Subgroups — #27271778

Order 13568.

202 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133123	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719908	C₄	1	1	yes	characteristic
5	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
6	C₂×#13133123	—	2	1	no	—
7	C₂×#13133123	—	2	1	no	—
8	C₂×#13133123	C₄	1	2	yes	complemented
9	C₁₆×Dic₅₃	—	2	1	no	—
10	C₁₆×Dic₅₃	—	2	1	no	—
11	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₃₂×C₅₃	—	2	1	no	—
13	C₂×C₃₂×C₅₃	—	2	1	no	—
14	C₂×C₃₂×C₅₃	C₄	1	2	yes	complemented
15	C₂²×#88322	C₈	1	1	yes	characteristic
16	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
17	C₂×#11719908	—	2	1	no	—
18	C₂×#11719908	—	2	1	no	—
19	C₂×#11719908	C₈	1	2	yes	—
20	C₈×Dic₅₃	—	2	1	no	—
21	C₈×Dic₅₃	—	2	1	no	—
22	#13133123	—	2	2	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
30	C₃₂×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂²×Dic₅₃	C₁₆	1	2	yes	—
33	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
38	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
39	C₄×Dic₅₃	—	2	1	no	—
40	C₄×Dic₅₃	—	2	1	no	—
41	C₂×#88322	—	2	1	no	—
42	C₂×#88322	—	2	1	no	—
43	C₂×#88322	C₁₆	1	2	yes	—
44	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	#11719908	—	2	2	no	—
49	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
55	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
56	C₂×Dic₅₃	—	2	2	no	—
57	C₂×Dic₅₃	—	2	2	no	—
58	C₂×Dic₅₃	C₃₂	1	4	yes	complemented
59	C₈×C₅₃	#95	1	1	yes	characteristic
60	C₈×C₅₃	#97	1	1	yes	characteristic
61	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
62	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
63	#88322	—	2	2	no	—
64	#7465	—	53	1	no	maximal, Sylow
65	C₂²×C₅₃	C₄×C₁₆	1	1	yes	characteristic
66	C₂²×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
67	C₂²×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
68	C₂²×C₅₃	C₂×C₃₂	1	2	yes	—
69	C₂²×C₅₃	M₆₄	1	2	yes	—
70	C₄×C₅₃	#336	1	1	yes	characteristic
71	C₄×C₅₃	#346	1	1	yes	characteristic
72	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
73	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
74	Dic₅₃	—	2	4	no	—
75	C₂²×C₃₂	—	53	1	no	—
76	C₂²×C₃₂	D₅₃	1	1	yes	characteristic
77	C₂×C₄×C₁₆	—	53	1	no	—
78	C₂×C₅₃	#1374	1	1	yes	characteristic
79	C₂×C₅₃	C₄×C₃₂	1	1	yes	characteristic
80	C₂×C₅₃	#1397	1	1	yes	characteristic
81	C₂×C₅₃	C₂²⋊C₃₂	1	2	yes	—
82	C₂×C₅₃	C₄⋊C₃₂	1	2	yes	—
83	C₂²×C₁₆	—	53	1	no	—
84	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
85	C₂×C₄×C₈	—	53	1	no	—
86	C₂×C₃₂	—	2	1	no	—
87	C₂×C₃₂	—	2	1	no	—
88	C₂×C₃₂	—	53	2	no	—
89	C₂×C₃₂	—	106	1	no	—
90	C₂×C₃₂	—	106	1	no	—
91	C₂×C₃₂	Dic₅₃	1	2	yes	complemented
92	C₄×C₁₆	—	106	1	no	—
93	C₄×C₁₆	—	106	1	no	—
94	C₅₃	#7465	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₈	—	53	1	no	—
96	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
97	C₂×C₄²	—	53	1	no	—
98	C₂×C₁₆	—	53	2	no	—
99	C₂×C₁₆	—	106	1	no	—
100	C₂×C₁₆	—	106	1	no	—
101	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
102	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
103	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
104	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
105	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
106	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
107	C₄×C₈	—	106	1	no	—
108	C₄×C₈	—	106	1	no	—
109	C₃₂	—	2	2	no	—
110	C₃₂	—	106	2	no	—
111	C₂²×C₄	—	53	2	no	—
112	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
113	C₂×C₈	—	53	2	no	—
114	C₂×C₈	—	106	1	no	—
115	C₂×C₈	—	106	1	no	—
116	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
117	C₂×C₈	#11719912	1	1	yes	characteristic
118	C₂×C₈	#11719912	1	1	yes	characteristic
119	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
120	C₂×C₈	#11719919	1	1	yes	characteristic
121	C₂×C₈	#11719921	1	1	yes	characteristic
122	C₄²	—	106	1	no	—
123	C₄²	—	106	1	no	—
124	C₁₆	—	106	2	no	—
125	C₁₆	#11719919	1	1	yes	characteristic
126	C₁₆	C₄⋊Dic₅₃	1	1	yes	characteristic
127	C₁₆	#11719921	1	1	yes	characteristic
128	C₁₆	C₂²⋊Dic₅₃	1	1	yes	characteristic
129	C₂³	C₁₆×D₅₃	1	1	yes	characteristic
130	C₂×C₄	—	53	4	no	—
131	C₂×C₄	—	106	2	no	—
132	C₂×C₄	—	106	2	no	—
133	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
134	C₂×C₄	#13133129	1	1	yes	characteristic
135	C₂×C₄	#13133146	1	1	yes	characteristic
136	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
137	C₂×C₄	#13133156	1	1	yes	characteristic
138	C₂×C₄	#13133156	1	1	yes	characteristic
139	C₈	—	106	2	no	—
140	C₈	#13133124	1	1	yes	characteristic
141	C₈	#13133127	1	1	yes	characteristic
142	C₈	#13133129	1	1	yes	characteristic
143	C₈	#13133146	1	1	yes	characteristic
144	C₂²	C₁₆×Dic₅₃	1	1	yes	characteristic
145	C₂²	#16810402	1	1	yes	characteristic
146	C₂²	#16810441	1	1	yes	characteristic
147	C₂²	C₃₂×D₅₃	1	2	yes	—
148	C₂²	#16810456	1	2	yes	—
149	C₄	—	106	4	no	—
150	C₄	#16810377	1	1	yes	characteristic
151	C₄	#16810398	1	1	yes	characteristic
152	C₄	#16810402	1	1	yes	characteristic
153	C₄	#16810441	1	1	yes	characteristic
154	C₂	#20272792	1	1	yes	characteristic
155	C₂	C₃₂×Dic₅₃	1	1	yes	characteristic
156	C₂	#20272848	1	1	yes	characteristic
157	C₂	#20272852	1	2	yes	—
158	C₂	#20272910	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₁₆	C₂×D₅₃
commutator	C₂×C₅₃	C₄×C₃₂
Fitting	C₂²×C₃₂×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₁₆
cosocle	C₂²×C₁₆×C₅₃	C₂²
hypercenter	C₂²×C₃₂	D₅₃
hypocenter	C₅₃	#7465
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

Complements

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

N	H ≅ G/N	type
8. C₂×#13133123	C₄	semidirect
14. C₂×C₃₂×C₅₃	C₄	semidirect
58. C₂×Dic₅₃	C₃₂	semidirect
91. C₂×C₃₂	Dic₅₃	semidirect
94. C₅₃	#7465	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133123	2
C₂×C₁₆×Dic₅₃	2
C₂²×C₃₂×C₅₃	2
#7465	53