Subgroups — C₂×2₋¹⁺⁴×Dic₅₃ #28290702

Order 13568.

11210 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×Dic₅₃	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16811106	C₂	1	10	yes	maximal, complemented
3	C₂×#16811101	C₂	1	5	yes	maximal, complemented
4	C₂×#41×Dic₅₃	C₂	1	10	yes	maximal, complemented
5	C₂²×Q₈×Dic₅₃	C₂	1	5	yes	maximal, complemented
6	C₂²×2₋¹⁺⁴×C₅₃	C₂	1	1	yes	maximal, characteristic
7	#16811106	C₂²	1	160	yes	complemented
8	#16811101	C₂²	1	80	yes	complemented
9	#41×Dic₅₃	C₂²	1	160	yes	complemented
10	C₂×Q₈×Dic₅₃	C₂²	1	10	yes	—
11	C₂×Q₈×Dic₅₃	C₂²	1	20	yes	—
12	C₂×Q₈×Dic₅₃	C₂²	1	10	yes	—
13	C₂×Q₈×Dic₅₃	C₂²	1	80	yes	complemented
14	C₂×D₄×Dic₅₃	C₂²	1	10	yes	—
15	C₂×D₄×Dic₅₃	C₂²	1	30	yes	—
16	C₂×#13133176	C₂²	1	30	yes	—
17	C₂²×C₄⋊Dic₅₃	C₂²	1	15	yes	—
18	C₂²×C₄×Dic₅₃	C₂²	1	15	yes	—
19	C₂×2₋¹⁺⁴×C₅₃	C₄	1	32	yes	complemented
20	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	16	yes	—
21	C₂²×#41×C₅₃	C₂²	1	10	yes	—
22	C₂³×Q₈×C₅₃	C₂²	1	5	yes	—
23	C₂³×Dic₅₃	C₂³	1	5	yes	—
24	C₂³×Dic₅₃	C₂³	1	15	yes	—
25	Q₈×Dic₅₃	C₂³	1	80	yes	—
26	Q₈×Dic₅₃	C₂³	1	160	yes	—
27	Q₈×Dic₅₃	C₂³	1	80	yes	—
28	D₄×Dic₅₃	C₂³	1	80	yes	—
29	D₄×Dic₅₃	C₂³	1	240	yes	—
30	#13133176	C₂³	1	240	yes	—
31	C₂×C₄⋊Dic₅₃	C₂³	1	30	yes	—
32	C₂×C₄⋊Dic₅₃	C₂³	1	120	yes	—
33	C₂×C₄⋊Dic₅₃	C₂³	1	10	yes	—
34	C₂×C₂²⋊Dic₅₃	C₂³	1	20	yes	—
35	C₂×C₄×Dic₅₃	C₂³	1	10	yes	—
36	C₂×C₄×Dic₅₃	C₂³	1	30	yes	—
37	C₂×C₄×Dic₅₃	C₂³	1	120	yes	—
38	2₋¹⁺⁴×C₅₃	C₂×C₄	1	256	yes	complemented
39	C₂×#41×C₅₃	C₂×C₄	1	160	yes	complemented
40	C₂×#41×C₅₃	C₂³	1	80	yes	—
41	C₂²×Q₈×C₅₃	C₂×C₄	1	80	yes	complemented
42	C₂²×Q₈×C₅₃	C₂³	1	10	yes	—
43	C₂²×Q₈×C₅₃	C₂³	1	40	yes	—
44	C₂²×D₄×C₅₃	C₂³	1	10	yes	—
45	C₂³×C₄×C₅₃	C₂³	1	15	yes	—
46	C₂⁴×C₅₃	C₂⁴	1	5	yes	—
47	#41×C₅₃	C₂²×C₄	1	640	yes	complemented
48	C₂×Q₈×C₅₃	C₂²×C₄	1	80	yes	—
49	C₂×Q₈×C₅₃	C₂²×C₄	1	320	yes	complemented
50	C₂×Q₈×C₅₃	C₂⁴	1	40	yes	—
51	C₂×D₄×C₅₃	C₂²×C₄	1	80	yes	—
52	C₂×D₄×C₅₃	C₂⁴	1	40	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	120	yes	—
