Subgroups — C₅₃×#34552 #27242064

Order 13568.

1962 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2402	C₂	1	6	yes	maximal, complemented
2	C₅₃×#2399	C₂	1	6	yes	maximal, complemented
3	C₅₃×#2396	C₂	1	12	yes	maximal
4	C₄²×Q₈×C₅₃	C₂	1	4	yes	maximal, complemented
5	C₄×#117×C₅₃	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₅₃	C₂²	1	12	yes	—
7	C₂×#117×C₅₃	C₂²	1	6	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#400	C₂²	1	12	yes	—
10	C₅₃×#399	C₂²	1	4	yes	—
11	C₅₃×#398	C₂²	1	24	yes	—
12	C₅₃×#396	C₂²	1	8	yes	—
13	C₅₃×#395	C₂²	1	12	yes	—
14	C₅₃×#393	C₂²	1	4	yes	—
15	C₄×C₄⋊C₄×C₅₃	C₂²	1	12	yes	—
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	12	yes	—
17	C₄×C₄⋊C₄×C₅₃	C₂²	1	24	yes	—
18	C₄×C₂²⋊C₄×C₅₃	C₂²	1	12	yes	—
19	C₅₃×#376	C₂²	1	3	yes	—
20	C₅₃×#376	C₂²	1	3	yes	—
21	C₄³×C₅₃	C₂²	1	6	yes	—
22	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
24	C₄×Q₈×C₅₃	—	2	48	no	—
25	#117×C₅₃	—	2	24	no	—
26	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
27	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
28	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
29	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
30	C₂×C₄²×C₅₃	Q₈	1	8	yes	complemented
31	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	24	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
34	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	#95×C₅₃	C₂³	1	12	yes	—
37	#95×C₅₃	C₂³	1	12	yes	—
38	#95×C₅₃	C₂³	1	12	yes	—
39	#95×C₅₃	C₂³	1	4	yes	—
40	#95×C₅₃	C₂³	1	8	yes	—
41	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
42	C₂×Q₈×C₅₃	—	2	24	no	—
43	C₂²×C₄×C₅₃	—	2	12	no	—
44	C₂²×C₄×C₅₃	C₂×Q₈	1	12	yes	—
45	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
46	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
47	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
48	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
49	C₄⋊C₄×C₅₃	—	2	24	no	—
50	C₄⋊C₄×C₅₃	—	2	48	no	—
51	C₄⋊C₄×C₅₃	—	2	24	no	—
52	C₂²⋊C₄×C₅₃	—	2	24	no	—
53	C₄²×C₅₃	—	2	12	no	—
54	C₄²×C₅₃	—	2	48	no	—
55	C₄²×C₅₃	C₂×Q₈	1	12	yes	—
56	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
57	C₄²×C₅₃	#41	1	24	yes	—
58	C₂³×C₅₃	—	2	6	no	—
59	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
60	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
61	Q₈×C₅₃	—	2	32	no	—
62	C₂×C₄×C₅₃	—	2	6	no	—
63	C₂×C₄×C₅₃	—	2	24	no	—
64	C₂×C₄×C₅₃	—	2	24	no	—
65	C₂×C₄×C₅₃	—	2	12	no	—
66	C₂×C₄×C₅₃	—	2	24	no	—
67	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
68	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
69	C₂×C₄×C₅₃	C₂×#41	1	24	yes	—
70	C₂×C₄×C₅₃	C₂×#41	1	12	yes	—
71	#34552	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂²×C₅₃	—	2	6	no	—
73	C₂²×C₅₃	—	4	4	no	—
74	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
75	C₂²×C₅₃	C₂²×#41	1	6	yes	—
76	C₄×C₅₃	—	2	6	no	—
77	C₄×C₅₃	—	2	24	no	—
78	C₄×C₅₃	—	4	12	no	—
79	C₄×C₅₃	#533	1	12	yes	—
80	#2402	C₂×C₅₃	1	6	yes	complemented
81	#2399	C₂×C₅₃	1	6	yes	complemented
82	#2396	C₂×C₅₃	1	12	yes	—
83	C₄²×Q₈	C₂×C₅₃	1	4	yes	complemented
84	C₄×#117	C₂×C₅₃	1	3	yes	—
85	C₂×C₅₃	—	4	2	no	—
86	C₂×C₅₃	C₂×#533	1	3	yes	—
87	C₂×C₅₃	#3444	1	4	yes	—
88	C₂×C₄×Q₈	C₂²×C₅₃	1	12	yes	—
89	C₂×#117	C₂²×C₅₃	1	6	yes	—
90	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
