Subgroups — C₅₃×#11355 #27218867

Order 13568.

686 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#382	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#343	C₂	1	4	yes	maximal
3	C₅₃×#1728	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1728	C₂	1	1	yes	maximal, characteristic
5	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
6	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#382	—	2	2	no	—
10	C₅₃×#382	—	2	2	no	—
11	C₂×#95×C₅₃	C₄	1	4	yes	—
12	C₅₃×#343	—	2	8	no	—
13	C₅₃×#336	—	2	2	no	—
14	C₅₃×#336	—	2	2	no	—
15	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
16	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₅₃	—	2	4	no	—
19	C₂×M₁₆×C₅₃	—	2	2	no	—
20	C₂×M₁₆×C₅₃	—	2	2	no	—
21	C₂×M₁₆×C₅₃	—	2	4	no	—
22	C₂×M₁₆×C₅₃	—	2	4	no	—
23	C₂×M₁₆×C₅₃	—	2	4	no	—
24	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
25	C₂×M₁₆×C₅₃	Q₈	1	4	yes	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
32	C₂×C₄²×C₅₃	—	2	4	no	—
33	C₂×C₄²×C₅₃	D₄	1	2	yes	—
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	#95×C₅₃	—	2	8	no	—
36	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	—	2	4	no	—
38	C₂²×C₄×C₅₃	—	2	4	no	—
39	C₂²×C₄×C₅₃	—	2	8	no	—
40	C₂²×C₄×C₅₃	—	4	4	no	—
41	C₂²×C₄×C₅₃	—	4	8	no	—
42	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
47	C₂²×C₄×C₅₃	#41	1	2	yes	—
48	C₂²×C₄×C₅₃	#41	1	4	yes	—
49	C₂²×C₄×C₅₃	#41	1	2	yes	—
50	M₁₆×C₅₃	—	2	8	no	—
51	M₁₆×C₅₃	—	2	8	no	—
52	M₁₆×C₅₃	—	4	4	no	—
53	M₁₆×C₅₃	—	4	4	no	—
54	C₂×C₈×C₅₃	—	2	4	no	—
55	C₂×C₈×C₅₃	—	2	4	no	—
56	C₂×C₈×C₅₃	—	4	2	no	—
57	C₂×C₈×C₅₃	—	4	2	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₄⋊C₄×C₅₃	—	4	4	no	—
61	C₄⋊C₄×C₅₃	—	4	2	no	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄²×C₅₃	—	2	4	no	—
64	C₄²×C₅₃	—	2	4	no	—
65	C₂³×C₅₃	—	2	4	no	—
66	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂³×C₅₃	#117	1	2	yes	—
68	C₂³×C₅₃	C₄×D₄	1	2	yes	—
69	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	2	8	no	—
72	C₂×C₄×C₅₃	—	4	8	no	—
73	C₂×C₄×C₅₃	—	4	4	no	—
74	C₂×C₄×C₅₃	—	4	8	no	—
75	C₂×C₄×C₅₃	—	8	8	no	—
76	C₂×C₄×C₅₃	#121	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
78	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
79	C₂×C₄×C₅₃	#123	1	2	yes	—
80	C₂×C₄×C₅₃	#123	1	4	yes	—
81	C₂×C₄×C₅₃	#123	1	2	yes	—
82	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#125	1	4	yes	—
85	C₂×C₄×C₅₃	#126	1	4	yes	—
86	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
88	C₈×C₅₃	—	4	4	no	—
89	C₈×C₅₃	—	4	4	no	—
90	#11355	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂²×C₅₃	—	2	4	no	—
92	C₂²×C₅₃	—	2	4	no	—
93	C₂²×C₅₃	—	2	4	no	—
94	C₂²×C₅₃	#381	1	1	yes	characteristic
95	C₂²×C₅₃	#387	1	4	yes	—
96	C₂²×C₅₃	#390	1	1	yes	characteristic
97	C₂²×C₅₃	#391	1	1	yes	characteristic
98	C₂²×C₅₃	#413	1	4	yes	—
99	C₄×C₅₃	—	4	4	no	—
100	C₄×C₅₃	—	8	4	no	—
101	C₄×C₅₃	#397	1	2	yes	—
102	C₄×C₅₃	#398	1	2	yes	—
103	C₄×C₅₃	#399	1	2	yes	—
104	C₄×C₅₃	#400	1	2	yes	—
105	C₂×#382	C₂×C₅₃	1	1	yes	characteristic
106	C₂×#343	C₂×C₅₃	1	4	yes	—
107	#1728	C₂×C₅₃	1	1	yes	characteristic
108	#1728	C₂×C₅₃	1	1	yes	characteristic
109	C₂×C₅₃	—	2	4	no	—
110	C₂×C₅₃	#1445	1	1	yes	characteristic
111	C₂×C₅₃	#1822	1	2	yes	—
112	C₂×C₅₃	#1967	1	2	yes	—
113	C₂×C₅₃	#1998	1	2	yes	—
