Subgroups — C₅₃×#3336 #20272273

Order 6784.

342 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	C₅₃×#552	C₂	1	2	yes	maximal
2	C₅₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#485	C₂	1	2	yes	maximal, complemented
4	C₅₃×#483	C₂	1	2	yes	maximal, complemented
5	C₅₃×#479	C₂	1	2	yes	maximal, complemented
6	C₅₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#107×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	#126×C₅₃	—	2	2	no	—
16	#125×C₅₃	—	2	2	no	—
17	#125×C₅₃	C₂²	1	2	yes	—
18	#123×C₅₃	—	2	2	no	—
19	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
20	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
21	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
22	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
23	#117×C₅₃	—	2	2	no	—
24	#117×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
26	#107×C₅₃	C₂²	1	2	yes	—
27	#107×C₅₃	C₂²	1	1	yes	characteristic
28	#107×C₅₃	C₂²	1	1	yes	characteristic
29	#107×C₅₃	C₂²	1	1	yes	characteristic
30	#106×C₅₃	C₂²	1	2	yes	—
31	#106×C₅₃	C₂²	1	2	yes	—
32	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
33	#103×C₅₃	C₂²	1	2	yes	—
34	#103×C₅₃	C₂²	1	2	yes	—
35	#103×C₅₃	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
40	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	Q₁₆×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₅₃	—	2	2	no	—
52	C₄⋊C₄×C₅₃	—	2	2	no	—
53	C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
58	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
59	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
60	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
61	C₂²⋊C₄×C₅₃	—	2	2	no	—
62	C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
65	C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
71	Q₈×C₅₃	—	2	2	no	—
72	Q₈×C₅₃	—	2	2	no	—
73	D₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	4	4	no	—
83	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₈×C₅₃	—	2	1	no	—
89	C₈×C₅₃	—	2	1	no	—
90	C₈×C₅₃	—	2	1	no	—
91	C₈×C₅₃	#41	1	2	yes	—
92	C₂²×C₅₃	—	2	1	no	—
93	C₂²×C₅₃	—	2	1	no	—
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	#135	1	2	yes	—
98	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	4	2	no	—
104	C₄×C₅₃	—	4	2	no	—
105	C₄×C₅₃	—	4	2	no	—
106	C₄×C₅₃	—	4	2	no	—
107	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
108	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
109	#3336	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₅₃	—	2	2	no	—
111	C₂×C₅₃	—	4	1	no	—
112	C₂×C₅₃	#547	1	1	yes	characteristic
113	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
114	C₂×C₅₃	#578	1	1	yes	characteristic
115	#552	C₂×C₅₃	1	2	yes	—
116	#499	C₂×C₅₃	1	1	yes	characteristic, complemented
117	#485	C₂×C₅₃	1	2	yes	complemented
118	#483	C₂×C₅₃	1	2	yes	complemented
119	#479	C₂×C₅₃	1	2	yes	complemented
120	#467	C₂×C₅₃	1	1	yes	characteristic, complemented
121	#467	C₂×C₅₃	1	1	yes	characteristic, complemented
122	C₄×Q₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
123	C₈×D₄	C₂×C₅₃	1	1	yes	characteristic
124	#427	C₂×C₅₃	1	1	yes	characteristic, complemented
125	C₂×#107	C₂×C₅₃	1	1	yes	characteristic, complemented
126	C₅₃	#3336	1	1	yes	characteristic, direct product factor, Sylow
127	C₂×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
128	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
129	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
130	#126	—	2	2	no	—
131	#125	—	2	2	no	—
132	#125	C₂²×C₅₃	1	2	yes	—
133	#123	—	2	2	no	—
134	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
135	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
136	C₄×Q₈	C₂²×C₅₃	1	2	yes	—
137	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
138	#117	—	2	2	no	—
139	#117	C₂²×C₅₃	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
141	#107	C₂²×C₅₃	1	1	yes	characteristic
142	#107	C₂²×C₅₃	1	1	yes	characteristic
143	#107	C₂²×C₅₃	1	2	yes	—
144	#107	C₂²×C₅₃	1	1	yes	characteristic
145	#106	C₂²×C₅₃	1	2	yes	—
146	#106	C₂²×C₅₃	1	2	yes	—
147	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
148	#103	C₂²×C₅₃	1	2	yes	—
149	#103	C₂²×C₅₃	1	2	yes	—
150	#103	C₂²×C₅₃	1	2	yes	—
151	C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
152	C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
153	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
154	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
155	C₂×D₄	D₄×C₅₃	1	1	yes	characteristic
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
159	Q₁₆	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
163	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
164	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
165	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
173	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
174	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
175	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
176	C₂²⋊C₄	—	2	2	no	—
177	C₂²⋊C₄	—	2	2	no	—
178	C₂²⋊C₄	—	2	2	no	—
179	C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
180	C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	C₂³×C₅₃	1	1	yes	characteristic
184	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
185	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
186	Q₈	—	2	2	no	—
187	Q₈	—	2	2	no	—
188	D₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
199	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
201	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
202	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	#41×C₅₃	1	2	yes	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	#135×C₅₃	1	2	yes	—
213	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	4	2	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	2	no	—
222	C₄	C₂×#41×C₅₃	1	1	yes	characteristic
223	C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	4	1	no	—
226	C₂	C₅₃×#547	1	1	yes	characteristic
227	C₂	C₂×#135×C₅₃	1	1	yes	characteristic
228	C₂	C₅₃×#578	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₅₃×#499	C₂	semidirect
3. C₅₃×#485	C₂	semidirect
4. C₅₃×#483	C₂	semidirect
5. C₅₃×#479	C₂	semidirect
6. C₅₃×#467	C₂	semidirect
7. C₅₃×#467	C₂	semidirect
8. C₄×Q₁₆×C₅₃	C₂	semidirect
10. C₅₃×#427	C₂	semidirect
11. C₂×#107×C₅₃	C₂	semidirect
109. #3336	C₅₃	direct (N × H)
116. #499	C₂×C₅₃	semidirect
117. #485	C₂×C₅₃	semidirect
118. #483	C₂×C₅₃	semidirect
119. #479	C₂×C₅₃	semidirect
120. #467	C₂×C₅₃	semidirect
121. #467	C₂×C₅₃	semidirect
122. C₄×Q₁₆	C₂×C₅₃	semidirect
124. #427	C₂×C₅₃	semidirect
125. C₂×#107	C₂×C₅₃	semidirect
126. C₅₃	#3336	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×#552	2
C₅₃×#499	2
C₅₃×#485	2
C₅₃×#483	2
C₅₃×#479	2
C₅₃×#467	2
C₄×Q₁₆×C₅₃	2
C₈×D₄×C₅₃	2
C₅₃×#427	2
C₂×#107×C₅₃	2
#3336	53

Subgroups — C53×#3336 #20272273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3336 #20272273