Subgroups — #27299097

Order 13568.

358 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
3	#20272793	C₂	1	4	yes	maximal
4	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂²×#11719908	C₂²	1	2	yes	—
7	#16810399	C₄	1	4	yes	complemented
8	C₄×#11719908	C₄	1	4	yes	complemented
9	C₂×#97×C₅₃	C₄	1	2	yes	—
10	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#376	C₄	1	1	yes	characteristic
14	C₂×#11719908	—	2	4	no	—
15	C₂×#11719908	—	2	4	no	—
16	C₂×#11719908	C₂×C₄	1	4	yes	—
17	C₂×#11719908	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	#97×C₅₃	—	2	4	no	—
24	#97×C₅₃	C₈	1	4	yes	—
25	C₄×C₈×C₅₃	—	2	4	no	—
26	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	#95×C₅₃	C₈	1	4	yes	—
29	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
30	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	C₄²	1	2	yes	—
35	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
37	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	#41	1	4	yes	—
40	C₄²×C₅₃	—	2	2	no	—
41	C₄²×C₅₃	—	2	2	no	—
42	C₄²×C₅₃	C₄²	1	2	yes	—
43	C₄²×C₅₃	C₂×C₈	1	2	yes	—
44	#11719908	—	2	4	no	—
45	#11719908	—	2	4	no	—
46	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	4	no	—
48	C₂×C₄×C₅₃	—	2	4	no	—
49	C₂×C₄×C₅₃	—	2	8	no	—
50	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
51	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#117	1	2	yes	—
53	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
55	C₈×C₅₃	—	2	4	no	—
56	C₈×C₅₃	#97	1	4	yes	—
57	C₈×C₅₃	#117	1	2	yes	—
58	C₈×C₅₃	#117	1	2	yes	—
59	#12592	—	53	1	no	maximal, Sylow
60	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
61	C₂²×C₅₃	#431	1	2	yes	—
62	C₂²×C₅₃	#504	1	4	yes	—
63	C₄×C₅₃	—	2	4	no	—
64	C₄×C₅₃	—	4	4	no	—
65	C₄×C₅₃	#376	1	1	yes	characteristic
66	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
67	C₄×C₅₃	#431	1	2	yes	—
68	C₂×#346	—	53	1	no	—
69	C₂×C₄×C₁₆	—	53	1	no	—
70	#1726	D₅₃	1	1	yes	characteristic
71	#1374	—	53	4	no	—
72	C₂×C₅₃	#1744	1	1	yes	characteristic
73	C₂×C₅₃	#2108	1	2	yes	—
74	C₂×C₅₃	#2164	1	4	yes	—
75	C₂²×C₁₆	—	53	2	no	—
76	C₂²×C₁₆	—	53	2	no	—
77	C₂×#97	Dic₅₃	1	2	yes	—
78	C₂×#97	Dic₅₃	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
80	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
81	#376	Dic₅₃	1	1	yes	characteristic
82	#346	—	53	4	no	—
83	C₄×C₁₆	—	53	4	no	—
84	C₅₃	#12592	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
86	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
87	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
88	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
89	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
90	C₂×C₁₆	—	53	4	no	—
91	C₂×C₁₆	—	53	4	no	—
92	C₂×C₁₆	—	106	4	no	—
93	C₂×C₁₆	—	106	4	no	—
94	#97	—	2	4	no	—
95	#97	#88322	1	4	yes	—
96	C₄×C₈	—	2	4	no	—
97	C₄×C₈	C₄×D₅₃	1	2	yes	—
98	C₄×C₈	C₄×D₅₃	1	2	yes	—
99	#95	#88322	1	4	yes	—
100	C₂²×C₄	C₂×#88322	1	4	yes	—
101	C₂²×C₄	C₂×#88322	1	2	yes	—
102	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	C₂×#88322	1	4	yes	—
106	C₂×C₈	#11719917	1	4	yes	—
107	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
108	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
109	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
110	C₂×C₈	#11719945	1	4	yes	—
111	C₄²	—	2	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	C₂×#88322	1	2	yes	—
114	C₄²	C₄×Dic₅₃	1	2	yes	—
115	C₁₆	—	106	4	no	—
116	C₁₆	—	106	4	no	—
117	C₂³	C₂²×#88322	1	1	yes	characteristic
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	8	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	C₄×#88322	1	4	yes	—
122	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
123	C₂×C₄	#13133176	1	2	yes	—
124	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
125	C₂×C₄	C₂×#11719917	1	2	yes	—
126	C₈	—	2	4	no	—
127	C₈	#13133128	1	4	yes	—
128	C₈	#13133174	1	2	yes	—
129	C₈	#13133174	1	2	yes	—
130	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
131	C₂²	#16810676	1	2	yes	—
132	C₂²	#16811040	1	4	yes	—
133	C₄	—	2	4	no	—
134	C₄	—	4	4	no	—
135	C₄	#16810469	1	1	yes	characteristic
136	C₄	C₂×#13133128	1	1	yes	characteristic
137	C₄	#16810676	1	2	yes	—
138	C₂	#20274461	1	1	yes	characteristic
139	C₂	#20276312	1	2	yes	—
140	C₂	#20276522	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#1726	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1726	D₅₃
hypocenter	C₅₃	#12592
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
7. #16810399	C₄	semidirect
8. C₄×#11719908	C₄	semidirect
84. C₅₃	#12592	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810399	2
C₂×C₄×#11719908	2
#20272793	2
C₅₃×#1726	2
#12592	53