Subgroups — C₂²×#41×Dic₅₃ #28290668

Order 13568.

12878 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×Dic₅₃	C₂	1	48	yes	maximal, direct product factor
2	C₂²×Q₈×Dic₅₃	C₂	1	2	yes	maximal, complemented
3	C₂²×D₄×Dic₅₃	C₂	1	6	yes	maximal, complemented
4	C₂²×#13133176	C₂	1	3	yes	maximal, complemented
5	C₂³×C₄×Dic₅₃	C₂	1	3	yes	maximal, complemented
6	C₂³×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₅₃	C₂²	1	6	yes	—
8	#41×Dic₅₃	C₂²	1	256	yes	direct product factor
9	C₂×Q₈×Dic₅₃	C₂²	1	48	yes	complemented
10	C₂×D₄×Dic₅₃	C₂²	1	144	yes	complemented
11	C₂×#13133176	C₂²	1	72	yes	complemented
12	C₂²×C₄⋊Dic₅₃	C₂²	1	6	yes	—
13	C₂²×C₂²⋊Dic₅₃	C₂²	1	6	yes	—
14	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×Dic₅₃	C₂²	1	3	yes	—
16	C₂²×C₄×Dic₅₃	C₂²	1	6	yes	—
17	C₂²×C₄×Dic₅₃	C₂²	1	72	yes	complemented
18	C₂²×#41×C₅₃	C₄	1	32	yes	complemented
19	C₂²×#41×C₅₃	C₂²	1	24	yes	—
20	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂³×D₄×C₅₃	C₂²	1	3	yes	—
22	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
23	C₂³×Dic₅₃	—	2	48	no	—
24	C₂³×Dic₅₃	C₂³	1	6	yes	—
25	C₂³×Dic₅₃	C₂³	1	72	yes	—
26	C₂³×Dic₅₃	C₂³	1	2	yes	—
27	Q₈×Dic₅₃	C₂³	1	128	yes	complemented
28	D₄×Dic₅₃	C₂³	1	384	yes	complemented
29	#13133176	C₂³	1	192	yes	complemented
30	C₂×C₄⋊Dic₅₃	C₂³	1	72	yes	—
31	C₂×C₂²⋊Dic₅₃	C₂³	1	72	yes	—
32	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
33	C₂×C₄×Dic₅₃	C₂³	1	36	yes	—
34	C₂×C₄×Dic₅₃	C₂³	1	72	yes	—
35	C₂×C₄×Dic₅₃	C₂³	1	192	yes	complemented
36	C₂⁵×C₅₃	C₂³	1	3	yes	—
37	C₂×#41×C₅₃	C₂×C₄	1	384	yes	complemented
38	C₂×#41×C₅₃	C₂³	1	64	yes	—
39	C₂²×Q₈×C₅₃	C₂×C₄	1	16	yes	complemented
40	C₂²×Q₈×C₅₃	C₂³	1	12	yes	—
41	C₂²×D₄×C₅₃	C₂×C₄	1	48	yes	complemented
42	C₂²×D₄×C₅₃	C₂³	1	36	yes	—
43	C₂³×C₄×C₅₃	C₂×C₄	1	48	yes	complemented
44	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
45	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	C₂³	1	36	yes	—
47	C₂⁴×C₅₃	—	2	24	no	—
48	C₂⁴×C₅₃	C₂²×C₄	1	24	yes	—
49	C₂⁴×C₅₃	C₂⁴	1	18	yes	—
50	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
51	#41×C₅₃	C₂²×C₄	1	512	yes	complemented
52	C₂×Q₈×C₅₃	C₂²×C₄	1	96	yes	complemented
53	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	—
54	C₂×D₄×C₅₃	C₂²×C₄	1	288	yes	complemented
55	C₂×D₄×C₅₃	C₂⁴	1	48	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	288	yes	complemented
59	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
