Subgroups — C₄×#16810380 #27289257

Order 13568.

954 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133176	C₂	1	2	yes	maximal
2	C₂×#16810380	C₂	1	4	yes	maximal
3	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133176	C₂²	1	4	yes	—
5	C₄×C₄⋊Dic₅₃	—	2	4	no	—
6	C₄×C₂²⋊Dic₅₃	—	2	4	no	—
7	#16810469	C₄	1	4	yes	complemented
8	C₄²×Dic₅₃	C₄	1	4	yes	complemented
9	#16810380	C₄	1	16	yes	direct product factor
10	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×M₁₆×C₅₃	—	2	4	no	—
13	C₂×#97×C₅₃	C₄	1	2	yes	complemented
14	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	complemented
15	#13133176	—	2	8	no	—
16	#13133176	C₂×C₄	1	16	yes	—
17	C₂×C₄⋊Dic₅₃	—	2	4	no	—
18	C₂×C₂²⋊Dic₅₃	—	2	4	no	—
19	C₂×C₄×Dic₅₃	—	2	8	no	—
20	C₂×C₄×Dic₅₃	C₂×C₄	1	8	yes	—
21	#13133124	—	2	8	no	—
22	C₂³×C₄×C₅₃	D₄	1	2	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₅₃	—	2	4	no	—
25	C₂×M₁₆×C₅₃	—	2	4	no	—
26	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	D₄	1	4	yes	—
33	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
34	#97×C₅₃	—	2	4	no	—
35	C₄×C₈×C₅₃	—	2	4	no	—
36	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	—	4	4	no	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	#41	1	4	yes	—
48	C₂²×Dic₅₃	—	2	8	no	—
49	M₁₆×C₅₃	—	2	16	no	—
50	C₂×C₈×C₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	—	2	4	no	—
52	C₂×C₈×C₅₃	C₄²	1	8	yes	complemented
53	C₄²×C₅₃	C₄²	1	2	yes	—
54	C₄²×C₅₃	C₄²	1	1	yes	characteristic
55	C₄²×C₅₃	C₄²	1	1	yes	characteristic
56	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
59	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
60	C₂²⋊Dic₅₃	—	2	16	no	—
61	C₄⋊Dic₅₃	—	2	16	no	—
62	C₄×Dic₅₃	—	2	8	no	—
63	C₄×Dic₅₃	—	4	16	no	—
64	C₄×Dic₅₃	C₄²	1	16	yes	complemented
65	C₂³×C₅₃	—	4	2	no	—
66	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
67	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂³×C₅₃	C₄×D₄	1	4	yes	—
69	C₂×C₄×C₅₃	—	2	8	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	4	4	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	#95	1	4	yes	—
74	C₂×C₄×C₅₃	#95	1	2	yes	—
75	C₂×C₄×C₅₃	#95	1	2	yes	—
76	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
77	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₅₃	#117	1	4	yes	—
82	C₂×C₄×C₅₃	#117	1	2	yes	—
83	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
84	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
85	C₂×Dic₅₃	—	2	16	no	—
86	C₂×Dic₅₃	—	4	16	no	—
87	C₈×C₅₃	—	2	8	no	—
88	C₄×H(3,ℤ/4)	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	—	2	4	no	—
90	C₂²×C₅₃	—	2	2	no	—
91	C₂²×C₅₃	—	4	4	no	—
92	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
93	C₂²×C₅₃	C₄×C₂²⋊C₄	1	4	yes	—
94	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	H(3,ℤ/4)	1	4	yes	complemented
98	C₄×C₅₃	C₄³	1	1	yes	characteristic
99	C₄×C₅₃	#376	1	1	yes	characteristic
100	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
101	C₄×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
102	Dic₅₃	—	4	16	no	—
103	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
104	C₄×#117	—	53	2	no	—
105	C₂×H(3,ℤ/4)	—	53	4	no	—
106	C₂×C₅₃	—	2	4	no	—
107	C₂×C₅₃	—	2	2	no	—
108	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
109	C₂×C₅₃	C₂×H(3,ℤ/4)	1	2	yes	—
110	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
111	C₂×#117	—	53	4	no	—
112	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
113	C₄×M₁₆	—	2	4	no	—
114	C₂×#97	Dic₅₃	1	2	yes	complemented
115	C₂×C₄×C₈	Dic₅₃	1	2	yes	complemented
116	C₄×C₄⋊C₄	—	106	4	no	—
117	C₄×C₂²⋊C₄	—	106	4	no	—
118	#376	—	53	4	no	—
119	C₄³	—	53	4	no	—
120	H(3,ℤ/4)	—	53	16	no	—
121	C₅₃	C₄×H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
