Subgroups — #27274535

Order 13568.

354 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274401	C₂	1	1	yes	maximal, characteristic
2	#20272998	C₂	1	1	yes	maximal, characteristic
3	#20272545	C₂	1	2	yes	maximal
4	#20272543	C₂	1	2	yes	maximal
5	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133128	C₂²	1	1	yes	characteristic
8	C₂×#13133127	C₄	1	2	yes	—
9	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
10	#16810469	C₂²	1	1	yes	characteristic
11	#16810344	C₄	1	2	yes	complemented
12	C₈×#88322	C₄	1	2	yes	complemented
13	C₅₃×#431	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#322	C₄	1	2	yes	complemented
16	C₈²×C₅₃	C₄	1	2	yes	complemented
17	C₂²×#88322	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
19	C₄⋊#88322	—	2	2	no	—
20	C₂²⋊#88322	—	2	2	no	—
21	#13133128	—	2	2	no	—
22	#13133127	C₈	1	8	yes	complemented
23	C₄×#88322	C₂×C₄	1	4	yes	—
24	C₈×Dic₅₃	—	2	4	no	—
25	#13133124	—	2	2	no	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	—	2	2	no	—
30	C₂²⋊C₈×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	#41	1	2	yes	—
38	C₂²×Dic₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	—	2	4	no	—
40	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
46	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₅₃	#41	1	2	yes	—
49	C₄×Dic₅₃	—	2	2	no	—
50	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
51	C₂×#88322	—	2	2	no	—
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	—	2	4	no	—
54	C₂×#88322	C₂×C₈	1	4	yes	—
55	C₂³×C₅₃	#117	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	4	2	no	—
57	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#117	1	2	yes	—
59	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#117	1	2	yes	—
61	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
63	C₂×Dic₅₃	—	2	4	no	—
64	C₂×Dic₅₃	—	4	4	no	—
65	C₈×C₅₃	—	4	4	no	—
66	C₈×C₅₃	C₄×C₈	1	2	yes	—
67	C₈×C₅₃	C₄×C₈	1	2	yes	—
68	C₈×C₅₃	#97	1	2	yes	—
69	C₈×C₅₃	#97	1	2	yes	—
70	#88322	—	2	4	no	—
71	#88322	—	4	4	no	—
72	#8138	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	—	2	2	no	—
74	C₂²×C₅₃	#376	1	1	yes	characteristic
75	C₂²×C₅₃	#431	1	2	yes	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
79	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
80	C₄×C₅₃	#431	1	2	yes	—
81	Dic₅₃	—	4	4	no	—
82	#1726	—	53	1	no	—
83	#1450	—	53	1	no	—
84	#1450	D₅₃	1	1	yes	characteristic
85	#1275	—	53	2	no	—
86	#1274	—	53	2	no	—
87	C₂×C₅₃	—	2	2	no	—
88	C₂×C₅₃	#1744	1	1	yes	characteristic
89	C₂×C₅₃	#1747	1	2	yes	—
90	#431	—	53	2	no	—
91	#431	C₂×D₅₃	1	2	yes	—
92	C₂×#97	—	53	2	no	—
93	C₂×#97	—	53	1	no	—
94	C₂×C₄×C₈	—	53	2	no	—
95	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
96	#376	—	53	1	no	—
97	#322	—	53	2	no	—
98	#322	Dic₅₃	1	2	yes	complemented
99	C₈²	—	53	2	no	—
100	C₈²	Dic₅₃	1	2	yes	complemented
101	C₅₃	#8138	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	—	53	2	no	—
103	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
104	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
105	C₂×C₄²	—	53	2	no	—
106	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	106	2	no	—
109	C₂²⋊C₈	—	2	2	no	—
110	C₂²⋊C₈	—	106	2	no	—
111	#97	—	53	8	no	—
112	#97	—	106	2	no	—
113	C₄×C₈	—	53	4	no	—
114	C₄×C₈	—	106	4	no	—
115	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
116	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
118	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
119	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
120	#95	—	106	2	no	—
121	C₂²×C₄	—	106	2	no	—
122	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
123	C₂²×C₄	#11719945	1	2	yes	—
124	C₂×C₈	—	2	4	no	—
125	C₂×C₈	—	53	4	no	—
126	C₂×C₈	—	106	2	no	—
127	C₂×C₈	—	106	2	no	—
128	C₂×C₈	—	106	4	no	—
129	C₂×C₈	C₈×D₅₃	1	4	yes	—
130	C₂×C₈	#11719912	1	4	yes	—
131	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
132	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
133	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
134	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
135	C₄²	—	53	4	no	—
136	C₄²	—	106	2	no	—
137	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
138	C₄²	#11719945	1	2	yes	—
139	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
140	C₂³	#13133174	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	106	4	no	—
143	C₂×C₄	—	212	4	no	—
144	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
145	C₂×C₄	#13133174	1	2	yes	—
146	C₂×C₄	#13133174	1	1	yes	characteristic
147	C₂×C₄	#13133176	1	2	yes	—
148	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
149	C₂×C₄	C₂×#11719912	1	2	yes	—
150	C₈	—	4	4	no	—
151	C₈	—	106	4	no	—
152	C₈	—	212	4	no	—
153	C₈	C₈×Dic₅₃	1	2	yes	—
154	C₈	C₈×Dic₅₃	1	2	yes	—
155	C₈	#13133127	1	2	yes	—
156	C₈	#13133127	1	2	yes	—
157	C₂²	—	2	2	no	—
158	C₂²	#16810469	1	1	yes	characteristic
159	C₂²	#16810674	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	212	4	no	—
163	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
164	C₄	C₂×#13133127	1	1	yes	characteristic
165	C₄	#16810674	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#20274460	1	1	yes	characteristic
168	C₂	#20274470	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1450	C₂
socle	C₂²×C₅₃	#376
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1450	D₅₃
hypocenter	C₅₃	#8138
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
11. #16810344	C₄	semidirect
12. C₈×#88322	C₄	semidirect
15. C₅₃×#322	C₄	semidirect
16. C₈²×C₅₃	C₄	semidirect
22. #13133127	C₈	semidirect
98. #322	Dic₅₃	semidirect
100. C₈²	Dic₅₃	semidirect
101. C₅₃	#8138	semidirect

Maximal subgroups

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

subgroup	index
#20274401	2
#20272998	2
#20272545	2
#20272543	2
C₅₃×#1450	2
#8138	53