Subgroups — C₅₃×#21433 #27228945

Order 13568.

668 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×#2298	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1924	C₂	1	4	yes	maximal
3	C₅₃×#1917	C₂	1	4	yes	maximal, complemented
4	C₅₃×#1768	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1750	C₂	1	2	yes	maximal
6	C₅₃×#1745	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
10	C₂×C₂²⋊C₈×C₅₃	C₂²	1	4	yes	—
11	C₂×#97×C₅₃	C₂²	1	4	yes	—
12	C₂×#97×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#388	C₄	1	4	yes	—
15	C₅₃×#387	C₄	1	4	yes	—
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#376	C₂²	1	1	yes	characteristic
19	C₄³×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#336	C₂²	1	4	yes	—
21	C₅₃×#336	C₂²	1	4	yes	—
22	C₅₃×#336	C₂²	1	4	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
25	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
26	#117×C₅₃	—	2	4	no	—
27	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	—	2	8	no	—
37	C₂²⋊C₈×C₅₃	—	2	8	no	—
38	#97×C₅₃	—	2	8	no	—
39	#97×C₅₃	—	2	4	no	—
40	C₄×C₈×C₅₃	—	2	4	no	—
41	#95×C₅₃	C₂×C₄	1	4	yes	—
42	#95×C₅₃	C₂×C₄	1	4	yes	—
43	#95×C₅₃	C₂×C₄	1	4	yes	—
44	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	4	no	—
46	C₂²×C₄×C₅₃	—	2	2	no	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₂×C₈×C₅₃	—	2	4	no	—
54	C₂×C₈×C₅₃	—	2	4	no	—
55	C₂×C₈×C₅₃	—	2	4	no	—
56	C₂×C₈×C₅₃	—	2	4	no	—
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	#41	1	8	yes	—
59	C₄⋊C₄×C₅₃	—	2	4	no	—
60	C₄⋊C₄×C₅₃	—	4	4	no	—
61	C₂²⋊C₄×C₅₃	—	2	4	no	—
62	C₂²⋊C₄×C₅₃	—	4	4	no	—
63	C₄²×C₅₃	—	2	4	no	—
64	C₄²×C₅₃	—	2	4	no	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	4	4	no	—
67	C₄²×C₅₃	#41	1	4	yes	—
68	C₂³×C₅₃	—	2	1	no	—
69	C₂³×C₅₃	—	2	2	no	—
70	C₂³×C₅₃	—	2	4	no	—
71	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	—	2	4	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	4	no	—
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	2	4	no	—
83	C₂×C₄×C₅₃	—	4	4	no	—
84	C₂×C₄×C₅₃	#117	1	4	yes	—
85	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
86	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
87	C₈×C₅₃	—	2	8	no	—
88	C₈×C₅₃	—	4	4	no	—
89	#21433	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×C₅₃	—	4	2	no	—
91	C₂²×C₅₃	—	4	4	no	—
92	C₂²×C₅₃	—	4	1	no	—
93	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
94	C₂²×C₅₃	C₄×#41	1	2	yes	—
95	C₂²×C₅₃	#568	1	4	yes	—
96	C₄×C₅₃	—	2	4	no	—
97	C₄×C₅₃	—	4	2	no	—
98	C₄×C₅₃	—	4	4	no	—
99	C₄×C₅₃	—	4	4	no	—
100	C₄×C₅₃	—	4	1	no	—
101	C₄×C₅₃	#529	1	4	yes	—
102	#2298	C₂×C₅₃	1	1	yes	characteristic
103	#1924	C₂×C₅₃	1	4	yes	—
104	#1917	C₂×C₅₃	1	4	yes	complemented
105	#1768	C₂×C₅₃	1	2	yes	complemented
106	#1750	C₂×C₅₃	1	2	yes	—
107	#1745	C₂×C₅₃	1	1	yes	characteristic, complemented
108	C₄×#97	C₂×C₅₃	1	1	yes	characteristic, complemented
109	C₂×C₅₃	—	8	1	no	—
110	C₂×C₅₃	#2298	1	1	yes	characteristic
111	C₂×C₅₃	#2923	1	2	yes	—
112	C₂×C₅₃	#2967	1	4	yes	—
113	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
114	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
115	C₂×C₂²⋊C₈	C₂²×C₅₃	1	4	yes	—
116	C₂×#97	C₂²×C₅₃	1	4	yes	—
117	C₂×#97	C₂²×C₅₃	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