122	C₂³×C₄	D₂₁₂	1	2	yes	—
123	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
124	C₂×M₁₆	—	2	4	no	—
125	C₂×M₁₆	—	2	4	no	—
126	C₂×M₁₆	C₄×D₅₃	1	8	yes	—
127	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
128	#117	—	53	16	no	—
129	#117	—	106	8	no	—
130	C₂×C₄⋊C₄	—	106	4	no	—
131	C₂×C₂²⋊C₄	—	106	4	no	—
132	C₂×C₄²	—	53	8	no	—
133	C₂×C₄²	—	106	8	no	—
134	C₂×C₄²	Dic₁₀₆	1	2	yes	—
135	C₂×C₄²	C₄×D₅₃	1	2	yes	—
136	C₂×C₄²	C₄×D₅₃	1	2	yes	—
137	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
138	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
139	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
140	#97	—	2	4	no	—
141	C₄×C₈	—	2	4	no	—
142	#95	—	106	8	no	—
143	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
144	C₂²×C₄	—	4	4	no	—
145	C₂²×C₄	—	106	8	no	—
146	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
147	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
148	C₂²×C₄	#11719921	1	4	yes	—
149	C₂²×C₄	#11719921	1	4	yes	—
150	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
153	C₂²×C₄	#11719945	1	4	yes	—
154	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
156	M₁₆	—	2	16	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	C₄×Dic₅₃	1	8	yes	complemented
160	C₄⋊C₄	—	106	16	no	—
161	C₂²⋊C₄	—	106	16	no	—
162	C₄²	—	53	16	no	—
163	C₄²	—	106	8	no	—
164	C₄²	—	212	16	no	—
165	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
166	C₄²	C₄×Dic₅₃	1	2	yes	—
167	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
168	C₄²	#11719919	1	4	yes	—
169	C₄²	#11719919	1	4	yes	—
170	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
171	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
172	C₂³	—	4	2	no	—
173	C₂³	C₂×#11719921	1	2	yes	—
174	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
175	C₂³	C₄×D₂₁₂	1	4	yes	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	106	16	no	—
181	C₂×C₄	—	212	16	no	—
182	C₂×C₄	#13133124	1	2	yes	—
183	C₂×C₄	#13133124	1	4	yes	—
184	C₂×C₄	#13133124	1	2	yes	—
185	C₂×C₄	C₄²×D₅₃	1	4	yes	—
186	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
188	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
189	C₂×C₄	C₂×#11719919	1	2	yes	—
190	C₂×C₄	#13133174	1	4	yes	—
191	C₂×C₄	#13133176	1	2	yes	—
192	C₂×C₄	C₄×C₂²⋊D₅₃	1	8	yes	—
193	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
194	C₈	—	2	8	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	4	4	no	—
198	C₂²	C₂×#13133124	1	1	yes	characteristic
199	C₂²	C₄×#11719921	1	4	yes	—
200	C₂²	C₄×C₄⋊Dic₅₃	1	2	yes	—
201	C₄	—	2	4	no	—
202	C₄	—	2	2	no	—
203	C₄	—	212	16	no	—
204	C₄	#16810380	1	4	yes	direct product factor
205	C₄	C₄²×Dic₅₃	1	1	yes	characteristic
206	C₄	#16810469	1	1	yes	characteristic
207	C₄	C₄×C₂²⋊Dic₅₃	1	2	yes	—
208	C₄	C₄×#11719919	1	4	yes	—
209	C₂	—	2	4	no	—
210	C₂	—	2	2	no	—
211	C₂	C₄×#13133124	1	1	yes	characteristic
212	C₂	C₂×#16810380	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄³
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	C₄×H(3,ℤ/4)
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
7. #16810469	C₄	semidirect
8. C₄²×Dic₅₃	C₄	semidirect
9. #16810380	C₄	direct (N × H)
13. C₂×#97×C₅₃	C₄	semidirect
14. C₂×C₄×C₈×C₅₃	C₄	semidirect
52. C₂×C₈×C₅₃	C₄²	semidirect
64. C₄×Dic₅₃	C₄²	semidirect
97. C₄×C₅₃	H(3,ℤ/4)	semidirect
114. C₂×#97	Dic₅₃	semidirect
115. C₂×C₄×C₈	Dic₅₃	semidirect
121. C₅₃	C₄×H(3,ℤ/4)	semidirect
159. C₂×C₈	C₄×Dic₅₃	semidirect
204. C₄	#16810380	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#13133176	2
C₂×#16810380	2
C₂×C₄×M₁₆×C₅₃	2
C₄×H(3,ℤ/4)	53

Subgroups — C4×#16810380 #27289257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810380 #27289257