Subgroups — C₂×D₁₆×Dic₅₃ #27537353

Order 13568.

806 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×Dic₅₃	C₂	1	2	yes	maximal, complemented
2	D₁₆×Dic₅₃	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810449	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16810424	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×D₄×Dic₅₃	—	2	2	no	—
8	C₂²×#11719908	C₂²	1	1	yes	characteristic
9	D₈×Dic₅₃	—	2	4	no	—
10	D₈×Dic₅₃	C₂²	1	8	yes	complemented
11	C₂×#13133151	C₂²	1	2	yes	—
12	C₂×#13133138	—	2	2	no	—
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810449	C₂²	1	4	yes	complemented
15	#16810424	C₂²	1	8	yes	complemented
16	C₁₆×Dic₅₃	C₂²	1	4	yes	complemented
17	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
18	C₂×D₁₆×C₅₃	C₄	1	8	yes	complemented
19	C₂×D₁₆×C₅₃	C₂²	1	4	yes	—
20	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₂³×Dic₅₃	—	4	2	no	—
22	C₂²×#88322	D₄	1	1	yes	characteristic
23	D₄×Dic₅₃	—	2	8	no	—
24	D₄×Dic₅₃	—	4	4	no	—
25	C₂×C₄⋊Dic₅₃	—	2	2	no	—
26	C₂×C₂²⋊Dic₅₃	—	4	2	no	—
27	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
28	C₂×#11719908	—	2	2	no	—
29	C₂×#11719908	C₂³	1	2	yes	—
30	#13133151	—	2	2	no	—
31	#13133151	C₂³	1	4	yes	—
32	#13133138	—	2	8	no	—
33	C₈×Dic₅₃	D₄	1	2	yes	—
34	C₈×Dic₅₃	C₂³	1	2	yes	—
35	C₂²×D₄×C₅₃	—	2	2	no	—
36	C₂×D₈×C₅₃	—	2	2	no	—
37	C₂×D₈×C₅₃	—	2	4	no	—
38	C₂×D₈×C₅₃	C₂×C₄	1	8	yes	complemented
39	C₂×D₈×C₅₃	C₂³	1	4	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	D₁₆×C₅₃	C₂×C₄	1	16	yes	complemented
42	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
43	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
44	C₂⁴×C₅₃	—	4	2	no	—
45	C₂×D₄×C₅₃	—	2	8	no	—
46	C₂×D₄×C₅₃	—	2	4	no	—
47	C₂×D₄×C₅₃	—	4	2	no	—
48	C₂×D₄×C₅₃	—	4	4	no	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₅₃	—	4	2	no	—
51	C₂²×Dic₅₃	—	4	8	no	—
52	C₂²×Dic₅₃	—	8	8	no	—
53	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
54	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
55	D₈×C₅₃	—	2	8	no	—
56	D₈×C₅₃	—	2	4	no	—
57	D₈×C₅₃	C₂²×C₄	1	8	yes	complemented
58	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	#41	1	2	yes	—
61	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
62	C₂²⋊Dic₅₃	—	4	8	no	—
63	C₄⋊Dic₅₃	—	2	4	no	—
64	C₄⋊Dic₅₃	—	4	2	no	—
65	C₄×Dic₅₃	D₈	1	2	yes	—
66	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
67	C₂×#88322	—	2	2	no	—
68	C₂×#88322	C₂×D₄	1	2	yes	—
69	C₁₆×C₅₃	C₂²×C₄	1	4	yes	complemented
70	#11719908	—	2	2	no	—
71	C₂³×C₅₃	—	4	4	no	—
72	C₂³×C₅₃	—	4	8	no	—
73	C₂³×C₅₃	—	8	4	no	—
74	C₂³×C₅₃	—	8	4	no	—
75	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
76	D₄×C₅₃	—	2	8	no	—
77	D₄×C₅₃	—	4	4	no	—
78	D₄×C₅₃	—	4	8	no	—
79	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#135	1	2	yes	—
82	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₅₃	—	2	2	no	—
84	C₂×Dic₅₃	—	4	4	no	—
85	C₂×Dic₅₃	—	8	4	no	—
86	C₂×Dic₅₃	—	8	16	no	—
