Subgroups — #27274515

Order 13568.

574 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	C₄²×#88322	C₂	1	2	yes	maximal
2	#20272542	C₂	1	4	yes	maximal
3	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#88322	C₄	1	4	yes	—
5	C₂×C₄×#88322	C₄	1	4	yes	—
6	C₂×C₄×#88322	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
10	C₄³×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×#88322	C₂×C₄	1	8	yes	—
12	C₄×#88322	—	2	8	no	—
13	C₄×#88322	—	2	8	no	—
14	C₄×#88322	C₈	1	16	yes	complemented
15	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
16	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₅₃	—	2	4	no	—
22	C₄×C₈×C₅₃	—	2	4	no	—
23	C₄×C₈×C₅₃	C₈	1	8	yes	complemented
24	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
26	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
28	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₅₃	—	2	4	no	—
30	C₂×C₈×C₅₃	—	2	4	no	—
31	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
32	C₄²×C₅₃	C₂×C₈	1	8	yes	—
33	C₄²×C₅₃	C₂×C₈	1	4	yes	—
34	C₄²×C₅₃	M₁₆	1	8	yes	—
35	C₄²×C₅₃	M₁₆	1	4	yes	—
36	C₄²×C₅₃	#41	1	4	yes	—
37	C₂×#88322	—	2	8	no	—
38	C₂×#88322	—	2	8	no	—
39	C₂×#88322	C₂×C₈	1	16	yes	—
40	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
42	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
43	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
44	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
45	C₂×C₄×C₅₃	#97	1	4	yes	—
46	C₂×C₄×C₅₃	#97	1	4	yes	—
47	C₂×C₄×C₅₃	#97	1	4	yes	—
48	C₂×C₄×C₅₃	#117	1	4	yes	—
49	C₂×C₄×C₅₃	#117	1	2	yes	—
50	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
51	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
53	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
54	C₈×C₅₃	—	2	8	no	—
55	#88322	—	2	16	no	—
56	#8131	—	53	1	no	maximal, Sylow
57	C₂²×C₅₃	#376	1	1	yes	characteristic
58	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
59	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
60	C₂²×C₅₃	C₂×#97	1	2	yes	—
61	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
62	C₄×C₅₃	C₈²	1	4	yes	—
63	C₄×C₅₃	#322	1	4	yes	—
64	C₄×C₅₃	#322	1	8	yes	—
65	C₄×C₅₃	#431	1	8	yes	—
66	C₄×C₅₃	#431	1	4	yes	—
67	C₄²×C₈	—	53	2	no	—
68	C₄²×C₈	D₅₃	1	1	yes	characteristic
69	#1273	—	53	4	no	—
70	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
71	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
72	C₂×C₅₃	C₂×#322	1	2	yes	—
73	C₂×C₅₃	#1744	1	2	yes	—
74	C₂×C₅₃	#1744	1	1	yes	characteristic
75	C₂×C₄×C₈	—	53	4	no	—
76	C₂×C₄×C₈	—	53	4	no	—
77	C₂×C₄×C₈	—	53	4	no	—
78	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
79	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
80	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
81	C₄³	C₂×D₅₃	1	1	yes	characteristic
82	C₅₃	#8131	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₈	—	53	8	no	—
84	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
85	C₂×C₄²	C₄×D₅₃	1	2	yes	—
86	C₂×C₄²	C₄×D₅₃	1	2	yes	—
87	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
88	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
89	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
90	C₄×C₈	—	2	4	no	—
91	C₄×C₈	—	2	4	no	—
92	C₄×C₈	—	53	16	no	—
93	C₄×C₈	—	106	8	no	—
94	C₄×C₈	—	106	8	no	—
95	C₄×C₈	#88322	1	8	yes	complemented
96	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
97	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
98	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
99	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
100	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	53	16	no	—
104	C₂×C₈	—	106	8	no	—
105	C₂×C₈	—	106	8	no	—
106	C₂×C₈	C₂×#88322	1	8	yes	—
107	C₄²	C₈×D₅₃	1	8	yes	—
108	C₄²	#11719912	1	8	yes	—
109	C₄²	C₂×#88322	1	4	yes	—
110	C₄²	#11719917	1	4	yes	—
111	C₄²	#11719945	1	4	yes	—
112	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
113	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
114	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
115	C₂×C₄	C₄×#88322	1	2	yes	—
116	C₂×C₄	C₄×#88322	1	2	yes	—
117	C₂×C₄	#13133127	1	4	yes	—
118	C₂×C₄	#13133127	1	4	yes	—
119	C₂×C₄	#13133128	1	4	yes	—
120	C₂×C₄	#13133174	1	4	yes	—
121	C₂×C₄	#13133176	1	2	yes	—
122	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
123	C₂×C₄	C₂²×#88322	1	2	yes	—
124	C₂×C₄	C₂×#11719912	1	4	yes	—
125	C₂×C₄	C₂×#11719917	1	2	yes	—
126	C₈	—	2	8	no	—
127	C₈	—	106	16	no	—
128	C₂²	#16810469	1	1	yes	characteristic
129	C₂²	C₂×C₈×Dic₅₃	1	2	yes	—
130	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
131	C₂²	C₂×#13133127	1	2	yes	—
132	C₂²	C₂×#13133128	1	1	yes	characteristic
133	C₄	C₈×#88322	1	4	yes	—
134	C₄	#16810344	1	4	yes	—
135	C₄	#16810345	1	8	yes	—
136	C₄	#16810674	1	8	yes	—
137	C₄	#16810676	1	4	yes	—
138	C₂	C₂×C₈×#88322	1	1	yes	characteristic
139	C₂	C₂×#16810344	1	1	yes	characteristic
140	C₂	C₂×#16810345	1	2	yes	—
141	C₂	#20274460	1	2	yes	—
142	C₂	#20274461	1	1	yes	characteristic

	subgroup	G/H
center	C₄³	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₈²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#8131
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
14. C₄×#88322	C₈	semidirect
23. C₄×C₈×C₅₃	C₈	semidirect
82. C₅₃	#8131	semidirect
95. C₄×C₈	#88322	semidirect

Maximal subgroups

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

subgroup	index
C₄²×#88322	2
#20272542	2
C₄²×C₈×C₅₃	2
#8131	53