Subgroups — C₅₃×#7027 #27214539

Order 13568.

338 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
5	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
6	C₂×#97×C₅₃	—	2	1	no	—
7	C₂×C₄×C₈×C₅₃	—	2	1	no	—
8	C₂×C₄×C₈×C₅₃	—	2	1	no	—
9	C₄³×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#336	—	2	1	no	—
11	C₅₃×#336	—	2	1	no	—
12	C₅₃×#336	C₄	1	2	yes	—
13	C₅₃×#336	C₄	1	2	yes	—
14	C₂²×C₈×C₅₃	—	2	2	no	—
15	C₂²×C₈×C₅₃	—	2	2	no	—
16	C₂²×C₈×C₅₃	—	2	2	no	—
17	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
24	C₄⋊C₈×C₅₃	—	2	4	no	—
25	C₄⋊C₈×C₅₃	—	2	2	no	—
26	C₄⋊C₈×C₅₃	C₈	1	4	yes	complemented
27	#97×C₅₃	—	4	1	no	—
28	#97×C₅₃	—	4	1	no	—
29	C₄×C₈×C₅₃	—	2	4	no	—
30	C₄×C₈×C₅₃	—	4	1	no	—
31	C₄×C₈×C₅₃	—	4	1	no	—
32	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	—	2	4	no	—
40	C₂×C₈×C₅₃	—	2	4	no	—
41	C₂×C₈×C₅₃	—	2	4	no	—
42	C₂×C₈×C₅₃	—	4	2	no	—
43	C₂×C₈×C₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	—	4	2	no	—
45	C₂×C₈×C₅₃	—	4	2	no	—
46	C₂×C₈×C₅₃	—	4	2	no	—
47	C₂×C₈×C₅₃	—	4	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	2	2	no	—
50	C₄²×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	—	4	1	no	—
52	C₄²×C₅₃	—	4	1	no	—
53	C₄²×C₅₃	C₂×C₈	1	2	yes	—
54	C₄²×C₅₃	M₁₆	1	2	yes	—
55	C₄²×C₅₃	D₈	1	1	yes	characteristic
56	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
57	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
58	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
59	C₂³×C₅₃	#95	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
75	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#104	1	2	yes	—
80	C₂×C₄×C₅₃	#104	1	2	yes	—
81	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
82	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#108	1	2	yes	—
85	C₈×C₅₃	—	4	4	no	—
86	C₈×C₅₃	—	4	4	no	—
87	C₈×C₅₃	—	4	4	no	—
88	#7027	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₅₃	#336	1	1	yes	characteristic
90	C₂²×C₅₃	#339	1	1	yes	characteristic
91	C₂²×C₅₃	#339	1	1	yes	characteristic
92	C₂²×C₅₃	#340	1	1	yes	characteristic
93	C₂²×C₅₃	#341	1	1	yes	characteristic
94	C₂²×C₅₃	#344	1	1	yes	characteristic
95	C₂²×C₅₃	#344	1	1	yes	characteristic
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	2	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
106	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
107	#1767	C₂×C₅₃	1	1	yes	characteristic
108	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
109	#1744	C₂×C₅₃	1	1	yes	characteristic
110	C₂×C₅₃	#1278	1	1	yes	characteristic
111	C₂×C₅₃	#1279	1	1	yes	characteristic
112	C₂×C₅₃	#1281	1	1	yes	characteristic
113	C₂×C₅₃	#1300	1	1	yes	characteristic
114	C₂×C₅₃	#1301	1	1	yes	characteristic
115	C₂×C₅₃	#1302	1	1	yes	characteristic
116	C₂×C₅₃	#1303	1	1	yes	characteristic
117	C₂×C₄⋊C₈	—	2	1	no	—
118	C₂×C₄⋊C₈	C₄×C₅₃	1	2	yes	—
119	C₂×#97	—	2	1	no	—
120	C₂×C₄×C₈	—	2	1	no	—
121	C₂×C₄×C₈	—	2	1	no	—
122	C₄³	C₂²×C₅₃	1	1	yes	characteristic
123	#336	—	2	1	no	—
124	#336	—	2	1	no	—
125	#336	C₄×C₅₃	1	2	yes	—
126	#336	C₄×C₅₃	1	2	yes	—
127	C₅₃	#7027	1	1	yes	characteristic, direct product factor, Sylow
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	2	2	no	—
