Subgroups — C₅₃×#19604 #27227116

Order 13568.

706 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3170	C₂	1	4	yes	maximal
2	C₅₃×#3160	C₂	1	2	yes	maximal
3	C₅₃×#1713	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1710	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1707	C₂	1	4	yes	maximal, complemented
6	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#533	C₂²	1	4	yes	—
8	C₅₃×#500	C₂²	1	4	yes	—
9	C₅₃×#500	C₂²	1	4	yes	—
10	C₅₃×#499	—	2	4	no	—
11	C₅₃×#498	C₂²	1	4	yes	—
12	C₅₃×#494	C₂²	1	4	yes	—
13	C₅₃×#487	C₂²	1	8	yes	—
14	C₅₃×#467	—	2	8	no	—
15	C₄×Q₁₆×C₅₃	—	2	8	no	—
16	C₅₃×#427	—	2	4	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
19	C₈²×C₅₃	C₂²	1	2	yes	—
20	C₈²×C₅₃	C₂²	1	1	yes	characteristic
21	C₈²×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×Q₁₆×C₅₃	—	2	8	no	—
23	C₂²×C₈×C₅₃	D₄	1	4	yes	—
24	C₂²×C₈×C₅₃	D₄	1	2	yes	—
25	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	—
26	#125×C₅₃	—	2	4	no	—
27	C₂²⋊Q₈×C₅₃	—	2	8	no	—
28	C₄×Q₈×C₅₃	—	2	8	no	—
29	#117×C₅₃	—	2	4	no	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	#107×C₅₃	—	2	8	no	—
32	#107×C₅₃	—	2	8	no	—
33	#106×C₅₃	—	2	8	no	—
34	#103×C₅₃	—	2	16	no	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₄×C₈×C₅₃	D₄	1	4	yes	—
37	C₄×C₈×C₅₃	C₂³	1	2	yes	—
38	C₄×C₈×C₅₃	C₂³	1	2	yes	—
39	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×Q₈×C₅₃	—	2	8	no	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	Q₁₆×C₅₃	—	4	16	no	—
45	C₂×C₈×C₅₃	—	2	8	no	—
46	C₂×C₈×C₅₃	Q₁₆	1	8	yes	complemented
47	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
48	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
49	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
51	C₄⋊C₄×C₅₃	—	2	8	no	—
52	C₄⋊C₄×C₅₃	—	2	8	no	—
53	C₄⋊C₄×C₅₃	—	4	8	no	—
54	C₂²⋊C₄×C₅₃	—	4	4	no	—
55	C₄²×C₅₃	—	4	4	no	—
56	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂×D₄	1	2	yes	—
58	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
59	C₂³×C₅₃	#127	1	1	yes	characteristic
60	Q₈×C₅₃	—	4	16	no	—
61	C₂×C₄×C₅₃	—	2	4	no	—
62	C₂×C₄×C₅₃	—	4	8	no	—
63	C₂×C₄×C₅₃	#127	1	2	yes	—
64	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
66	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
68	C₈×C₅₃	—	2	4	no	—
69	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
70	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
71	C₈×C₅₃	#135	1	8	yes	—
72	#19604	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂²×C₅₃	—	2	2	no	—
74	C₂²×C₅₃	#494	1	2	yes	—
75	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	8	8	no	—
78	C₄×C₅₃	#494	1	2	yes	—
79	C₄×C₅₃	C₂²×Q₁₆	1	2	yes	—
80	C₄×C₅₃	C₂×#135	1	4	yes	—
81	#3170	C₂×C₅₃	1	4	yes	—
82	#3160	C₂×C₅₃	1	2	yes	—
83	#1713	C₂×C₅₃	1	2	yes	complemented
84	#1710	C₂×C₅₃	1	2	yes	complemented
85	#1707	C₂×C₅₃	1	4	yes	complemented
86	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic
87	C₂×C₅₃	—	2	2	no	—
88	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
89	C₂×C₅₃	#3147	1	2	yes	—
90	#533	C₂²×C₅₃	1	4	yes	—
91	#500	C₂²×C₅₃	1	4	yes	—
92	#500	C₂²×C₅₃	1	4	yes	—
93	#499	—	2	4	no	—