60	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	48	yes	—
62	C₂²×Dic₅₃	—	2	24	no	—
63	C₂²×Dic₅₃	—	2	288	no	—
64	C₂²×Dic₅₃	#41	1	16	yes	direct product factor
65	C₂²×Dic₅₃	C₂⁴	1	36	yes	—
66	C₂²×Dic₅₃	C₂⁴	1	96	yes	—
67	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
68	C₂²⋊Dic₅₃	C₂⁴	1	96	yes	—
69	C₄⋊Dic₅₃	C₂⁴	1	96	yes	—
70	C₄×Dic₅₃	C₂⁴	1	16	yes	—
71	C₄×Dic₅₃	C₂⁴	1	48	yes	—
72	C₄×Dic₅₃	C₂⁴	1	96	yes	—
73	C₂³×C₅₃	—	2	96	no	—
74	C₂³×C₅₃	—	2	72	no	—
75	C₂³×C₅₃	C₂³×C₄	1	72	yes	—
76	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
77	C₂³×C₅₃	C₂×#41	1	8	yes	—
78	C₂³×C₅₃	C₂⁵	1	12	yes	—
79	C₂³×C₅₃	C₂⁵	1	3	yes	—
80	Q₈×C₅₃	C₂³×C₄	1	64	yes	complemented
81	D₄×C₅₃	C₂³×C₄	1	192	yes	complemented
82	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
83	C₂×C₄×C₅₃	C₂³×C₄	1	72	yes	—
84	C₂×C₄×C₅₃	C₂³×C₄	1	192	yes	complemented
85	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
86	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
87	C₂×Dic₅₃	—	2	72	no	—
88	C₂×Dic₅₃	—	2	192	no	—
89	C₂×Dic₅₃	C₂×#41	1	48	yes	direct product factor
90	C₂×Dic₅₃	C₂⁵	1	24	yes	—
91	C₂×Dic₅₃	C₂⁵	1	8	yes	—
92	C₂²×C₄×#41	—	53	1	no	maximal, Sylow
93	C₂²×C₅₃	—	2	24	no	—
94	C₂²×C₅₃	—	2	144	no	—
95	C₂²×C₅₃	C₄×#41	1	16	yes	complemented
96	C₂²×C₅₃	C₂⁴×C₄	1	24	yes	—
97	C₂²×C₅₃	C₂⁴×C₄	1	6	yes	—
98	C₂²×C₅₃	C₂²×#41	1	12	yes	—
99	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
100	C₄×C₅₃	C₂⁴×C₄	1	8	yes	—
101	C₄×C₅₃	C₂⁴×C₄	1	24	yes	—
102	Dic₅₃	—	2	24	no	—
103	Dic₅₃	C₂²×#41	1	16	yes	direct product factor
104	C₂³×#41	D₅₃	1	1	yes	characteristic
105	C₂×C₄×#41	—	53	48	no	—
106	C₂²×C₄×Q₈	—	53	2	no	—
107	C₂²×C₄×D₄	—	53	6	no	—
108	C₂²×#117	—	53	3	no	—
109	C₂³×C₄²	—	53	3	no	—
110	C₂×C₅₃	—	2	24	no	—
111	C₂×C₅₃	C₂×C₄×#41	1	12	yes	complemented
112	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
113	C₂×C₅₃	C₂³×#41	1	2	yes	—
114	C₂²×#41	Dic₅₃	1	32	yes	direct product factor
115	C₂²×#41	C₂×D₅₃	1	24	yes	—
116	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic
117	C₂³×D₄	C₂×D₅₃	1	3	yes	—
118	C₂⁴×C₄	—	53	6	no	—
119	C₂⁴×C₄	C₂×D₅₃	1	3	yes	—
120	C₄×#41	—	53	256	no	—
121	C₂×C₄×Q₈	—	53	48	no	—
122	C₂×C₄×D₄	—	53	144	no	—
123	C₂×#117	—	53	72	no	—
124	C₂²×C₄⋊C₄	—	53	6	no	—
125	C₂²×C₂²⋊C₄	—	53	6	no	—
126	C₂²×C₄²	—	53	6	no	—
127	C₂²×C₄²	—	53	72	no	—
128	C₂²×C₄²	—	53	1	no	—
129	C₂²×C₄²	—	53	3	no	—
