Subgroups — C₅₃×#2992 #20271929

Order 6784.

356 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×#529	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#446	C₂	1	2	yes	maximal, complemented
3	C₅₃×#435	C₂	1	2	yes	maximal, complemented
4	C₅₃×#435	C₂	1	2	yes	maximal, complemented
5	C₈×D₄×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₅₃×#433	C₂	1	2	yes	maximal, complemented
7	C₅₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#424	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
14	#126×C₅₃	C₄	1	2	yes	—
15	#125×C₅₃	C₄	1	1	yes	characteristic
16	#124×C₅₃	C₄	1	1	yes	characteristic
17	#123×C₅₃	C₄	1	2	yes	—
18	C₂²⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
19	#121×C₅₃	C₄	1	1	yes	characteristic
20	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄×D₄×C₅₃	C₂²	1	2	yes	—
22	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
23	#117×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
26	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
27	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
29	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
30	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
32	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
33	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
34	#97×C₅₃	C₂²	1	2	yes	—
35	#97×C₅₃	C₂²	1	2	yes	—
36	C₄×C₈×C₅₃	C₂²	1	2	yes	—
37	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
44	M₁₆×C₅₃	—	4	2	no	—
45	C₂×C₈×C₅₃	—	4	1	no	—
46	C₂×C₈×C₅₃	—	4	2	no	—
47	C₂×C₈×C₅₃	—	4	1	no	—
48	C₂×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂×C₈×C₅₃	C₂³	1	2	yes	—
50	C₂×C₈×C₅₃	C₂³	1	2	yes	—
51	C₂×C₈×C₅₃	C₂³	1	2	yes	—
52	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
53	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
54	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
56	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
58	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
60	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
61	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
62	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
63	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂³	1	2	yes	—
66	C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	Q₈×C₅₃	—	4	1	no	—
71	D₄×C₅₃	—	2	4	no	—
72	D₄×C₅₃	—	4	1	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	4	1	no	—
79	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₈×C₅₃	—	2	2	no	—
89	C₈×C₅₃	—	2	2	no	—
90	C₈×C₅₃	—	2	2	no	—
91	C₈×C₅₃	—	2	2	no	—
92	C₂²×C₅₃	—	2	1	no	—
93	C₂²×C₅₃	—	2	1	no	—
94	C₂²×C₅₃	—	2	2	no	—
95	C₂²×C₅₃	—	2	2	no	—
96	C₂²×C₅₃	—	2	2	no	—
97	C₂²×C₅₃	#131	1	2	yes	—
98	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	2	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	2	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
108	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
109	#2992	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₅₃	—	2	2	no	—
111	C₂×C₅₃	—	4	2	no	—
112	C₂×C₅₃	#520	1	1	yes	characteristic
113	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
114	C₂×C₅₃	#568	1	1	yes	characteristic
115	#529	C₂×C₅₃	1	1	yes	characteristic
116	#446	C₂×C₅₃	1	2	yes	complemented
117	#435	C₂×C₅₃	1	2	yes	complemented
118	#435	C₂×C₅₃	1	2	yes	complemented
119	C₈×D₄	C₂×C₅₃	1	2	yes	complemented
120	#433	C₂×C₅₃	1	2	yes	complemented
121	#432	C₂×C₅₃	1	1	yes	characteristic, complemented
122	#424	C₂×C₅₃	1	2	yes	complemented
123	C₂×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
124	C₅₃	#2992	1	1	yes	characteristic, direct product factor, Sylow
125	C₂×M₁₆	C₂²×C₅₃	1	2	yes	—
126	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
128	C₂²×C₈	C₂²×C₅₃	1	2	yes	—
129	#126	C₄×C₅₃	1	2	yes	—
130	#125	C₄×C₅₃	1	1	yes	characteristic
131	#124	C₄×C₅₃	1	1	yes	characteristic
132	#123	C₄×C₅₃	1	2	yes	—
133	C₂²⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
134	#121	C₄×C₅₃	1	1	yes	characteristic
135	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
136	C₄×D₄	C₂²×C₅₃	1	2	yes	—
137	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
138	#117	C₂²×C₅₃	1	2	yes	—
139	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
140	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
141	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
142	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
143	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
144	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
145	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
146	C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
147	C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
148	C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
149	#97	C₂²×C₅₃	1	2	yes	—
150	#97	C₂²×C₅₃	1	2	yes	—
151	C₄×C₈	C₂²×C₅₃	1	2	yes	—
152	C₂×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
153	C₂×D₄	C₂×C₄×C₅₃	1	2	yes	—
154	C₂×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
159	M₁₆	—	4	2	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	C₂³×C₅₃	1	2	yes	—
164	C₂×C₈	C₂³×C₅₃	1	2	yes	—
165	C₂×C₈	C₂³×C₅₃	1	2	yes	—
166	C₂×C₈	C₂³×C₅₃	1	2	yes	—
167	C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
168	C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
169	C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
170	C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
171	C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
172	C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
173	C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
174	C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
175	C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
176	C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
177	C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
178	C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
179	C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
180	C₄²	C₂³×C₅₃	1	2	yes	—
181	C₄²	C₂³×C₅₃	1	1	yes	characteristic
182	C₄²	C₂³×C₅₃	1	1	yes	characteristic
183	C₂³	C₂²×C₄×C₅₃	1	2	yes	—
184	C₂³	C₂²×C₄×C₅₃	1	1	yes	characteristic
185	Q₈	—	4	1	no	—
186	D₄	—	2	4	no	—
187	D₄	—	4	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
196	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
197	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
198	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
199	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
200	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
201	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
202	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	#131×C₅₃	1	2	yes	—
213	C₂²	C₂³×C₄×C₅₃	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	2	no	—
222	C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
223	C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	4	2	no	—
226	C₂	C₅₃×#520	1	1	yes	characteristic
227	C₂	C₂×#131×C₅₃	1	1	yes	characteristic
228	C₂	C₅₃×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂³×C₄
prime radical	#121×C₅₃	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#446	C₂	semidirect
3. C₅₃×#435	C₂	semidirect
4. C₅₃×#435	C₂	semidirect
5. C₈×D₄×C₅₃	C₂	semidirect
6. C₅₃×#433	C₂	semidirect
7. C₅₃×#432	C₂	semidirect
8. C₅₃×#424	C₂	semidirect
9. C₂×C₄⋊C₈×C₅₃	C₂	semidirect
109. #2992	C₅₃	direct (N × H)
116. #446	C₂×C₅₃	semidirect
117. #435	C₂×C₅₃	semidirect
118. #435	C₂×C₅₃	semidirect
119. C₈×D₄	C₂×C₅₃	semidirect
120. #433	C₂×C₅₃	semidirect
121. #432	C₂×C₅₃	semidirect
122. #424	C₂×C₅₃	semidirect
123. C₂×C₄⋊C₈	C₂×C₅₃	semidirect
124. C₅₃	#2992	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×#529	2
C₅₃×#446	2
C₅₃×#435	2
C₈×D₄×C₅₃	2
C₅₃×#433	2
C₅₃×#432	2
C₅₃×#424	2
C₂×C₄⋊C₈×C₅₃	2
#2992	53

Subgroups — C53×#2992 #20271929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2992 #20271929