Subgroups — C₂³×#13133174 #28290650

Order 13568.

14334 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133174	C₂	1	56	yes	maximal, direct product factor
3	C₂³×#11719919	C₂	1	2	yes	maximal, complemented
4	C₂³×#11719921	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₅₃	C₂²	1	2	yes	—
9	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×D₅₃	C₄	1	32	yes	complemented
11	C₂³×C₄×D₅₃	C₂²	1	28	yes	—
12	C₂×#13133174	C₂²	1	448	yes	direct product factor
13	C₂²×#11719919	C₂²	1	56	yes	complemented
14	C₂²×#11719921	C₂²	1	56	yes	—
15	C₂²×C₄×Dic₅₃	C₂²	1	28	yes	complemented
16	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
17	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₅₃	C₂²	1	28	yes	complemented
19	C₂⁴×D₅₃	—	2	16	no	—
20	C₂⁴×D₅₃	C₂×C₄	1	16	yes	—
21	C₂⁴×D₅₃	C₂³	1	14	yes	—
22	C₂³×Dic₅₃	—	2	16	no	—
23	C₂³×Dic₅₃	C₂×C₄	1	16	yes	—
24	C₂³×Dic₅₃	C₂³	1	28	yes	—
25	C₂³×Dic₅₃	C₂³	1	14	yes	—
26	C₂²×C₄×D₅₃	C₂×C₄	1	448	yes	complemented
27	C₂²×C₄×D₅₃	C₂³	1	112	yes	—
28	#13133174	C₂³	1	512	yes	direct product factor
29	C₂×#11719919	C₂³	1	224	yes	complemented
30	C₂×#11719921	C₂³	1	224	yes	—
31	C₂×C₄×Dic₅₃	C₂³	1	112	yes	complemented
32	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₅₃	—	2	16	no	—
34	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	—
35	C₂³×C₄×C₅₃	C₂³	1	28	yes	—
36	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	112	yes	complemented
38	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
39	C₂⁴×C₅₃	#41	1	16	yes	—
40	C₂⁴×C₅₃	C₂⁴	1	7	yes	—
41	C₂³×D₅₃	—	2	128	no	—
42	C₂³×D₅₃	—	2	112	no	—
43	C₂³×D₅₃	C₂²×C₄	1	112	yes	—
44	C₂³×D₅₃	C₂⁴	1	28	yes	—
45	C₂²×C₄×C₅₃	—	2	112	no	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	112	yes	—
47	C₂²×C₄×C₅₃	C₂⁴	1	56	yes	—
48	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
49	C₂²×Dic₅₃	—	2	112	no	—
50	C₂²×Dic₅₃	C₂²×C₄	1	112	yes	—
51	C₂²×Dic₅₃	C₂⁴	1	56	yes	—
52	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
53	C₂×C₄×D₅₃	C₂²×C₄	1	896	yes	complemented
54	C₂×C₄×D₅₃	C₂⁴	1	64	yes	—
55	C₄²×C₅₃	C₂⁴	1	64	yes	complemented
56	#11719921	C₂⁴	1	128	yes	—
57	#11719919	C₂⁴	1	128	yes	complemented
58	C₄×Dic₅₃	C₂⁴	1	64	yes	complemented
59	C₂³×C₅₃	#117	1	64	yes	complemented
60	C₂³×C₅₃	C₂³×C₄	1	28	yes	—
61	C₂³×C₅₃	C₂×#41	1	56	yes	—
62	C₂³×C₅₃	C₂⁵	1	7	yes	—
63	C₂²×D₅₃	—	2	448	no	—
64	C₂²×D₅₃	—	2	112	no	—
65	C₂²×D₅₃	C₂³×C₄	1	112	yes	—
66	C₂²×D₅₃	C₂⁵	1	8	yes	—
67	C₂×C₄×C₅₃	—	2	112	no	—
68	C₂×C₄×C₅₃	C₂³×C₄	1	112	yes	—
69	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
