Subgroups — #27273462

Order 13568.

518 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	#20274461	C₂	1	2	yes	maximal
2	C₂×#16810345	C₂	1	2	yes	maximal
3	C₂×#16810344	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
5	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	#16810378	C₄	1	8	yes	—
9	#16810345	C₄	1	8	yes	complemented
10	#16810344	C₄	1	4	yes	complemented
11	C₈×#88322	C₄	1	4	yes	complemented
12	C₂×#97×C₅₃	C₄	1	4	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	C₂×C₄	1	4	yes	—
17	C₂²×#88322	C₂×C₄	1	4	yes	—
18	C₄×#88322	C₂×C₄	1	4	yes	—
19	C₄×#88322	C₂×C₄	1	4	yes	—
20	C₄×#88322	C₂×C₄	1	4	yes	—
21	C₄×#88322	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	#97×C₅₃	—	2	8	no	—
28	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
34	C₂×C₈×C₅₃	—	2	4	no	—
35	C₂×C₈×C₅₃	C₄²	1	4	yes	—
36	C₂×C₈×C₅₃	C₄²	1	4	yes	—
37	C₂×C₈×C₅₃	M₁₆	1	8	yes	—
38	C₄²×C₅₃	—	2	4	no	—
39	C₄²×C₅₃	—	2	8	no	—
40	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
43	C₂×#88322	—	2	8	no	—
44	C₂×#88322	—	2	8	no	—
45	C₂×#88322	C₄²	1	8	yes	—
46	C₂×#88322	C₄²	1	8	yes	—
47	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	4	no	—
49	C₂×C₄×C₅₃	—	2	8	no	—
50	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
55	C₂×C₄×C₅₃	#131	1	8	yes	—
56	C₈×C₅₃	—	2	4	no	—
57	C₈×C₅₃	#97	1	8	yes	—
58	#88322	—	2	8	no	—
59	#88322	—	2	8	no	—
60	#7844	—	53	1	no	maximal, Sylow
61	C₂²×C₅₃	C₄³	1	1	yes	characteristic
62	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
63	C₂²×C₅₃	#405	1	4	yes	—
64	C₄×C₅₃	—	4	4	no	—
65	C₄×C₅₃	C₂×#97	1	2	yes	—
66	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
67	C₄×C₅₃	#405	1	4	yes	—
68	C₄×C₅₃	#405	1	4	yes	—
69	#1744	—	53	2	no	—
70	C₄×#97	D₅₃	1	1	yes	characteristic
71	C₂×#322	—	53	1	no	—
72	C₂×#322	—	53	2	no	—
73	C₂×C₈²	—	53	1	no	—
74	C₂×C₅₃	#1453	1	4	yes	—
75	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
76	C₂×C₅₃	#1726	1	2	yes	—
77	C₂×#97	Dic₅₃	1	4	yes	—
78	C₂×C₄×C₈	—	53	2	no	—
79	C₂×C₄×C₈	—	53	2	no	—
80	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
81	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
82	C₄³	C₂×D₅₃	1	1	yes	characteristic
83	#336	—	53	8	no	—
84	#322	—	53	4	no	—
85	#322	—	53	8	no	—
86	C₈²	—	53	4	no	—
87	C₅₃	#7844	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	53	4	no	—
89	C₂²×C₈	—	53	4	no	—
90	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
91	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
92	C₂×C₄²	C₄×D₅₃	1	4	yes	—
93	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
94	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
95	#97	—	2	8	no	—
96	C₄×C₈	—	53	4	no	—
97	C₄×C₈	—	53	4	no	—
98	C₄×C₈	—	53	4	no	—
99	C₄×C₈	—	53	4	no	—
100	C₄×C₈	C₄×D₅₃	1	2	yes	—
101	C₄×C₈	C₄×D₅₃	1	4	yes	—
102	C₄×C₈	C₄×D₅₃	1	2	yes	—
103	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
104	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
105	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	—	53	8	no	—
108	C₂×C₈	—	53	8	no	—
109	C₂×C₈	—	106	8	no	—
110	C₂×C₈	—	106	8	no	—
111	C₂×C₈	#11719917	1	8	yes	—
112	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
113	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
114	C₄²	—	2	4	no	—
115	C₄²	—	2	8	no	—
116	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
117	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
118	C₄²	C₂×C₄×D₅₃	1	2	yes	—
119	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	8	no	—
122	C₂×C₄	C₄²×D₅₃	1	2	yes	—
123	C₂×C₄	C₄²×D₅₃	1	2	yes	—
124	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
126	C₂×C₄	C₂×#11719917	1	4	yes	—
127	C₂×C₄	#13133225	1	8	yes	—
128	C₈	—	2	4	no	—
129	C₈	—	106	8	no	—
130	C₈	—	106	8	no	—
131	C₈	#13133128	1	8	yes	—
132	C₂²	C₄²×Dic₅₃	1	1	yes	characteristic
133	C₂²	C₄×#11719917	1	2	yes	—
134	C₂²	#16810581	1	4	yes	—
135	C₄	—	4	4	no	—
136	C₄	C₂×#13133128	1	2	yes	—
137	C₄	C₄×#11719917	1	2	yes	—
138	C₄	#16810580	1	4	yes	—
139	C₄	#16810580	1	4	yes	—
140	C₂	#20273012	1	4	yes	—
141	C₂	C₄×#13133128	1	1	yes	characteristic
142	C₂	#20274403	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₄³
Fitting	C₄×#97×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#97	D₅₃
hypocenter	C₅₃	#7844
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
9. #16810345	C₄	semidirect
10. #16810344	C₄	semidirect
11. C₈×#88322	C₄	semidirect
87. C₅₃	#7844	semidirect

Maximal subgroups

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

subgroup	index
#20274461	2
C₂×#16810345	2
C₂×#16810344	2
C₂×C₈×#88322	2
C₄×#97×C₅₃	2
#7844	53