Subgroups — #346×Dic₅₃ #27298912

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810398	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₅₃	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
13	#16810399	C₄	1	4	yes	complemented
14	#16810398	C₄	1	8	yes	complemented
15	C₁₆×Dic₅₃	C₄	1	8	yes	complemented
16	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#346	C₄	1	4	yes	complemented
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₈	1	4	yes	—
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#11719908	C₂×C₄	1	4	yes	—
25	C₂×#11719908	C₂×C₄	1	4	yes	—
26	C₂×#11719908	C₂×C₄	1	4	yes	—
27	C₄×#88322	C₈	1	4	yes	—
28	C₈×Dic₅₃	C₈	1	4	yes	—
29	C₈×Dic₅₃	C₈	1	4	yes	—
30	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
31	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
36	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
37	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
43	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₅₃	C₄²	1	2	yes	—
45	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₄²	1	2	yes	—
52	C₄²×C₅₃	C₂×C₈	1	2	yes	—
53	C₄×Dic₅₃	—	2	8	no	—
54	C₄×Dic₅₃	C₄²	1	2	yes	—
55	C₄×Dic₅₃	C₄²	1	2	yes	—
56	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
57	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
58	C₂×#88322	C₄²	1	2	yes	—
59	C₂×#88322	C₄²	1	2	yes	—
60	C₂×#88322	C₂×C₈	1	2	yes	—
61	C₂×#88322	C₂×C₈	1	2	yes	—
62	C₂×#88322	C₂×C₈	1	2	yes	—
63	C₂×#88322	C₂×C₈	1	2	yes	—
64	C₁₆×C₅₃	C₄²	1	8	yes	complemented
65	#11719908	C₄²	1	8	yes	complemented
66	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
69	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×Dic₅₃	—	2	4	no	—
76	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
77	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
78	C₂×Dic₅₃	M₃₂	1	8	yes	—
79	C₈×C₅₃	C₄×C₈	1	4	yes	—
80	C₈×C₅₃	C₂×C₄²	1	2	yes	—
81	C₈×C₅₃	C₂×C₄²	1	2	yes	—
82	#88322	C₄×C₈	1	4	yes	—
83	#88322	C₄×C₈	1	4	yes	—
84	C₄×#346	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
86	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
88	C₂²×C₅₃	C₂×M₃₂	1	4	yes	—
89	C₄×C₅₃	—	2	4	no	—
90	C₄×C₅₃	C₄³	1	1	yes	characteristic
91	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
92	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
93	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
94	Dic₅₃	—	2	4	no	—
95	Dic₅₃	#346	1	8	yes	direct product factor
96	C₂×#346	—	53	2	no	—
97	C₂×#346	—	53	1	no	—
98	C₂×#346	D₅₃	1	1	yes	characteristic
99	C₂×C₄×C₁₆	—	53	2	no	—
100	C₄²×C₈	—	53	1	no	—
101	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
102	C₂×C₅₃	C₂×#346	1	2	yes	—
103	C₂×C₅₃	C₄×M₃₂	1	2	yes	—
104	C₂×C₅₃	C₄×M₃₂	1	2	yes	—
105	C₂²×C₁₆	—	53	2	no	—
106	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
107	C₂×C₄×C₈	—	53	1	no	—
108	C₂×C₄×C₈	—	53	1	no	—
109	C₂×C₄×C₈	—	53	1	no	—
110	C₂×C₄×C₈	—	53	1	no	—
111	C₂×C₄×C₈	—	53	1	no	—
112	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
113	C₄³	—	53	1	no	—
114	#346	—	53	8	no	—
115	#346	—	53	4	no	—
116	#346	Dic₅₃	1	4	yes	direct product factor
117	C₄×C₁₆	—	53	8	no	—
118	C₅₃	C₄×#346	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₈	—	53	1	no	—
120	C₂²×C₈	—	53	1	no	—
121	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
123	C₂×C₄²	—	53	1	no	—
124	C₂×C₄²	—	53	1	no	—