119	#388	C₄×C₅₃	1	4	yes	—
120	#387	C₄×C₅₃	1	4	yes	—
121	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
122	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
123	#376	C₂²×C₅₃	1	1	yes	characteristic
124	C₄³	C₂²×C₅₃	1	1	yes	characteristic
125	#336	C₂²×C₅₃	1	4	yes	—
126	#336	C₂²×C₅₃	1	4	yes	—
127	#336	C₂²×C₅₃	1	4	yes	—
128	C₅₃	#21433	1	1	yes	characteristic, direct product factor, Sylow
129	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
130	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
131	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
132	#117	—	2	4	no	—
133	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
134	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
135	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
136	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
137	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
138	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
139	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
140	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
141	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
142	C₄⋊C₈	—	2	8	no	—
143	C₂²⋊C₈	—	2	8	no	—
144	#97	—	2	8	no	—
145	#97	—	2	4	no	—
146	C₄×C₈	—	2	4	no	—
147	#95	C₂×C₄×C₅₃	1	4	yes	—
148	#95	C₂×C₄×C₅₃	1	4	yes	—
149	#95	C₂×C₄×C₅₃	1	4	yes	—
150	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
154	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
155	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
156	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
157	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
158	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	#41×C₅₃	1	8	yes	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₂²⋊C₄	—	2	4	no	—
168	C₂²⋊C₄	—	4	4	no	—
169	C₄²	—	2	4	no	—
170	C₄²	—	2	4	no	—
171	C₄²	—	2	2	no	—
172	C₄²	—	4	4	no	—
173	C₄²	#41×C₅₃	1	4	yes	—
174	C₂³	—	2	1	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	4	no	—
177	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	#117×C₅₃	1	4	yes	—
191	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
192	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
193	C₈	—	2	8	no	—
194	C₈	—	4	4	no	—
195	C₂²	—	4	4	no	—
196	C₂²	—	4	1	no	—
197	C₂²	—	4	2	no	—
198	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
199	C₂²	C₄×#41×C₅₃	1	2	yes	—
200	C₂²	C₅₃×#568	1	4	yes	—
201	C₄	—	2	4	no	—
202	C₄	—	4	4	no	—
203	C₄	—	4	4	no	—
204	C₄	—	4	1	no	—
205	C₄	—	4	2	no	—
206	C₄	C₅₃×#529	1	4	yes	—
207	C₂	—	8	1	no	—
208	C₂	C₅₃×#2298	1	1	yes	characteristic
209	C₂	C₅₃×#2923	1	2	yes	—
210	C₂	C₅₃×#2967	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1917	C₂	semidirect
4. C₅₃×#1768	C₂	semidirect
6. C₅₃×#1745	C₂	semidirect
7. C₄×#97×C₅₃	C₂	semidirect
89. #21433	C₅₃	direct (N × H)
104. #1917	C₂×C₅₃	semidirect
105. #1768	C₂×C₅₃	semidirect
107. #1745	C₂×C₅₃	semidirect
108. C₄×#97	C₂×C₅₃	semidirect
128. C₅₃	#21433	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×#2298	2
C₅₃×#1924	2
C₅₃×#1917	2
C₅₃×#1768	2
C₅₃×#1750	2
C₅₃×#1745	2
C₄×#97×C₅₃	2
#21433	53

Subgroups — C53×#21433 #27228945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21433 #27228945