70	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
71	C₂×Dic₅₃	—	2	112	no	—
72	C₂×Dic₅₃	C₂³×C₄	1	112	yes	—
73	C₂×Dic₅₃	C₂⁵	1	16	yes	—
74	C₂×Dic₅₃	C₂⁵	1	8	yes	—
75	C₄×D₅₃	C₂³×C₄	1	256	yes	complemented
76	C₂³×#117	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	C₂×#117	1	112	yes	complemented
78	C₂²×C₅₃	C₂⁴×C₄	1	14	yes	—
79	C₂²×C₅₃	C₂²×#41	1	28	yes	—
80	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
81	C₂×D₅₃	—	2	16	no	—
82	C₂×D₅₃	—	2	224	no	—
83	C₂×D₅₃	C₂⁴×C₄	1	16	yes	—
84	C₄×C₅₃	—	2	16	no	—
85	C₄×C₅₃	C₂⁴×C₄	1	16	yes	—
86	Dic₅₃	—	2	16	no	—
87	Dic₅₃	C₂⁴×C₄	1	16	yes	—
88	C₂⁵×C₄	—	53	1	no	—
89	C₂²×#117	—	53	56	no	—
90	C₂³×C₄⋊C₄	—	53	2	no	—
91	C₂³×C₂²⋊C₄	—	53	2	no	—
92	C₂³×C₄²	—	53	1	no	—
93	C₂³×C₄²	D₅₃	1	1	yes	characteristic, complemented
94	C₂×C₅₃	C₂²×#117	1	28	yes	complemented
95	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
96	C₂×C₅₃	C₂³×#41	1	2	yes	—
97	D₅₃	—	2	16	no	—
98	C₂⁶	—	53	1	no	—
99	C₂⁴×C₄	—	53	1	no	—
100	C₂⁴×C₄	—	53	32	no	—
101	C₂⁴×C₄	—	53	28	no	—
102	C₂⁴×C₄	—	53	2	no	—
103	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
104	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
105	C₂×#117	—	53	448	no	—
106	C₂²×C₄⋊C₄	—	53	56	no	—
107	C₂²×C₂²⋊C₄	—	53	56	no	—
108	C₂²×C₄²	—	53	28	no	—
109	C₂²×C₄²	C₂×D₅₃	1	28	yes	complemented
110	C₅₃	C₂³×#117	1	1	yes	characteristic, complemented, Sylow
111	C₂⁵	—	53	16	no	—
112	C₂⁵	—	53	14	no	—
113	C₂⁵	—	106	16	no	—
114	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
115	C₂³×C₄	—	2	16	no	—
116	C₂³×C₄	—	53	16	no	—
117	C₂³×C₄	—	53	14	no	—
118	C₂³×C₄	—	53	448	no	—
119	C₂³×C₄	—	53	112	no	—
120	C₂³×C₄	—	53	28	no	—
121	C₂³×C₄	—	106	16	no	—
122	C₂³×C₄	C₄×D₅₃	1	16	yes	—
123	C₂³×C₄	C₂²×D₅₃	1	28	yes	—
124	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
125	#117	—	53	512	no	—
126	C₂×C₄⋊C₄	—	53	224	no	—
127	C₂×C₂²⋊C₄	—	53	224	no	—
128	C₂×C₄²	—	53	112	no	—
129	C₂×C₄²	C₂²×D₅₃	1	112	yes	complemented
130	C₂⁴	—	53	112	no	—
131	C₂⁴	—	53	28	no	—
132	C₂⁴	—	106	128	no	—
133	C₂⁴	—	106	112	no	—
134	C₂⁴	C₂×C₄×D₅₃	1	8	yes	—
135	C₂⁴	#11719945	1	16	yes	—
136	C₂⁴	C₂³×D₅₃	1	7	yes	—
137	C₂²×C₄	—	2	112	no	—
138	C₂²×C₄	—	53	112	no	—
139	C₂²×C₄	—	53	28	no	—
140	C₂²×C₄	—	53	896	no	—
141	C₂²×C₄	—	53	64	no	—
142	C₂²×C₄	—	53	56	no	—
143	C₂²×C₄	—	106	112	no	—
144	C₂²×C₄	C₂×C₄×D₅₃	1	112	yes	—
145	C₂²×C₄	C₂³×D₅₃	1	56	yes	—
146	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
147	C₄⋊C₄	—	53	128	no	—
148	C₂²⋊C₄	—	53	128	no	—
149	C₄²	—	53	64	no	—
