Subgroups — C₂²×C₅₃×#490 #27268119

Order 13568.

2830 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#490	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#102×C₅₃	C₂	1	4	yes	maximal, complemented
5	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄×C₅₃	—	2	2	no	—
7	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
8	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×#127×C₅₃	—	2	6	no	—
10	C₂×#127×C₅₃	C₂²	1	12	yes	—
11	C₂×#125×C₅₃	—	2	6	no	—
12	C₂×#125×C₅₃	C₂²	1	12	yes	complemented
13	C₂²×C₄⋊C₄×C₅₃	—	2	2	no	—
14	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#490	C₂²	1	64	yes	direct product factor
17	C₂×#102×C₅₃	C₂²	1	48	yes	complemented
18	C₂×#97×C₅₃	C₂²	1	12	yes	complemented
19	C₂⁵×C₅₃	—	2	2	no	—
20	C₂²×D₄×C₅₃	—	2	6	no	—
21	C₂²×D₄×C₅₃	—	2	24	no	—
22	C₂²×D₄×C₅₃	—	2	16	no	—
23	C₂²×D₄×C₅₃	—	4	16	no	—
24	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
25	C₂³×C₄×C₅₃	—	2	2	no	—
26	C₂³×C₄×C₅₃	D₄	1	2	yes	—
27	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	—	2	8	no	—
29	C₂²×C₈×C₅₃	C₂³	1	12	yes	—
30	#127×C₅₃	—	2	24	no	—
31	#127×C₅₃	C₂³	1	16	yes	—
32	#125×C₅₃	—	2	24	no	—
33	#125×C₅₃	C₂³	1	16	yes	complemented
34	C₂×C₄⋊C₄×C₅₃	—	2	24	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
36	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
37	C₂×C₄²×C₅₃	D₄	1	6	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
39	#102×C₅₃	C₂³	1	64	yes	complemented
40	#97×C₅₃	C₂³	1	16	yes	complemented
41	C₂⁴×C₅₃	—	2	12	no	—
42	C₂⁴×C₅₃	—	4	8	no	—
43	C₂⁴×C₅₃	—	4	16	no	—
44	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×D₄×C₅₃	—	2	12	no	—
46	C₂×D₄×C₅₃	—	2	32	no	—
47	C₂×D₄×C₅₃	—	2	48	no	—
48	C₂×D₄×C₅₃	—	4	96	no	—
49	C₂×D₄×C₅₃	—	4	24	no	—
50	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
51	C₂²×C₄×C₅₃	—	2	8	no	—
52	C₂²×C₄×C₅₃	—	2	12	no	—
53	C₂²×C₄×C₅₃	—	4	8	no	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₅₃	#41	1	8	yes	—
57	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
58	C₂×C₈×C₅₃	—	2	24	no	—
59	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
60	C₄⋊C₄×C₅₃	—	2	32	no	—
61	C₄⋊C₄×C₅₃	—	2	12	no	—
62	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
63	C₄²×C₅₃	C₂×D₄	1	12	yes	—
64	C₄²×C₅₃	C₂⁴	1	4	yes	—
65	C₂³×C₅₃	—	2	8	no	—
66	C₂³×C₅₃	—	4	24	no	—
67	C₂³×C₅₃	—	4	48	no	—
68	C₂³×C₅₃	—	8	32	no	—
69	C₂³×C₅₃	#124	1	4	yes	—
70	C₂³×C₅₃	AGL(1,ℤ/8)	1	8	yes	—
71	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
72	D₄×C₅₃	—	2	16	no	—
73	D₄×C₅₃	—	4	64	no	—
74	D₄×C₅₃	—	4	24	no	—
75	C₂×C₄×C₅₃	—	2	24	no	—
76	C₂×C₄×C₅₃	—	2	8	no	—
77	C₂×C₄×C₅₃	—	4	24	no	—
78	C₂×C₄×C₅₃	#124	1	12	yes	—
79	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
