Subgroups — C₅₃×#12900 #27220412

Order 13568.

382 conjugacy classes of subgroups.

The subgroup lattice has 148 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	4	yes	maximal
2	C₅₃×#2150	C₂	1	2	yes	maximal
3	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₅₃	C₂²	1	4	yes	—
5	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₄³×C₅₃	C₄	1	1	yes	characteristic
10	C₅₃×#364	—	2	8	no	—
11	C₅₃×C₄⋊C₁₆	—	2	4	no	—
12	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
15	C₂×C₄²×C₅₃	—	2	2	no	—
16	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	M₃₂×C₅₃	—	2	8	no	—
19	C₂×C₁₆×C₅₃	—	2	4	no	—
20	C₄×C₈×C₅₃	—	2	4	no	—
21	C₄×C₈×C₅₃	C₈	1	2	yes	—
22	C₄×C₈×C₅₃	C₈	1	2	yes	—
23	C₄×C₈×C₅₃	D₄	1	4	yes	—
24	C₄×C₈×C₅₃	D₄	1	2	yes	—
25	C₄×C₈×C₅₃	Q₈	1	4	yes	—
26	C₄×C₈×C₅₃	Q₈	1	2	yes	—
27	C₂²×C₄×C₅₃	—	2	2	no	—
28	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
29	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
33	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
34	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
35	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
36	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
39	C₄²×C₅₃	—	2	8	no	—
40	C₄²×C₅₃	—	2	2	no	—
41	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
42	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₁₆×C₅₃	—	4	4	no	—
46	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	8	no	—
48	C₂×C₄×C₅₃	—	2	4	no	—
49	C₂×C₄×C₅₃	—	2	4	no	—
50	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
51	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
56	C₈×C₅₃	—	2	2	no	—
57	C₈×C₅₃	C₄⋊C₈	1	8	yes	—
58	C₈×C₅₃	#127	1	1	yes	characteristic
59	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
60	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
61	#12900	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂²×C₅₃	—	2	2	no	—
63	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂²×C₅₃	#423	1	2	yes	—
65	C₄×C₅₃	—	2	8	no	—
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	#364	1	4	yes	complemented
68	C₄×C₅₃	#384	1	1	yes	characteristic
69	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
70	C₄×C₅₃	#423	1	1	yes	characteristic
71	C₂×#364	C₂×C₅₃	1	4	yes	—
72	#2150	C₂×C₅₃	1	2	yes	—
73	C₄²×C₈	C₂×C₅₃	1	1	yes	characteristic
74	C₂×C₅₃	—	2	2	no	—
75	C₂×C₅₃	#1842	1	1	yes	characteristic
76	C₂×C₅₃	C₂×#364	1	2	yes	—
77	C₂×M₃₂	C₂²×C₅₃	1	4	yes	—
78	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
79	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
81	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
82	C₄³	C₄×C₅₃	1	1	yes	characteristic
83	#364	—	2	8	no	—
84	C₄⋊C₁₆	—	2	4	no	—
85	C₅₃	#12900	1	1	yes	characteristic, direct product factor, Sylow
86	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
87	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
88	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
89	C₂×C₄²	—	2	2	no	—
90	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
91	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
92	M₃₂	—	2	8	no	—
93	C₂×C₁₆	—	2	4	no	—
94	C₄×C₈	—	2	4	no	—
95	C₄×C₈	C₈×C₅₃	1	2	yes	—
96	C₄×C₈	C₈×C₅₃	1	2	yes	—
97	C₄×C₈	D₄×C₅₃	1	2	yes	—
98	C₄×C₈	D₄×C₅₃	1	4	yes	—
99	C₄×C₈	Q₈×C₅₃	1	4	yes	—
100	C₄×C₈	Q₈×C₅₃	1	2	yes	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	M₁₆×C₅₃	1	2	yes	—
103	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
106	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
107	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
108	C₂×C₈	M₁₆×C₅₃	1	4	yes	—
109	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
110	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
111	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
112	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
113	C₄²	—	2	8	no	—
114	C₄²	—	2	2	no	—
115	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
116	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
117	C₄²	C₂×C₈×C₅₃	1	1	yes	characteristic
118	C₄²	C₂×C₈×C₅₃	1	1	yes	characteristic
119	C₁₆	—	4	4	no	—
120	C₂³	C₂×M₁₆×C₅₃	1	1	yes	characteristic
121	C₂×C₄	—	2	8	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
125	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
126	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
127	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
128	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
129	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
130	C₈	—	2	2	no	—
131	C₈	C₄⋊C₈×C₅₃	1	8	yes	—
132	C₈	#127×C₅₃	1	1	yes	characteristic
133	C₈	C₄⋊Q₈×C₅₃	1	2	yes	—
134	C₈	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
135	C₂²	—	2	2	no	—
136	C₂²	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
137	C₂²	C₅₃×#423	1	2	yes	—
138	C₄	—	2	8	no	—
139	C₄	—	2	2	no	—
140	C₄	C₅₃×#364	1	4	yes	complemented
141	C₄	C₅₃×#384	1	1	yes	characteristic
142	C₄	C₂×C₄⋊C₈×C₅₃	1	2	yes	—
143	C₄	C₅₃×#423	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	C₅₃×#1842	1	1	yes	characteristic
146	C₂	C₂×C₅₃×#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
61. #12900	C₅₃	direct (N × H)
67. C₄×C₅₃	#364	semidirect
85. C₅₃	#12900	direct (N × H)
140. C₄	C₅₃×#364	semidirect

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×#2150	2
C₄²×C₈×C₅₃	2
#12900	53

Subgroups — C53×#12900 #27220412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12900 #27220412