Subgroups — C₅₃×#9635 #27217147

Order 13568.

260 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	C₅₃×#1574	C₂	1	2	yes	maximal
2	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1299	C₂	1	2	yes	maximal
6	C₅₃×#431	C₂²	1	2	yes	—
7	C₅₃×#431	C₂²	1	1	yes	characteristic
8	C₅₃×#423	C₂²	1	2	yes	—
9	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#335	—	2	2	no	—
12	C₅₃×#335	C₄	1	2	yes	—
13	C₅₃×#334	—	2	2	no	—
14	C₅₃×#334	—	2	1	no	—
15	C₅₃×#334	—	2	1	no	—
16	C₅₃×#334	C₄	1	2	yes	—
17	C₅₃×#322	—	2	1	no	—
18	C₅₃×#322	—	2	1	no	—
19	C₂×M₁₆×C₅₃	—	2	1	no	—
20	C₂×M₁₆×C₅₃	—	2	2	no	—
21	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₈×C₅₃	—	2	2	no	—
25	C₄⋊C₈×C₅₃	—	2	2	no	—
26	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
27	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
28	C₂²⋊C₈×C₅₃	—	2	2	no	—
29	C₂²⋊C₈×C₅₃	—	2	1	no	—
30	#97×C₅₃	Q₈	1	2	yes	—
31	C₄×C₈×C₅₃	—	2	1	no	—
32	C₄×C₈×C₅₃	—	2	1	no	—
33	C₄×C₈×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
41	M₁₆×C₅₃	—	4	2	no	—
42	M₁₆×C₅₃	—	8	2	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	4	1	no	—
48	C₂×C₈×C₅₃	—	4	2	no	—
49	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	C₂³×C₅₃	#125	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₈×C₅₃	—	2	1	no	—
69	C₈×C₅₃	—	2	1	no	—
70	C₈×C₅₃	—	2	1	no	—
71	C₈×C₅₃	—	4	2	no	—
72	C₈×C₅₃	—	4	4	no	—
73	C₈×C₅₃	—	4	2	no	—
74	C₈×C₅₃	—	8	2	no	—
75	C₈×C₅₃	#108	1	2	yes	—
76	#9635	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	—	2	1	no	—
80	C₂²×C₅₃	#389	1	1	yes	characteristic
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	—	2	1	no	—
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
85	C₄×C₅₃	#430	1	1	yes	characteristic
86	C₄×C₅₃	#444	1	2	yes	—
87	C₄×C₅₃	#471	1	2	yes	—
88	#1574	C₂×C₅₃	1	2	yes	—
89	#1566	C₂×C₅₃	1	1	yes	characteristic
90	#1564	C₂×C₅₃	1	1	yes	characteristic
91	#1451	C₂×C₅₃	1	1	yes	characteristic
92	#1299	C₂×C₅₃	1	2	yes	—
93	C₂×C₅₃	—	4	1	no	—
94	C₂×C₅₃	#1946	1	1	yes	characteristic
95	C₂×C₅₃	#1950	1	1	yes	characteristic
96	C₂×C₅₃	#1951	1	1	yes	characteristic
97	#431	C₂²×C₅₃	1	2	yes	—
98	#431	C₂²×C₅₃	1	1	yes	characteristic
99	#423	C₂²×C₅₃	1	2	yes	—
100	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
102	#335	—	2	2	no	—
103	#335	C₄×C₅₃	1	2	yes	—
104	#334	—	2	2	no	—
105	#334	—	2	1	no	—
106	#334	—	2	1	no	—
107	#334	C₄×C₅₃	1	2	yes	—
108	#322	—	2	1	no	—
109	#322	—	2	1	no	—
110	C₅₃	#9635	1	1	yes	characteristic, direct product factor, Sylow
111	C₂×M₁₆	—	2	1	no	—
112	C₂×M₁₆	—	2	2	no	—
113	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
114	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
115	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
119	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₂²⋊C₈	—	2	1	no	—
122	#97	Q₈×C₅₃	1	2	yes	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
127	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
128	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
129	C₄×C₈	D₄×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	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
133	M₁₆	—	4	2	no	—
134	M₁₆	—	8	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
142	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
143	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
144	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
145	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
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	C₂³	#125×C₅₃	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
155	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
156	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
157	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
158	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
159	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	4	2	no	—
164	C₈	—	4	4	no	—
165	C₈	—	4	2	no	—
166	C₈	—	8	2	no	—
167	C₈	#108×C₅₃	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	C₅₃×#389	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	C₂×#108×C₅₃	1	1	yes	characteristic
176	C₄	C₅₃×#430	1	1	yes	characteristic
177	C₄	C₅₃×#444	1	2	yes	—
178	C₄	C₅₃×#471	1	2	yes	—
179	C₂	—	4	1	no	—
180	C₂	C₅₃×#1946	1	1	yes	characteristic
181	C₂	C₅₃×#1950	1	1	yes	characteristic
182	C₂	C₅₃×#1951	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#121
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	#125
LCS	G ▷ C₄² ▷ C₂² ▷ 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. #9635	C₅₃	direct (N × H)
110. C₅₃	#9635	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×#1574	2
C₅₃×#1566	2
C₅₃×#1564	2
C₅₃×#1451	2
C₅₃×#1299	2
#9635	53

Subgroups — C53×#9635 #27217147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9635 #27217147