80	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
81	C₂×C₄×C₅₃	C₂×#41	1	12	yes	—
82	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
83	C₈×C₅₃	—	2	8	no	—
84	C₂²×#490	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₅₃	—	4	8	no	—
86	C₂²×C₅₃	—	4	16	no	—
87	C₂²×C₅₃	—	8	48	no	—
88	C₂²×C₅₃	#490	1	16	yes	direct product factor
89	C₂²×C₅₃	C₂×#124	1	6	yes	—
90	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	12	yes	—
91	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
92	C₄×C₅₃	—	2	8	no	—
93	C₄×C₅₃	—	4	8	no	—
94	C₄×C₅₃	C₂×#124	1	6	yes	—
95	C₄×C₅₃	C₂²×#41	1	2	yes	—
96	C₂²×#127	C₂×C₅₃	1	1	yes	characteristic
97	C₂²×#125	C₂×C₅₃	1	1	yes	characteristic, complemented
98	C₂×#490	C₂×C₅₃	1	24	yes	direct product factor
99	C₂²×#102	C₂×C₅₃	1	4	yes	complemented
100	C₂²×#97	C₂×C₅₃	1	1	yes	characteristic, complemented
101	C₂×C₅₃	—	8	8	no	—
102	C₂×C₅₃	C₂×#490	1	12	yes	direct product factor
103	C₂×C₅₃	C₂²×#124	1	1	yes	characteristic
104	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	2	yes	—
105	C₂³×D₄	—	2	2	no	—
106	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
107	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
108	C₂×#127	—	2	6	no	—
109	C₂×#127	C₂²×C₅₃	1	12	yes	—
110	C₂×#125	—	2	6	no	—
111	C₂×#125	C₂²×C₅₃	1	12	yes	complemented
112	C₂²×C₄⋊C₄	—	2	2	no	—
113	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
114	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
115	#490	C₂²×C₅₃	1	64	yes	direct product factor
116	C₂×#102	C₂²×C₅₃	1	48	yes	complemented
117	C₂×#97	C₂²×C₅₃	1	12	yes	complemented
118	C₅₃	C₂²×#490	1	1	yes	characteristic, direct product factor, Sylow
119	C₂⁵	—	2	2	no	—
120	C₂²×D₄	—	2	6	no	—
121	C₂²×D₄	—	2	24	no	—
122	C₂²×D₄	—	2	16	no	—
123	C₂²×D₄	—	4	16	no	—
124	C₂²×D₄	C₂³×C₅₃	1	12	yes	—
125	C₂³×C₄	—	2	2	no	—
126	C₂³×C₄	D₄×C₅₃	1	2	yes	—
127	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
128	C₂²×C₈	—	2	8	no	—
129	C₂²×C₈	C₂³×C₅₃	1	12	yes	—
130	#127	—	2	24	no	—
131	#127	C₂³×C₅₃	1	16	yes	—
132	#125	—	2	24	no	—
133	#125	C₂³×C₅₃	1	16	yes	complemented
134	C₂×C₄⋊C₄	—	2	24	no	—
135	C₂×C₄⋊C₄	—	2	6	no	—
136	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
137	C₂×C₄²	D₄×C₅₃	1	6	yes	—
138	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
139	#102	C₂³×C₅₃	1	64	yes	complemented
140	#97	C₂³×C₅₃	1	16	yes	complemented
141	C₂⁴	—	2	12	no	—
142	C₂⁴	—	4	8	no	—
143	C₂⁴	—	4	16	no	—
144	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
145	C₂×D₄	—	2	12	no	—
146	C₂×D₄	—	2	32	no	—
147	C₂×D₄	—	2	48	no	—
148	C₂×D₄	—	4	96	no	—
149	C₂×D₄	—	4	24	no	—
150	C₂×D₄	C₂⁴×C₅₃	1	8	yes	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	2	12	no	—
153	C₂²×C₄	—	4	8	no	—
154	C₂²×C₄	C₂×D₄×C₅₃	1	12	yes	—
