Subgroups — C₅₃×#12060 #27219572

Order 13568.

204 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2183	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₈×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1367	C₂	1	1	yes	maximal, characteristic
5	C₅₃×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₅₃×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1312	C₂	1	1	yes	maximal, characteristic
8	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
11	C₅₃×C₄⋊C₁₆	—	2	1	no	—
12	C₅₃×C₄⋊C₁₆	—	2	1	no	—
13	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₅₃×#346	—	2	1	no	—
16	C₄×C₁₆×C₅₃	—	2	1	no	—
17	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#335	C₄	1	1	yes	characteristic
20	C₅₃×Q₈⋊C₈	C₄	1	1	yes	characteristic
21	C₅₃×Q₈⋊C₈	C₄	1	1	yes	characteristic
22	C₈²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₁₆×C₅₃	C₈	1	1	yes	characteristic
24	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	—	2	1	no	—
27	C₂×C₁₆×C₅₃	—	2	1	no	—
28	C₂×C₁₆×C₅₃	—	2	1	no	—
29	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
31	#107×C₅₃	C₈	1	1	yes	characteristic
32	C₄⋊C₈×C₅₃	—	2	1	no	—
33	C₄⋊C₈×C₅₃	—	2	1	no	—
34	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	#103×C₅₃	C₈	1	1	yes	characteristic
37	#103×C₅₃	C₈	1	1	yes	characteristic
38	C₄×C₈×C₅₃	—	2	1	no	—
39	C₄×C₈×C₅₃	—	2	1	no	—
40	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
45	Q₁₆×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₅₃	—	2	1	no	—
54	C₄⋊C₄×C₅₃	—	2	1	no	—
55	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
56	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
57	C₄²×C₅₃	—	2	1	no	—
58	C₄²×C₅₃	—	2	1	no	—
59	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₁₆×C₅₃	—	2	1	no	—
61	C₁₆×C₅₃	—	2	1	no	—
62	C₁₆×C₅₃	—	4	1	no	—
63	C₁₆×C₅₃	—	4	1	no	—
64	C₁₆×C₅₃	—	4	1	no	—
65	Q₈×C₅₃	—	2	1	no	—
66	Q₈×C₅₃	—	2	1	no	—
67	Q₈×C₅₃	—	2	1	no	—
68	Q₈×C₅₃	M₃₂	1	2	yes	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
76	C₈×C₅₃	—	2	2	no	—
77	C₈×C₅₃	—	2	1	no	—
78	C₈×C₅₃	—	2	1	no	—
79	C₈×C₅₃	—	2	1	no	—
80	C₈×C₅₃	—	4	1	no	—
81	C₈×C₅₃	—	4	1	no	—
82	C₈×C₅₃	#137	1	1	yes	characteristic
83	C₈×C₅₃	#137	1	1	yes	characteristic
84	#12060	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
86	C₄×C₅₃	—	2	2	no	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	—	4	1	no	—
89	C₄×C₅₃	—	4	1	no	—
90	C₄×C₅₃	—	4	1	no	—
91	C₄×C₅₃	#441	1	1	yes	characteristic
92	C₄×C₅₃	#444	1	1	yes	characteristic
93	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
94	C₄×C₅₃	#504	1	1	yes	characteristic
95	#2183	C₂×C₅₃	1	1	yes	characteristic
96	Q₈×C₁₆	C₂×C₅₃	1	1	yes	characteristic
97	C₈×Q₁₆	C₂×C₅₃	1	1	yes	characteristic
98	#1367	C₂×C₅₃	1	1	yes	characteristic
99	Q₈⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
100	Q₈⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
101	#1312	C₂×C₅₃	1	1	yes	characteristic
102	C₂×C₅₃	#1579	1	1	yes	characteristic
103	C₂×C₅₃	#2168	1	1	yes	characteristic
104	C₂×C₅₃	#2170	1	1	yes	characteristic
105	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic
106	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic
107	C₄×Q₁₆	C₄×C₅₃	1	1	yes	characteristic
108	C₄⋊C₁₆	—	2	1	no	—
109	C₄⋊C₁₆	—	2	1	no	—
110	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
111	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
112	#346	—	2	1	no	—
113	C₄×C₁₆	—	2	1	no	—
114	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
115	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
116	#335	C₄×C₅₃	1	1	yes	characteristic
117	Q₈⋊C₈	C₄×C₅₃	1	1	yes	characteristic
118	Q₈⋊C₈	C₄×C₅₃	1	1	yes	characteristic
119	C₈²	C₂²×C₅₃	1	1	yes	characteristic
120	C₅₃	#12060	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×Q₁₆	C₈×C₅₃	1	1	yes	characteristic
122	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
123	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	—	2	1	no	—
126	C₂×C₁₆	—	2	1	no	—
127	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
128	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
129	#107	C₈×C₅₃	1	1	yes	characteristic
130	C₄⋊C₈	—	2	1	no	—
131	C₄⋊C₈	—	2	1	no	—
132	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
133	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
134	#103	C₈×C₅₃	1	1	yes	characteristic
135	#103	C₈×C₅₃	1	1	yes	characteristic
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
139	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
140	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
141	C₂×Q₈	C₂×C₈×C₅₃	1	1	yes	characteristic
142	C₂×Q₈	C₂×C₈×C₅₃	1	1	yes	characteristic
143	Q₁₆	—	2	2	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
148	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
149	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
150	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
151	C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
154	C₄⋊C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
155	C₄²	—	2	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
158	C₁₆	—	2	1	no	—
159	C₁₆	—	2	1	no	—
160	C₁₆	—	4	1	no	—
161	C₁₆	—	4	1	no	—
162	C₁₆	—	4	1	no	—
163	Q₈	—	2	1	no	—
164	Q₈	—	2	1	no	—
165	Q₈	—	2	1	no	—
166	Q₈	M₃₂×C₅₃	1	2	yes	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
172	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
173	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	4	1	no	—
179	C₈	—	4	1	no	—
180	C₈	#137×C₅₃	1	1	yes	characteristic
181	C₈	#137×C₅₃	1	1	yes	characteristic
182	C₂²	C₈×D₄×C₅₃	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	4	1	no	—
187	C₄	—	4	1	no	—
188	C₄	C₅₃×#441	1	1	yes	characteristic
189	C₄	C₅₃×#444	1	1	yes	characteristic
190	C₄	C₂×M₃₂×C₅₃	1	1	yes	characteristic
191	C₄	C₅₃×#504	1	1	yes	characteristic
192	C₂	C₅₃×#1579	1	1	yes	characteristic
193	C₂	C₅₃×#2168	1	1	yes	characteristic
194	C₂	C₅₃×#2170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle, prime radical	C₂²×C₅₃	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
84. #12060	C₅₃	direct (N × H)
120. C₅₃	#12060	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×#2183	2
Q₈×C₁₆×C₅₃	2
C₈×Q₁₆×C₅₃	2
C₅₃×#1367	2
C₅₃×Q₈⋊C₁₆	2
C₅₃×#1312	2
#12060	53

Subgroups — C53×#12060 #27219572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12060 #27219572