Subgroups — C₅₃×#13182 #27220694

Order 13568.

280 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2161	C₂	1	2	yes	maximal
2	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1772	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1389	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
7	C₂×#108×C₅₃	—	2	1	no	—
8	C₂×#108×C₅₃	C₂²	1	2	yes	—
9	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#427	C₂²	1	1	yes	characteristic
11	C₅₃×#423	—	2	1	no	—
12	C₅₃×#405	—	2	1	no	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#368	—	2	4	no	—
15	C₅₃×#364	C₄	1	2	yes	complemented
16	C₅₃×#364	C₄	1	2	yes	complemented
17	C₅₃×#339	—	2	1	no	—
18	C₂×M₁₆×C₅₃	—	2	1	no	—
19	C₂×M₁₆×C₅₃	—	2	2	no	—
20	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	#117×C₅₃	—	2	1	no	—
23	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
24	M₃₂×C₅₃	—	2	2	no	—
25	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₁₆×C₅₃	—	2	2	no	—
27	#108×C₅₃	—	2	2	no	—
28	#108×C₅₃	—	2	1	no	—
29	#108×C₅₃	—	2	2	no	—
30	#108×C₅₃	—	4	1	no	—
31	#108×C₅₃	—	4	1	no	—
32	#108×C₅₃	D₄	1	2	yes	—
33	#107×C₅₃	—	2	1	no	—
34	#106×C₅₃	Q₈	1	2	yes	—
35	C₄⋊C₈×C₅₃	—	4	2	no	—
36	#97×C₅₃	—	4	1	no	—
37	C₄×C₈×C₅₃	—	4	1	no	—
38	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₅₃	—	2	2	no	—
45	M₁₆×C₅₃	—	4	1	no	—
46	M₁₆×C₅₃	—	4	2	no	—
47	M₁₆×C₅₃	—	4	2	no	—
48	C₂×C₈×C₅₃	—	4	1	no	—
49	C₂×C₈×C₅₃	—	4	2	no	—
50	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
53	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
54	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
55	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
61	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₂²⋊C₄×C₅₃	—	4	1	no	—
64	C₄²×C₅₃	—	2	1	no	—
65	C₄²×C₅₃	—	4	1	no	—
66	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
68	C₁₆×C₅₃	—	2	4	no	—
69	C₂³×C₅₃	#137	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	—	4	2	no	—
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	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	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₈×C₅₃	—	4	2	no	—
81	C₈×C₅₃	—	8	2	no	—
82	C₈×C₅₃	#107	1	1	yes	characteristic
83	C₈×C₅₃	#107	1	1	yes	characteristic
84	C₈×C₅₃	#107	1	1	yes	characteristic
85	C₈×C₅₃	#107	1	1	yes	characteristic
86	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
87	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
88	C₈×C₅₃	#125	1	1	yes	characteristic
89	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
90	#13182	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂²×C₅₃	—	4	1	no	—
92	C₂²×C₅₃	#428	1	1	yes	characteristic
93	C₂²×C₅₃	#462	1	1	yes	characteristic
94	C₂²×C₅₃	#474	1	1	yes	characteristic
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	—	4	1	no	—
97	C₄×C₅₃	—	8	2	no	—
98	C₄×C₅₃	#389	1	1	yes	characteristic
99	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
100	C₄×C₅₃	#459	1	1	yes	characteristic
101	C₄×C₅₃	#477	1	1	yes	characteristic
102	#2161	C₂×C₅₃	1	2	yes	—
103	C₂×#364	C₂×C₅₃	1	1	yes	characteristic
104	#1848	C₂×C₅₃	1	1	yes	characteristic
105	#1772	C₂×C₅₃	1	1	yes	characteristic
106	#1389	C₂×C₅₃	1	2	yes	—
107	C₂×C₅₃	—	2	1	no	—
108	C₂×C₅₃	—	2	1	no	—
109	C₂×C₅₃	—	2	1	no	—
110	C₂×C₅₃	#1947	1	1	yes	characteristic
111	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
112	C₂×#108	—	2	1	no	—
113	C₂×#108	C₂²×C₅₃	1	2	yes	—
114	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
115	#427	C₂²×C₅₃	1	1	yes	characteristic
116	#423	—	2	1	no	—
117	#405	—	2	1	no	—
118	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
119	#368	—	2	4	no	—
120	#364	C₄×C₅₃	1	2	yes	complemented
121	#364	C₄×C₅₃	1	2	yes	complemented
122	#339	—	2	1	no	—
123	C₅₃	#13182	1	1	yes	characteristic, direct product factor, Sylow
124	C₂×M₁₆	—	2	1	no	—
125	C₂×M₁₆	—	2	2	no	—
126	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
127	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
128	#117	—	2	1	no	—
129	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
130	M₃₂	—	2	2	no	—
131	M₃₂	C₂×C₄×C₅₃	1	4	yes	—
132	C₂×C₁₆	—	2	2	no	—
133	#108	—	2	2	no	—
134	#108	—	2	1	no	—
135	#108	—	2	2	no	—
136	#108	—	4	1	no	—
137	#108	—	4	1	no	—
138	#108	D₄×C₅₃	1	2	yes	—
139	#107	—	2	1	no	—
140	#106	Q₈×C₅₃	1	2	yes	—
141	C₄⋊C₈	—	4	2	no	—
142	#97	—	4	1	no	—
143	C₄×C₈	—	4	1	no	—
144	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
145	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
146	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
147	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	4	1	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	4	2	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
157	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
158	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
159	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
160	C₂×C₈	Q₁₆×C₅₃	1	1	yes	characteristic
161	C₂×C₈	Q₁₆×C₅₃	1	1	yes	characteristic
162	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
163	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
164	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
165	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
166	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
167	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
168	C₄⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
173	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
174	C₁₆	—	2	4	no	—
175	C₂³	#137×C₅₃	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
181	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
182	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
183	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
184	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
185	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
186	C₈	—	4	2	no	—
187	C₈	—	8	2	no	—
188	C₈	#107×C₅₃	1	1	yes	characteristic
189	C₈	#107×C₅₃	1	1	yes	characteristic
190	C₈	#107×C₅₃	1	1	yes	characteristic
191	C₈	#107×C₅₃	1	1	yes	characteristic
192	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
193	C₈	C₄×Q₈×C₅₃	1	1	yes	characteristic
194	C₈	#125×C₅₃	1	1	yes	characteristic
195	C₈	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
196	C₂²	—	4	1	no	—
197	C₂²	C₅₃×#428	1	1	yes	characteristic
198	C₂²	C₅₃×#462	1	1	yes	characteristic
199	C₂²	C₅₃×#474	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	—	8	2	no	—
203	C₄	C₅₃×#389	1	1	yes	characteristic
204	C₄	C₂×#107×C₅₃	1	1	yes	characteristic
205	C₄	C₅₃×#459	1	1	yes	characteristic
206	C₄	C₅₃×#477	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	C₅₃×#1947	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	#459
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#1947
prime radical	C₂³×C₅₃	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₅₃×#364	C₄	semidirect
16. C₅₃×#364	C₄	semidirect
90. #13182	C₅₃	direct (N × H)
120. #364	C₄×C₅₃	semidirect
121. #364	C₄×C₅₃	semidirect
123. C₅₃	#13182	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×#2161	2
C₂×C₅₃×#364	2
C₅₃×#1848	2
C₅₃×#1772	2
C₅₃×#1389	2
#13182	53

Subgroups — C53×#13182 #27220694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13182 #27220694