114	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
115	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
116	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
117	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
118	#382	—	2	2	no	—
119	#382	—	2	2	no	—
120	C₂×#95	C₄×C₅₃	1	4	yes	—
121	#343	—	2	8	no	—
122	#336	—	2	2	no	—
123	#336	—	2	2	no	—
124	C₅₃	#11355	1	1	yes	characteristic, direct product factor, Sylow
125	C₂³×C₄	C₂×C₄×C₅₃	1	4	yes	—
126	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
127	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
128	C₂×M₁₆	—	2	2	no	—
129	C₂×M₁₆	—	2	2	no	—
130	C₂×M₁₆	—	2	4	no	—
131	C₂×M₁₆	—	2	4	no	—
132	C₂×M₁₆	—	2	4	no	—
133	C₂×M₁₆	—	2	4	no	—
134	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
135	C₂×M₁₆	Q₈×C₅₃	1	4	yes	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	4	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	4	no	—
142	C₂×C₄²	—	2	4	no	—
143	C₂×C₄²	D₄×C₅₃	1	2	yes	—
144	C₂×C₄²	D₄×C₅₃	1	2	yes	—
145	#95	—	2	8	no	—
146	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	—	4	4	no	—
151	C₂²×C₄	—	4	8	no	—
152	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
153	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
155	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
157	C₂²×C₄	#41×C₅₃	1	2	yes	—
158	C₂²×C₄	#41×C₅₃	1	4	yes	—
159	C₂²×C₄	#41×C₅₃	1	2	yes	—
160	M₁₆	—	2	8	no	—
161	M₁₆	—	2	8	no	—
162	M₁₆	—	4	4	no	—
163	M₁₆	—	4	4	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₄⋊C₄	—	4	4	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄²	—	2	4	no	—
174	C₄²	—	2	4	no	—
175	C₂³	—	2	4	no	—
176	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
177	C₂³	#117×C₅₃	1	2	yes	—
178	C₂³	C₄×D₄×C₅₃	1	2	yes	—
179	C₂³	C₄×Q₈×C₅₃	1	2	yes	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	—	8	8	no	—
186	C₂×C₄	#121×C₅₃	1	2	yes	—
187	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
188	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
189	C₂×C₄	#123×C₅₃	1	2	yes	—
190	C₂×C₄	#123×C₅₃	1	4	yes	—
191	C₂×C₄	#123×C₅₃	1	2	yes	—
192	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
193	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
194	C₂×C₄	#125×C₅₃	1	4	yes	—
195	C₂×C₄	#126×C₅₃	1	4	yes	—
196	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
197	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
198	C₈	—	4	4	no	—
199	C₈	—	4	4	no	—
200	C₂²	—	2	4	no	—
201	C₂²	—	2	4	no	—
202	C₂²	—	2	4	no	—
203	C₂²	C₅₃×#381	1	1	yes	characteristic
204	C₂²	C₅₃×#387	1	4	yes	—
205	C₂²	C₅₃×#390	1	1	yes	characteristic
206	C₂²	C₅₃×#391	1	1	yes	characteristic
207	C₂²	C₅₃×#413	1	4	yes	—
208	C₄	—	4	4	no	—
209	C₄	—	8	4	no	—
210	C₄	C₅₃×#397	1	2	yes	—
211	C₄	C₅₃×#398	1	2	yes	—
212	C₄	C₅₃×#399	1	2	yes	—
213	C₄	C₅₃×#400	1	2	yes	—
214	C₂	—	2	4	no	—
215	C₂	C₅₃×#1445	1	1	yes	characteristic
216	C₂	C₅₃×#1822	1	2	yes	—
217	C₂	C₅₃×#1967	1	2	yes	—
218	C₂	C₅₃×#1998	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₄×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
90. #11355	C₅₃	direct (N × H)
124. C₅₃	#11355	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#382	2
C₂×C₅₃×#343	2
C₅₃×#1728	2
#11355	53

Subgroups — C53×#11355 #27218867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11355 #27218867