Subgroups — C₅₃×#9609 #27217121

Order 13568.

294 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1565	C₂	1	2	yes	maximal
3	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1283	C₂	1	2	yes	maximal
6	C₅₃×#431	C₂²	1	1	yes	characteristic
7	C₅₃×#431	C₂²	1	1	yes	characteristic
8	C₂×#108×C₅₃	C₄	1	2	yes	—
9	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
10	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#341	C₄	1	1	yes	characteristic
13	C₅₃×#341	C₄	1	1	yes	characteristic
14	C₅₃×#335	—	2	2	no	—
15	C₅₃×#334	—	2	2	no	—
16	C₅₃×#334	—	2	2	no	—
17	C₅₃×#322	—	2	2	no	—
18	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₅₃	—	2	2	no	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	#108×C₅₃	—	2	4	no	—
25	C₄⋊C₈×C₅₃	—	2	2	no	—
26	C₄⋊C₈×C₅₃	—	2	2	no	—
27	C₄⋊C₈×C₅₃	—	2	1	no	—
28	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
29	C₂²⋊C₈×C₅₃	—	2	1	no	—
30	C₂²⋊C₈×C₅₃	Q₈	1	2	yes	—
31	#97×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	—	2	1	no	—
33	C₄×C₈×C₅₃	—	2	1	no	—
34	C₄×C₈×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₄×C₈×C₅₃	Q₈	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
40	M₁₆×C₅₃	—	2	2	no	—
41	M₁₆×C₅₃	—	2	2	no	—
42	M₁₆×C₅₃	—	2	2	no	—
43	M₁₆×C₅₃	M₁₆	1	4	yes	—
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	—	8	2	no	—
52	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₅₃	#41	1	1	yes	characteristic
58	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	4	1	no	—
61	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
68	C₂×C₄×C₅₃	#131	1	2	yes	—
69	C₈×C₅₃	—	2	2	no	—
70	C₈×C₅₃	—	2	2	no	—
71	C₈×C₅₃	—	2	4	no	—
72	C₈×C₅₃	—	4	2	no	—
73	C₈×C₅₃	—	4	2	no	—
74	C₈×C₅₃	—	4	2	no	—
75	C₈×C₅₃	—	8	2	no	—
76	#9609	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×C₅₃	—	2	1	no	—
78	C₂²×C₅₃	—	2	1	no	—
79	C₂²×C₅₃	#389	1	1	yes	characteristic
80	C₂²×C₅₃	#446	1	2	yes	—
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	—	2	1	no	—
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	#423	1	1	yes	characteristic
85	C₄×C₅₃	#424	1	1	yes	characteristic
86	C₄×C₅₃	#435	1	2	yes	—
87	C₄×C₅₃	#471	1	2	yes	—
88	C₄×#108	C₂×C₅₃	1	1	yes	characteristic
89	#1565	C₂×C₅₃	1	2	yes	—
90	C₂×#334	C₂×C₅₃	1	1	yes	characteristic
91	#1452	C₂×C₅₃	1	1	yes	characteristic
92	#1283	C₂×C₅₃	1	2	yes	—
93	C₂×C₅₃	—	2	2	no	—
94	C₂×C₅₃	#1940	1	1	yes	characteristic
95	C₂×C₅₃	#1951	1	1	yes	characteristic
96	C₂×C₅₃	#1953	1	1	yes	characteristic
97	#431	C₂²×C₅₃	1	1	yes	characteristic
98	#431	C₂²×C₅₃	1	1	yes	characteristic
99	C₂×#108	C₄×C₅₃	1	2	yes	—
100	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
101	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
102	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
103	#341	C₄×C₅₃	1	1	yes	characteristic
104	#341	C₄×C₅₃	1	1	yes	characteristic
105	#335	—	2	2	no	—
106	#334	—	2	2	no	—
107	#334	—	2	2	no	—
108	#322	—	2	2	no	—
109	C₅₃	#9609	1	1	yes	characteristic, direct product factor, Sylow
110	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
111	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
114	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
115	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
116	#108	—	2	4	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	1	no	—
120	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
121	C₂²⋊C₈	—	2	1	no	—
122	C₂²⋊C₈	Q₈×C₅₃	1	2	yes	—
123	#97	—	2	2	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	2	1	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	D₄×C₅₃	1	2	yes	—
128	C₄×C₈	Q₈×C₅₃	1	2	yes	—
129	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
130	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
131	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
132	M₁₆	—	2	2	no	—
133	M₁₆	—	2	2	no	—
134	M₁₆	—	2	2	no	—
135	M₁₆	M₁₆×C₅₃	1	4	yes	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	8	2	no	—
144	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
145	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
146	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
147	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
148	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
149	C₄²	#41×C₅₃	1	1	yes	characteristic
150	C₂³	C₄×Q₈×C₅₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
154	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
155	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
156	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
157	C₂×C₄	#125×C₅₃	1	1	yes	characteristic
158	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
159	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
160	C₂×C₄	#131×C₅₃	1	2	yes	—
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	—	2	4	no	—
164	C₈	—	4	2	no	—
165	C₈	—	4	2	no	—
166	C₈	—	4	2	no	—
167	C₈	—	8	2	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	C₅₃×#389	1	1	yes	characteristic
171	C₂²	C₅₃×#446	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	C₅₃×#423	1	1	yes	characteristic
176	C₄	C₅₃×#424	1	1	yes	characteristic
177	C₄	C₅₃×#435	1	2	yes	—
178	C₄	C₅₃×#471	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	C₅₃×#1940	1	1	yes	characteristic
181	C₂	C₅₃×#1951	1	1	yes	characteristic
182	C₂	C₅₃×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#121
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
76. #9609	C₅₃	direct (N × H)
109. C₅₃	#9609	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#108×C₅₃	2
C₅₃×#1565	2
C₂×C₅₃×#334	2
C₅₃×#1452	2
C₅₃×#1283	2
#9609	53

Subgroups — C53×#9609 #27217121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9609 #27217121