Subgroups — #27293075

Order 13568.

330 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133156	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272997	C₂	1	1	yes	maximal, characteristic
4	#20272631	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1311	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133156	C₄	1	2	yes	—
9	C₂×C₁₆×D₅₃	C₄	1	2	yes	—
10	#16810674	C₄	1	2	yes	—
11	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#16810398	C₄	1	2	yes	—
13	C₄×#11719908	C₂²	1	1	yes	characteristic
14	C₄×#11719908	C₂²	1	1	yes	characteristic
15	C₁₆×Dic₅₃	C₄	1	2	yes	—
16	#16810345	C₄	1	2	yes	—
17	C₈×#88322	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₈²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
24	#13133156	C₈	1	8	yes	complemented
25	C₂×#11719908	C₂×C₄	1	2	yes	—
26	C₂×#11719908	C₂×C₄	1	2	yes	—
27	C₁₆×D₅₃	C₈	1	8	yes	complemented
28	#13133146	C₈	1	4	yes	—
29	#13133129	C₈	1	4	yes	—
30	C₄×#88322	C₂×C₄	1	1	yes	characteristic
31	C₄×#88322	C₂×C₄	1	2	yes	—
32	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
39	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
45	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
46	C₂×#88322	C₄²	1	1	yes	characteristic
47	C₂×#88322	C₄²	1	1	yes	characteristic
48	C₂×#88322	C₂×C₈	1	4	yes	—
49	C₈×D₅₃	C₂×C₈	1	4	yes	—
50	C₈×D₅₃	C₂×C₈	1	4	yes	—
51	C₁₆×C₅₃	C₂×C₈	1	4	yes	—
52	C₁₆×C₅₃	C₂×C₈	1	4	yes	—
53	#11719908	C₂×C₈	1	4	yes	—
54	#11719908	C₂×C₈	1	4	yes	—
55	C₂²×D₅₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
58	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
59	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
60	C₄×D₅₃	—	2	4	no	—
61	C₄×D₅₃	C₄×C₈	1	2	yes	—
62	C₄×D₅₃	C₄×C₈	1	2	yes	—
63	C₈×C₅₃	—	2	4	no	—
64	C₈×C₅₃	C₂²×C₈	1	2	yes	—
65	C₈×C₅₃	C₂²×C₈	1	2	yes	—
66	#88322	—	2	4	no	—
67	#88322	C₄×C₈	1	2	yes	—
68	#88322	C₄×C₈	1	2	yes	—
69	#11399	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
71	C₂×D₅₃	—	2	2	no	—
72	C₂×D₅₃	C₈²	1	2	yes	—
73	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
74	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
75	C₄×C₅₃	#504	1	4	yes	—
76	Dic₅₃	—	2	2	no	—
77	Dic₅₃	C₈²	1	2	yes	—
78	C₄×M₃₂	—	53	1	no	—
79	C₂×C₄×C₁₆	—	53	1	no	—
80	#1450	—	53	1	no	—
81	#1311	—	53	1	no	—
82	#1311	D₅₃	1	1	yes	characteristic, complemented
83	C₈×C₁₆	—	53	1	no	—
84	C₈×C₁₆	—	53	1	no	—
85	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
86	C₂×C₅₃	#2108	1	2	yes	—
87	D₅₃	—	2	2	no	—
88	C₂×M₃₂	—	53	2	no	—
89	C₂²×C₁₆	—	53	2	no	—
90	#431	—	53	2	no	—
91	C₂×C₄×C₈	—	53	1	no	—
92	#346	—	53	2	no	—
93	C₄×C₁₆	—	53	2	no	—
94	C₄×C₁₆	—	53	1	no	—
95	C₄×C₁₆	—	53	1	no	—
96	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
97	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
98	#322	—	53	2	no	—
99	C₈²	—	53	1	no	—
100	C₈²	C₂×D₅₃	1	1	yes	characteristic
101	C₅₃	#11399	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	—	53	1	no	—
103	C₂²×C₈	—	53	1	no	—
104	C₂×C₄²	—	53	1	no	—
105	M₃₂	—	53	8	no	—
106	C₂×C₁₆	—	53	2	no	—
107	C₂×C₁₆	—	53	2	no	—
108	C₂×C₁₆	—	53	8	no	—
109	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
110	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
111	C₄⋊C₈	—	53	4	no	—
112	C₂²⋊C₈	—	53	4	no	—
113	C₄×C₈	—	53	1	no	—
114	C₄×C₈	—	53	1	no	—
115	C₄×C₈	—	53	2	no	—
116	C₄×C₈	—	53	1	no	—
117	C₄×C₈	C₄×D₅₃	1	2	yes	—
118	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
119	C₂²×C₄	—	53	1	no	—
120	C₂²×C₄	—	53	2	no	—
121	C₂×C₈	—	53	1	no	—
122	C₂×C₈	—	53	1	no	—
123	C₂×C₈	—	53	4	no	—
124	C₂×C₈	—	53	4	no	—
125	C₂×C₈	—	53	4	no	—
126	C₂×C₈	C₈×D₅₃	1	4	yes	—
127	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
128	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
129	C₄²	—	53	1	no	—
130	C₄²	—	53	2	no	—
131	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
132	C₁₆	—	53	4	no	—
133	C₁₆	—	53	4	no	—
134	C₁₆	C₈×D₅₃	1	4	yes	—
135	C₁₆	C₈×D₅₃	1	4	yes	—
136	C₂³	—	53	1	no	—
137	C₂×C₄	—	53	1	no	—
138	C₂×C₄	—	53	2	no	—
139	C₂×C₄	—	53	2	no	—
140	C₂×C₄	—	53	2	no	—
141	C₂×C₄	—	106	4	no	—
142	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
143	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	53	2	no	—
146	C₈	—	53	2	no	—
147	C₈	—	106	4	no	—
148	C₈	C₂×C₈×D₅₃	1	2	yes	—
149	C₈	C₂×C₈×D₅₃	1	2	yes	—
150	C₂²	—	53	2	no	—
151	C₂²	—	106	2	no	—
152	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
153	C₄	—	53	2	no	—
154	C₄	—	106	2	no	—
155	C₄	C₄×C₈×D₅₃	1	1	yes	characteristic
156	C₄	C₄×C₈×D₅₃	1	1	yes	characteristic
157	C₄	#16811041	1	4	yes	—
158	C₂	—	106	2	no	—
159	C₂	C₈²×D₅₃	1	1	yes	characteristic
160	C₂	#20276313	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₅₃
commutator	C₂×C₅₃	C₂×C₈²
Fitting	C₅₃×#1311	C₂
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂²×D₅₃	C₄×C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1311	D₅₃
hypocenter	C₅₃	#11399
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ #1311

Complements

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

N	H ≅ G/N	type
4. #20272631	C₂	semidirect
5. C₈×#11719908	C₂	semidirect
6. C₁₆×#88322	C₂	semidirect
7. C₅₃×#1311	C₂	semidirect
24. #13133156	C₈	semidirect
27. C₁₆×D₅₃	C₈	semidirect
82. #1311	D₅₃	semidirect
101. C₅₃	#11399	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133156	2
C₄×C₁₆×D₅₃	2
#20272997	2
#20272631	2
C₈×#11719908	2
C₁₆×#88322	2
C₅₃×#1311	2
#11399	53