Subgroups — #27299090

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272792	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×C₄×#88322	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
9	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
10	C₄×#11719908	C₄	1	4	yes	complemented
11	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₅₃	C₄	1	4	yes	complemented
14	C₂²×#88322	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
16	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
17	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
18	C₂×#11719908	—	2	4	no	—
19	C₂×#11719908	C₂×C₄	1	4	yes	—
20	C₄×#88322	C₈	1	4	yes	—
21	C₈×Dic₅₃	—	2	4	no	—
22	C₈×Dic₅₃	—	2	4	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	—	2	4	no	—
27	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
34	C₂×C₈×C₅₃	C₄²	1	2	yes	—
35	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
36	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	#41	1	4	yes	—
39	C₄²×C₅₃	C₄²	1	2	yes	—
40	C₄²×C₅₃	C₂×C₈	1	2	yes	—
41	C₄×Dic₅₃	—	2	4	no	—
42	C₄×Dic₅₃	—	2	2	no	—
43	C₄×Dic₅₃	—	2	2	no	—
44	C₄×Dic₅₃	C₁₆	1	8	yes	complemented
45	C₂×#88322	—	2	2	no	—
46	C₂×#88322	—	2	2	no	—
47	C₂×#88322	C₂×C₈	1	4	yes	—
48	C₁₆×C₅₃	—	2	4	no	—
49	#11719908	—	2	4	no	—
50	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
54	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
55	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#117	1	2	yes	—
57	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
59	C₂×Dic₅₃	—	2	4	no	—
60	C₂×Dic₅₃	—	2	4	no	—
61	C₂×Dic₅₃	C₂×C₁₆	1	8	yes	—
62	C₈×C₅₃	#97	1	4	yes	—
63	C₈×C₅₃	#117	1	2	yes	—
64	C₈×C₅₃	#117	1	2	yes	—
65	#88322	—	2	4	no	—
66	#12590	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
68	C₂²×C₅₃	#431	1	2	yes	—
69	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
70	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
71	C₄×C₅₃	C₄×C₁₆	1	4	yes	—
72	C₄×C₅₃	#346	1	4	yes	—
73	C₄×C₅₃	#376	1	1	yes	characteristic
74	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
75	C₄×C₅₃	#431	1	2	yes	—
76	Dic₅₃	—	2	8	no	—
77	C₂×C₄×C₁₆	—	53	1	no	—
78	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
79	C₄²×C₈	—	53	1	no	—
80	#1374	—	53	4	no	—
81	C₂×C₅₃	#1744	1	1	yes	characteristic
82	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
83	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
84	C₂×C₅₃	#2162	1	4	yes	—
85	C₂²×C₁₆	—	53	2	no	—
86	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
87	C₂×C₄×C₈	—	53	1	no	—
88	C₂×C₄×C₈	—	53	2	no	—
89	C₂×C₄×C₈	—	53	2	no	—
90	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
91	C₄³	—	53	1	no	—
92	C₄×C₁₆	—	53	4	no	—
93	C₄×C₁₆	Dic₅₃	1	4	yes	complemented
94	C₅₃	#12590	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₈	—	53	2	no	—
96	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
98	C₂×C₄²	—	53	2	no	—
99	C₂×C₄²	—	53	2	no	—
100	C₂×C₄²	—	53	2	no	—
101	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	—	53	4	no	—
104	C₂×C₁₆	—	106	4	no	—
105	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
106	C₄×C₈	—	53	4	no	—
107	C₄×C₈	—	106	4	no	—
108	C₄×C₈	—	106	4	no	—
109	C₄×C₈	C₄×D₅₃	1	2	yes	—
110	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
111	C₂²×C₄	—	53	4	no	—
112	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
113	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
114	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
115	C₂×C₈	—	53	4	no	—
116	C₂×C₈	—	106	2	no	—
117	C₂×C₈	—	106	2	no	—
118	C₂×C₈	#11719912	1	4	yes	—
119	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
120	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
121	C₂×C₈	#11719945	1	4	yes	—
122	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
123	C₄²	—	53	8	no	—
124	C₄²	—	106	4	no	—
125	C₄²	—	106	2	no	—
126	C₄²	—	106	2	no	—
127	C₄²	C₈×D₅₃	1	2	yes	—
128	C₄²	C₄×Dic₅₃	1	2	yes	—
129	C₁₆	—	2	4	no	—
130	C₁₆	—	106	4	no	—
131	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
132	C₂×C₄	—	53	8	no	—
133	C₂×C₄	—	106	4	no	—
134	C₂×C₄	—	106	4	no	—
135	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
136	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
137	C₂×C₄	C₁₆×D₅₃	1	4	yes	—
138	C₂×C₄	#13133156	1	4	yes	—
139	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
140	C₂×C₄	#13133174	1	2	yes	—
141	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
142	C₂×C₄	C₂×#11719912	1	2	yes	—
143	C₈	—	106	4	no	—
144	C₈	#13133127	1	4	yes	—
145	C₈	#13133174	1	2	yes	—
146	C₈	#13133176	1	2	yes	—
147	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
148	C₂²	#16810674	1	2	yes	—
149	C₂²	C₂×C₁₆×D₅₃	1	2	yes	—
150	C₂²	C₂×#13133156	1	2	yes	—
151	C₄	—	106	8	no	—
152	C₄	C₁₆×Dic₅₃	1	4	yes	—
153	C₄	#16810398	1	4	yes	—
154	C₄	#16810469	1	1	yes	characteristic
155	C₄	C₂×#13133127	1	1	yes	characteristic
156	C₄	#16810674	1	2	yes	—
157	C₂	#20274460	1	1	yes	characteristic
158	C₂	C₂×C₁₆×Dic₅₃	1	1	yes	characteristic
159	C₂	C₂×#16810398	1	1	yes	characteristic
160	C₂	#20276512	1	4	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#12590
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. C₄×#11719908	C₄	semidirect
13. C₄×C₁₆×C₅₃	C₄	semidirect
44. C₄×Dic₅₃	C₁₆	semidirect
93. C₄×C₁₆	Dic₅₃	semidirect
94. C₅₃	#12590	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719908	2
C₄×C₈×Dic₅₃	2
#20272792	2
C₂×C₄×C₁₆×C₅₃	2
#12590	53