91	#400	C₂²×C₅₃	1	12	yes	—
92	#399	C₂²×C₅₃	1	4	yes	—
93	#398	C₂²×C₅₃	1	24	yes	—
94	#396	C₂²×C₅₃	1	8	yes	—
95	#395	C₂²×C₅₃	1	12	yes	—
96	#393	C₂²×C₅₃	1	4	yes	—
97	C₄×C₄⋊C₄	C₂²×C₅₃	1	12	yes	—
98	C₄×C₄⋊C₄	C₂²×C₅₃	1	24	yes	—
99	C₄×C₄⋊C₄	C₂²×C₅₃	1	12	yes	—
100	C₄×C₂²⋊C₄	C₂²×C₅₃	1	12	yes	—
101	#376	C₂²×C₅₃	1	3	yes	—
102	#376	C₂²×C₅₃	1	3	yes	—
103	C₄³	C₂²×C₅₃	1	6	yes	—
104	C₅₃	#34552	1	1	yes	characteristic, direct product factor, Sylow
105	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
106	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
107	C₄×Q₈	—	2	48	no	—
108	#117	—	2	24	no	—
109	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
110	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
111	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
112	C₂×C₂²⋊C₄	C₂³×C₅₃	1	12	yes	—
113	C₂×C₄²	Q₈×C₅₃	1	8	yes	complemented
114	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
115	C₂×C₄²	C₂³×C₅₃	1	24	yes	—
116	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
117	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
118	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
119	#95	C₂³×C₅₃	1	12	yes	—
120	#95	C₂³×C₅₃	1	12	yes	—
121	#95	C₂³×C₅₃	1	12	yes	—
122	#95	C₂³×C₅₃	1	4	yes	—
123	#95	C₂³×C₅₃	1	8	yes	—
124	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
125	C₂×Q₈	—	2	24	no	—
126	C₂²×C₄	—	2	12	no	—
127	C₂²×C₄	C₂×Q₈×C₅₃	1	12	yes	—
128	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
129	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
130	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
131	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
132	C₄⋊C₄	—	2	24	no	—
133	C₄⋊C₄	—	2	48	no	—
134	C₄⋊C₄	—	2	24	no	—
135	C₂²⋊C₄	—	2	24	no	—
136	C₄²	—	2	12	no	—
137	C₄²	—	2	48	no	—
138	C₄²	C₂×Q₈×C₅₃	1	12	yes	—
139	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
140	C₄²	#41×C₅₃	1	24	yes	—
141	C₂³	—	2	6	no	—
142	C₂³	C₂²×Q₈×C₅₃	1	2	yes	—
143	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
144	Q₈	—	2	32	no	—
145	C₂×C₄	—	2	24	no	—
146	C₂×C₄	—	2	6	no	—
147	C₂×C₄	—	2	24	no	—
148	C₂×C₄	—	2	12	no	—
149	C₂×C₄	—	2	24	no	—
150	C₂×C₄	C₂²×Q₈×C₅₃	1	6	yes	—
151	C₂×C₄	C₂²×Q₈×C₅₃	1	6	yes	—
152	C₂×C₄	C₂×#41×C₅₃	1	24	yes	—
153	C₂×C₄	C₂×#41×C₅₃	1	12	yes	—
154	C₂²	—	2	6	no	—
155	C₂²	—	4	4	no	—
156	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
157	C₂²	C₂²×#41×C₅₃	1	6	yes	—
158	C₄	—	2	6	no	—
159	C₄	—	2	24	no	—
160	C₄	—	4	12	no	—
161	C₄	C₅₃×#533	1	12	yes	—
162	C₂	—	4	2	no	—
163	C₂	C₂×C₅₃×#533	1	3	yes	—
164	C₂	C₅₃×#3444	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₅₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
socle	C₂³×C₅₃	C₂⁵
prime radical	C₂⁴×C₅₃	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#2402	C₂	semidirect
2. C₅₃×#2399	C₂	semidirect
4. C₄²×Q₈×C₅₃	C₂	semidirect
30. C₂×C₄²×C₅₃	Q₈	semidirect
71. #34552	C₅₃	direct (N × H)
80. #2402	C₂×C₅₃	semidirect
81. #2399	C₂×C₅₃	semidirect
83. C₄²×Q₈	C₂×C₅₃	semidirect
104. C₅₃	#34552	direct (N × H)
113. C₂×C₄²	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2402	2
C₅₃×#2399	2
C₅₃×#2396	2
C₄²×Q₈×C₅₃	2
C₄×#117×C₅₃	2
#34552	53

Subgroups — C53×#34552 #27242064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34552 #27242064