	1	1	yes	characteristic
122	C₂×C₄	—	106	4	no	—
123	C₂×C₄	—	106	8	no	—
124	C₂×C₄	C₄²×D₅₃	1	2	yes	—
125	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
128	C₂×C₄	M₁₆×D₅₃	1	8	yes	—
129	C₂×C₄	C₂×#11719912	1	4	yes	—
130	C₈	—	2	8	no	—
131	C₈	—	53	16	no	—
132	C₈	—	106	4	no	—
133	C₈	#13133127	1	8	yes	complemented
134	C₂²	C₄²×Dic₅₃	1	1	yes	characteristic
135	C₂²	C₄×#11719912	1	2	yes	—
136	C₂²	#16810575	1	4	yes	—
137	C₄	—	212	4	no	—
138	C₄	#97×D₅₃	1	4	yes	—
139	C₄	C₂×#13133127	1	2	yes	—
140	C₄	C₄×#11719912	1	2	yes	—
141	C₄	M₁₆×Dic₅₃	1	4	yes	—
142	C₂	#20273019	1	4	yes	—
143	C₂	#97×Dic₅₃	1	2	yes	—
144	C₂	C₄×#13133127	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₄³
Fitting	C₂×C₅₃×#322	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#7846
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
11. #16810345	C₄	semidirect
12. C₈×#88322	C₄	semidirect
15. C₅₃×#322	C₄	semidirect
58. C₈×C₅₃	#97	semidirect
60. #88322	#97	semidirect
86. #322	Dic₅₃	semidirect
88. C₅₃	#7846	semidirect
133. C₈	#13133127	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133127	2
C₂×#16810345	2
C₂×C₈×#88322	2
C₂×C₅₃×#322	2
#7846	53