Subgroups — C₂²×#16810486 #27542356

Order 13568.

8062 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊Dic₅₃	C₂	1	2	yes	maximal
2	C₂×#16810486	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133124	C₂	1	4	yes	maximal, complemented
4	C₂⁵×C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₅₃	C₂²	1	4	yes	—
6	C₂²×C₂²⋊Dic₅₃	—	2	16	no	—
7	C₂²×C₂²⋊Dic₅₃	C₂²	1	24	yes	—
8	#16810486	C₂²	1	64	yes	direct product factor
9	C₂×#13133124	C₂²	1	48	yes	complemented
10	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₅₃	—	2	16	no	—
12	C₂⁴×C₄×C₅₃	C₄	1	16	yes	complemented
13	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₅₃	C₂²	1	12	yes	—
15	C₂³×Dic₅₃	—	2	16	no	—
16	C₂³×Dic₅₃	—	2	32	no	—
17	C₂³×Dic₅₃	C₂³	1	24	yes	—
18	C₂×C₂²⋊Dic₅₃	—	2	96	no	—
19	C₂×C₂²⋊Dic₅₃	C₂³	1	32	yes	—
20	#13133124	C₂³	1	64	yes	complemented
21	C₂⁵×C₅₃	—	2	16	no	—
22	C₂⁵×C₅₃	C₂×C₄	1	8	yes	—
23	C₂⁵×C₅₃	D₄	1	16	yes	—
24	C₂⁵×C₅₃	C₂³	1	6	yes	—
25	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	—	2	128	no	—
27	C₂³×C₄×C₅₃	—	2	16	no	—
28	C₂³×C₄×C₅₃	—	2	96	no	—
29	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	—
30	C₂³×C₄×C₅₃	C₂×C₄	1	96	yes	complemented
31	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
32	C₂³×C₄×C₅₃	C₂³	1	16	yes	—
33	C₂⁴×C₅₃	—	2	64	no	—
34	C₂⁴×C₅₃	—	2	128	no	—
35	C₂⁴×C₅₃	—	2	48	no	—
36	C₂⁴×C₅₃	C₂²⋊C₄	1	64	yes	—
37	C₂⁴×C₅₃	C₂²×C₄	1	24	yes	—
38	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
39	C₂⁴×C₅₃	C₂×D₄	1	48	yes	—
40	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
41	C₂⁴×C₅₃	#41	1	16	yes	—
42	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
43	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
44	C₂²×C₄×C₅₃	—	2	64	no	—
45	C₂²×C₄×C₅₃	—	2	384	no	—
46	C₂²×C₄×C₅₃	—	2	48	no	—
47	C₂²×C₄×C₅₃	—	2	64	no	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	48	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	64	yes	complemented
50	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
51	C₂²×Dic₅₃	—	2	48	no	—
52	C₂²×Dic₅₃	—	2	96	no	—
53	C₂²×Dic₅₃	—	4	64	no	—
54	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
55	C₂²⋊Dic₅₃	—	2	64	no	—
56	C₂³×C₅₃	—	2	96	no	—
57	C₂³×C₅₃	—	2	192	no	—
58	C₂³×C₅₃	—	2	16	no	—
59	C₂³×C₅₃	—	2	256	no	—
60	C₂³×C₅₃	C₂×C₂²⋊C₄	1	96	yes	—
61	C₂³×C₅₃	C₂×C₂²⋊C₄	1	16	yes	—
62	C₂³×C₅₃	#117	1	32	yes	—
63	C₂³×C₅₃	#123	1	64	yes	—
64	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
65	C₂³×C₅₃	C₂³×C₄	1	6	yes	—
66	C₂³×C₅₃	C₂²×D₄	1	16	yes	—