131	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
132	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
133	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
134	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
135	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
136	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
137	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
138	C₄⋊C₈	—	2	4	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	C₈×C₅₃	1	4	yes	complemented
141	#97	—	4	1	no	—
142	#97	—	4	1	no	—
143	C₄×C₈	—	2	4	no	—
144	C₄×C₈	—	4	1	no	—
145	C₄×C₈	—	4	1	no	—
146	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
147	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
148	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
149	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
150	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
151	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
152	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	4	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	4	1	no	—
166	C₄²	—	4	1	no	—
167	C₄²	C₂×C₈×C₅₃	1	2	yes	—
168	C₄²	M₁₆×C₅₃	1	2	yes	—
169	C₄²	D₈×C₅₃	1	1	yes	characteristic
170	C₄²	SD₁₆×C₅₃	1	1	yes	characteristic
171	C₄²	SD₁₆×C₅₃	1	1	yes	characteristic
172	C₄²	Q₁₆×C₅₃	1	1	yes	characteristic
173	C₂³	#95×C₅₃	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
187	C₂×C₄	#97×C₅₃	1	1	yes	characteristic
188	C₂×C₄	C₂²⋊C₈×C₅₃	1	2	yes	—
189	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
190	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
191	C₂×C₄	#103×C₅₃	1	1	yes	characteristic
192	C₂×C₄	#103×C₅₃	1	1	yes	characteristic
193	C₂×C₄	#104×C₅₃	1	2	yes	—
194	C₂×C₄	#104×C₅₃	1	2	yes	—
195	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
196	C₂×C₄	#106×C₅₃	1	1	yes	characteristic
197	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
198	C₂×C₄	#108×C₅₃	1	2	yes	—
199	C₈	—	4	4	no	—
200	C₈	—	4	4	no	—
201	C₈	—	4	4	no	—
202	C₂²	C₅₃×#336	1	1	yes	characteristic
203	C₂²	C₅₃×#339	1	1	yes	characteristic
204	C₂²	C₅₃×#339	1	1	yes	characteristic
205	C₂²	C₅₃×#340	1	1	yes	characteristic
206	C₂²	C₅₃×#341	1	1	yes	characteristic
207	C₂²	C₅₃×#344	1	1	yes	characteristic
208	C₂²	C₅₃×#344	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	2	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	C₅₃×D₄⋊C₈	1	2	yes	—
219	C₄	C₅₃×Q₈⋊C₈	1	2	yes	—
220	C₂	C₅₃×#1278	1	1	yes	characteristic
221	C₂	C₅₃×#1279	1	1	yes	characteristic
222	C₂	C₅₃×#1281	1	1	yes	characteristic
223	C₂	C₅₃×#1300	1	1	yes	characteristic
224	C₂	C₅₃×#1301	1	1	yes	characteristic
225	C₂	C₅₃×#1302	1	1	yes	characteristic
226	C₂	C₅₃×#1303	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈×C₅₃
Frattini, cosocle	C₄³	C₂²×C₅₃
socle, prime radical	C₂³×C₅₃	#95
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
26. C₄⋊C₈×C₅₃	C₈	semidirect
88. #7027	C₅₃	direct (N × H)
127. C₅₃	#7027	direct (N × H)
140. C₄⋊C₈	C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#1767	2
C₄×C₄⋊C₈×C₅₃	2
C₅₃×#1744	2
#7027	53

Subgroups — C53×#7027 #27214539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7027 #27214539