130	C₅₃	C₂²×C₄×#41	1	1	yes	characteristic, complemented, Sylow
131	C₂⁵	C₂²×D₅₃	1	3	yes	—
132	C₂×#41	C₂×Dic₅₃	1	384	yes	direct product factor
133	C₂×#41	C₂²×D₅₃	1	64	yes	—
134	C₂²×Q₈	C₂×Dic₅₃	1	16	yes	complemented
135	C₂²×Q₈	C₂²×D₅₃	1	12	yes	—
136	C₂²×D₄	C₂×Dic₅₃	1	48	yes	complemented
137	C₂²×D₄	C₂²×D₅₃	1	36	yes	—
138	C₂³×C₄	—	53	2	no	—
139	C₂³×C₄	—	53	6	no	—
140	C₂³×C₄	—	53	72	no	—
141	C₂³×C₄	—	106	48	no	—
142	C₂³×C₄	C₂×Dic₅₃	1	48	yes	complemented
143	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
144	C₂³×C₄	C₂²×D₅₃	1	36	yes	—
145	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
146	C₄×Q₈	—	53	128	no	—
147	C₄×D₄	—	53	384	no	—
148	#117	—	53	192	no	—
149	C₂×C₄⋊C₄	—	53	72	no	—
150	C₂×C₂²⋊C₄	—	53	72	no	—
151	C₂×C₄²	—	53	36	no	—
152	C₂×C₄²	—	53	72	no	—
153	C₂×C₄²	—	53	192	no	—
154	C₂×C₄²	—	53	12	no	—
155	C₂⁴	—	2	24	no	—
156	C₂⁴	C₂²×Dic₅₃	1	24	yes	—
157	C₂⁴	C₂³×D₅₃	1	18	yes	—
158	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
159	#41	C₂²×Dic₅₃	1	512	yes	direct product factor
160	C₂×Q₈	C₂²×Dic₅₃	1	96	yes	complemented
161	C₂×Q₈	C₂³×D₅₃	1	16	yes	—
162	C₂×D₄	C₂²×Dic₅₃	1	288	yes	complemented
163	C₂×D₄	C₂³×D₅₃	1	48	yes	—
164	C₂²×C₄	—	53	16	no	—
165	C₂²×C₄	—	53	12	no	—
166	C₂²×C₄	—	53	36	no	—
167	C₂²×C₄	—	53	96	no	—
168	C₂²×C₄	—	106	288	no	—
169	C₂²×C₄	—	106	24	no	—
170	C₂²×C₄	C₂²×Dic₅₃	1	8	yes	—
171	C₂²×C₄	C₂²×Dic₅₃	1	24	yes	—
172	C₂²×C₄	C₂²×Dic₅₃	1	288	yes	complemented
173	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
174	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
175	C₂²×C₄	C₂³×D₅₃	1	48	yes	—
176	C₄⋊C₄	—	53	96	no	—
177	C₂²⋊C₄	—	53	96	no	—
178	C₄²	—	53	16	no	—
179	C₄²	—	53	48	no	—
180	C₄²	—	53	96	no	—
181	C₂³	—	2	96	no	—
182	C₂³	—	2	72	no	—
183	C₂³	C₂³×Dic₅₃	1	72	yes	—
184	C₂³	C₂³×Dic₅₃	1	4	yes	—
185	C₂³	#41×D₅₃	1	8	yes	—
186	C₂³	C₂⁴×D₅₃	1	12	yes	—
187	C₂³	C₂⁴×D₅₃	1	3	yes	—
188	Q₈	C₂³×Dic₅₃	1	64	yes	complemented
189	D₄	C₂³×Dic₅₃	1	192	yes	complemented
190	C₂×C₄	—	53	48	no	—
191	C₂×C₄	—	53	8	no	—
192	C₂×C₄	—	53	24	no	—
193	C₂×C₄	—	106	192	no	—
194	C₂×C₄	—	106	72	no	—
195	C₂×C₄	C₂³×Dic₅₃	1	24	yes	—
196	C₂×C₄	C₂³×Dic₅₃	1	72	yes	—
197	C₂×C₄	C₂³×Dic₅₃	1	192	yes	complemented
198	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
199	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
200	C₂²	—	2	144	no	—
201	C₂²	—	2	24	no	—
202	C₂²	#41×Dic₅₃	1	16	yes	direct product factor
