Subgroups — C₅₃×#8615 #27216127

Order 13568.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1801	C₂	1	2	yes	maximal
3	C₅₃×#1566	C₂	1	2	yes	maximal
4	C₅₃×#1292	C₂	1	2	yes	maximal
5	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₈×Q₈×C₅₃	—	2	2	no	—
7	C₈×Q₈×C₅₃	—	2	2	no	—
8	C₅₃×#423	C₂²	1	4	yes	—
9	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#101×C₅₃	—	2	2	no	—
12	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#343	—	2	2	no	—
15	C₅₃×#334	—	2	2	no	—
16	C₅₃×#334	—	2	2	no	—
17	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂×M₁₆×C₅₃	—	2	4	no	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×Q₈×C₅₃	—	2	2	no	—
23	C₄×Q₈×C₅₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	—	2	4	no	—
29	C₄⋊C₈×C₅₃	—	2	2	no	—
30	C₄⋊C₈×C₅₃	—	2	4	no	—
31	C₄⋊C₈×C₅₃	—	2	4	no	—
32	#101×C₅₃	—	4	4	no	—
33	C₄×C₈×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	C₂×Q₈×C₅₃	—	2	2	no	—
38	C₂×Q₈×C₅₃	—	2	2	no	—
39	C₂×Q₈×C₅₃	SD₁₆	1	4	yes	—
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	M₁₆×C₅₃	—	4	4	no	—
46	M₁₆×C₅₃	—	4	4	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	4	1	no	—
50	C₂×C₈×C₅₃	—	4	2	no	—
51	C₂×C₈×C₅₃	—	4	4	no	—
52	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₄⋊C₄×C₅₃	—	2	4	no	—
57	C₄²×C₅₃	—	2	2	no	—
58	C₄²×C₅₃	—	2	2	no	—
59	C₄²×C₅₃	C₂×D₄	1	2	yes	—
60	C₄²×C₅₃	#41	1	2	yes	—
61	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	Q₈×C₅₃	—	2	8	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	4	1	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	#106	1	4	yes	—
71	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#121	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₅₃	#135	1	2	yes	—
77	C₈×C₅₃	—	2	1	no	—
78	C₈×C₅₃	—	2	2	no	—
79	C₈×C₅₃	—	4	2	no	—
80	C₈×C₅₃	—	8	4	no	—
81	C₈×C₅₃	#101	1	2	yes	complemented
82	#8615	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂²×C₅₃	—	2	1	no	—
84	C₂²×C₅₃	—	2	1	no	—
85	C₂²×C₅₃	#380	1	1	yes	characteristic
86	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
87	C₂²×C₅₃	#427	1	1	yes	characteristic
88	C₄×C₅₃	—	2	1	no	—
89	C₄×C₅₃	—	2	4	no	—
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	—	4	2	no	—
92	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
93	C₄×C₅₃	#413	1	1	yes	characteristic
94	C₄×C₅₃	#465	1	4	yes	—
95	C₂×C₈×Q₈	C₂×C₅₃	1	1	yes	characteristic
96	#1801	C₂×C₅₃	1	2	yes	—
97	#1566	C₂×C₅₃	1	2	yes	—
98	#1292	C₂×C₅₃	1	2	yes	—
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	—	2	1	no	—
101	C₂×C₅₃	#1801	1	1	yes	characteristic
102	C₂×C₅₃	#1807	1	1	yes	characteristic
103	C₂×C₅₃	#1813	1	1	yes	characteristic
104	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
105	C₈×Q₈	—	2	2	no	—
106	C₈×Q₈	—	2	2	no	—
107	#423	C₂²×C₅₃	1	4	yes	—
108	C₂×C₄⋊C₈	C₄×C₅₃	1	2	yes	—
109	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
110	C₂×#101	—	2	2	no	—
111	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
112	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
113	#343	—	2	2	no	—
114	#334	—	2	2	no	—
115	#334	—	2	2	no	—
116	C₅₃	#8615	1	1	yes	characteristic, direct product factor, Sylow
117	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
118	C₂×M₁₆	—	2	4	no	—
119	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
120	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
121	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
122	C₄×Q₈	—	2	2	no	—
123	C₄×Q₈	—	2	2	no	—
124	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
125	C₂×C₄⋊C₄	Q₈×C₅₃	1	2	yes	—
126	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
127	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
128	C₄⋊C₈	—	2	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	4	no	—
131	C₄⋊C₈	—	2	4	no	—
132	#101	—	4	4	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	D₄×C₅₃	1	2	yes	—
136	C₄×C₈	D₄×C₅₃	1	2	yes	—
137	C₂×Q₈	—	2	2	no	—
138	C₂×Q₈	—	2	2	no	—
139	C₂×Q₈	SD₁₆×C₅₃	1	4	yes	—
140	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
141	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
142	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
143	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
144	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
145	M₁₆	—	4	4	no	—
146	M₁₆	—	4	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	4	4	no	—
152	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
153	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
154	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	2	2	no	—
159	C₄²	C₂×D₄×C₅₃	1	2	yes	—
160	C₄²	#41×C₅₃	1	2	yes	—
161	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
162	Q₈	—	2	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	#106×C₅₃	1	4	yes	—
171	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
172	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
173	C₂×C₄	#121×C₅₃	1	2	yes	—
174	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
175	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
176	C₂×C₄	#135×C₅₃	1	2	yes	—
177	C₈	—	2	1	no	—
178	C₈	—	2	2	no	—
179	C₈	—	4	2	no	—
180	C₈	—	8	4	no	—
181	C₈	#101×C₅₃	1	2	yes	complemented
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	C₅₃×#380	1	1	yes	characteristic
185	C₂²	C₂×#106×C₅₃	1	1	yes	characteristic
186	C₂²	C₅₃×#427	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	—	4	2	no	—
191	C₄	C₂×#101×C₅₃	1	1	yes	characteristic
192	C₄	C₅₃×#413	1	1	yes	characteristic
193	C₄	C₅₃×#465	1	4	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	C₅₃×#1801	1	1	yes	characteristic
197	C₂	C₅₃×#1807	1	1	yes	characteristic
198	C₂	C₅₃×#1813	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#380
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ 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
81. C₈×C₅₃	#101	semidirect
82. #8615	C₅₃	direct (N × H)
116. C₅₃	#8615	direct (N × H)
181. C₈	#101×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₅₃	2
C₅₃×#1801	2
C₅₃×#1566	2
C₅₃×#1292	2
#8615	53

Subgroups — C53×#8615 #27216127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8615 #27216127