Subgroups — #27271790

Order 13568.

178 conjugacy classes of subgroups.

The subgroup lattice has 152 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	C₂²×#13133123	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic
4	#16810457	—	2	1	no	—
5	#16810457	C₄	1	2	yes	complemented
6	C₂×#13133123	—	2	1	no	—
7	C₂×#13133123	—	2	1	no	—
8	C₂×#13133123	—	2	1	no	—
9	C₂×#13133123	C₄	1	2	yes	complemented
10	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
11	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#405	C₄	1	1	yes	characteristic
13	C₅₃×#346	—	2	1	no	—
14	C₄×C₁₆×C₅₃	—	2	1	no	—
15	#13133123	—	2	2	no	—
16	#13133123	—	2	2	no	—
17	C₂×M₁₆×C₅₃	C₈	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	#117×C₅₃	C₈	1	1	yes	characteristic
20	M₃₂×C₅₃	—	2	1	no	—
21	M₃₂×C₅₃	C₈	1	2	yes	—
22	C₂×C₁₆×C₅₃	—	2	1	no	—
23	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
29	#97×C₅₃	—	2	1	no	—
30	C₄×C₈×C₅₃	—	2	1	no	—
31	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
32	M₁₆×C₅₃	—	2	1	no	—
33	M₁₆×C₅₃	C₁₆	1	2	yes	—
34	C₂×C₈×C₅₃	—	2	1	no	—
35	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
41	C₄⋊C₄×C₅₃	C₁₆	1	2	yes	—
42	C₂²⋊C₄×C₅₃	—	2	1	no	—
43	C₄²×C₅₃	—	2	1	no	—
44	C₁₆×C₅₃	—	2	2	no	—
45	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂³×C₅₃	M₃₂	1	1	yes	characteristic
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
56	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
57	C₈×C₅₃	—	2	2	no	—
58	C₈×C₅₃	#95	1	1	yes	characteristic
59	C₈×C₅₃	#97	1	1	yes	characteristic
60	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
61	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
62	#7470	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	#346	1	1	yes	characteristic
65	C₂²×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
66	C₂²×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
67	C₄×C₅₃	—	4	2	no	—
68	C₄×C₅₃	#336	1	1	yes	characteristic
69	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
70	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
71	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
72	C₂×M₆₄	—	53	1	no	—
73	C₂²×C₃₂	—	53	1	no	—
74	#2108	D₅₃	1	1	yes	characteristic
75	C₂×C₅₃	—	2	1	no	—
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	#1374	1	1	yes	characteristic
78	C₂×C₅₃	#1401	1	2	yes	—
79	C₂×M₃₂	Dic₅₃	1	1	yes	characteristic
80	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
81	#405	Dic₅₃	1	1	yes	characteristic
82	M₆₄	—	53	2	no	—
83	M₆₄	—	106	1	no	—
84	C₂×C₃₂	—	53	2	no	—
85	C₂×C₃₂	—	106	1	no	—
86	C₂×C₃₂	—	106	1	no	—
87	C₂×C₃₂	—	106	1	no	—
88	#346	—	2	1	no	—
89	C₄×C₁₆	—	2	1	no	—
90	C₅₃	#7470	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₁₆	#88322	1	1	yes	characteristic
92	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
93	#117	#88322	1	1	yes	characteristic
94	M₃₂	—	2	1	no	—
95	M₃₂	#88322	1	2	yes	—
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
98	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
99	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
100	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
101	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
103	#97	—	2	1	no	—
104	C₄×C₈	—	2	1	no	—
105	C₃₂	—	106	2	no	—
106	C₃₂	—	106	2	no	—
107	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
108	M₁₆	—	2	1	no	—
109	M₁₆	#11719908	1	2	yes	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	C₂×#88322	1	1	yes	characteristic
112	C₂×C₈	#11719917	1	1	yes	characteristic
113	C₂×C₈	#11719917	1	1	yes	characteristic
114	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
115	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
117	C₄⋊C₄	#11719908	1	2	yes	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₁₆	—	2	2	no	—
121	C₁₆	#11719919	1	1	yes	characteristic
122	C₁₆	#11719919	1	1	yes	characteristic
123	C₁₆	#11719921	1	1	yes	characteristic
124	C₁₆	#11719921	1	1	yes	characteristic
125	C₂³	#13133157	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₄×#88322	1	1	yes	characteristic
128	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
129	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
130	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
131	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
132	C₂×C₄	#13133157	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	#13133124	1	1	yes	characteristic
135	C₈	#13133128	1	1	yes	characteristic
136	C₈	C₂²⋊#88322	1	1	yes	characteristic
137	C₈	C₄⋊#88322	1	1	yes	characteristic
138	C₂²	—	2	2	no	—
139	C₂²	#16810399	1	1	yes	characteristic
140	C₂²	C₂²⋊#11719908	1	1	yes	characteristic
141	C₂²	C₄⋊#11719908	1	1	yes	characteristic
142	C₄	—	4	2	no	—
143	C₄	#16810378	1	1	yes	characteristic
144	C₄	C₄×#11719908	1	1	yes	characteristic
145	C₄	C₂²⋊#11719908	1	1	yes	characteristic
146	C₄	C₄⋊#11719908	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#20272793	1	1	yes	characteristic
150	C₂	#20272857	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₄×C₁₆
Frattini	C₂²×C₁₆	C₂×D₅₃
Fitting	C₅₃×#2108	C₂
socle	C₂²×C₅₃	C₂²⋊C₁₆
prime radical	C₂³×C₅₃	M₃₂
cosocle	C₂²×C₁₆×C₅₃	C₂²
hypercenter	#2108	D₅₃
hypocenter	C₅₃	#7470
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
5. #16810457	C₄	semidirect
9. C₂×#13133123	C₄	semidirect
90. C₅₃	#7470	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
C₂²×#13133123	2
C₅₃×#2108	2
#7470	53