Subgroups — C₅₃×#21735 #27229247

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 198 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×#2960	C₂	1	2	yes	maximal
3	C₅₃×#1995	C₂	1	2	yes	maximal
4	C₅₃×#1946	C₂	1	4	yes	maximal
5	C₅₃×#1849	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1772	C₂	1	4	yes	maximal
7	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#533	C₂²	1	2	yes	—
9	C₅₃×#501	C₄	1	8	yes	complemented
10	C₅₃×#446	—	2	4	no	—
11	C₈×Q₈×C₅₃	—	2	4	no	—
12	C₂×#108×C₅₃	C₂²	1	4	yes	—
13	C₅₃×#427	C₂²	1	4	yes	—
14	C₅₃×#424	C₂²	1	4	yes	—
15	C₅₃×#423	C₂²	1	2	yes	—
16	C₅₃×#405	C₂²	1	4	yes	—
17	C₂×#97×C₅₃	C₂²	1	2	yes	—
18	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#376	C₂²	1	1	yes	characteristic
22	C₅₃×#339	C₂²	1	8	yes	—
23	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
24	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
26	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
28	#125×C₅₃	C₂×C₄	1	4	yes	—
29	C₂²⋊Q₈×C₅₃	—	2	4	no	—
30	C₄×Q₈×C₅₃	—	2	4	no	—
31	#117×C₅₃	C₂³	1	4	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	#108×C₅₃	—	2	8	no	—
35	#107×C₅₃	C₂×C₄	1	8	yes	—
36	#106×C₅₃	C₂×C₄	1	8	yes	—
37	C₄⋊C₈×C₅₃	—	2	8	no	—
38	C₄⋊C₈×C₅₃	—	2	4	no	—
39	#97×C₅₃	—	2	4	no	—
40	#97×C₅₃	—	2	4	no	—
41	#97×C₅₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₅₃	—	2	4	no	—
43	C₄×C₈×C₅₃	D₄	1	2	yes	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₄×C₈×C₅₃	Q₈	1	8	yes	complemented
46	#95×C₅₃	—	2	2	no	—
47	C₂×Q₈×C₅₃	—	2	4	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	#41	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
51	M₁₆×C₅₃	—	2	8	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	2	4	no	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
59	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
60	C₂×C₈×C₅₃	#41	1	4	yes	—
61	C₄⋊C₄×C₅₃	—	2	8	no	—
62	C₄⋊C₄×C₅₃	—	2	4	no	—
63	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
64	C₂²⋊C₄×C₅₃	—	2	4	no	—
65	C₄²×C₅₃	—	2	4	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
68	C₄²×C₅₃	C₂×D₄	1	2	yes	—
69	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
70	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
71	Q₈×C₅₃	—	4	8	no	—
72	C₂×C₄×C₅₃	—	2	8	no	—
73	C₂×C₄×C₅₃	—	2	4	no	—
74	C₂×C₄×C₅₃	—	4	2	no	—
75	C₂×C₄×C₅₃	—	4	4	no	—
76	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
77	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
81	C₈×C₅₃	—	2	2	no	—
82	C₈×C₅₃	—	4	4	no	—
83	C₈×C₅₃	C₄×Q₈	1	8	yes	—
84	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
85	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
86	#21735	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₅₃	—	2	2	no	—
88	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
89	C₂²×C₅₃	#533	1	2	yes	—
90	C₄×C₅₃	—	2	4	no	—
91	C₄×C₅₃	—	4	4	no	—
92	C₄×C₅₃	—	4	8	no	—
93	C₄×C₅₃	C₂×C₄×Q₈	1	2	yes	—
94	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
95	C₄×C₅₃	#532	1	1	yes	characteristic
96	#3162	C₂×C₅₃	1	1	yes	characteristic
97	#2960	C₂×C₅₃	1	2	yes	—
98	#1995	C₂×C₅₃	1	2	yes	—
99	#1946	C₂×C₅₃	1	4	yes	—
100	#1849	C₂×C₅₃	1	1	yes	characteristic
101	#1772	C₂×C₅₃	1	4	yes	—
102	#1726	C₂×C₅₃	1	1	yes	characteristic
103	C₂×C₅₃	—	2	2	no	—
104	C₂×C₅₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₅₃	#2955	1	2	yes	—
106	#533	C₂²×C₅₃	1	2	yes	—
107	#501	C₄×C₅₃	1	8	yes	complemented
108	#446	—	2	4	no	—
109	C₈×Q₈	—	2	4	no	—
