Subgroups — C₅₃×#22259 #27229771

Order 13568.

594 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×#3162	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2961	C₂	1	2	yes	maximal
3	C₅₃×#1994	C₂	1	2	yes	maximal
4	C₅₃×#1922	C₂	1	8	yes	maximal
5	C₅₃×#1838	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1745	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#533	C₂²	1	2	yes	—
8	C₅₃×#501	C₄	1	8	yes	complemented
9	C₅₃×#446	—	2	8	no	—
10	C₅₃×#427	C₂²	1	4	yes	—
11	C₅₃×#424	C₂²	1	4	yes	—
12	C₅₃×#423	C₂²	1	2	yes	—
13	C₅₃×#405	C₂²	1	4	yes	—
14	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#341	C₂²	1	4	yes	—
18	C₅₃×#339	C₂²	1	8	yes	—
19	C₅₃×#336	C₂²	1	4	yes	—
20	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
21	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
22	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
23	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
24	#125×C₅₃	C₂×C₄	1	4	yes	—
25	C₂²⋊Q₈×C₅₃	—	2	4	no	—
26	C₄×Q₈×C₅₃	—	2	4	no	—
27	#117×C₅₃	C₂³	1	4	yes	—
28	C₂×C₄²×C₅₃	—	2	2	no	—
29	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	#107×C₅₃	C₂×C₄	1	8	yes	—
32	#106×C₅₃	C₂×C₄	1	8	yes	—
33	C₄⋊C₈×C₅₃	—	2	8	no	—
34	C₄⋊C₈×C₅₃	—	2	4	no	—
35	#97×C₅₃	—	2	4	no	—
36	#97×C₅₃	C₂×C₄	1	4	yes	—
37	#97×C₅₃	D₄	1	4	yes	—
38	C₄×C₈×C₅₃	—	2	4	no	—
39	C₂×Q₈×C₅₃	—	2	4	no	—
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	#41	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
43	M₁₆×C₅₃	—	2	8	no	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₂×C₈×C₅₃	—	2	4	no	—
46	C₂×C₈×C₅₃	—	2	4	no	—
47	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
48	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
49	C₄⋊C₄×C₅₃	—	2	8	no	—
50	C₄⋊C₄×C₅₃	—	2	4	no	—
51	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
52	C₂²⋊C₄×C₅₃	—	2	4	no	—
53	C₄²×C₅₃	—	2	4	no	—
54	C₄²×C₅₃	—	2	4	no	—
55	C₄²×C₅₃	—	4	4	no	—
56	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
57	C₄²×C₅₃	C₂×D₄	1	2	yes	—
58	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
59	Q₈×C₅₃	—	8	4	no	—
60	C₂×C₄×C₅₃	—	2	4	no	—
61	C₂×C₄×C₅₃	—	2	8	no	—
62	C₂×C₄×C₅₃	—	4	2	no	—
63	C₂×C₄×C₅₃	—	4	2	no	—
64	C₂×C₄×C₅₃	—	4	2	no	—
65	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
66	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
69	C₈×C₅₃	—	2	4	no	—
70	C₈×C₅₃	—	2	4	no	—
71	C₈×C₅₃	—	4	4	no	—
72	#22259	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂²×C₅₃	—	2	2	no	—
74	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
75	C₂²×C₅₃	#541	1	2	yes	—
76	C₄×C₅₃	—	2	4	no	—
77	C₄×C₅₃	—	4	4	no	—
78	C₄×C₅₃	—	4	8	no	—
79	C₄×C₅₃	#519	1	2	yes	—
80	C₄×C₅₃	#536	1	2	yes	—
81	#3162	C₂×C₅₃	1	1	yes	characteristic
82	#2961	C₂×C₅₃	1	2	yes	—
83	#1994	C₂×C₅₃	1	2	yes	—
84	#1922	C₂×C₅₃	1	8	yes	—
85	#1838	C₂×C₅₃	1	1	yes	characteristic
86	#1745	C₂×C₅₃	1	1	yes	characteristic
87	C₂×C₅₃	—	2	2	no	—
88	C₂×C₅₃	#2327	1	1	yes	characteristic
89	C₂×C₅₃	#2956	1	2	yes	—
90	#533	C₂²×C₅₃	1	2	yes	—
91	#501	C₄×C₅₃	1	8	yes	complemented
92	#446	—	2	8	no	—
93	#427	C₂²×C₅₃	1	4	yes	—
94	#424	C₂²×C₅₃	1	4	yes	—
95	#423	C₂²×C₅₃	1	2	yes	—
96	#405	C₂²×C₅₃	1	4	yes	—
97	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
98	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
99	C₄³	C₂²×C₅₃	1	1	yes	characteristic
100	#341	C₂²×C₅₃	1	4	yes	—
101	#339	C₂²×C₅₃	1	8	yes	—
102	#336	C₂²×C₅₃	1	4	yes	—
103	C₅₃	#22259	1	1	yes	characteristic, direct product factor, Sylow
104	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
105	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
106	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
107	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
108	#125	C₂×C₄×C₅₃	1	4	yes	—
109	C₂²⋊Q₈	—	2	4	no	—
110	C₄×Q₈	—	2	4	no	—
111	#117	C₂³×C₅₃	1	4	yes	—
112	C₂×C₄²	—	2	2	no	—
113	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
114	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
115	#107	C₂×C₄×C₅₃	1	8	yes	—
116	#106	C₂×C₄×C₅₃	1	8	yes	—
117	C₄⋊C₈	—	2	8	no	—
118	C₄⋊C₈	—	2	4	no	—
119	#97	—	2	4	no	—
120	#97	C₂×C₄×C₅₃	1	4	yes	—
121	#97	D₄×C₅₃	1	4	yes	—
122	C₄×C₈	—	2	4	no	—
123	C₂×Q₈	—	2	4	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	#41×C₅₃	1	2	yes	—
126	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
127	M₁₆	—	2	8	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
132	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
133	C₄⋊C₄	—	2	8	no	—
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
136	C₂²⋊C₄	—	2	4	no	—
137	C₄²	—	2	4	no	—
138	C₄²	—	2	4	no	—
139	C₄²	—	4	4	no	—
140	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
141	C₄²	C₂×D₄×C₅₃	1	2	yes	—
142	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
143	Q₈	—	8	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
150	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
151	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
152	C₂×C₄	2₋¹⁺⁴×C₅₃	1	4	yes	—
153	C₈	—	2	4	no	—
154	C₈	—	2	4	no	—
155	C₈	—	4	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
158	C₂²	C₅₃×#541	1	2	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	4	4	no	—
161	C₄	—	4	8	no	—
162	C₄	C₅₃×#519	1	2	yes	—
163	C₄	C₅₃×#536	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	C₅₃×#2327	1	1	yes	characteristic
166	C₂	C₅₃×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅₃×#501	C₄	semidirect
72. #22259	C₅₃	direct (N × H)
91. #501	C₄×C₅₃	semidirect
103. C₅₃	#22259	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3162	2
C₅₃×#2961	2
C₅₃×#1994	2
C₅₃×#1922	2
C₅₃×#1838	2
C₅₃×#1745	2
#22259	53

Subgroups — C53×#22259 #27229771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22259 #27229771