54	C₂²×C₄×C₅₃	C₂⁴	1	10	yes	—
55	C₂²×C₄×C₅₃	C₂⁴	1	60	yes	—
56	C₂²×Dic₅₃	—	2	20	no	—
57	C₂²×Dic₅₃	—	4	30	no	—
58	C₂²×Dic₅₃	C₂⁴	1	5	yes	—
59	C₂²×Dic₅₃	C₂⁴	1	20	yes	—
60	C₂²×Dic₅₃	C₂⁴	1	10	yes	—
61	C₂²×Dic₅₃	C₂⁴	1	60	yes	—
62	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
63	C₂²⋊Dic₅₃	C₂⁴	1	80	yes	—
64	C₄⋊Dic₅₃	C₂⁴	1	120	yes	—
65	C₄⋊Dic₅₃	C₂⁴	1	40	yes	—
66	C₄×Dic₅₃	C₂⁴	1	40	yes	—
67	C₄×Dic₅₃	C₂⁴	1	120	yes	—
68	C₂³×C₅₃	—	2	10	no	—
69	C₂³×C₅₃	—	2	10	no	—
70	C₂³×C₅₃	C₂³×C₄	1	20	yes	—
71	C₂³×C₅₃	C₂⁵	1	10	yes	—
72	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
73	Q₈×C₅₃	C₂³×C₄	1	160	yes	—
74	D₄×C₅₃	C₂³×C₄	1	160	yes	—
75	C₂×C₄×C₅₃	C₂³×C₄	1	40	yes	—
76	C₂×C₄×C₅₃	C₂³×C₄	1	240	yes	—
77	C₂×C₄×C₅₃	C₂⁵	1	20	yes	—
78	C₂×Dic₅₃	—	2	10	no	—
79	C₂×Dic₅₃	—	2	20	no	—
80	C₂×Dic₅₃	—	2	40	no	—
81	C₂×Dic₅₃	—	4	60	no	—
82	C₂×Dic₅₃	2₋¹⁺⁴	1	4	yes	direct product factor
83	C₂×Dic₅₃	C₂⁵	1	20	yes	—
84	C₂×Dic₅₃	C₂⁵	1	10	yes	—
85	C₂×Dic₅₃	C₂⁵	1	2	yes	—
86	C₂×C₄×2₋¹⁺⁴	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	—	2	10	no	—
88	C₂²×C₅₃	—	2	10	no	—
89	C₂²×C₅₃	—	2	40	no	—
90	C₂²×C₅₃	C₂⁴×C₄	1	20	yes	—
91	C₂²×C₅₃	C₂⁴×C₄	1	2	yes	—
92	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
93	C₂²×C₅₃	#585	1	2	yes	—
94	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
95	C₄×C₅₃	C₂⁴×C₄	1	40	yes	—
96	Dic₅₃	—	2	10	no	—
97	Dic₅₃	—	2	20	no	—
98	Dic₅₃	C₂×2₋¹⁺⁴	1	4	yes	direct product factor
99	C₂²×2₋¹⁺⁴	D₅₃	1	1	yes	characteristic
100	C₄×2₋¹⁺⁴	—	53	32	no	—
101	C₂×#520	—	53	10	no	—
102	C₂×#519	—	53	5	no	—
103	C₂×C₄×#41	—	53	10	no	—
104	C₂²×C₄×Q₈	—	53	5	no	—
105	C₂×C₅₃	—	2	20	no	—
106	C₂×C₅₃	C₄×2₋¹⁺⁴	1	4	yes	complemented
107	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
108	C₂×C₅₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
109	C₂×C₅₃	C₂×#585	1	1	yes	characteristic
110	C₂×2₋¹⁺⁴	Dic₅₃	1	32	yes	direct product factor
111	C₂×2₋¹⁺⁴	C₂×D₅₃	1	16	yes	—
112	C₂²×#41	C₂×D₅₃	1	10	yes	—
113	C₂³×Q₈	C₂×D₅₃	1	5	yes	—