125	C₂×C₄²	—	53	4	no	—
126	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
127	C₂×C₁₆	—	53	4	no	—
128	C₂×C₁₆	—	53	4	no	—
129	C₂×C₁₆	—	53	4	no	—
130	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
131	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
132	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
133	C₄×C₈	—	53	4	no	—
134	C₄×C₈	—	53	4	no	—
135	C₄×C₈	—	53	4	no	—
136	C₄×C₈	—	53	4	no	—
137	C₄×C₈	—	53	4	no	—
138	C₄×C₈	C₄×D₅₃	1	2	yes	—
139	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
140	C₂²×C₄	—	53	2	no	—
141	C₂²×C₄	—	53	2	no	—
142	C₂²×C₄	C₈×D₅₃	1	2	yes	—
143	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
144	C₂×C₈	—	53	2	no	—
145	C₂×C₈	—	53	2	no	—
146	C₂×C₈	—	53	2	no	—
147	C₂×C₈	—	53	2	no	—
148	C₂×C₈	—	53	2	no	—
149	C₂×C₈	—	53	2	no	—
150	C₂×C₈	C₈×D₅₃	1	2	yes	—
151	C₂×C₈	C₈×D₅₃	1	2	yes	—
152	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
153	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
154	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
155	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
156	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
157	C₄²	—	53	2	no	—
158	C₄²	—	53	2	no	—
159	C₄²	—	53	2	no	—
160	C₄²	—	53	2	no	—
161	C₄²	—	106	8	no	—
162	C₄²	C₈×D₅₃	1	2	yes	—
163	C₄²	C₄×Dic₅₃	1	2	yes	—
164	C₁₆	—	53	8	no	—
165	C₁₆	C₄×Dic₅₃	1	8	yes	complemented
166	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	53	8	no	—
169	C₂×C₄	—	53	4	no	—
170	C₂×C₄	—	53	4	no	—
171	C₂×C₄	—	106	4	no	—
172	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
173	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
174	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
176	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
177	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
178	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
179	C₈	—	53	4	no	—
180	C₈	—	53	4	no	—
181	C₈	C₈×Dic₅₃	1	4	yes	—
182	C₈	C₄²×D₅₃	1	2	yes	—
183	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
184	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
185	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
186	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
187	C₂²	M₃₂×D₅₃	1	4	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	53	8	no	—
190	C₄	—	106	4	no	—
191	C₄	C₄²×Dic₅₃	1	1	yes	characteristic
192	C₄	C₄×C₈×D₅₃	1	1	yes	characteristic
193	C₄	C₄×C₈×D₅₃	1	1	yes	characteristic
194	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
195	C₂	C₄×C₈×Dic₅₃	1	1	yes	characteristic
196	C₂	#346×D₅₃	1	2	yes	—
197	C₂	M₃₂×Dic₅₃	1	2	yes	—
198	C₂	#20276308	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂×C₅₃	C₄²×C₈
Fitting	C₂×C₅₃×#346	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	C₄×#346
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
13. #16810399	C₄	semidirect
14. #16810398	C₄	semidirect
15. C₁₆×Dic₅₃	C₄	semidirect
18. C₅₃×#346	C₄	semidirect
64. C₁₆×C₅₃	C₄²	semidirect
65. #11719908	C₄²	semidirect
95. Dic₅₃	#346	direct (N × H)
116. #346	Dic₅₃	direct (N × H)
118. C₅₃	C₄×#346	semidirect
165. C₁₆	C₄×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810399	2
C₂×#16810398	2
C₂×C₁₆×Dic₅₃	2
C₄×C₈×Dic₅₃	2
C₂×C₅₃×#346	2
C₄×#346	53

Subgroups — #346×Dic53 #27298912

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #346×Dic₅₃ #27298912