Subgroups — #27271779

Order 13568.

202 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133123	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133123	—	2	2	no	—
4	C₂×#13133123	—	2	2	no	—
5	C₂×#13133123	C₄	1	4	yes	complemented
6	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₄×C₁₆×C₅₃	—	2	2	no	—
10	#13133123	—	2	4	no	—
11	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
12	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	—	2	1	no	—
15	C₂×C₁₆×C₅₃	—	2	1	no	—
16	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
17	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
20	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
21	C₄×C₈×C₅₃	—	2	1	no	—
22	C₄×C₈×C₅₃	—	2	1	no	—
23	C₂²×C₄×C₅₃	C₁₆	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
25	C₂×C₈×C₅₃	—	2	1	no	—
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	C₁₆	1	2	yes	—
28	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
29	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
34	C₄²×C₅₃	—	2	1	no	—
35	C₄²×C₅₃	—	2	1	no	—
36	C₁₆×C₅₃	—	2	2	no	—
37	C₁₆×C₅₃	C₂²⋊C₄	1	2	yes	—
38	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	C₃₂	1	4	yes	complemented
43	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
48	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
49	C₈×C₅₃	—	2	2	no	—
50	C₈×C₅₃	#95	1	1	yes	characteristic
51	C₈×C₅₃	#97	1	1	yes	characteristic
52	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
53	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
54	#7465	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	C₄×C₁₆	1	1	yes	characteristic
56	C₂²×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
57	C₂²×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
58	C₂²×C₅₃	C₂×C₃₂	1	2	yes	—
59	C₂²×C₅₃	M₆₄	1	2	yes	—
60	C₄×C₅₃	—	2	4	no	—
61	C₄×C₅₃	#336	1	1	yes	characteristic
62	C₄×C₅₃	#346	1	1	yes	characteristic
63	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
64	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
65	C₂²×C₃₂	—	53	2	no	—
66	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
67	C₂×C₅₃	#1374	1	1	yes	characteristic
68	C₂×C₅₃	C₄×C₃₂	1	1	yes	characteristic
69	C₂×C₅₃	#1397	1	1	yes	characteristic
70	C₂×C₅₃	C₂²⋊C₃₂	1	2	yes	—
71	C₂×C₅₃	C₄⋊C₃₂	1	2	yes	—
72	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
73	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
74	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
75	C₂×C₃₂	—	53	4	no	—
76	C₂×C₃₂	—	106	2	no	—
77	C₂×C₃₂	—	106	2	no	—
78	C₄×C₁₆	—	2	2	no	—
79	C₅₃	#7465	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	#88322	1	1	yes	characteristic
81	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
82	C₂×C₄²	#88322	1	1	yes	characteristic
83	C₂×C₁₆	—	2	1	no	—
84	C₂×C₁₆	—	2	1	no	—
85	C₂×C₁₆	#88322	1	2	yes	—
86	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
87	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
88	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
89	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₃₂	—	106	4	no	—
93	C₂²×C₄	#11719908	1	2	yes	—
94	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	#11719908	1	2	yes	—
98	C₂×C₈	C₂×#88322	1	1	yes	characteristic
99	C₂×C₈	#11719917	1	1	yes	characteristic
100	C₂×C₈	#11719917	1	1	yes	characteristic
101	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
102	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
103	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
104	C₄²	—	2	1	no	—
105	C₄²	—	2	1	no	—
106	C₁₆	—	2	2	no	—
107	C₁₆	#11719919	1	2	yes	—
108	C₁₆	#11719921	1	2	yes	—
109	C₂³	C₂×#11719908	1	1	yes	characteristic
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	#13133123	1	4	yes	complemented
113	C₂×C₄	C₄×#88322	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
115	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
116	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
117	C₂×C₄	#13133157	1	1	yes	characteristic
118	C₂×C₄	#13133157	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	#13133124	1	1	yes	characteristic
121	C₈	#13133128	1	1	yes	characteristic
122	C₈	C₂²⋊#88322	1	1	yes	characteristic
123	C₈	C₄⋊#88322	1	1	yes	characteristic
124	C₂²	C₄×#11719908	1	1	yes	characteristic
125	C₂²	C₂²⋊#11719908	1	1	yes	characteristic
126	C₂²	C₄⋊#11719908	1	1	yes	characteristic
127	C₂²	C₂×#13133123	1	2	yes	—
128	C₂²	#16810457	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	#16810378	1	1	yes	characteristic
131	C₄	#16810399	1	1	yes	characteristic
132	C₄	C₂²⋊#11719908	1	1	yes	characteristic
133	C₄	C₄⋊#11719908	1	1	yes	characteristic
134	C₂	#20272793	1	1	yes	characteristic
135	C₂	C₄×#13133123	1	1	yes	characteristic
136	C₂	#20272849	1	1	yes	characteristic
137	C₂	C₂²⋊#13133123	1	2	yes	—
138	C₂	C₄⋊#13133123	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₁₆	C₂×D₅₃
commutator	C₂×C₅₃	C₄×C₃₂
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₁₆
cosocle	C₂²×C₁₆×C₅₃	C₂²
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#7465
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
5. C₂×#13133123	C₄	semidirect
42. C₂×C₄×C₅₃	C₃₂	semidirect
79. C₅₃	#7465	semidirect
112. C₂×C₄	#13133123	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133123	2
C₂×C₄×C₁₆×C₅₃	2
#7465	53