114	#520	—	53	160	no	—
115	#519	—	53	80	no	—
116	C₄×#41	—	53	160	no	—
117	C₂×C₄×Q₈	—	53	10	no	—
118	C₂×C₄×Q₈	—	53	20	no	—
119	C₂×C₄×Q₈	—	53	10	no	—
120	C₂×C₄×Q₈	—	53	80	no	—
121	C₂×C₄×D₄	—	53	10	no	—
122	C₂×C₄×D₄	—	53	30	no	—
123	C₂×#117	—	53	30	no	—
124	C₂²×C₄⋊C₄	—	53	15	no	—
125	C₂²×C₄²	—	53	15	no	—
126	C₅₃	C₂×C₄×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
127	2₋¹⁺⁴	C₂×Dic₅₃	1	256	yes	direct product factor
128	C₂×#41	C₂×Dic₅₃	1	160	yes	complemented
129	C₂×#41	C₂²×D₅₃	1	80	yes	—
130	C₂²×Q₈	C₂×Dic₅₃	1	80	yes	complemented
131	C₂²×Q₈	C₂²×D₅₃	1	10	yes	—
132	C₂²×Q₈	C₂²×D₅₃	1	40	yes	—
133	C₂²×D₄	C₂²×D₅₃	1	10	yes	—
134	C₂³×C₄	—	53	5	no	—
135	C₂³×C₄	—	53	15	no	—
136	C₂³×C₄	C₂²×D₅₃	1	15	yes	—
137	C₄×Q₈	—	53	80	no	—
138	C₄×Q₈	—	53	160	no	—
139	C₄×Q₈	—	53	80	no	—
140	C₄×D₄	—	53	80	no	—
141	C₄×D₄	—	53	240	no	—
142	#117	—	53	240	no	—
143	C₂×C₄⋊C₄	—	53	30	no	—
144	C₂×C₄⋊C₄	—	53	120	no	—
145	C₂×C₄⋊C₄	—	53	10	no	—
146	C₂×C₂²⋊C₄	—	53	20	no	—
147	C₂×C₄²	—	53	30	no	—
148	C₂×C₄²	—	53	10	no	—
149	C₂×C₄²	—	53	120	no	—
150	C₂⁴	C₂³×D₅₃	1	5	yes	—
151	#41	C₂²×Dic₅₃	1	640	yes	complemented
152	C₂×Q₈	C₂²×Dic₅₃	1	80	yes	—
153	C₂×Q₈	C₂²×Dic₅₃	1	320	yes	complemented
154	C₂×Q₈	C₂³×D₅₃	1	40	yes	—
155	C₂×D₄	C₂²×Dic₅₃	1	80	yes	—
156	C₂×D₄	C₂³×D₅₃	1	40	yes	—
157	C₂²×C₄	—	53	1	no	—
158	C₂²×C₄	—	53	20	no	—
159	C₂²×C₄	—	53	5	no	—
160	C₂²×C₄	—	53	10	no	—
161	C₂²×C₄	—	53	60	no	—
162	C₂²×C₄	—	106	20	no	—
163	C₂²×C₄	—	212	30	no	—
164	C₂²×C₄	C₂²×Dic₅₃	1	120	yes	—
165	C₂²×C₄	C₂³×D₅₃	1	60	yes	—
166	C₂²×C₄	C₂³×D₅₃	1	10	yes	—
167	C₄⋊C₄	—	53	120	no	—
168	C₄⋊C₄	—	53	40	no	—
169	C₂²⋊C₄	—	53	80	no	—
170	C₄²	—	53	40	no	—
171	C₄²	—	53	120	no	—
172	C₂³	—	2	10	no	—
173	C₂³	—	2	10	no	—
174	C₂³	C₂³×Dic₅₃	1	20	yes	—
175	C₂³	C₂⁴×D₅₃	1	10	yes	—
176	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
177	Q₈	C₂³×Dic₅₃	1	160	yes	—
178	D₄	C₂³×Dic₅₃	1	160	yes	—
179	C₂×C₄	—	53	4	no	—
180	C₂×C₄	—	53	2	no	—
181	C₂×C₄	—	53	10	no	—
182	C₂×C₄	—	53	20	no	—
183	C₂×C₄	—	106	40	no	—