67	C₂³×C₅₃	C₂²×D₄	1	12	yes	—
68	C₂³×C₅₃	C₂×#41	1	24	yes	—
69	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	96	no	—
71	C₂×C₄×C₅₃	—	2	128	no	—
72	C₂×C₄×C₅₃	—	2	16	no	—
73	C₂×C₄×C₅₃	C₂³×C₄	1	16	yes	—
74	C₂×Dic₅₃	—	2	16	no	—
75	C₂×Dic₅₃	—	2	32	no	—
76	C₂×Dic₅₃	—	4	96	no	—
77	C₂²×#381	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	—	2	16	no	—
79	C₂²×C₅₃	—	2	32	no	—
80	C₂²×C₅₃	—	2	192	no	—
81	C₂²×C₅₃	#381	1	64	yes	complemented
82	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	16	yes	—
83	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	12	yes	—
84	C₂²×C₅₃	C₂×#117	1	24	yes	—
85	C₂²×C₅₃	C₂×#123	1	48	yes	—
86	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
87	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
88	C₂²×C₅₃	C₂²×#41	1	4	yes	—
89	C₄×C₅₃	—	2	16	no	—
90	Dic₅₃	—	4	16	no	—
91	C₂⁵×C₄	D₅₃	1	1	yes	characteristic
92	C₂³×C₂²⋊C₄	—	53	2	no	—
93	C₂×#381	—	53	24	no	—
94	C₂²×#95	—	53	4	no	—
95	C₂×C₅₃	—	2	16	no	—
96	C₂×C₅₃	C₂×#381	1	24	yes	complemented
97	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₅₃	C₂²×#117	1	2	yes	—
99	C₂×C₅₃	C₂²×#123	1	4	yes	—
100	C₂⁶	C₂×D₅₃	1	1	yes	characteristic
101	C₂⁴×C₄	—	2	16	no	—
102	C₂⁴×C₄	—	53	4	no	—
103	C₂⁴×C₄	Dic₅₃	1	16	yes	complemented
104	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
105	C₂⁴×C₄	C₂×D₅₃	1	12	yes	—
106	C₂²×C₂²⋊C₄	—	53	24	no	—
107	C₂²×C₂²⋊C₄	—	106	16	no	—
108	#381	—	53	64	no	—
109	C₂×#95	—	53	48	no	—
110	C₅₃	C₂²×#381	1	1	yes	characteristic, complemented, Sylow
111	C₂⁵	—	2	16	no	—
112	C₂⁵	C₂×Dic₅₃	1	8	yes	—
113	C₂⁵	C₂²⋊D₅₃	1	16	yes	—
114	C₂⁵	C₂²×D₅₃	1	6	yes	—
115	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
116	C₂³×C₄	—	2	16	no	—
117	C₂³×C₄	—	2	96	no	—
118	C₂³×C₄	—	2	128	no	—
119	C₂³×C₄	—	53	24	no	—
120	C₂³×C₄	—	106	32	no	—
121	C₂³×C₄	—	106	16	no	—
122	C₂³×C₄	C₂×Dic₅₃	1	16	yes	—
123	C₂³×C₄	C₂×Dic₅₃	1	96	yes	complemented
124	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
125	C₂³×C₄	C₂²×D₅₃	1	16	yes	—
126	C₂×C₂²⋊C₄	—	53	32	no	—
127	C₂×C₂²⋊C₄	—	106	96	no	—
128	#95	—	53	64	no	—
129	C₂⁴	—	2	64	no	—
130	C₂⁴	—	2	128	no	—
131	C₂⁴	—	2	48	no	—
132	C₂⁴	C₂²⋊Dic₅₃	1	64	yes	—
133	C₂⁴	#11719945	1	16	yes	—
134	C₂⁴	C₂²×Dic₅₃	1	24	yes	—
135	C₂⁴	C₂²×Dic₅₃	1	4	yes	—
136	C₂⁴	C₂×C₂²⋊D₅₃	1	48	yes	—
137	C₂⁴	C₂×C₂²⋊D₅₃	1	8	yes	—
138	C₂⁴	C₂³×D₅₃	1	4	yes	—
139	C₂⁴	C₂³×D₅₃	1	3	yes	—
140	C₂²×C₄	—	2	64	no	—
141	C₂²×C₄	—	2	384	no	—
142	C₂²×C₄	—	2	48	no	—
143	C₂²×C₄	—	2	64	no	—
