Subgroups — C₅₃×#12881 #27220393

Order 13568.

298 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#364	C₂	1	2	yes	maximal
2	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1377	C₂	1	2	yes	maximal
4	C₅₃×#1375	C₂	1	2	yes	maximal
5	C₂×M₃₂×C₅₃	C₂²	1	4	yes	—
6	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
7	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#376	C₄	1	1	yes	characteristic
13	C₅₃×#364	—	2	4	no	—
14	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂³	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	C₂×C₄	1	1	yes	characteristic
21	M₃₂×C₅₃	—	2	4	no	—
22	M₃₂×C₅₃	—	2	4	no	—
23	C₂×C₁₆×C₅₃	—	2	4	no	—
24	#97×C₅₃	—	2	2	no	—
25	#97×C₅₃	—	2	2	no	—
26	#97×C₅₃	—	2	2	no	—
27	C₄×C₈×C₅₃	—	2	2	no	—
28	C₄×C₈×C₅₃	C₈	1	4	yes	—
29	C₄×C₈×C₅₃	D₄	1	2	yes	—
30	C₄×C₈×C₅₃	Q₈	1	2	yes	—
31	#95×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	—	2	1	no	—
33	C₂²×C₄×C₅₃	—	2	1	no	—
34	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	—	2	1	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
44	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
45	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	#41	1	2	yes	—
48	C₂×C₈×C₅₃	#41	1	2	yes	—
49	C₄²×C₅₃	—	2	1	no	—
50	C₄²×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	—	2	1	no	—
52	C₄²×C₅₃	—	2	1	no	—
53	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₄²×C₅₃	C₂×C₈	1	2	yes	—
55	C₁₆×C₅₃	—	4	4	no	—
56	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	4	2	no	—
61	C₂×C₄×C₅₃	—	4	2	no	—
62	C₂×C₄×C₅₃	—	4	2	no	—
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
69	C₈×C₅₃	—	2	2	no	—
70	C₈×C₅₃	—	2	2	no	—
71	C₈×C₅₃	C₄⋊C₈	1	4	yes	—
72	C₈×C₅₃	#124	1	2	yes	—
73	C₈×C₅₃	#125	1	2	yes	—
74	#12881	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂²×C₅₃	—	2	2	no	—
76	C₂²×C₅₃	#423	1	1	yes	characteristic
77	C₂²×C₅₃	#431	1	1	yes	characteristic
78	C₂²×C₅₃	#432	1	1	yes	characteristic
79	C₄×C₅₃	—	4	2	no	—
80	C₄×C₅₃	—	4	2	no	—
81	C₄×C₅₃	—	4	2	no	—
82	C₄×C₅₃	#382	1	1	yes	characteristic
83	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₄×C₅₃	#431	1	1	yes	characteristic
85	C₄×C₅₃	#432	1	1	yes	characteristic
86	C₂×#364	C₂×C₅₃	1	2	yes	—
87	#1726	C₂×C₅₃	1	1	yes	characteristic
88	#1377	C₂×C₅₃	1	2	yes	—
89	#1375	C₂×C₅₃	1	2	yes	—
90	C₂×C₅₃	—	2	1	no	—
91	C₂×C₅₃	—	2	1	no	—
92	C₂×C₅₃	#1831	1	1	yes	characteristic
93	C₂×C₅₃	#2166	1	2	yes	—
94	C₂×M₃₂	C₂²×C₅₃	1	4	yes	—
95	C₂×#97	C₄×C₅₃	1	1	yes	characteristic
96	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
97	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
98	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
99	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
101	#376	C₄×C₅₃	1	1	yes	characteristic
102	#364	—	2	4	no	—
103	C₅₃	#12881	1	1	yes	characteristic, direct product factor, Sylow
104	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
105	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
106	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
107	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
108	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
109	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
110	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
111	M₃₂	—	2	4	no	—
112	M₃₂	—	2	4	no	—
113	C₂×C₁₆	—	2	4	no	—
114	#97	—	2	2	no	—
115	#97	—	2	2	no	—
116	#97	—	2	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	C₈×C₅₃	1	4	yes	—
119	C₄×C₈	D₄×C₅₃	1	2	yes	—
120	C₄×C₈	Q₈×C₅₃	1	2	yes	—
121	#95	—	2	2	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	M₁₆×C₅₃	1	2	yes	—
125	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
131	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
132	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
133	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
134	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
135	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
136	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
137	C₂×C₈	#41×C₅₃	1	2	yes	—
138	C₂×C₈	#41×C₅₃	1	2	yes	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
144	C₄²	C₂×C₈×C₅₃	1	2	yes	—
145	C₁₆	—	4	4	no	—
146	C₂³	C₂×M₁₆×C₅₃	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
154	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
155	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
156	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
157	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
158	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	2	2	no	—
161	C₈	C₄⋊C₈×C₅₃	1	4	yes	—
162	C₈	#124×C₅₃	1	2	yes	—
163	C₈	#125×C₅₃	1	2	yes	—
164	C₂²	—	2	2	no	—
165	C₂²	C₅₃×#423	1	1	yes	characteristic
166	C₂²	C₅₃×#431	1	1	yes	characteristic
167	C₂²	C₅₃×#432	1	1	yes	characteristic
168	C₄	—	4	2	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	2	no	—
171	C₄	C₅₃×#382	1	1	yes	characteristic
172	C₄	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
173	C₄	C₅₃×#431	1	1	yes	characteristic
174	C₄	C₅₃×#432	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	C₅₃×#1831	1	1	yes	characteristic
178	C₂	C₅₃×#2166	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	#117
commutator	C₂×C₄	C₂²×C₈×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	#432
prime radical	C₂³×C₅₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
74. #12881	C₅₃	direct (N × H)
103. C₅₃	#12881	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×#364	2
C₅₃×#1726	2
C₅₃×#1377	2
C₅₃×#1375	2
#12881	53

Subgroups — C53×#12881 #27220393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12881 #27220393