Subgroups — C₅₃×#20797 #27228309

Order 13568.

1126 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#427	C₂	1	4	yes	maximal
2	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2285	C₂	1	2	yes	maximal
4	C₅₃×#1846	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1845	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1844	C₂	1	2	yes	maximal, complemented
7	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×#117×C₅₃	C₂²	1	4	yes	—
9	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#427	—	2	16	no	—
11	C₂×#107×C₅₃	C₂²	1	8	yes	—
12	C₂×#106×C₅₃	C₂²	1	8	yes	—
13	C₂×C₄×C₈×C₅₃	C₄	1	8	yes	complemented
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#384	C₂²	1	4	yes	—
18	C₅₃×#382	C₂²	1	4	yes	—
19	C₅₃×#380	—	2	4	no	—
20	C₄×C₄⋊C₄×C₅₃	—	2	4	no	—
21	C₂³×C₄×C₅₃	D₄	1	2	yes	—
22	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₅₃	D₄	1	8	yes	—
25	C₂²×C₈×C₅₃	Q₈	1	8	yes	—
26	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
28	#117×C₅₃	—	2	16	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
30	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
31	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
32	C₂×C₄²×C₅₃	—	2	4	no	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#107×C₅₃	—	2	16	no	—
39	#106×C₅₃	—	2	16	no	—
40	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
43	#95×C₅₃	—	2	8	no	—
44	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	8	no	—
46	C₂²×C₄×C₅₃	—	2	8	no	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
51	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
54	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₄⋊C₄	1	32	yes	complemented
57	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
59	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
60	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
61	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
62	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
63	C₄⋊C₄×C₅₃	—	2	16	no	—
64	C₄⋊C₄×C₅₃	—	4	16	no	—
65	C₂²⋊C₄×C₅₃	—	4	8	no	—
66	C₄²×C₅₃	—	4	8	no	—
67	C₄²×C₅₃	C₄⋊C₄	1	8	yes	—
68	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
69	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
71	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
72	C₂³×C₅₃	—	2	4	no	—
73	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
74	C₂³×C₅₃	#127	1	2	yes	—
75	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
76	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	2	16	no	—
79	C₂×C₄×C₅₃	—	4	8	no	—
80	C₂×C₄×C₅₃	—	4	16	no	—
81	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
82	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
83	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
85	C₂×C₄×C₅₃	#127	1	2	yes	—
86	C₂×C₄×C₅₃	C₄⋊Q₈	1	8	yes	—
87	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
88	C₂×C₄×C₅₃	#135	1	8	yes	—
89	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
92	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
93	C₈×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
94	C₈×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
95	#20797	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂²×C₅₃	—	2	4	no	—
97	C₂²×C₅₃	—	2	4	no	—
98	C₂²×C₅₃	—	2	4	no	—
99	C₂²×C₅₃	#384	1	4	yes	—
100	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
102	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	C₂²×C₅₃	C₂×#135	1	4	yes	—
104	C₄×C₅₃	—	2	4	no	—
105	C₄×C₅₃	—	8	8	no	—
106	C₄×C₅₃	#384	1	4	yes	—
107	C₄×C₅₃	#427	1	8	yes	complemented
108	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
109	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
110	C₂×#427	C₂×C₅₃	1	4	yes	—
111	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
112	#2285	C₂×C₅₃	1	2	yes	—
113	#1846	C₂×C₅₃	1	4	yes	complemented
114	#1845	C₂×C₅₃	1	2	yes	complemented
115	#1844	C₂×C₅₃	1	2	yes	complemented
116	C₂×C₅₃	—	2	4	no	—
117	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
118	C₂×C₅₃	C₂×#427	1	2	yes	—
119	C₂×C₅₃	#3147	1	2	yes	—
120	C₂×C₅₃	#3160	1	2	yes	—
121	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
122	C₂×#117	C₂²×C₅₃	1	4	yes	—
123	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
124	#427	—	2	16	no	—
125	C₂×#107	C₂²×C₅₃	1	8	yes	—
126	C₂×#106	C₂²×C₅₃	1	8	yes	—
127	C₂×C₄×C₈	C₄×C₅₃	1	8	yes	complemented
128	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
131	#384	C₂²×C₅₃	1	4	yes	—
132	#382	C₂²×C₅₃	1	4	yes	—
133	#380	—	2	4	no	—
134	C₄×C₄⋊C₄	—	2	4	no	—
