Subgroups — C₅₃×#3077 #20272014

Order 6784.

440 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#478	C₂	1	4	yes	maximal, complemented
4	C₅₃×#476	C₂	1	2	yes	maximal, complemented
5	C₅₃×#474	C₂	1	2	yes	maximal, complemented
6	C₅₃×#428	C₂	1	2	yes	maximal, complemented
7	C₅₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#418	C₂	1	2	yes	maximal
9	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₅₃	—	2	1	no	—
11	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
13	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
14	#125×C₅₃	C₂²	1	2	yes	complemented
15	C₂²⋊Q₈×C₅₃	—	2	1	no	—
16	C₂²⋊Q₈×C₅₃	—	2	1	no	—
17	C₄×Q₈×C₅₃	—	2	1	no	—
18	C₄×Q₈×C₅₃	—	2	1	no	—
19	C₄×D₄×C₅₃	—	2	1	no	—
20	C₄×D₄×C₅₃	—	2	1	no	—
21	C₄×D₄×C₅₃	C₂²	1	2	yes	—
22	C₄×D₄×C₅₃	C₂²	1	2	yes	—
23	#117×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
26	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	#107×C₅₃	C₂²	1	4	yes	complemented
28	#106×C₅₃	C₂²	1	4	yes	complemented
29	C₄⋊C₈×C₅₃	C₂²	1	2	yes	complemented
30	C₄⋊C₈×C₅₃	C₂²	1	2	yes	complemented
31	#102×C₅₃	C₂²	1	4	yes	—
32	#102×C₅₃	C₂²	1	4	yes	—
33	C₂⁴×C₅₃	—	2	1	no	—
34	C₂×Q₈×C₅₃	—	2	1	no	—
35	C₂×Q₈×C₅₃	—	2	1	no	—
36	C₂×D₄×C₅₃	—	2	1	no	—
37	C₂×D₄×C₅₃	—	2	2	no	—
38	C₂×D₄×C₅₃	Q₈	1	4	yes	complemented
39	C₂×D₄×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₅₃	—	2	1	no	—
41	C₂²×C₄×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	4	4	no	—
44	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	M₁₆×C₅₃	—	2	4	no	—
48	C₂×C₈×C₅₃	C₂³	1	4	yes	—
49	C₄⋊C₄×C₅₃	—	2	1	no	—
50	C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₄⋊C₄×C₅₃	—	2	2	no	—
52	C₄⋊C₄×C₅₃	—	2	1	no	—
53	C₄⋊C₄×C₅₃	—	2	1	no	—
54	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
55	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
56	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
57	C₂²⋊C₄×C₅₃	—	2	2	no	—
58	C₂²⋊C₄×C₅₃	—	2	2	no	—
59	C₂²⋊C₄×C₅₃	—	2	2	no	—
60	C₄²×C₅₃	—	2	2	no	—
61	C₄²×C₅₃	D₄	1	1	yes	characteristic
62	C₄²×C₅₃	D₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	C₂³×C₅₃	—	2	4	no	—
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	—	4	2	no	—
68	C₂³×C₅₃	—	4	2	no	—
69	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
70	Q₈×C₅₃	—	4	2	no	—
71	Q₈×C₅₃	—	4	2	no	—
72	D₄×C₅₃	—	2	4	no	—
73	D₄×C₅₃	—	2	2	no	—
74	D₄×C₅₃	C₂×Q₈	1	4	yes	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	4	1	no	—
80	C₂×C₄×C₅₃	—	4	4	no	—
81	C₂×C₄×C₅₃	—	4	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	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
89	C₂×C₄×C₅₃	#41	1	2	yes	—
90	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
91	C₈×C₅₃	—	2	4	no	—
92	C₂²×C₅₃	—	2	1	no	—
93	C₂²×C₅₃	—	2	1	no	—
94	C₂²×C₅₃	—	2	4	no	—
95	C₂²×C₅₃	—	4	8	no	—
96	C₂²×C₅₃	—	4	2	no	—
97	C₂²×C₅₃	—	4	2	no	—
98	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
99	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
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	C₂²⋊Q₈	1	2	yes	—
107	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
108	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
109	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
110	#3077	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂×C₅₃	—	2	2	no	—
112	C₂×C₅₃	—	4	4	no	—
113	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₅₃	#575	1	1	yes	characteristic
116	#533	C₂×C₅₃	1	1	yes	characteristic, complemented
117	C₂×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic
118	#478	C₂×C₅₃	1	4	yes	complemented
119	#476	C₂×C₅₃	1	2	yes	complemented
120	#474	C₂×C₅₃	1	2	yes	complemented
121	#428	C₂×C₅₃	1	2	yes	complemented
122	#423	C₂×C₅₃	1	1	yes	characteristic, complemented
123	#418	C₂×C₅₃	1	2	yes	—
124	C₅₃	#3077	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
126	C₂³×C₄	—	2	1	no	—
127	C₂×M₁₆	C₂²×C₅₃	1	2	yes	—
128	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