184	C₂×C₄	—	106	10	no	—
185	C₂×C₄	—	106	20	no	—
186	C₂×C₄	—	212	60	no	—
187	C₂×C₄	C₂³×Dic₅₃	1	40	yes	—
188	C₂×C₄	C₂³×Dic₅₃	1	240	yes	—
189	C₂×C₄	C₂⁴×D₅₃	1	20	yes	—
190	C₂²	—	2	40	no	—
191	C₂²	—	2	10	no	—
192	C₂²	—	2	10	no	—
193	C₂²	C₂⁴×Dic₅₃	1	20	yes	—
194	C₂²	C₂⁴×Dic₅₃	1	2	yes	—
195	C₂²	2₋¹⁺⁴×D₅₃	1	2	yes	—
196	C₂²	#16811460	1	2	yes	—
197	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
198	C₄	—	53	4	no	—
199	C₄	—	106	10	no	—
200	C₄	—	106	20	no	—
201	C₄	C₂⁴×Dic₅₃	1	40	yes	—
202	C₂	—	2	20	no	—
203	C₂	2₋¹⁺⁴×Dic₅₃	1	4	yes	direct product factor
204	C₂	C₂⁵×Dic₅₃	1	1	yes	characteristic
205	C₂	C₂×2₋¹⁺⁴×D₅₃	1	1	yes	characteristic
206	C₂	C₂×#16811460	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting, prime radical	C₂²×2₋¹⁺⁴×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂²×2₋¹⁺⁴	D₅₃
hypocenter	C₅₃	C₂×C₄×2₋¹⁺⁴
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×Dic₅₃	C₂	direct (N × H)
2. C₂×#16811106	C₂	semidirect
3. C₂×#16811101	C₂	semidirect
4. C₂×#41×Dic₅₃	C₂	semidirect
5. C₂²×Q₈×Dic₅₃	C₂	semidirect
7. #16811106	C₂²	semidirect
8. #16811101	C₂²	semidirect
9. #41×Dic₅₃	C₂²	semidirect
13. C₂×Q₈×Dic₅₃	C₂²	semidirect
19. C₂×2₋¹⁺⁴×C₅₃	C₄	semidirect
38. 2₋¹⁺⁴×C₅₃	C₂×C₄	semidirect
39. C₂×#41×C₅₃	C₂×C₄	semidirect
41. C₂²×Q₈×C₅₃	C₂×C₄	semidirect
47. #41×C₅₃	C₂²×C₄	semidirect
49. C₂×Q₈×C₅₃	C₂²×C₄	semidirect
82. C₂×Dic₅₃	2₋¹⁺⁴	direct (N × H)
98. Dic₅₃	C₂×2₋¹⁺⁴	direct (N × H)
106. C₂×C₅₃	C₄×2₋¹⁺⁴	semidirect
110. C₂×2₋¹⁺⁴	Dic₅₃	direct (N × H)
126. C₅₃	C₂×C₄×2₋¹⁺⁴	semidirect
127. 2₋¹⁺⁴	C₂×Dic₅₃	direct (N × H)
128. C₂×#41	C₂×Dic₅₃	semidirect
130. C₂²×Q₈	C₂×Dic₅₃	semidirect
151. #41	C₂²×Dic₅₃	semidirect
153. C₂×Q₈	C₂²×Dic₅₃	semidirect
203. C₂	2₋¹⁺⁴×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
2₋¹⁺⁴×Dic₅₃	2
C₂×#16811106	2
C₂×#16811101	2
C₂×#41×Dic₅₃	2
C₂²×Q₈×Dic₅₃	2
C₂²×2₋¹⁺⁴×C₅₃	2
C₂×C₄×2₋¹⁺⁴	53

Subgroups — C2×2−1+4×Dic53 #28290702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×2₋¹⁺⁴×Dic₅₃ #28290702