Subgroups — #27299091

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 144 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	2	yes	maximal
2	#20272793	C₂	1	4	yes	maximal
3	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719908	C₂²	1	4	yes	—
5	C₄×#11719908	C₄	1	8	yes	complemented
6	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₄³×C₅₃	C₄	1	1	yes	characteristic
11	C₂×#11719908	—	2	8	no	—
12	C₂×#11719908	C₂×C₄	1	8	yes	—
13	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	C₈	1	2	yes	—
17	C₂×C₄²×C₅₃	C₈	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₅₃	—	2	4	no	—
21	C₄×C₈×C₅₃	—	2	4	no	—
22	C₄×C₈×C₅₃	C₈	1	4	yes	—
23	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
25	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	C₄²	1	2	yes	—
31	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
33	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
34	C₂×C₈×C₅₃	#41	1	4	yes	—
35	C₄²×C₅₃	—	2	4	no	—
36	C₄²×C₅₃	—	2	2	no	—
37	C₄²×C₅₃	—	2	2	no	—
38	C₄²×C₅₃	C₁₆	1	8	yes	complemented
39	C₄²×C₅₃	C₄²	1	2	yes	—
40	C₄²×C₅₃	C₂×C₈	1	2	yes	—
41	#11719908	—	2	8	no	—
42	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
43	C₂×C₄×C₅₃	—	2	4	no	—
44	C₂×C₄×C₅₃	—	2	4	no	—
45	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×C₁₆	1	8	yes	—
48	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
49	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
50	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#117	1	2	yes	—
52	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
54	C₈×C₅₃	—	2	4	no	—
55	C₈×C₅₃	#97	1	4	yes	—
56	C₈×C₅₃	#117	1	4	yes	—
57	#12590	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
59	C₂²×C₅₃	#431	1	2	yes	—
60	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
61	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
62	C₄×C₅₃	—	2	8	no	—
63	C₄×C₅₃	C₄×C₁₆	1	4	yes	—
64	C₄×C₅₃	#346	1	4	yes	—
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×C₄×C₁₆	—	53	2	no	—
69	C₄²×C₈	D₅₃	1	1	yes	characteristic
70	#1374	—	53	4	no	—
71	C₂×C₅₃	#1744	1	1	yes	characteristic
72	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
73	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
74	C₂×C₅₃	#2162	1	4	yes	—
75	C₂²×C₁₆	—	53	4	no	—
76	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
77	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
79	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
80	C₄³	Dic₅₃	1	1	yes	characteristic
81	C₄×C₁₆	—	53	8	no	—
82	C₅₃	#12590	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
84	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
85	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
86	C₂×C₄²	#88322	1	2	yes	—
87	C₂×C₄²	#88322	1	2	yes	—
88	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
89	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
90	C₂×C₁₆	—	53	8	no	—
91	C₂×C₁₆	—	106	8	no	—
92	C₄×C₈	—	2	4	no	—
93	C₄×C₈	—	2	4	no	—
94	C₄×C₈	#88322	1	4	yes	—
95	C₄×C₈	C₄×D₅₃	1	4	yes	—
96	C₂²×C₄	C₂×#88322	1	4	yes	—
97	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
98	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
99	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	C₂×#88322	1	4	yes	—
103	C₂×C₈	#11719917	1	4	yes	—
104	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
105	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
106	C₂×C₈	#11719945	1	4	yes	—
107	C₄²	—	2	4	no	—
108	C₄²	—	2	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	#11719908	1	8	yes	complemented
111	C₄²	C₂×#88322	1	2	yes	—
112	C₄²	C₄×Dic₅₃	1	2	yes	—
113	C₁₆	—	106	8	no	—
114	C₂³	C₂²×#88322	1	1	yes	characteristic
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	C₄×#88322	1	2	yes	—
118	C₂×C₄	C₄×#88322	1	2	yes	—
119	C₂×C₄	C₂×#11719908	1	8	yes	—
120	C₂×C₄	C₂×#11719908	1	4	yes	—
121	C₂×C₄	#13133157	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	4	yes	—
129	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
130	C₂²	#16810676	1	2	yes	—
131	C₂²	C₂²×#11719908	1	2	yes	—
132	C₂²	C₂×#13133157	1	2	yes	—
133	C₄	—	2	8	no	—
134	C₄	C₄×#11719908	1	4	yes	—
135	C₄	#16810399	1	4	yes	—
136	C₄	#16810469	1	1	yes	characteristic
137	C₄	C₂×#13133128	1	1	yes	characteristic
138	C₄	#16810676	1	2	yes	—
139	C₂	#20274461	1	1	yes	characteristic
140	C₂	C₂×C₄×#11719908	1	1	yes	characteristic
141	C₂	C₂×#16810399	1	1	yes	characteristic
142	C₂	#20276514	1	4	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#12590
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
5. C₄×#11719908	C₄	semidirect
38. C₄²×C₅₃	C₁₆	semidirect
82. C₅₃	#12590	semidirect
110. C₄²	#11719908	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
#20272793	2
C₄²×C₈×C₅₃	2
#12590	53