155	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
156	C₂²×C₄	#41×C₅₃	1	8	yes	—
157	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
158	C₂×C₈	—	2	24	no	—
159	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
160	C₄⋊C₄	—	2	32	no	—
161	C₄⋊C₄	—	2	12	no	—
162	C₄⋊C₄	C₂⁴×C₅₃	1	8	yes	—
163	C₄²	C₂×D₄×C₅₃	1	12	yes	—
164	C₄²	C₂⁴×C₅₃	1	4	yes	—
165	C₂³	—	2	8	no	—
166	C₂³	—	4	24	no	—
167	C₂³	—	4	48	no	—
168	C₂³	—	8	32	no	—
169	C₂³	#124×C₅₃	1	4	yes	—
170	C₂³	AGL(1,ℤ/8)×C₅₃	1	8	yes	—
171	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
172	D₄	—	2	16	no	—
173	D₄	—	4	64	no	—
174	D₄	—	4	24	no	—
175	C₂×C₄	—	2	24	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	4	24	no	—
178	C₂×C₄	#124×C₅₃	1	12	yes	—
179	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
180	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
181	C₂×C₄	C₂×#41×C₅₃	1	12	yes	—
182	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
183	C₈	—	2	8	no	—
184	C₂²	—	4	16	no	—
185	C₂²	—	4	8	no	—
186	C₂²	—	8	48	no	—
187	C₂²	C₅₃×#490	1	16	yes	direct product factor
188	C₂²	C₂×#124×C₅₃	1	6	yes	—
189	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	12	yes	—
190	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
191	C₄	—	2	8	no	—
192	C₄	—	4	8	no	—
193	C₄	C₂×#124×C₅₃	1	6	yes	—
194	C₄	C₂²×#41×C₅₃	1	2	yes	—
195	C₂	—	8	8	no	—
196	C₂	C₂×C₅₃×#490	1	12	yes	direct product factor
197	C₂	C₂²×#124×C₅₃	1	1	yes	characteristic
198	C₂	C₂²×AGL(1,ℤ/8)×C₅₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₂²×#127×C₅₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#125×C₅₃	C₂	semidirect
3. C₂×C₅₃×#490	C₂	direct (N × H)
4. C₂²×#102×C₅₃	C₂	semidirect
5. C₂²×#97×C₅₃	C₂	semidirect
12. C₂×#125×C₅₃	C₂²	semidirect
16. C₅₃×#490	C₂²	direct (N × H)
17. C₂×#102×C₅₃	C₂²	semidirect
18. C₂×#97×C₅₃	C₂²	semidirect
33. #125×C₅₃	C₂³	semidirect
39. #102×C₅₃	C₂³	semidirect
40. #97×C₅₃	C₂³	semidirect
84. C₂²×#490	C₅₃	direct (N × H)
88. C₂²×C₅₃	#490	direct (N × H)
97. C₂²×#125	C₂×C₅₃	semidirect
98. C₂×#490	C₂×C₅₃	direct (N × H)
99. C₂²×#102	C₂×C₅₃	semidirect
100. C₂²×#97	C₂×C₅₃	semidirect
102. C₂×C₅₃	C₂×#490	direct (N × H)
111. C₂×#125	C₂²×C₅₃	semidirect
115. #490	C₂²×C₅₃	direct (N × H)
116. C₂×#102	C₂²×C₅₃	semidirect
117. C₂×#97	C₂²×C₅₃	semidirect
118. C₅₃	C₂²×#490	direct (N × H)
133. #125	C₂³×C₅₃	semidirect
139. #102	C₂³×C₅₃	semidirect
140. #97	C₂³×C₅₃	semidirect
187. C₂²	C₅₃×#490	direct (N × H)
196. C₂	C₂×C₅₃×#490	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#127×C₅₃	2
C₂²×#125×C₅₃	2
C₂×C₅₃×#490	2
C₂²×#102×C₅₃	2
C₂²×#97×C₅₃	2
C₂²×#490	53

Subgroups — C22×C53×#490 #27268119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#490 #27268119