Subgroups — C₅₃×#8592 #27216104

Order 13568.

644 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1831	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₃×#331	C₂	1	2	yes	maximal
4	C₅₃×#1296	C₂	1	2	yes	maximal, complemented
5	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×C₄×D₄×C₅₃	—	2	2	no	—
7	C₂×C₄×D₄×C₅₃	C₄	1	2	yes	—
8	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
9	C₅₃×#390	—	2	2	no	—
10	C₅₃×#390	C₄	1	2	yes	—
11	C₅₃×#384	C₂²	1	1	yes	characteristic
12	C₄³×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#336	—	2	2	no	—
14	C₅₃×#336	—	2	2	no	—
15	C₅₃×#331	—	2	4	no	—
16	C₂²×D₄×C₅₃	—	2	2	no	—
17	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₅₃	—	2	2	no	—
19	C₂²×C₈×C₅₃	—	2	4	no	—
20	#127×C₅₃	—	2	2	no	—
21	#127×C₅₃	D₄	1	2	yes	—
22	#127×C₅₃	Q₈	1	2	yes	—
23	C₄×D₄×C₅₃	—	4	8	no	—
24	C₄×D₄×C₅₃	—	4	4	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
28	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	D₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	4	no	—
35	C₄⋊C₈×C₅₃	—	2	4	no	—
36	C₂⁴×C₅₃	—	2	2	no	—
37	C₂×D₄×C₅₃	—	2	2	no	—
38	C₂×D₄×C₅₃	—	2	4	no	—
39	C₂×D₄×C₅₃	—	2	4	no	—
40	C₂×D₄×C₅₃	—	4	2	no	—
41	C₂×D₄×C₅₃	—	4	2	no	—
42	C₂×D₄×C₅₃	—	4	4	no	—
43	C₂×D₄×C₅₃	—	4	4	no	—
44	C₂×D₄×C₅₃	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₄×C₅₃	—	4	8	no	—
47	C₂²×C₄×C₅₃	—	4	4	no	—
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₅₃	—	4	4	no	—
54	C₂×C₈×C₅₃	—	4	4	no	—
55	C₂×C₈×C₅₃	—	4	4	no	—
56	C₄⋊C₄×C₅₃	—	4	2	no	—
57	C₄⋊C₄×C₅₃	—	4	4	no	—
58	C₂²⋊C₄×C₅₃	—	4	4	no	—
59	C₂²⋊C₄×C₅₃	—	4	8	no	—
60	C₄²×C₅₃	—	2	2	no	—
61	C₄²×C₅₃	—	2	4	no	—
62	C₄²×C₅₃	—	2	4	no	—
63	C₄²×C₅₃	—	4	1	no	—
64	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₅₃	#41	1	2	yes	—
67	C₂³×C₅₃	—	2	4	no	—
68	C₂³×C₅₃	—	4	2	no	—
69	C₂³×C₅₃	—	4	2	no	—
70	C₂³×C₅₃	—	4	4	no	—
71	C₂³×C₅₃	—	4	4	no	—
72	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	D₄×C₅₃	—	4	4	no	—
74	D₄×C₅₃	—	4	4	no	—
75	D₄×C₅₃	—	4	8	no	—
76	D₄×C₅₃	—	4	8	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	4	4	no	—
87	C₂×C₄×C₅₃	—	8	8	no	—
88	C₂×C₄×C₅₃	#100	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#121	1	2	yes	—
92	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
93	C₂×C₄×C₅₃	#135	1	4	yes	—
94	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
95	C₈×C₅₃	—	8	4	no	—
96	#8592	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂²×C₅₃	—	4	4	no	—
98	C₂²×C₅₃	—	4	8	no	—
99	C₂²×C₅₃	—	8	2	no	—
100	C₂²×C₅₃	—	8	4	no	—
101	C₂²×C₅₃	—	8	2	no	—
102	C₂²×C₅₃	#380	1	1	yes	characteristic
103	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
104	C₂²×C₅₃	#413	1	1	yes	characteristic
105	C₂²×C₅₃	#416	1	2	yes	—
106	C₂²×C₅₃	#418	1	2	yes	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	—	4	1	no	—
111	C₄×C₅₃	—	4	2	no	—
112	C₄×C₅₃	—	8	2	no	—
113	C₄×C₅₃	#463	1	4	yes	—
114	C₄×C₅₃	#478	1	4	yes	—
115	C₄×#127	C₂×C₅₃	1	1	yes	characteristic
116	#1831	C₂×C₅₃	1	2	yes	complemented
117	C₂×#331	C₂×C₅₃	1	2	yes	—
118	#1296	C₂×C₅₃	1	2	yes	complemented
119	C₂×C₅₃	—	8	4	no	—
120	C₂×C₅₃	#1543	1	1	yes	characteristic
121	C₂×C₅₃	#1553	1	2	yes	—
122	C₂×C₅₃	#1555	1	1	yes	characteristic
123	C₂×C₅₃	#1800	1	1	yes	characteristic
124	C₂×C₅₃	#1804	1	2	yes	—
125	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
126	C₂×C₄×D₄	—	2	2	no	—
127	C₂×C₄×D₄	C₄×C₅₃	1	2	yes	—