150	C₄²	C₂³×D₅₃	1	64	yes	complemented
151	C₂³	—	53	112	no	—
152	C₂³	—	53	8	no	—
153	C₂³	—	106	448	no	—
154	C₂³	—	106	112	no	—
155	C₂³	#13133174	1	64	yes	direct product factor
156	C₂³	C₂²×C₄×D₅₃	1	28	yes	—
157	C₂³	C₂×#11719945	1	56	yes	—
158	C₂³	C₂⁴×D₅₃	1	7	yes	—
159	C₂×C₄	—	2	112	no	—
160	C₂×C₄	—	53	112	no	—
161	C₂×C₄	—	53	8	no	—
162	C₂×C₄	—	53	256	no	—
163	C₂×C₄	—	53	16	no	—
164	C₂×C₄	—	106	112	no	—
165	C₂×C₄	C₂²×C₄×D₅₃	1	112	yes	—
166	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
167	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
168	C₂²	—	53	16	no	—
169	C₂²	—	106	224	no	—
170	C₂²	—	106	16	no	—
171	C₂²	C₂×#13133174	1	112	yes	direct product factor
172	C₂²	C₂³×C₄×D₅₃	1	14	yes	—
173	C₂²	C₂²×#11719945	1	28	yes	—
174	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
175	C₄	—	2	16	no	—
176	C₄	—	53	16	no	—
177	C₄	—	106	16	no	—
178	C₄	C₂³×C₄×D₅₃	1	16	yes	—
179	C₂	—	106	16	no	—
180	C₂	C₂²×#13133174	1	28	yes	direct product factor
181	C₂	C₂⁴×C₄×D₅₃	1	1	yes	characteristic
182	C₂	C₂³×#11719945	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₄²×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
prime radical	C₂⁵×D₅₃	C₂²
cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂³×C₄²	D₅₃
hypocenter	C₅₃	C₂³×#117
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×#13133174	C₂	direct (N × H)
3. C₂³×#11719919	C₂	semidirect
5. C₂³×C₄×Dic₅₃	C₂	semidirect
6. C₂³×C₄²×C₅₃	C₂	semidirect
10. C₂³×C₄×D₅₃	C₄	semidirect
12. C₂×#13133174	C₂²	direct (N × H)
13. C₂²×#11719919	C₂²	semidirect
15. C₂²×C₄×Dic₅₃	C₂²	semidirect
18. C₂²×C₄²×C₅₃	C₂²	semidirect
26. C₂²×C₄×D₅₃	C₂×C₄	semidirect
28. #13133174	C₂³	direct (N × H)
29. C₂×#11719919	C₂³	semidirect
31. C₂×C₄×Dic₅₃	C₂³	semidirect
37. C₂×C₄²×C₅₃	C₂³	semidirect
53. C₂×C₄×D₅₃	C₂²×C₄	semidirect
55. C₄²×C₅₃	C₂⁴	semidirect
57. #11719919	C₂⁴	semidirect
58. C₄×Dic₅₃	C₂⁴	semidirect
59. C₂³×C₅₃	#117	semidirect
75. C₄×D₅₃	C₂³×C₄	semidirect
77. C₂²×C₅₃	C₂×#117	semidirect
93. C₂³×C₄²	D₅₃	semidirect
94. C₂×C₅₃	C₂²×#117	semidirect
109. C₂²×C₄²	C₂×D₅₃	semidirect
110. C₅₃	C₂³×#117	semidirect
129. C₂×C₄²	C₂²×D₅₃	semidirect
150. C₄²	C₂³×D₅₃	semidirect
155. C₂³	#13133174	direct (N × H)
171. C₂²	C₂×#13133174	direct (N × H)
180. C₂	C₂²×#13133174	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄×D₅₃	2
C₂²×#13133174	2
C₂³×#11719919	2
C₂³×#11719921	2
C₂³×C₄×Dic₅₃	2
C₂³×C₄²×C₅₃	2
C₂³×#117	53

Subgroups — C23×#13133174 #28290650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133174 #28290650