Subgroups — #368×Dic₅₃ #27299506

Order 13568.

314 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810453	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#108×Dic₅₃	C₂	1	2	yes	maximal
4	#20272808	C₂	1	2	yes	maximal
5	C₂×C₅₃×#368	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	1	yes	characteristic
7	C₂×#13133153	C₂²	1	2	yes	—
8	M₁₆×Dic₅₃	—	2	2	no	—
9	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
10	#16810453	C₄	1	4	yes	complemented
11	C₁₆×Dic₅₃	C₄	1	4	yes	—
12	#16810387	—	2	2	no	—
13	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#108×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#368	C₄	1	4	yes	complemented
16	C₂×#11719917	—	2	2	no	—
17	C₂²×#88322	D₄	1	1	yes	characteristic
18	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
19	C₂×#11719908	—	2	1	no	—
20	C₂×#11719908	—	2	1	no	—
21	C₂×#11719908	C₂×C₄	1	2	yes	—
22	#13133153	—	2	2	no	—
23	#13133153	C₂×C₄	1	4	yes	—
24	#13133127	—	4	2	no	—
25	C₈×Dic₅₃	—	4	2	no	—
26	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
27	C₈×Dic₅₃	Q₈	1	2	yes	—
28	C₂×M₁₆×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
33	#108×C₅₃	—	2	2	no	—
34	#108×C₅₃	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₅₃	Q₁₆	1	2	yes	—
37	M₁₆×C₅₃	—	2	4	no	—
38	M₁₆×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	—	4	2	no	—
40	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
45	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
46	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
47	C₄×Dic₅₃	D₈	1	1	yes	characteristic
48	C₄×Dic₅₃	D₈	1	1	yes	characteristic
49	#11719917	—	2	4	no	—
50	#11719917	—	4	2	no	—
51	C₂×#88322	—	2	1	no	—
52	C₂×#88322	—	2	1	no	—
53	C₂×#88322	—	4	2	no	—
54	C₂×#88322	C₄⋊C₄	1	2	yes	—
55	C₁₆×C₅₃	C₄²	1	4	yes	—
56	#11719908	—	2	2	no	—
57	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
64	C₂×Dic₅₃	—	2	4	no	—
65	C₂×Dic₅₃	#107	1	2	yes	—
66	C₂×Dic₅₃	#107	1	2	yes	—
67	C₈×C₅₃	—	4	4	no	—
68	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₅₃	#117	1	1	yes	characteristic
70	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
71	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
72	#88322	—	2	2	no	—
73	#88322	—	4	4	no	—
74	C₄×#368	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	—	2	2	no	—
76	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
77	C₂²×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
78	C₂²×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
79	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₅₃	#427	1	1	yes	characteristic
81	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
82	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
83	Dic₅₃	—	2	2	no	—
84	Dic₅₃	#368	1	4	yes	direct product factor
85	C₂×#368	—	53	1	no	—
86	C₂×#368	D₅₃	1	1	yes	characteristic
87	C₂×C₄×C₁₆	—	53	1	no	—
88	C₄×#108	—	53	2	no	—
89	#1381	—	53	2	no	—
90	C₂×C₅₃	—	2	1	no	—
91	C₂×C₅₃	—	2	1	no	—
92	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
93	C₂×C₅₃	C₂×#368	1	2	yes	—
94	C₂²×C₁₆	—	53	1	no	—
95	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
96	C₂×#108	—	53	2	no	—
97	C₂×#108	C₂×D₅₃	1	2	yes	—
98	C₄×M₁₆	—	106	2	no	—
99	C₂×C₄×C₈	—	53	1	no	—
100	#368	—	53	4	no	—
101	#368	Dic₅₃	1	4	yes	direct product factor
102	C₄×C₁₆	—	53	4	no	—
103	#341	—	106	2	no	—
104	C₅₃	C₄×#368	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₁₆	—	2	2	no	—
106	C₂×M₁₆	—	106	2	no	—
107	C₂²×C₈	—	53	1	no	—
108	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
109	C₂×C₄²	—	53	1	no	—
110	C₂×C₁₆	—	53	2	no	—
111	C₂×C₁₆	—	106	1	no	—
112	C₂×C₁₆	—	106	1	no	—
113	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
114	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
115	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
116	#108	—	2	2	no	—
117	#108	—	53	4	no	—
118	#108	—	106	2	no	—
119	#108	C₂×Dic₅₃	1	4	yes	—
120	#97	—	212	2	no	—
121	C₄×C₈	—	53	2	no	—
122	C₄×C₈	—	53	2	no	—
123	C₄×C₈	—	212	2	no	—
124	C₂²×C₄	—	53	2	no	—
125	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
126	M₁₆	—	2	4	no	—
127	M₁₆	—	4	2	no	—
128	M₁₆	—	106	4	no	—
129	M₁₆	—	212	2	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	53	2	no	—
132	C₂×C₈	—	106	1	no	—
133	C₂×C₈	—	106	1	no	—
134	C₂×C₈	—	212	2	no	—
135	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
137	C₂×C₈	#11719947	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
139	C₂×C₈	#11719949	1	1	yes	characteristic
140	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
141	C₄²	—	53	2	no	—
142	C₄²	—	53	1	no	—
143	C₄²	—	53	1	no	—
144	C₁₆	—	106	2	no	—
145	C₁₆	C₄×Dic₅₃	1	4	yes	—
146	C₂³	Q₁₆×D₅₃	1	1	yes	characteristic
147	C₂×C₄	—	53	2	no	—
148	C₂×C₄	—	53	2	no	—
149	C₂×C₄	—	106	4	no	—
150	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
151	C₂×C₄	#13133179	1	1	yes	characteristic
152	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
153	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
154	C₂×C₄	#13133236	1	1	yes	characteristic
155	C₂×C₄	#13133240	1	1	yes	characteristic
156	C₈	—	4	4	no	—
157	C₈	—	106	2	no	—
158	C₈	—	212	4	no	—
159	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
160	C₈	#13133175	1	1	yes	characteristic
161	C₈	#13133182	1	1	yes	characteristic
162	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	#107×D₅₃	1	1	yes	characteristic
165	C₂²	#16810717	1	1	yes	characteristic
166	C₂²	Q₁₆×Dic₅₃	1	1	yes	characteristic
167	C₄	—	53	4	no	—
168	C₄	—	106	2	no	—
169	C₄	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
170	C₄	#16810660	1	1	yes	characteristic
171	C₄	#16810705	1	1	yes	characteristic
172	C₄	D₈×Dic₅₃	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#107×Dic₅₃	1	1	yes	characteristic
176	C₂	#368×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#368	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#368	D₅₃
hypocenter	C₅₃	C₄×#368
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
10. #16810453	C₄	semidirect
15. C₅₃×#368	C₄	semidirect
84. Dic₅₃	#368	direct (N × H)
101. #368	Dic₅₃	direct (N × H)
104. C₅₃	C₄×#368	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810453	2
C₂×C₁₆×Dic₅₃	2
#108×Dic₅₃	2
#20272808	2
C₂×C₅₃×#368	2
C₄×#368	53

Subgroups — #368×Dic53 #27299506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #368×Dic₅₃ #27299506