128	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
129	#390	—	2	2	no	—
130	#390	C₄×C₅₃	1	2	yes	—
131	#384	C₂²×C₅₃	1	1	yes	characteristic
132	C₄³	C₂²×C₅₃	1	1	yes	characteristic
133	#336	—	2	2	no	—
134	#336	—	2	2	no	—
135	#331	—	2	4	no	—
136	C₅₃	#8592	1	1	yes	characteristic, direct product factor, Sylow
137	C₂²×D₄	—	2	2	no	—
138	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
139	C₂³×C₄	—	2	2	no	—
140	C₂²×C₈	—	2	4	no	—
141	#127	—	2	2	no	—
142	#127	D₄×C₅₃	1	2	yes	—
143	#127	Q₈×C₅₃	1	2	yes	—
144	C₄×D₄	—	4	8	no	—
145	C₄×D₄	—	4	4	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
148	C₂×C₂²⋊C₄	—	2	4	no	—
149	C₂×C₂²⋊C₄	—	2	2	no	—
150	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
151	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
152	C₂×C₄²	D₄×C₅₃	1	2	yes	—
153	C₂×C₄²	D₄×C₅₃	1	2	yes	—
154	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
155	C₄⋊C₈	—	2	4	no	—
156	C₄⋊C₈	—	2	4	no	—
157	C₂⁴	—	2	2	no	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	2	no	—
163	C₂×D₄	—	4	4	no	—
164	C₂×D₄	—	4	4	no	—
165	C₂×D₄	C₄⋊C₄×C₅₃	1	4	yes	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	4	8	no	—
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
171	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
174	C₂×C₈	—	4	4	no	—
175	C₂×C₈	—	4	4	no	—
176	C₂×C₈	—	4	4	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	4	no	—
179	C₂²⋊C₄	—	4	4	no	—
180	C₂²⋊C₄	—	4	8	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	4	no	—
183	C₄²	—	2	4	no	—
184	C₄²	—	4	1	no	—
185	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
186	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
187	C₄²	#41×C₅₃	1	2	yes	—
188	C₂³	—	2	4	no	—
189	C₂³	—	4	2	no	—
190	C₂³	—	4	2	no	—
191	C₂³	—	4	4	no	—
192	C₂³	—	4	4	no	—
193	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
194	D₄	—	4	4	no	—
195	D₄	—	4	4	no	—
196	D₄	—	4	8	no	—
197	D₄	—	4	8	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	8	no	—
209	C₂×C₄	#100×C₅₃	1	2	yes	—
210	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
211	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
212	C₂×C₄	#121×C₅₃	1	2	yes	—
213	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
214	C₂×C₄	#135×C₅₃	1	4	yes	—
215	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
216	C₈	—	8	4	no	—
217	C₂²	—	4	4	no	—
218	C₂²	—	4	8	no	—
219	C₂²	—	8	2	no	—
220	C₂²	—	8	4	no	—
221	C₂²	—	8	2	no	—
222	C₂²	C₅₃×#380	1	1	yes	characteristic
223	C₂²	C₂×#100×C₅₃	1	1	yes	characteristic
224	C₂²	C₅₃×#413	1	1	yes	characteristic
225	C₂²	C₅₃×#416	1	2	yes	—
226	C₂²	C₅₃×#418	1	2	yes	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	8	2	no	—
233	C₄	C₅₃×#463	1	4	yes	—
234	C₄	C₅₃×#478	1	4	yes	—
235	C₂	—	8	4	no	—
236	C₂	C₅₃×#1543	1	1	yes	characteristic
237	C₂	C₅₃×#1553	1	2	yes	—
238	C₂	C₅₃×#1555	1	1	yes	characteristic
239	C₂	C₅₃×#1800	1	1	yes	characteristic
240	C₂	C₅₃×#1804	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1831	C₂	semidirect
4. C₅₃×#1296	C₂	semidirect
96. #8592	C₅₃	direct (N × H)
116. #1831	C₂×C₅₃	semidirect
118. #1296	C₂×C₅₃	semidirect
136. C₅₃	#8592	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#127×C₅₃	2
C₅₃×#1831	2
C₂×C₅₃×#331	2
C₅₃×#1296	2
#8592	53

Subgroups — C53×#8592 #27216104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8592 #27216104