Subgroups — #27304279

Order 13568.

214 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810457	C₂	1	1	yes	maximal, characteristic
2	#20276311	C₂	1	1	yes	maximal, characteristic
3	#20272850	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₄	1	1	yes	characteristic
6	#16810578	C₄	1	1	yes	characteristic
7	#16810457	C₄	1	2	yes	complemented
8	#16810457	C₂²	1	2	yes	—
9	C₂×#13133123	C₄	1	2	yes	complemented
10	#16810398	C₂²	1	2	yes	—
11	C₁₆×Dic₅₃	C₄	1	2	yes	—
12	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×M₆₄	C₄	1	2	yes	complemented
14	C₅₃×M₆₄	C₂²	1	2	yes	—
15	C₂×C₃₂×C₅₃	C₄	1	2	yes	complemented
16	C₂×#11719917	C₈	1	1	yes	characteristic
17	#13133176	C₈	1	1	yes	characteristic
18	#13133157	C₈	1	2	yes	—
19	#13133157	C₈	1	2	yes	—
20	C₂×#11719908	C₂×C₄	1	2	yes	—
21	#13133127	C₈	1	2	yes	—
22	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
23	#13133123	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
31	C₃₂×C₅₃	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²⋊Dic₅₃	—	2	1	no	—
40	C₄⋊Dic₅₃	—	2	1	no	—
41	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
42	#11719917	—	2	1	no	—
43	#11719917	—	2	1	no	—
44	C₂×#88322	C₂×C₈	1	2	yes	—
45	C₁₆×C₅₃	C₄²	1	1	yes	characteristic
46	C₁₆×C₅₃	C₄²	1	1	yes	characteristic
47	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	#11719908	C₂×C₈	1	4	yes	—
50	C₂³×C₅₃	M₃₂	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
54	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
55	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	2	no	—
58	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
59	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
60	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
61	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
62	#88322	—	2	2	no	—
63	#13551	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	—	4	1	no	—
65	C₂²×C₅₃	#346	1	1	yes	characteristic
66	C₂²×C₅₃	#346	1	1	yes	characteristic
67	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
68	C₄×C₅₃	#346	1	1	yes	characteristic
69	C₄×C₅₃	#346	1	1	yes	characteristic
70	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
71	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
72	Dic₅₃	—	4	2	no	—
73	C₂×M₆₄	—	53	1	no	—
74	C₂×M₆₄	D₅₃	1	1	yes	characteristic
75	#2108	—	53	1	no	—
76	#1398	—	53	4	no	—
77	C₂×C₅₃	—	2	1	no	—
78	C₂×C₅₃	—	2	1	no	—
79	C₂×C₅₃	—	2	1	no	—
80	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
81	C₂×M₃₂	—	53	1	no	—
82	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
83	#405	—	53	1	no	—
84	M₆₄	—	53	2	no	—
85	M₆₄	—	53	2	no	—
86	M₆₄	Dic₅₃	1	2	yes	complemented
87	M₆₄	C₂×D₅₃	1	2	yes	—
88	C₂×C₃₂	—	53	2	no	—
89	C₂×C₃₂	Dic₅₃	1	2	yes	complemented
90	#346	—	53	2	no	—
91	C₄×C₁₆	—	53	2	no	—
92	C₅₃	#13551	1	1	yes	characteristic, complemented, Sylow
93	C₂×M₁₆	—	53	1	no	—
94	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
95	#117	—	53	1	no	—
96	M₃₂	—	53	2	no	—
97	M₃₂	—	53	2	no	—
98	C₂×C₁₆	—	53	2	no	—
99	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
100	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
101	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
102	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
103	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
104	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
105	#97	—	53	2	no	—
106	C₄×C₈	—	53	2	no	—
107	C₃₂	—	53	4	no	—
108	C₃₂	C₂×Dic₅₃	1	4	yes	—
109	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
110	M₁₆	—	106	1	no	—
111	M₁₆	—	106	1	no	—
112	C₂×C₈	—	53	2	no	—
113	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
114	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
115	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
116	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
117	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
118	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
119	C₄⋊C₄	—	106	1	no	—
120	C₂²⋊C₄	—	106	1	no	—
121	C₄²	—	53	2	no	—
122	C₁₆	—	53	4	no	—
123	C₁₆	C₄×Dic₅₃	1	1	yes	characteristic
124	C₁₆	C₄×Dic₅₃	1	1	yes	characteristic
125	C₁₆	C₂×C₄×D₅₃	1	1	yes	characteristic
126	C₁₆	C₂²×Dic₅₃	1	1	yes	characteristic
127	C₂³	#13133156	1	1	yes	characteristic
128	C₂×C₄	—	106	2	no	—
129	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
130	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
131	C₂×C₄	#13133156	1	1	yes	characteristic
132	C₂×C₄	#13133156	1	1	yes	characteristic
133	C₂×C₄	#13133156	1	1	yes	characteristic
134	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
135	C₈	—	106	2	no	—
136	C₈	C₈×Dic₅₃	1	1	yes	characteristic
137	C₈	C₈×Dic₅₃	1	1	yes	characteristic
138	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
139	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
140	C₂²	—	4	1	no	—
141	C₂²	#16810398	1	1	yes	characteristic
142	C₂²	#16810398	1	1	yes	characteristic
143	C₂²	C₂×#13133156	1	1	yes	characteristic
144	C₄	—	212	2	no	—
145	C₄	#16810398	1	1	yes	characteristic
146	C₄	#16810398	1	1	yes	characteristic
147	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
148	C₄	C₂×#13133156	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	—	2	1	no	—
152	C₂	C₂×#16810398	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₄×Dic₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×M₆₄	C₂
socle	C₂×C₅₃	C₂×#346
prime radical	C₂³×C₅₃	M₃₂
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×M₆₄	D₅₃
hypocenter	C₅₃	#13551
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
3. #20272850	C₂	semidirect
7. #16810457	C₄	semidirect
9. C₂×#13133123	C₄	semidirect
13. C₅₃×M₆₄	C₄	semidirect
15. C₂×C₃₂×C₅₃	C₄	semidirect
86. M₆₄	Dic₅₃	semidirect
89. C₂×C₃₂	Dic₅₃	semidirect
92. C₅₃	#13551	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810457	2
#20276311	2
#20272850	2
C₂×C₅₃×M₆₄	2
#13551	53