Subgroups — C₅₃×#12009 #27219521

Order 13568.

242 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1372	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1371	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
8	C₅₃×#431	—	2	1	no	—
9	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#346	C₄	1	2	yes	—
12	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#335	—	2	1	no	—
14	C₅₃×#335	C₂²	1	2	yes	—
15	C₅₃×#335	C₂²	1	1	yes	characteristic
16	C₅₃×#335	C₂²	1	1	yes	characteristic
17	C₅₃×#334	—	2	1	no	—
18	C₂²×C₈×C₅₃	—	4	1	no	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
22	M₃₂×C₅₃	—	2	2	no	—
23	M₃₂×C₅₃	C₈	1	4	yes	complemented
24	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
26	C₄⋊C₈×C₅₃	—	2	1	no	—
27	C₄⋊C₈×C₅₃	—	2	1	no	—
28	C₄⋊C₈×C₅₃	—	2	2	no	—
29	C₄⋊C₈×C₅₃	—	4	1	no	—
30	C₂²⋊C₈×C₅₃	—	4	1	no	—
31	C₄×C₈×C₅₃	—	4	1	no	—
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	4	1	no	—
41	C₂×C₈×C₅₃	—	4	1	no	—
42	C₂×C₈×C₅₃	—	4	2	no	—
43	C₂×C₈×C₅₃	—	8	2	no	—
44	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
48	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₅₃	D₈	1	1	yes	characteristic
52	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
53	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₁₆×C₅₃	—	2	2	no	—
55	C₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
56	C₂³×C₅₃	#107	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	4	1	no	—
60	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
61	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#108	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₈×C₅₃	—	2	1	no	—
69	C₈×C₅₃	—	2	1	no	—
70	C₈×C₅₃	—	8	1	no	—
71	C₈×C₅₃	—	8	2	no	—
72	C₈×C₅₃	—	8	1	no	—
73	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
74	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
75	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
76	#12009	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	#335	1	2	yes	—
80	C₂²×C₅₃	C₂×#107	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	#335	1	2	yes	—
85	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
87	C₄×C₅₃	#509	1	1	yes	characteristic
88	C₄×C₅₃	#510	1	1	yes	characteristic
89	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic
90	#1564	C₂×C₅₃	1	1	yes	characteristic
91	C₂×#335	C₂×C₅₃	1	1	yes	characteristic
92	#1372	C₂×C₅₃	1	2	yes	complemented
93	#1371	C₂×C₅₃	1	2	yes	complemented
94	C₂×C₅₃	—	2	2	no	—
95	C₂×C₅₃	C₂×#335	1	1	yes	characteristic
96	C₂×C₅₃	#2159	1	1	yes	characteristic
97	C₂×C₅₃	#2161	1	1	yes	characteristic
98	C₂×M₃₂	C₄×C₅₃	1	1	yes	characteristic
99	C₂×M₃₂	C₄×C₅₃	1	1	yes	characteristic
100	#431	—	2	1	no	—
101	C₂×C₄⋊C₈	—	2	1	no	—
102	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
103	#346	C₄×C₅₃	1	2	yes	—
104	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
105	#335	—	2	1	no	—
106	#335	C₂²×C₅₃	1	2	yes	—
107	#335	C₂²×C₅₃	1	1	yes	characteristic
108	#335	C₂²×C₅₃	1	1	yes	characteristic
109	#334	—	2	1	no	—
110	C₅₃	#12009	1	1	yes	characteristic, direct product factor, Sylow
111	C₂²×C₈	—	4	1	no	—
112	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
113	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
114	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
115	M₃₂	—	2	2	no	—
116	M₃₂	C₈×C₅₃	1	4	yes	complemented
117	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
118	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
119	C₄⋊C₈	—	2	1	no	—
120	C₄⋊C₈	—	2	1	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	4	1	no	—
123	C₂²⋊C₈	—	4	1	no	—
124	C₄×C₈	—	4	1	no	—
125	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
126	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
127	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
128	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
129	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
130	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
131	C₂²×C₄	Q₁₆×C₅₃	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	8	2	no	—
137	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
138	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
139	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
140	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
141	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
142	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
143	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
144	C₄²	D₈×C₅₃	1	1	yes	characteristic
145	C₄²	Q₁₆×C₅₃	1	1	yes	characteristic
146	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
147	C₁₆	—	2	2	no	—
148	C₁₆	C₂×C₈×C₅₃	1	4	yes	complemented
149	C₂³	#107×C₅₃	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
154	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
155	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
156	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
157	C₂×C₄	#108×C₅₃	1	2	yes	—
158	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
159	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
160	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	8	1	no	—
164	C₈	—	8	2	no	—
165	C₈	—	8	1	no	—
166	C₈	C₄⋊C₈×C₅₃	1	2	yes	—
167	C₈	C₂²×C₈×C₅₃	1	1	yes	characteristic
168	C₈	C₂×M₁₆×C₅₃	1	1	yes	characteristic
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	C₅₃×#335	1	2	yes	—
172	C₂²	C₂×#107×C₅₃	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	C₅₃×#335	1	2	yes	—
177	C₄	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
178	C₄	C₂×#108×C₅₃	1	1	yes	characteristic
179	C₄	C₅₃×#509	1	1	yes	characteristic
180	C₄	C₅₃×#510	1	1	yes	characteristic
181	C₂	—	2	2	no	—
182	C₂	C₂×C₅₃×#335	1	1	yes	characteristic
183	C₂	C₅₃×#2159	1	1	yes	characteristic
184	C₂	C₅₃×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×D₈
commutator	C₈	C₂²×C₈×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1372	C₂	semidirect
5. C₅₃×#1371	C₂	semidirect
23. M₃₂×C₅₃	C₈	semidirect
55. C₁₆×C₅₃	C₂×C₈	semidirect
76. #12009	C₅₃	direct (N × H)
92. #1372	C₂×C₅₃	semidirect
93. #1371	C₂×C₅₃	semidirect
110. C₅₃	#12009	direct (N × H)
116. M₃₂	C₈×C₅₃	semidirect
148. C₁₆	C₂×C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₃₂×C₅₃	2
C₅₃×#1564	2
C₂×C₅₃×#335	2
C₅₃×#1372	2
C₅₃×#1371	2
#12009	53

Subgroups — C53×#12009 #27219521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12009 #27219521