Subgroups — #27293083

Order 13568.

298 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
2	#20272998	C₂	1	1	yes	maximal, characteristic
3	#20272634	C₂	1	4	yes	maximal, complemented
4	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₄	1	2	yes	—
6	#16810676	C₄	1	2	yes	—
7	C₄×#11719908	C₄	1	2	yes	—
8	C₄×#11719908	C₂²	1	2	yes	—
9	#16810344	C₄	1	2	yes	—
10	C₈×#88322	C₂²	1	2	yes	—
11	C₂×M₃₂×C₅₃	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#346	C₄	1	2	yes	—
14	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂×#11719908	—	2	4	no	—
16	C₂×#11719908	C₂×C₄	1	4	yes	—
17	C₄⋊#88322	C₈	1	4	yes	—
18	C₂²⋊#88322	C₈	1	4	yes	—
19	C₄×#88322	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	M₃₂×C₅₃	—	2	4	no	—
24	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
33	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
34	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
35	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₅₃	C₄²	1	1	yes	characteristic
38	C₄²×C₅₃	C₂×C₈	1	2	yes	—
39	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×#88322	C₂×C₈	1	8	yes	—
41	C₁₆×C₅₃	—	2	4	no	—
42	#11719908	—	2	4	no	—
43	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	4	no	—
45	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#97	1	2	yes	—
48	C₂×C₄×C₅₃	#97	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
51	C₈×C₅₃	#97	1	2	yes	—
52	C₈×C₅₃	#97	1	2	yes	—
53	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
54	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
55	#88322	—	2	8	no	—
56	#11401	—	53	1	no	maximal, Sylow
57	C₂²×C₅₃	—	2	2	no	—
58	C₂²×C₅₃	#322	1	2	yes	—
59	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	#322	1	2	yes	—
62	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
63	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
64	C₄×C₅₃	#504	1	4	yes	—
65	C₄×M₃₂	D₅₃	1	1	yes	characteristic
66	C₂×C₄×C₁₆	—	53	1	no	—
67	#1450	—	53	1	no	—
68	#1312	—	53	4	no	—
69	C₂×C₅₃	—	2	2	no	—
70	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
71	C₂×C₅₃	#2108	1	2	yes	—
72	C₂×M₃₂	Dic₅₃	1	2	yes	—
73	C₂²×C₁₆	—	53	2	no	—
74	#431	—	53	2	no	—
75	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
76	#346	Dic₅₃	1	2	yes	—
77	C₄×C₁₆	—	53	2	no	—
78	C₄×C₁₆	—	53	2	no	—
79	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
80	#322	—	53	2	no	—
81	C₈²	—	53	2	no	—
82	C₅₃	#11401	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
84	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
85	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
86	M₃₂	—	2	4	no	—
87	C₂×C₁₆	—	53	4	no	—
88	C₂×C₁₆	—	106	4	no	—
89	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
90	C₄⋊C₈	—	53	4	no	—
91	C₂²⋊C₈	—	53	4	no	—
92	C₄×C₈	—	53	4	no	—
93	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
94	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
95	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
96	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
97	C₂²×C₄	C₈×D₅₃	1	2	yes	—
98	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
99	C₂×C₈	—	53	8	no	—
100	C₂×C₈	#11719912	1	4	yes	—
101	C₂×C₈	#11719917	1	4	yes	—
102	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
103	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
104	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
105	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
106	C₄²	C₈×D₅₃	1	2	yes	—
107	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
108	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
109	C₁₆	—	2	4	no	—
110	C₁₆	—	106	4	no	—
111	C₂³	C₈×Dic₅₃	1	1	yes	characteristic
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
114	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
115	C₂×C₄	#13133127	1	2	yes	—
116	C₂×C₄	#13133128	1	2	yes	—
117	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
118	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
119	C₈	—	106	8	no	—
120	C₈	#13133127	1	2	yes	—
121	C₈	#13133128	1	2	yes	—
122	C₈	C₂×#11719912	1	2	yes	—
123	C₈	C₂×#11719917	1	2	yes	—
124	C₂²	—	2	2	no	—
125	C₂²	#16810345	1	2	yes	—
126	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	#16810345	1	2	yes	—
129	C₄	C₂×#13133127	1	1	yes	characteristic
130	C₄	C₂×#13133128	1	1	yes	characteristic
131	C₄	#16811041	1	4	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#16810345	1	1	yes	characteristic
134	C₂	#20276314	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₄×M₃₂×C₅₃	C₂
prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#11401
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20272634	C₂	semidirect
82. C₅₃	#11401	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719908	2
#20272998	2
#20272634	2
C₄×M₃₂×C₅₃	2
#11401	53