Subgroups — C₅₃×#9693 #27217205

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2862	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#333	C₂	1	2	yes	maximal
5	C₂×C₅₃×#332	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₅₃	—	2	1	no	—
11	C₅₃×#391	—	2	2	no	—
12	C₅₃×#391	C₄	1	4	yes	—
13	C₅₃×#382	—	2	1	no	—
14	C₅₃×#382	—	2	1	no	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#336	—	2	1	no	—
17	C₅₃×#336	—	2	2	no	—
18	C₅₃×#333	—	2	4	no	—
19	C₅₃×#332	—	2	4	no	—
20	C₂²×Q₈×C₅₃	—	2	1	no	—
21	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	—	2	2	no	—
24	C₄⋊Q₈×C₅₃	—	2	2	no	—
25	C₄⋊Q₈×C₅₃	—	2	2	no	—
26	C₄⋊Q₈×C₅₃	—	2	2	no	—
27	C₄⋊Q₈×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	—	2	2	no	—
38	C₄⋊C₈×C₅₃	—	2	2	no	—
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
41	C₄×C₈×C₅₃	—	4	2	no	—
42	#95×C₅₃	—	2	4	no	—
43	#95×C₅₃	—	2	4	no	—
44	C₂×Q₈×C₅₃	—	2	4	no	—
45	C₂×Q₈×C₅₃	—	4	4	no	—
46	C₂×Q₈×C₅₃	—	4	1	no	—
47	C₂×Q₈×C₅₃	—	4	1	no	—
48	C₂×Q₈×C₅₃	—	4	4	no	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	—	2	4	no	—
57	C₂×C₈×C₅₃	—	4	2	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	—	4	4	no	—
61	C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄⋊C₄×C₅₃	—	4	2	no	—
64	C₄⋊C₄×C₅₃	—	4	1	no	—
65	C₄⋊C₄×C₅₃	—	4	1	no	—
66	C₄²×C₅₃	—	2	4	no	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	—	4	2	no	—
69	C₄²×C₅₃	SD₁₆	1	2	yes	—
70	C₄²×C₅₃	Q₁₆	1	2	yes	—
71	C₄²×C₅₃	C₂×D₄	1	2	yes	—
72	C₄²×C₅₃	#41	1	2	yes	—
73	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	Q₈×C₅₃	—	8	4	no	—
75	Q₈×C₅₃	—	8	2	no	—
76	Q₈×C₅₃	—	8	2	no	—
77	C₂×C₄×C₅₃	—	2	2	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	—	2	2	no	—
83	C₂×C₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	4	2	no	—
85	C₂×C₄×C₅₃	—	4	4	no	—
86	C₂×C₄×C₅₃	—	4	4	no	—
87	C₂×C₄×C₅₃	#103	1	4	yes	—
88	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₅₃	#121	1	2	yes	—
90	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#135	1	4	yes	—
95	C₂×C₄×C₅₃	#137	1	2	yes	—
96	C₈×C₅₃	—	4	4	no	—
97	C₈×C₅₃	—	8	2	no	—
98	#9693	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₅₃	#390	1	1	yes	characteristic
100	C₂²×C₅₃	#413	1	2	yes	—
101	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
102	C₂²×C₅₃	#416	1	2	yes	—
103	C₂²×C₅₃	#419	1	1	yes	characteristic
104	C₄×C₅₃	—	2	4	no	—
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	4	2	no	—
107	C₄×C₅₃	—	8	2	no	—
108	C₄×C₅₃	—	8	2	no	—
109	C₄×C₅₃	#461	1	2	yes	—
110	C₄×C₅₃	#462	1	2	yes	—
111	C₄×C₅₃	#488	1	2	yes	—
112	C₄×C₅₃	#495	1	2	yes	—
113	#2862	C₂×C₅₃	1	1	yes	characteristic
114	#1831	C₂×C₅₃	1	1	yes	characteristic
115	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
116	C₂×#333	C₂×C₅₃	1	2	yes	—
117	C₂×#332	C₂×C₅₃	1	2	yes	—
118	C₂×C₅₃	#1542	1	2	yes	—
119	C₂×C₅₃	#1554	1	2	yes	—
120	C₂×C₅₃	#1958	1	1	yes	characteristic
121	C₂×C₅₃	#1967	1	1	yes	characteristic
122	C₂×C₅₃	#1981	1	1	yes	characteristic
123	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
124	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
125	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
126	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
127	C₂×C₄×C₈	—	2	1	no	—
128	#391	—	2	2	no	—
129	#391	C₄×C₅₃	1	4	yes	—
130	#382	—	2	1	no	—
131	#382	—	2	1	no	—
132	C₄³	C₂²×C₅₃	1	1	yes	characteristic
133	#336	—	2	2	no	—
134	#336	—	2	1	no	—
135	#333	—	2	4	no	—
136	#332	—	2	4	no	—
137	C₅₃	#9693	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×Q₈	—	2	1	no	—
139	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	2	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
150	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
151	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
152	C₂×C₄²	D₄×C₅₃	1	2	yes	—
153	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
154	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
159	C₄×C₈	—	4	2	no	—
160	#95	—	2	4	no	—
161	#95	—	2	4	no	—
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	4	4	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	—	4	1	no	—
166	C₂×Q₈	—	4	4	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
170	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
171	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	4	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₄²	—	2	4	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	4	2	no	—
187	C₄²	SD₁₆×C₅₃	1	2	yes	—
188	C₄²	Q₁₆×C₅₃	1	2	yes	—
189	C₄²	C₂×D₄×C₅₃	1	2	yes	—
190	C₄²	#41×C₅₃	1	2	yes	—
191	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
192	Q₈	—	8	4	no	—
193	Q₈	—	8	2	no	—
194	Q₈	—	8	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	#103×C₅₃	1	4	yes	—
206	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
207	C₂×C₄	#121×C₅₃	1	2	yes	—
208	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
209	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
210	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
211	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
212	C₂×C₄	#135×C₅₃	1	4	yes	—
213	C₂×C₄	#137×C₅₃	1	2	yes	—
214	C₈	—	4	4	no	—
215	C₈	—	8	2	no	—
216	C₂²	C₅₃×#390	1	1	yes	characteristic
217	C₂²	C₅₃×#413	1	2	yes	—
218	C₂²	C₂×#103×C₅₃	1	1	yes	characteristic
219	C₂²	C₅₃×#416	1	2	yes	—
220	C₂²	C₅₃×#419	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	4	no	—
223	C₄	—	4	2	no	—
224	C₄	—	8	2	no	—
225	C₄	—	8	2	no	—
226	C₄	C₅₃×#461	1	2	yes	—
227	C₄	C₅₃×#462	1	2	yes	—
228	C₄	C₅₃×#488	1	2	yes	—
229	C₄	C₅₃×#495	1	2	yes	—
230	C₂	C₅₃×#1542	1	2	yes	—
231	C₂	C₅₃×#1554	1	2	yes	—
232	C₂	C₅₃×#1958	1	1	yes	characteristic
233	C₂	C₅₃×#1967	1	1	yes	characteristic
234	C₂	C₅₃×#1981	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
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
98. #9693	C₅₃	direct (N × H)
137. C₅₃	#9693	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×#2862	2
C₅₃×#1831	2
C₄×C₄⋊C₈×C₅₃	2
C₂×C₅₃×#333	2
C₂×C₅₃×#332	2
#9693	53

Subgroups — C53×#9693 #27217205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9693 #27217205