135	C₅₃	#20797	1	1	yes	characteristic, direct product factor, Sylow
136	C₂³×C₄	D₄×C₅₃	1	2	yes	—
137	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
138	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
139	C₂²×C₈	D₄×C₅₃	1	8	yes	—
140	C₂²×C₈	Q₈×C₅₃	1	8	yes	—
141	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
142	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
143	#117	—	2	16	no	—
144	C₂×C₄⋊C₄	—	2	8	no	—
145	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
146	C₂×C₂²⋊C₄	—	2	4	no	—
147	C₂×C₄²	—	2	4	no	—
148	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
149	C₂×C₄²	D₄×C₅₃	1	2	yes	—
150	C₂×C₄²	Q₈×C₅₃	1	4	yes	—
151	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
152	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
153	#107	—	2	16	no	—
154	#106	—	2	16	no	—
155	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
156	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
157	C₄×C₈	C₂×C₄×C₅₃	1	8	yes	—
158	#95	—	2	8	no	—
159	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	C₄⋊C₄×C₅₃	1	8	yes	—
163	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
164	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
165	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
167	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
168	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
169	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
170	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
171	C₂×C₈	C₄⋊C₄×C₅₃	1	32	yes	complemented
172	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
173	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
174	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
175	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
176	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
177	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
178	C₄⋊C₄	—	2	16	no	—
179	C₄⋊C₄	—	4	16	no	—
180	C₂²⋊C₄	—	4	8	no	—
181	C₄²	—	4	8	no	—
182	C₄²	C₄⋊C₄×C₅₃	1	8	yes	—
183	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
184	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
185	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
186	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
187	C₂³	—	2	4	no	—
188	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
189	C₂³	#127×C₅₃	1	2	yes	—
190	C₂³	C₄⋊Q₈×C₅₃	1	2	yes	—
191	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	16	no	—
194	C₂×C₄	—	4	8	no	—
195	C₂×C₄	—	4	16	no	—
196	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
197	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	8	yes	—
198	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
199	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
200	C₂×C₄	#127×C₅₃	1	2	yes	—
201	C₂×C₄	C₄⋊Q₈×C₅₃	1	8	yes	—
202	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
203	C₂×C₄	#135×C₅₃	1	8	yes	—
204	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
205	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
206	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
207	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
208	C₈	C₂×C₄⋊C₄×C₅₃	1	8	yes	—
209	C₈	C₂×C₄⋊C₄×C₅₃	1	8	yes	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	4	no	—
213	C₂²	C₅₃×#384	1	4	yes	—
214	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
215	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
216	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
217	C₂²	C₂×#135×C₅₃	1	4	yes	—
218	C₄	—	2	4	no	—
219	C₄	—	8	8	no	—
220	C₄	C₅₃×#384	1	4	yes	—
221	C₄	C₅₃×#427	1	8	yes	complemented
222	C₄	C₂²×C₄⋊C₄×C₅₃	1	2	yes	—
223	C₄	C₂×C₄⋊Q₈×C₅₃	1	2	yes	—
224	C₂	—	2	4	no	—
225	C₂	C₂×C₅₃×#384	1	1	yes	characteristic
226	C₂	C₂×C₅₃×#427	1	2	yes	—
227	C₂	C₅₃×#3147	1	2	yes	—
228	C₂	C₅₃×#3160	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1846	C₂	semidirect
5. C₅₃×#1845	C₂	semidirect
6. C₅₃×#1844	C₂	semidirect
13. C₂×C₄×C₈×C₅₃	C₄	semidirect
56. C₂×C₈×C₅₃	C₄⋊C₄	semidirect
95. #20797	C₅₃	direct (N × H)
107. C₄×C₅₃	#427	semidirect
113. #1846	C₂×C₅₃	semidirect
114. #1845	C₂×C₅₃	semidirect
115. #1844	C₂×C₅₃	semidirect
127. C₂×C₄×C₈	C₄×C₅₃	semidirect
135. C₅₃	#20797	direct (N × H)
171. C₂×C₈	C₄⋊C₄×C₅₃	semidirect
221. C₄	C₅₃×#427	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#427	2
C₂²×C₄×C₈×C₅₃	2
C₅₃×#2285	2
C₅₃×#1846	2
C₅₃×#1845	2
C₅₃×#1844	2
#20797	53

Subgroups — C53×#20797 #27228309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#20797 #27228309