87	C₂×Dic₅₃	D₁₆	1	4	yes	direct product factor
88	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
89	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
90	C₈×C₅₃	C₄×D₄	1	2	yes	—
91	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
92	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
93	#88322	—	2	2	no	—
94	C₂×C₄×D₁₆	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	—	4	8	no	—
96	C₂²×C₅₃	—	8	4	no	—
97	C₂²×C₅₃	—	8	4	no	—
98	C₂²×C₅₃	—	8	16	no	—
99	C₂²×C₅₃	C₄×D₈	1	2	yes	—
100	C₂²×C₅₃	C₂×D₁₆	1	2	yes	—
101	C₂²×C₅₃	#508	1	2	yes	—
102	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
103	C₄×C₅₃	C₄×D₈	1	2	yes	—
104	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
105	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
106	Dic₅₃	—	2	2	no	—
107	Dic₅₃	—	8	4	no	—
108	Dic₅₃	C₂×D₁₆	1	4	yes	direct product factor
109	C₂²×D₁₆	D₅₃	1	1	yes	characteristic
110	C₂×C₄×D₈	—	53	2	no	—
111	C₄×D₁₆	—	53	8	no	—
112	C₂×#366	—	53	1	no	—
113	C₂×#357	—	53	2	no	—
114	C₂×C₄×C₁₆	—	53	1	no	—
115	C₂×C₅₃	—	8	8	no	—
116	C₂×C₅₃	C₄×D₁₆	1	4	yes	complemented
117	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
118	C₂×C₅₃	C₂²×D₁₆	1	1	yes	characteristic
119	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
120	C₂²×D₈	C₂×D₅₃	1	2	yes	—
121	C₂×C₄×D₄	—	106	2	no	—
122	C₂×D₁₆	Dic₅₃	1	8	yes	direct product factor
123	C₂×D₁₆	C₂×D₅₃	1	4	yes	—
124	C₂²×C₁₆	—	53	1	no	—
125	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
126	C₄×D₈	—	53	8	no	—
127	C₄×D₈	—	106	4	no	—
128	C₂×#107	—	53	2	no	—
129	C₂×#102	—	106	2	no	—
130	C₂×C₄×C₈	—	53	1	no	—
131	#366	—	53	4	no	—
132	#357	—	53	8	no	—
133	C₄×C₁₆	—	53	4	no	—
134	C₅₃	C₂×C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
135	C₂²×D₄	—	2	2	no	—
136	C₂³×C₄	—	212	2	no	—
137	C₂×D₈	—	2	2	no	—
138	C₂×D₈	—	2	4	no	—
139	C₂×D₈	C₂×Dic₅₃	1	8	yes	complemented
140	C₂×D₈	C₂²×D₅₃	1	4	yes	—
141	C₂²×C₈	—	53	1	no	—
142	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
143	C₄×D₄	—	106	8	no	—
144	C₄×D₄	—	212	4	no	—
145	C₂×C₄⋊C₄	—	106	2	no	—
146	C₂×C₂²⋊C₄	—	212	2	no	—
147	C₂×C₄²	—	53	1	no	—
148	D₁₆	C₂×Dic₅₃	1	16	yes	direct product factor
149	C₂×C₁₆	—	53	2	no	—
150	C₂×C₁₆	—	106	2	no	—
151	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	complemented
152	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
153	#107	—	53	4	no	—
154	#107	—	106	2	no	—
155	#102	—	106	8	no	—
156	C₄×C₈	—	53	2	no	—
157	C₄×C₈	—	53	2	no	—
158	C₂⁴	—	4	2	no	—
159	C₂×D₄	—	2	8	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂²×C₄	—	53	1	no	—
164	C₂²×C₄	—	53	1	no	—
165	C₂²×C₄	—	424	8	no	—
166	C₂²×C₄	—	212	8	no	—
167	C₂²×C₄	—	212	2	no	—
168	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
169	D₈	—	2	8	no	—
170	D₈	—	2	4	no	—
171	D₈	C₂²×Dic₅₃	1	8	yes	complemented
172	C₂×C₈	—	53	2	no	—
173	C₂×C₈	—	106	2	no	—
174	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