129	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
130	#125	C₂²×C₅₃	1	2	yes	complemented
131	C₂²⋊Q₈	—	2	1	no	—
132	C₂²⋊Q₈	—	2	1	no	—
133	C₄×Q₈	—	2	1	no	—
134	C₄×Q₈	—	2	1	no	—
135	C₄×D₄	—	2	1	no	—
136	C₄×D₄	—	2	1	no	—
137	C₄×D₄	C₂²×C₅₃	1	2	yes	—
138	C₄×D₄	C₂²×C₅₃	1	2	yes	—
139	#117	C₂²×C₅₃	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
141	C₂×C₂²⋊C₄	—	2	1	no	—
142	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
143	#107	C₂²×C₅₃	1	4	yes	complemented
144	#106	C₂²×C₅₃	1	4	yes	complemented
145	C₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
146	C₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
147	#102	C₂²×C₅₃	1	4	yes	—
148	#102	C₂²×C₅₃	1	4	yes	—
149	C₂⁴	—	2	1	no	—
150	C₂×Q₈	—	2	1	no	—
151	C₂×Q₈	—	2	1	no	—
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	Q₈×C₅₃	1	4	yes	complemented
155	C₂×D₄	C₂³×C₅₃	1	2	yes	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	4	4	no	—
160	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
161	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
162	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
163	M₁₆	—	2	4	no	—
164	C₂×C₈	C₂³×C₅₃	1	4	yes	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
171	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
172	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
173	C₂²⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	—	2	2	no	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	D₄×C₅₃	1	1	yes	characteristic
178	C₄²	D₄×C₅₃	1	1	yes	characteristic
179	C₄²	C₂³×C₅₃	1	1	yes	characteristic
180	C₄²	C₂³×C₅₃	1	1	yes	characteristic
181	C₂³	—	2	4	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	4	2	no	—
184	C₂³	—	4	2	no	—
185	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
186	Q₈	—	4	2	no	—
187	Q₈	—	4	2	no	—
188	D₄	—	2	4	no	—
189	D₄	—	2	2	no	—
190	D₄	C₂×Q₈×C₅₃	1	4	yes	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	4	no	—
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	C₂×D₄×C₅₃	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
204	C₂×C₄	C₂×Q₈×C₅₃	1	2	yes	—
205	C₂×C₄	#41×C₅₃	1	2	yes	—
206	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
207	C₈	—	2	4	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	4	8	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	C₂²⋊Q₈×C₅₃	1	2	yes	—
215	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	2	no	—
222	C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
223	C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
224	C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
225	C₄	C₂×#41×C₅₃	1	1	yes	characteristic
226	C₂	—	2	2	no	—
227	C₂	—	4	4	no	—
228	C₂	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
229	C₂	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
230	C₂	C₅₃×#575	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₅₃×#533	C₂	semidirect
3. C₅₃×#478	C₂	semidirect
4. C₅₃×#476	C₂	semidirect
5. C₅₃×#474	C₂	semidirect
6. C₅₃×#428	C₂	semidirect
7. C₅₃×#423	C₂	semidirect
12. C₄⋊Q₈×C₅₃	C₂²	semidirect
13. C₄⋊Q₈×C₅₃	C₂²	semidirect
14. #125×C₅₃	C₂²	semidirect
27. #107×C₅₃	C₂²	semidirect
28. #106×C₅₃	C₂²	semidirect
29. C₄⋊C₈×C₅₃	C₂²	semidirect
30. C₄⋊C₈×C₅₃	C₂²	semidirect
38. C₂×D₄×C₅₃	Q₈	semidirect
110. #3077	C₅₃	direct (N × H)
116. #533	C₂×C₅₃	semidirect
118. #478	C₂×C₅₃	semidirect
119. #476	C₂×C₅₃	semidirect
120. #474	C₂×C₅₃	semidirect
121. #428	C₂×C₅₃	semidirect
122. #423	C₂×C₅₃	semidirect
124. C₅₃	#3077	direct (N × H)
128. C₄⋊Q₈	C₂²×C₅₃	semidirect
129. C₄⋊Q₈	C₂²×C₅₃	semidirect
130. #125	C₂²×C₅₃	semidirect
143. #107	C₂²×C₅₃	semidirect
144. #106	C₂²×C₅₃	semidirect
145. C₄⋊C₈	C₂²×C₅₃	semidirect
146. C₄⋊C₈	C₂²×C₅₃	semidirect
154. C₂×D₄	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#533	2
C₂×C₄×D₄×C₅₃	2
C₅₃×#478	2
C₅₃×#476	2
C₅₃×#474	2
C₅₃×#428	2
C₅₃×#423	2
C₅₃×#418	2
#3077	53

Subgroups — C53×#3077 #20272014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3077 #20272014