Subgroups — #27299103

Order 13568.

354 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	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133127	C₂	1	1	yes	maximal, characteristic
3	#20272794	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂×#13133127	C₄	1	2	yes	—
7	C₂×#13133127	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₄	1	1	yes	characteristic
9	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
10	#16810399	C₄	1	2	yes	complemented
11	C₄×#11719908	C₄	1	2	yes	complemented
12	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#346	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₅₃	C₄	1	2	yes	complemented
16	C₂²×#88322	C₂×C₄	1	2	yes	—
17	C₂×C₄×Dic₅₃	—	2	2	no	—
18	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
19	#13133157	—	2	4	no	—
20	C₂×#11719908	C₂×C₄	1	4	yes	—
21	#13133127	—	2	4	no	—
22	#13133127	C₈	1	8	yes	—
23	C₄×#88322	—	2	2	no	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₅₃	—	2	4	no	—
28	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	#41	1	4	yes	—
42	C₄²×C₅₃	C₄²	1	1	yes	characteristic
43	C₄²×C₅₃	C₄²	1	1	yes	characteristic
44	C₄²×C₅₃	M₁₆	1	2	yes	—
45	C₄×Dic₅₃	—	2	2	no	—
46	C₄×Dic₅₃	—	4	4	no	—
47	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
48	C₂×#88322	—	2	2	no	—
49	C₂×#88322	—	2	2	no	—
50	C₂×#88322	C₂×C₈	1	4	yes	—
51	C₁₆×C₅₃	—	2	4	no	—
52	#11719908	—	2	4	no	—
53	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	2	4	no	—
55	C₂×C₄×C₅₃	#97	1	2	yes	—
56	C₂×C₄×C₅₃	#97	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#117	1	2	yes	—
59	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
61	C₂×Dic₅₃	—	2	4	no	—
62	C₂×Dic₅₃	—	4	4	no	—
63	C₈×C₅₃	C₄×C₈	1	2	yes	—
64	C₈×C₅₃	C₄×C₈	1	2	yes	—
65	C₈×C₅₃	#117	1	2	yes	—
66	C₈×C₅₃	#117	1	2	yes	—
67	#88322	—	2	4	no	—
68	#12594	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	—	2	2	no	—
70	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
71	C₂²×C₅₃	#431	1	2	yes	—
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	#347	1	4	yes	—
74	C₄×C₅₃	#376	1	1	yes	characteristic
75	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
76	C₄×C₅₃	#431	1	2	yes	—
77	Dic₅₃	—	4	4	no	—
78	C₄×M₃₂	—	53	1	no	—
79	C₄×M₃₂	D₅₃	1	1	yes	characteristic
80	C₄×#97	—	53	1	no	—
81	#1375	—	53	4	no	—
82	C₂×C₅₃	—	2	2	no	—
83	C₂×C₅₃	#1744	1	1	yes	characteristic
84	C₂×C₅₃	C₂×#347	1	2	yes	—
85	C₂×M₃₂	—	53	2	no	—
86	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
87	C₂×#97	—	53	2	no	—
88	C₂×#97	—	53	2	no	—
89	C₂×C₄×C₈	—	53	1	no	—
90	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
91	C₄³	—	53	1	no	—
92	#346	—	53	2	no	—
93	#346	Dic₅₃	1	2	yes	complemented
94	C₄×C₁₆	—	53	2	no	—
95	C₄×C₁₆	Dic₅₃	1	2	yes	complemented
96	C₅₃	#12594	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₈	—	53	2	no	—
98	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
99	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
100	C₂×C₄²	—	53	2	no	—
101	C₂×C₄²	—	106	2	no	—
102	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
103	M₃₂	—	2	4	no	—
104	M₃₂	—	106	4	no	—
105	C₂×C₁₆	—	53	4	no	—
106	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
107	#97	—	53	8	no	—
108	#97	—	106	4	no	—
109	C₄×C₈	—	106	2	no	—
110	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
111	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
113	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
114	C₂²×C₄	—	106	2	no	—
115	C₂²×C₄	#11719912	1	2	yes	—
116	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
117	C₂×C₈	—	53	4	no	—
118	C₂×C₈	—	106	2	no	—
119	C₂×C₈	—	106	2	no	—
120	C₂×C₈	C₈×D₅₃	1	4	yes	—
121	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
122	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
123	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
124	C₂×C₈	#11719945	1	4	yes	—
125	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
126	C₄²	—	53	4	no	—
127	C₄²	—	106	2	no	—
128	C₄²	—	212	4	no	—
129	C₄²	#11719912	1	2	yes	—
130	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
131	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
132	C₁₆	—	2	4	no	—
133	C₁₆	—	106	4	no	—
134	C₂³	C₂×#11719912	1	1	yes	characteristic
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	106	4	no	—
137	C₂×C₄	—	212	4	no	—
138	C₂×C₄	#13133127	1	2	yes	—
139	C₂×C₄	#13133127	1	2	yes	—
140	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
141	C₂×C₄	#13133174	1	2	yes	—
142	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
143	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
144	C₈	—	106	4	no	—
145	C₈	C₈×Dic₅₃	1	2	yes	—
146	C₈	C₈×Dic₅₃	1	2	yes	—
147	C₈	#13133174	1	2	yes	—
148	C₈	#13133176	1	2	yes	—
149	C₂²	—	2	2	no	—
150	C₂²	C₂×#13133127	1	1	yes	characteristic
151	C₂²	#16810674	1	2	yes	—
152	C₄	—	2	2	no	—
153	C₄	—	212	4	no	—
154	C₄	#16810400	1	4	yes	—
155	C₄	#16810469	1	1	yes	characteristic
156	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
157	C₄	#16810674	1	2	yes	—
158	C₂	—	2	2	no	—
159	C₂	#20274460	1	1	yes	characteristic
160	C₂	C₂×#16810400	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#12594
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
10. #16810399	C₄	semidirect
11. C₄×#11719908	C₄	semidirect
14. C₅₃×#346	C₄	semidirect
15. C₄×C₁₆×C₅₃	C₄	semidirect
93. #346	Dic₅₃	semidirect
95. C₄×C₁₆	Dic₅₃	semidirect
96. C₅₃	#12594	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133157	2
C₄×#13133127	2
#20272794	2
C₄×M₃₂×C₅₃	2
#12594	53