144	C₂²×C₄	—	53	16	no	—
145	C₂²×C₄	—	106	48	no	—
146	C₂²×C₄	—	106	96	no	—
147	C₂²×C₄	—	212	64	no	—
148	C₂²×C₄	C₂²×Dic₅₃	1	48	yes	—
149	C₂²×C₄	C₂²×Dic₅₃	1	64	yes	complemented
150	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
151	C₂²⋊C₄	—	106	64	no	—
152	C₂³	—	2	96	no	—
153	C₂³	—	2	192	no	—
154	C₂³	—	2	16	no	—
155	C₂³	—	2	256	no	—
156	C₂³	C₂×C₂²⋊Dic₅₃	1	96	yes	—
157	C₂³	C₂×C₂²⋊Dic₅₃	1	16	yes	—
158	C₂³	#13133176	1	32	yes	—
159	C₂³	#13133200	1	64	yes	—
160	C₂³	C₂³×Dic₅₃	1	8	yes	—
161	C₂³	C₂³×Dic₅₃	1	6	yes	—
162	C₂³	C₂²×C₂²⋊D₅₃	1	16	yes	—
163	C₂³	C₂²×C₂²⋊D₅₃	1	12	yes	—
164	C₂³	C₂×#11719945	1	24	yes	—
165	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
166	C₂×C₄	—	2	96	no	—
167	C₂×C₄	—	2	128	no	—
168	C₂×C₄	—	2	16	no	—
169	C₂×C₄	—	106	16	no	—
170	C₂×C₄	—	106	32	no	—
171	C₂×C₄	—	212	96	no	—
172	C₂×C₄	C₂³×Dic₅₃	1	16	yes	—
173	C₂²	—	2	192	no	—
174	C₂²	—	2	16	no	—
175	C₂²	—	2	32	no	—
176	C₂²	#16810486	1	64	yes	direct product factor
177	C₂²	C₂²×C₂²⋊Dic₅₃	1	16	yes	—
178	C₂²	C₂²×C₂²⋊Dic₅₃	1	12	yes	—
179	C₂²	C₂×#13133176	1	24	yes	—
180	C₂²	C₂×#13133200	1	48	yes	—
181	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
182	C₂²	C₂³×C₂²⋊D₅₃	1	2	yes	—
183	C₂²	C₂²×#11719945	1	4	yes	—
184	C₄	—	2	16	no	—
185	C₄	—	212	16	no	—
186	C₂	—	2	16	no	—
187	C₂	C₂×#16810486	1	24	yes	direct product factor
188	C₂	C₂³×C₂²⋊Dic₅₃	1	1	yes	characteristic
189	C₂	C₂²×#13133176	1	2	yes	—
190	C₂	C₂²×#13133200	1	4	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂⁵×C₄×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
prime radical	C₂⁶×C₅₃	C₂²
cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₂⁵×C₄	D₅₃
hypocenter	C₅₃	C₂²×#381
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#16810486	C₂	direct (N × H)
3. C₂²×#13133124	C₂	semidirect
8. #16810486	C₂²	direct (N × H)
9. C₂×#13133124	C₂²	semidirect
12. C₂⁴×C₄×C₅₃	C₄	semidirect
20. #13133124	C₂³	semidirect
30. C₂³×C₄×C₅₃	C₂×C₄	semidirect
49. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
81. C₂²×C₅₃	#381	semidirect
96. C₂×C₅₃	C₂×#381	semidirect
103. C₂⁴×C₄	Dic₅₃	semidirect
110. C₅₃	C₂²×#381	semidirect
123. C₂³×C₄	C₂×Dic₅₃	semidirect
149. C₂²×C₄	C₂²×Dic₅₃	semidirect
176. C₂²	#16810486	direct (N × H)
187. C₂	C₂×#16810486	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₂²⋊Dic₅₃	2
C₂×#16810486	2
C₂²×#13133124	2
C₂⁵×C₄×C₅₃	2
C₂²×#381	53

Subgroups — C22×#16810486 #27542356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810486 #27542356