175	C₂×C₈	#11719947	1	2	yes	—
176	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
177	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
178	C₄⋊C₄	—	106	4	no	—
179	C₄⋊C₄	—	212	2	no	—
180	C₂²⋊C₄	—	212	8	no	—
181	C₄²	—	53	2	no	—
182	C₄²	—	53	2	no	—
183	C₁₆	—	106	2	no	—
184	C₁₆	C₂²×Dic₅₃	1	4	yes	complemented
185	C₂³	—	4	4	no	—
186	C₂³	—	4	8	no	—
187	C₂³	—	8	4	no	—
188	C₂³	—	8	4	no	—
189	C₂³	D₈×D₅₃	1	1	yes	characteristic
190	D₄	—	2	8	no	—
191	D₄	—	4	4	no	—
192	D₄	—	4	8	no	—
193	C₂×C₄	—	53	4	no	—
194	C₂×C₄	—	53	2	no	—
195	C₂×C₄	—	53	2	no	—
196	C₂×C₄	—	106	2	no	—
197	C₂×C₄	—	424	16	no	—
198	C₂×C₄	—	424	4	no	—
199	C₂×C₄	—	212	4	no	—
200	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
201	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
202	C₂×C₄	#13133236	1	2	yes	—
203	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
204	C₈	—	106	2	no	—
205	C₈	D₄×Dic₅₃	1	2	yes	—
206	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
207	C₈	C₂×#11719947	1	1	yes	characteristic
208	C₂²	—	4	8	no	—
209	C₂²	—	8	4	no	—
210	C₂²	—	8	16	no	—
211	C₂²	—	8	4	no	—
212	C₂²	D₈×Dic₅₃	1	2	yes	—
213	C₂²	D₁₆×D₅₃	1	2	yes	—
214	C₂²	#16811052	1	2	yes	—
215	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
216	C₄	—	53	4	no	—
217	C₄	—	106	2	no	—
218	C₄	—	424	4	no	—
219	C₄	D₈×Dic₅₃	1	2	yes	—
220	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
221	C₄	C₂×#13133236	1	1	yes	characteristic
222	C₂	—	8	8	no	—
223	C₂	D₁₆×Dic₅₃	1	4	yes	direct product factor
224	C₂	C₂×D₈×Dic₅₃	1	1	yes	characteristic
225	C₂	C₂×D₁₆×D₅₃	1	1	yes	characteristic
226	C₂	C₂×#16811052	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting, prime radical	C₂²×D₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×D₁₆	D₅₃
hypocenter	C₅₃	C₂×C₄×D₁₆
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₈×Dic₅₃	C₂	semidirect
2. D₁₆×Dic₅₃	C₂	direct (N × H)
3. C₂×#16810449	C₂	semidirect
4. C₂×#16810424	C₂	semidirect
5. C₂×C₁₆×Dic₅₃	C₂	semidirect
10. D₈×Dic₅₃	C₂²	semidirect
14. #16810449	C₂²	semidirect
15. #16810424	C₂²	semidirect
16. C₁₆×Dic₅₃	C₂²	semidirect
18. C₂×D₁₆×C₅₃	C₄	semidirect
38. C₂×D₈×C₅₃	C₂×C₄	semidirect
41. D₁₆×C₅₃	C₂×C₄	semidirect
42. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
57. D₈×C₅₃	C₂²×C₄	semidirect
69. C₁₆×C₅₃	C₂²×C₄	semidirect
87. C₂×Dic₅₃	D₁₆	direct (N × H)
108. Dic₅₃	C₂×D₁₆	direct (N × H)
116. C₂×C₅₃	C₄×D₁₆	semidirect
122. C₂×D₁₆	Dic₅₃	direct (N × H)
134. C₅₃	C₂×C₄×D₁₆	semidirect
139. C₂×D₈	C₂×Dic₅₃	semidirect
148. D₁₆	C₂×Dic₅₃	direct (N × H)
151. C₂×C₁₆	C₂×Dic₅₃	semidirect
171. D₈	C₂²×Dic₅₃	semidirect
184. C₁₆	C₂²×Dic₅₃	semidirect
223. C₂	D₁₆×Dic₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₈×Dic₅₃	2
D₁₆×Dic₅₃	2
C₂×#16810449	2
C₂×#16810424	2
C₂×C₁₆×Dic₅₃	2
C₂²×D₁₆×C₅₃	2
C₂×C₄×D₁₆	53

Subgroups — C2×D16×Dic53 #27537353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₆×Dic₅₃ #27537353