Subgroups — #27300481

Order 13568.

282 conjugacy classes of subgroups.

The subgroup lattice has 168 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	#20274887	C₂	1	1	yes	maximal, characteristic
3	#20272808	C₂	1	4	yes	maximal
4	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×#13133153	—	2	1	no	—
7	C₂×#13133153	C₂²	1	2	yes	—
8	#16810647	—	2	1	no	—
9	C₄×#11719908	C₄	1	2	yes	—
10	C₄×#11719908	C₄	1	2	yes	—
11	C₂×#108×C₅₃	—	2	1	no	—
12	C₂×#108×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#423	—	2	1	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#11719917	—	2	2	no	—
16	C₂×#11719908	—	2	2	no	—
17	C₂×#11719908	—	2	2	no	—
18	C₂×#11719908	C₂×C₄	1	4	yes	—
19	#13133153	—	2	4	no	—
20	#13133153	—	4	2	no	—
21	C₄⋊#88322	—	4	2	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
26	#108×C₅₃	—	2	4	no	—
27	#108×C₅₃	—	4	2	no	—
28	C₄⋊C₈×C₅₃	—	4	2	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	M₁₆×C₅₃	—	4	2	no	—
36	M₁₆×C₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₅₃	#41	1	2	yes	—
45	C₂×C₈×C₅₃	#41	1	2	yes	—
46	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₅₃	D₈	1	2	yes	—
49	#11719917	—	4	2	no	—
50	#11719917	—	4	2	no	—
51	C₂×#88322	—	4	2	no	—
52	#11719908	—	2	4	no	—
53	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	2	4	no	—
55	C₂×C₄×C₅₃	#107	1	2	yes	—
56	C₂×C₄×C₅₃	#107	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₅₃	—	8	2	no	—
64	C₈×C₅₃	#106	1	2	yes	—
65	C₈×C₅₃	#106	1	2	yes	—
66	C₈×C₅₃	#117	1	2	yes	—
67	C₈×C₅₃	#125	1	2	yes	—
68	#88322	—	8	2	no	—
69	#12891	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	—	2	2	no	—
71	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
72	C₂²×C₅₃	#487	1	1	yes	characteristic
73	C₂²×C₅₃	#487	1	1	yes	characteristic
74	C₄×C₅₃	—	2	2	no	—
75	C₄×C₅₃	#368	1	4	yes	—
76	C₄×C₅₃	#382	1	1	yes	characteristic
77	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
78	C₄×C₅₃	#486	1	1	yes	characteristic
79	C₄×C₅₃	#486	1	1	yes	characteristic
80	C₂×C₄×C₁₆	—	53	1	no	—
81	#1848	—	53	1	no	—
82	#1848	D₅₃	1	1	yes	characteristic
83	#1381	—	53	4	no	—
84	C₂×C₅₃	—	2	1	no	—
85	C₂×C₅₃	—	2	1	no	—
86	C₂×C₅₃	#1834	1	1	yes	characteristic
87	C₂×C₅₃	C₂×#368	1	2	yes	—
88	C₂²×C₁₆	—	53	2	no	—
89	C₂×#108	—	2	1	no	—
90	C₂×#108	—	53	2	no	—
91	C₂×#108	—	106	1	no	—
92	C₂×#108	C₂×D₅₃	1	2	yes	—
93	#423	—	2	1	no	—
94	#423	—	106	1	no	—
95	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
96	C₄×C₁₆	—	53	2	no	—
97	C₄×C₁₆	—	53	2	no	—
98	C₅₃	#12891	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	106	2	no	—
101	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
103	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
104	C₂×C₁₆	—	53	4	no	—
105	C₂×C₁₆	—	106	2	no	—
106	C₂×C₁₆	—	106	2	no	—
107	#108	—	2	4	no	—
108	#108	—	4	2	no	—
109	#108	—	106	4	no	—
110	#108	—	212	2	no	—
111	C₄⋊C₈	—	4	2	no	—
112	C₄⋊C₈	—	212	2	no	—
113	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
114	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
115	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
116	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
117	C₂²×C₄	#11719925	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
119	M₁₆	—	4	2	no	—
120	M₁₆	—	4	2	no	—
121	M₁₆	—	212	2	no	—
122	M₁₆	—	212	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	212	2	no	—
125	C₂×C₈	#11719919	1	1	yes	characteristic
126	C₂×C₈	#11719919	1	1	yes	characteristic
127	C₂×C₈	#11719923	1	2	yes	—
128	C₂×C₈	#11719924	1	2	yes	—
129	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
131	C₂×C₈	#11719947	1	2	yes	—
132	C₂×C₈	#11719949	1	2	yes	—
133	C₄²	#11719919	1	1	yes	characteristic
134	C₄²	#11719919	1	1	yes	characteristic
135	C₄²	#11719922	1	2	yes	—
136	C₁₆	—	106	4	no	—
137	C₂³	C₂×#11719925	1	1	yes	characteristic
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	#13133150	1	2	yes	—
140	C₂×C₄	#13133150	1	2	yes	—
141	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
142	C₂×C₄	#13133204	1	1	yes	characteristic
143	C₂×C₄	#13133206	1	1	yes	characteristic
144	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
145	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
146	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
147	C₈	—	8	2	no	—
148	C₈	—	424	2	no	—
149	C₈	#13133148	1	2	yes	—
150	C₈	#13133148	1	2	yes	—
151	C₈	#13133175	1	2	yes	—
152	C₈	#13133208	1	2	yes	—
153	C₂²	—	2	2	no	—
154	C₂²	C₂×#13133150	1	1	yes	characteristic
155	C₂²	#16810973	1	1	yes	characteristic
156	C₂²	#16810975	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	#16810452	1	4	yes	—
159	C₄	#16810488	1	1	yes	characteristic
160	C₄	C₂×#13133148	1	1	yes	characteristic
161	C₄	#16810969	1	1	yes	characteristic
162	C₄	#16810971	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	#20274842	1	1	yes	characteristic
166	C₂	C₂×#16810452	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1848	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1848	D₅₃
hypocenter	C₅₃	#12891
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
98. C₅₃	#12891	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
#20274887	2
#20272808	2
C₅₃×#1848	2
#12891	53