110	C₂×#108	C₂²×C₅₃	1	4	yes	—
111	#427	C₂²×C₅₃	1	4	yes	—
112	#424	C₂²×C₅₃	1	4	yes	—
113	#423	C₂²×C₅₃	1	2	yes	—
114	#405	C₂²×C₅₃	1	4	yes	—
115	C₂×#97	C₂²×C₅₃	1	2	yes	—
116	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
119	#376	C₂²×C₅₃	1	1	yes	characteristic
120	#339	C₂²×C₅₃	1	8	yes	—
121	C₅₃	#21735	1	1	yes	characteristic, direct product factor, Sylow
122	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
123	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
124	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
125	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
126	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
127	#125	C₂×C₄×C₅₃	1	4	yes	—
128	C₂²⋊Q₈	—	2	4	no	—
129	C₄×Q₈	—	2	4	no	—
130	#117	C₂³×C₅₃	1	4	yes	—
131	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
132	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
133	#108	—	2	8	no	—
134	#107	C₂×C₄×C₅₃	1	8	yes	—
135	#106	C₂×C₄×C₅₃	1	8	yes	—
136	C₄⋊C₈	—	2	8	no	—
137	C₄⋊C₈	—	2	4	no	—
138	#97	—	2	4	no	—
139	#97	—	2	4	no	—
140	#97	C₂×C₄×C₅₃	1	4	yes	—
141	C₄×C₈	—	2	4	no	—
142	C₄×C₈	D₄×C₅₃	1	2	yes	—
143	C₄×C₈	D₄×C₅₃	1	2	yes	—
144	C₄×C₈	Q₈×C₅₃	1	8	yes	complemented
145	#95	—	2	2	no	—
146	C₂×Q₈	—	2	4	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	#41×C₅₃	1	2	yes	—
149	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
150	M₁₆	—	2	8	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
154	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
155	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
156	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
157	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
158	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
159	C₂×C₈	#41×C₅₃	1	4	yes	—
160	C₄⋊C₄	—	2	8	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	2	2	no	—
166	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
167	C₄²	C₂×D₄×C₅₃	1	2	yes	—
168	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
169	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
170	Q₈	—	4	8	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
176	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
177	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
178	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
179	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	4	4	no	—
182	C₈	C₄×Q₈×C₅₃	1	8	yes	—
183	C₈	C₄⋊Q₈×C₅₃	1	2	yes	—
184	C₈	C₄⋊Q₈×C₅₃	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
187	C₂²	C₅₃×#533	1	2	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	4	4	no	—
190	C₄	—	4	8	no	—
191	C₄	C₂×C₄×Q₈×C₅₃	1	2	yes	—
192	C₄	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
193	C₄	C₅₃×#532	1	1	yes	characteristic
194	C₂	—	2	2	no	—
195	C₂	C₄×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
196	C₂	C₅₃×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₅₃	C₂×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×C₈×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₅₃×#501	C₄	semidirect
45. C₄×C₈×C₅₃	Q₈	semidirect
86. #21735	C₅₃	direct (N × H)
107. #501	C₄×C₅₃	semidirect
121. C₅₃	#21735	direct (N × H)
144. C₄×C₈	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×#3162	2
C₅₃×#2960	2
C₅₃×#1995	2
C₅₃×#1946	2
C₅₃×#1849	2
C₅₃×#1772	2
C₅₃×#1726	2
#21735	53

Subgroups — C53×#21735 #27229247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21735 #27229247