Subgroups — C₅₃×#9599 #27217111

Order 13568.

378 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1939	C₂	1	2	yes	maximal
2	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1279	C₂	1	2	yes	maximal
6	C₂×#107×C₅₃	C₄	1	2	yes	—
7	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
8	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#340	C₄	1	1	yes	characteristic
14	C₅₃×#340	C₄	1	1	yes	characteristic
15	C₅₃×#336	—	2	2	no	—
16	C₅₃×#334	—	2	2	no	—
17	C₅₃×#322	—	2	2	no	—
18	C₂²×C₈×C₅₃	—	2	2	no	—
19	C₂²×C₈×C₅₃	—	2	1	no	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	D₄	1	2	yes	—
23	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅₃	—	2	2	no	—
26	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	#107×C₅₃	—	2	4	no	—
28	C₄⋊C₈×C₅₃	—	2	2	no	—
29	C₄⋊C₈×C₅₃	—	2	2	no	—
30	C₄⋊C₈×C₅₃	—	4	4	no	—
31	C₄×C₈×C₅₃	—	2	1	no	—
32	C₄×C₈×C₅₃	—	2	1	no	—
33	C₄×C₈×C₅₃	—	2	1	no	—
34	C₄×C₈×C₅₃	D₄	1	2	yes	—
35	C₄×C₈×C₅₃	Q₈	1	2	yes	—
36	C₄×C₈×C₅₃	Q₈	1	2	yes	—
37	#95×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂×C₈×C₅₃	—	2	4	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	4	2	no	—
51	C₂×C₈×C₅₃	—	4	2	no	—
52	C₂×C₈×C₅₃	—	4	2	no	—
53	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
59	C₄²×C₅₃	—	4	2	no	—
60	C₄²×C₅₃	—	4	2	no	—
61	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₅₃	#41	1	1	yes	characteristic
65	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
80	C₂×C₄×C₅₃	#131	1	2	yes	—
81	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
82	C₂×C₄×C₅₃	#137	1	2	yes	—
83	C₈×C₅₃	—	2	4	no	—
84	C₈×C₅₃	—	2	2	no	—
85	C₈×C₅₃	—	2	2	no	—
86	C₈×C₅₃	—	4	2	no	—
87	C₈×C₅₃	—	8	2	no	—
88	#9599	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₅₃	#389	1	1	yes	characteristic
90	C₂²×C₅₃	#435	1	2	yes	—
91	C₂²×C₅₃	#441	1	1	yes	characteristic
92	C₂²×C₅₃	#442	1	1	yes	characteristic
93	C₂²×C₅₃	#443	1	2	yes	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	—	4	4	no	—
97	C₄×C₅₃	—	8	2	no	—
98	C₄×C₅₃	#423	1	1	yes	characteristic
99	C₄×C₅₃	#424	1	1	yes	characteristic
100	C₄×C₅₃	#446	1	2	yes	—
101	C₄×C₅₃	#468	1	2	yes	—
102	C₄×C₅₃	#501	1	2	yes	—
103	#1939	C₂×C₅₃	1	2	yes	—
104	C₄×#107	C₂×C₅₃	1	1	yes	characteristic
105	C₂×#334	C₂×C₅₃	1	1	yes	characteristic
106	C₂×#322	C₂×C₅₃	1	1	yes	characteristic
107	#1279	C₂×C₅₃	1	2	yes	—
108	C₂×C₅₃	#1586	1	2	yes	—
109	C₂×C₅₃	#1587	1	2	yes	—
110	C₂×C₅₃	#1940	1	1	yes	characteristic
111	C₂×C₅₃	#1943	1	1	yes	characteristic
112	C₂×C₅₃	#1945	1	1	yes	characteristic
113	C₂×#107	C₄×C₅₃	1	2	yes	—
114	C₂×C₄⋊C₈	—	2	2	no	—
115	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
116	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
119	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
120	#340	C₄×C₅₃	1	1	yes	characteristic
121	#340	C₄×C₅₃	1	1	yes	characteristic
122	#336	—	2	2	no	—
123	#334	—	2	2	no	—
124	#322	—	2	2	no	—
125	C₅₃	#9599	1	1	yes	characteristic, direct product factor, Sylow
126	C₂²×C₈	—	2	1	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
129	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
130	C₂²×C₈	D₄×C₅₃	1	2	yes	—
131	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
132	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
133	C₂×C₄²	—	2	2	no	—
134	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
135	#107	—	2	4	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	4	4	no	—
139	C₄×C₈	—	2	1	no	—
140	C₄×C₈	—	2	1	no	—
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	D₄×C₅₃	1	2	yes	—
143	C₄×C₈	Q₈×C₅₃	1	2	yes	—
144	C₄×C₈	Q₈×C₅₃	1	2	yes	—
145	#95	—	2	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
150	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
162	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	M₁₆×C₅₃	1	4	yes	—
167	C₄²	—	4	2	no	—
168	C₄²	—	4	2	no	—
169	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
170	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
171	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
172	C₄²	#41×C₅₃	1	1	yes	characteristic
173	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
182	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
183	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
184	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
185	C₂×C₄	#125×C₅₃	1	1	yes	characteristic
186	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
187	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
188	C₂×C₄	#131×C₅₃	1	2	yes	—
189	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
190	C₂×C₄	#137×C₅₃	1	2	yes	—
191	C₈	—	2	4	no	—
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	4	2	no	—
195	C₈	—	8	2	no	—
196	C₂²	C₅₃×#389	1	1	yes	characteristic
197	C₂²	C₅₃×#435	1	2	yes	—
198	C₂²	C₅₃×#441	1	1	yes	characteristic
199	C₂²	C₅₃×#442	1	1	yes	characteristic
200	C₂²	C₅₃×#443	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	4	4	no	—
204	C₄	—	8	2	no	—
205	C₄	C₅₃×#423	1	1	yes	characteristic
206	C₄	C₅₃×#424	1	1	yes	characteristic
207	C₄	C₅₃×#446	1	2	yes	—
208	C₄	C₅₃×#468	1	2	yes	—
209	C₄	C₅₃×#501	1	2	yes	—
210	C₂	C₅₃×#1586	1	2	yes	—
211	C₂	C₅₃×#1587	1	2	yes	—
212	C₂	C₅₃×#1940	1	1	yes	characteristic
213	C₂	C₅₃×#1943	1	1	yes	characteristic
214	C₂	C₅₃×#1945	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
88. #9599	C₅₃	direct (N × H)
125. C₅₃	#9599	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×#1939	2
C₄×#107×C₅₃	2
C₂×C₅₃×#334	2
C₂×C₅₃×#322	2
C₅₃×#1279	2
#9599	53

Subgroups — C53×#9599 #27217111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9599 #27217111