94	#498	C₂²×C₅₃	1	4	yes	—
95	#494	C₂²×C₅₃	1	4	yes	—
96	#487	C₂²×C₅₃	1	8	yes	—
97	#467	—	2	8	no	—
98	C₄×Q₁₆	—	2	8	no	—
99	#427	—	2	4	no	—
100	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
102	C₈²	C₂²×C₅₃	1	2	yes	—
103	C₈²	C₂²×C₅₃	1	1	yes	characteristic
104	C₈²	C₂²×C₅₃	1	1	yes	characteristic
105	C₅₃	#19604	1	1	yes	characteristic, direct product factor, Sylow
106	C₂×Q₁₆	—	2	8	no	—
107	C₂²×C₈	D₄×C₅₃	1	4	yes	—
108	C₂²×C₈	D₄×C₅₃	1	2	yes	—
109	C₄⋊Q₈	C₂³×C₅₃	1	8	yes	—
110	#125	—	2	4	no	—
111	C₂²⋊Q₈	—	2	8	no	—
112	C₄×Q₈	—	2	8	no	—
113	#117	—	2	4	no	—
114	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
115	#107	—	2	8	no	—
116	#107	—	2	8	no	—
117	#106	—	2	8	no	—
118	#103	—	2	16	no	—
119	C₄×C₈	D₄×C₅₃	1	4	yes	—
120	C₄×C₈	D₄×C₅₃	1	2	yes	—
121	C₄×C₈	C₂³×C₅₃	1	2	yes	—
122	C₄×C₈	C₂³×C₅₃	1	2	yes	—
123	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
124	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
125	C₂×Q₈	—	2	8	no	—
126	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
127	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
128	Q₁₆	—	4	16	no	—
129	C₂×C₈	—	2	8	no	—
130	C₂×C₈	Q₁₆×C₅₃	1	8	yes	complemented
131	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
132	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
133	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
134	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
135	C₄⋊C₄	—	2	8	no	—
136	C₄⋊C₄	—	2	8	no	—
137	C₄⋊C₄	—	4	8	no	—
138	C₂²⋊C₄	—	4	4	no	—
139	C₄²	—	4	4	no	—
140	C₄²	C₂×D₄×C₅₃	1	2	yes	—
141	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
142	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
143	C₂³	#127×C₅₃	1	1	yes	characteristic
144	Q₈	—	4	16	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	4	8	no	—
147	C₂×C₄	#127×C₅₃	1	2	yes	—
148	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
149	C₂×C₄	C₂×Q₁₆×C₅₃	1	4	yes	—
150	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
151	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
152	C₈	—	2	4	no	—
153	C₈	C₂×Q₁₆×C₅₃	1	4	yes	—
154	C₈	C₂×Q₁₆×C₅₃	1	4	yes	—
155	C₈	#135×C₅₃	1	8	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	C₅₃×#494	1	2	yes	—
158	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	8	8	no	—
161	C₄	C₅₃×#494	1	2	yes	—
162	C₄	C₂²×Q₁₆×C₅₃	1	2	yes	—
163	C₄	C₂×#135×C₅₃	1	4	yes	—
164	C₂	—	2	2	no	—
165	C₂	C₂×C₅₃×#494	1	1	yes	characteristic
166	C₂	C₅₃×#3147	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×#127
prime radical	C₂³×C₅₃	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1713	C₂	semidirect
4. C₅₃×#1710	C₂	semidirect
5. C₅₃×#1707	C₂	semidirect
46. C₂×C₈×C₅₃	Q₁₆	semidirect
72. #19604	C₅₃	direct (N × H)
83. #1713	C₂×C₅₃	semidirect
84. #1710	C₂×C₅₃	semidirect
85. #1707	C₂×C₅₃	semidirect
105. C₅₃	#19604	direct (N × H)
130. C₂×C₈	Q₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3170	2
C₅₃×#3160	2
C₅₃×#1713	2
C₅₃×#1710	2
C₅₃×#1707	2
C₂×C₈²×C₅₃	2
#19604	53

Subgroups — C53×#19604 #27227116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19604 #27227116