203	C₂²	C₂⁴×Dic₅₃	1	24	yes	—
204	C₂²	C₂⁴×Dic₅₃	1	6	yes	—
205	C₂²	C₂×#41×D₅₃	1	12	yes	—
206	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
207	C₄	—	53	16	no	—
208	C₄	—	106	24	no	—
209	C₄	C₂⁴×Dic₅₃	1	8	yes	—
210	C₄	C₂⁴×Dic₅₃	1	24	yes	—
211	C₂	—	2	24	no	—
212	C₂	C₂×#41×Dic₅₃	1	12	yes	direct product factor
213	C₂	C₂⁵×Dic₅₃	1	1	yes	characteristic
214	C₂	C₂²×#41×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting, prime radical	C₂³×#41×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂³×#41	D₅₃
hypocenter	C₅₃	C₂²×C₄×#41
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#41×Dic₅₃	C₂	direct (N × H)
2. C₂²×Q₈×Dic₅₃	C₂	semidirect
3. C₂²×D₄×Dic₅₃	C₂	semidirect
4. C₂²×#13133176	C₂	semidirect
5. C₂³×C₄×Dic₅₃	C₂	semidirect
8. #41×Dic₅₃	C₂²	direct (N × H)
9. C₂×Q₈×Dic₅₃	C₂²	semidirect
10. C₂×D₄×Dic₅₃	C₂²	semidirect
11. C₂×#13133176	C₂²	semidirect
17. C₂²×C₄×Dic₅₃	C₂²	semidirect
18. C₂²×#41×C₅₃	C₄	semidirect
27. Q₈×Dic₅₃	C₂³	semidirect
28. D₄×Dic₅₃	C₂³	semidirect
29. #13133176	C₂³	semidirect
35. C₂×C₄×Dic₅₃	C₂³	semidirect
37. C₂×#41×C₅₃	C₂×C₄	semidirect
39. C₂²×Q₈×C₅₃	C₂×C₄	semidirect
41. C₂²×D₄×C₅₃	C₂×C₄	semidirect
43. C₂³×C₄×C₅₃	C₂×C₄	semidirect
51. #41×C₅₃	C₂²×C₄	semidirect
52. C₂×Q₈×C₅₃	C₂²×C₄	semidirect
54. C₂×D₄×C₅₃	C₂²×C₄	semidirect
58. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
64. C₂²×Dic₅₃	#41	direct (N × H)
80. Q₈×C₅₃	C₂³×C₄	semidirect
81. D₄×C₅₃	C₂³×C₄	semidirect
84. C₂×C₄×C₅₃	C₂³×C₄	semidirect
89. C₂×Dic₅₃	C₂×#41	direct (N × H)
95. C₂²×C₅₃	C₄×#41	semidirect
103. Dic₅₃	C₂²×#41	direct (N × H)
111. C₂×C₅₃	C₂×C₄×#41	semidirect
114. C₂²×#41	Dic₅₃	direct (N × H)
130. C₅₃	C₂²×C₄×#41	semidirect
132. C₂×#41	C₂×Dic₅₃	direct (N × H)
134. C₂²×Q₈	C₂×Dic₅₃	semidirect
136. C₂²×D₄	C₂×Dic₅₃	semidirect
142. C₂³×C₄	C₂×Dic₅₃	semidirect
159. #41	C₂²×Dic₅₃	direct (N × H)
160. C₂×Q₈	C₂²×Dic₅₃	semidirect
162. C₂×D₄	C₂²×Dic₅₃	semidirect
172. C₂²×C₄	C₂²×Dic₅₃	semidirect
188. Q₈	C₂³×Dic₅₃	semidirect
189. D₄	C₂³×Dic₅₃	semidirect
197. C₂×C₄	C₂³×Dic₅₃	semidirect
202. C₂²	#41×Dic₅₃	direct (N × H)
212. C₂	C₂×#41×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×#41×Dic₅₃	2
C₂²×Q₈×Dic₅₃	2
C₂²×D₄×Dic₅₃	2
C₂²×#13133176	2
C₂³×C₄×Dic₅₃	2
C₂³×#41×C₅₃	2
C₂²×C₄×#41	53

Subgroups — C22×#41×Dic53 #28290668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#41×Dic₅₃ #28290668