Subgroups — C₅₃×#9636 #27217148

Order 13568.

260 conjugacy classes of subgroups.

The subgroup lattice has 194 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×#1567	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	—	2	1	no	—
13	C₅₃×#335	—	2	1	no	—
14	C₅₃×#335	C₄	1	2	yes	—
15	C₅₃×#334	—	2	2	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	1	yes	characteristic
28	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂²⋊C₈×C₅₃	—	2	2	no	—
30	C₂²⋊C₈×C₅₃	—	2	1	no	—
31	#97×C₅₃	Q₈	1	2	yes	—
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	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
42	M₁₆×C₅₃	—	4	2	no	—
43	M₁₆×C₅₃	—	8	2	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	—	2	4	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	4	1	no	—
50	C₂×C₈×C₅₃	—	4	2	no	—
51	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂³×C₅₃	#125	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
71	C₈×C₅₃	—	2	1	no	—
72	C₈×C₅₃	—	2	1	no	—
73	C₈×C₅₃	—	2	1	no	—
74	C₈×C₅₃	—	4	2	no	—
75	C₈×C₅₃	—	4	4	no	—
76	C₈×C₅₃	—	4	1	no	—
77	C₈×C₅₃	—	4	1	no	—
78	C₈×C₅₃	—	8	2	no	—
79	C₈×C₅₃	#108	1	2	yes	—
80	#9636	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂²×C₅₃	—	2	1	no	—
82	C₂²×C₅₃	—	2	1	no	—
83	C₂²×C₅₃	—	2	1	no	—
84	C₂²×C₅₃	#389	1	1	yes	characteristic
85	C₄×C₅₃	—	2	1	no	—
86	C₄×C₅₃	—	2	1	no	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
89	C₄×C₅₃	#430	1	1	yes	characteristic
90	C₄×C₅₃	#444	1	2	yes	—
91	C₄×C₅₃	#472	1	1	yes	characteristic
92	C₄×C₅₃	#473	1	1	yes	characteristic
93	#1574	C₂×C₅₃	1	2	yes	—
94	#1567	C₂×C₅₃	1	1	yes	characteristic
95	#1564	C₂×C₅₃	1	1	yes	characteristic
96	#1451	C₂×C₅₃	1	1	yes	characteristic
97	#1299	C₂×C₅₃	1	2	yes	—
98	C₂×C₅₃	—	4	1	no	—
99	C₂×C₅₃	#1946	1	1	yes	characteristic
100	C₂×C₅₃	#1950	1	1	yes	characteristic
101	C₂×C₅₃	#1952	1	1	yes	characteristic
102	#431	C₂²×C₅₃	1	2	yes	—
103	#431	C₂²×C₅₃	1	1	yes	characteristic
104	#423	C₂²×C₅₃	1	2	yes	—
105	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
107	#335	—	2	2	no	—
108	#335	—	2	1	no	—
109	#335	—	2	1	no	—
110	#335	C₄×C₅₃	1	2	yes	—
111	#334	—	2	2	no	—
112	#334	C₄×C₅₃	1	2	yes	—
113	#322	—	2	1	no	—
114	#322	—	2	1	no	—
115	C₅₃	#9636	1	1	yes	characteristic, direct product factor, Sylow
116	C₂×M₁₆	—	2	1	no	—
117	C₂×M₁₆	—	2	2	no	—
118	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
119	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
120	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
124	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
125	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
126	C₂²⋊C₈	—	2	2	no	—
127	C₂²⋊C₈	—	2	1	no	—
128	#97	Q₈×C₅₃	1	2	yes	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	—	2	1	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
133	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
134	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
135	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
137	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
138	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
139	M₁₆	—	4	2	no	—
140	M₁₆	—	8	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
149	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
150	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
151	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
152	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
153	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
154	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
155	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
156	C₂³	#125×C₅₃	1	1	yes	characteristic
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
163	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
164	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
165	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
166	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
167	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	4	2	no	—
172	C₈	—	4	4	no	—
173	C₈	—	4	1	no	—
174	C₈	—	4	1	no	—
175	C₈	—	8	2	no	—
176	C₈	#108×C₅₃	1	2	yes	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	1	no	—
180	C₂²	C₅₃×#389	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	C₂×#108×C₅₃	1	1	yes	characteristic
185	C₄	C₅₃×#430	1	1	yes	characteristic
186	C₄	C₅₃×#444	1	2	yes	—
187	C₄	C₅₃×#472	1	1	yes	characteristic
188	C₄	C₅₃×#473	1	1	yes	characteristic
189	C₂	—	4	1	no	—
190	C₂	C₅₃×#1946	1	1	yes	characteristic
191	C₂	C₅₃×#1950	1	1	yes	characteristic
192	C₂	C₅₃×#1952	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
80. #9636	C₅₃	direct (N × H)
115. C₅₃	#9636	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×#1567	2
C₅₃×#1564	2
C₅₃×#1451	2
C₅₃×#1299	2
#9636	53

Subgroups — C53×#9636 #27217148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9636 #27217148