Subgroups — C₅₃×#12879 #27220391

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1374	C₂	1	4	yes	maximal
4	C₂²×C₁₆×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×C₄⋊C₁₆	—	2	4	no	—
11	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	C₈	1	2	yes	—
15	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₅₃	—	2	4	no	—
20	C₂×C₁₆×C₅₃	—	2	8	no	—
21	C₄×C₈×C₅₃	—	2	4	no	—
22	C₄×C₈×C₅₃	—	2	4	no	—
23	C₄×C₈×C₅₃	—	2	2	no	—
24	C₄×C₈×C₅₃	D₄	1	2	yes	—
25	C₄×C₈×C₅₃	Q₈	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
28	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
29	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₅₃	M₁₆	1	4	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	#41	1	4	yes	—
39	C₄²×C₅₃	—	2	4	no	—
40	C₄²×C₅₃	—	2	2	no	—
41	C₄²×C₅₃	—	2	2	no	—
42	C₄²×C₅₃	—	2	2	no	—
43	C₄²×C₅₃	C₁₆	1	4	yes	complemented
44	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₄²×C₅₃	M₁₆	1	2	yes	—
46	C₁₆×C₅₃	—	4	4	no	—
47	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
56	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
58	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#117	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
63	C₈×C₅₃	—	2	2	no	—
64	C₈×C₅₃	—	2	4	no	—
65	C₈×C₅₃	#124	1	2	yes	—
66	C₈×C₅₃	#125	1	2	yes	—
67	#12879	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₅₃	#431	1	2	yes	—
70	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
71	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
72	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
73	C₄×C₅₃	—	2	2	no	—
74	C₄×C₅₃	—	2	4	no	—
75	C₄×C₅₃	—	2	4	no	—
76	C₄×C₅₃	C₄⋊C₁₆	1	4	yes	—
77	C₄×C₅₃	#382	1	1	yes	characteristic
78	C₄×C₅₃	#423	1	1	yes	characteristic
79	C₄×C₅₃	#432	1	2	yes	—
80	C₂×C₄⋊C₁₆	C₂×C₅₃	1	2	yes	—
81	C₄²×C₈	C₂×C₅₃	1	1	yes	characteristic
82	#1374	C₂×C₅₃	1	4	yes	—
83	C₂×C₅₃	#1831	1	1	yes	characteristic
84	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
85	C₂×C₅₃	#2150	1	1	yes	characteristic
86	C₂×C₅₃	#2162	1	2	yes	—
87	C₂×C₅₃	#2163	1	2	yes	—
88	C₂²×C₁₆	C₂²×C₅₃	1	4	yes	—
89	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
90	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
91	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
92	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
93	C₄³	C₄×C₅₃	1	1	yes	characteristic
94	C₄⋊C₁₆	—	2	4	no	—
95	C₅₃	#12879	1	1	yes	characteristic, direct product factor, Sylow
96	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
97	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
98	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
99	C₂×C₄²	C₈×C₅₃	1	2	yes	—
100	C₂×C₄²	C₈×C₅₃	1	1	yes	characteristic
101	C₂×C₄²	C₈×C₅₃	1	1	yes	characteristic
102	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
103	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
104	C₂×C₁₆	—	2	4	no	—
105	C₂×C₁₆	—	2	8	no	—
106	C₄×C₈	—	2	4	no	—
107	C₄×C₈	—	2	4	no	—
108	C₄×C₈	—	2	2	no	—
109	C₄×C₈	D₄×C₅₃	1	2	yes	—
110	C₄×C₈	Q₈×C₅₃	1	2	yes	—
111	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
113	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
114	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
115	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
120	C₂×C₈	M₁₆×C₅₃	1	4	yes	—
121	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
122	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
123	C₂×C₈	#41×C₅₃	1	4	yes	—
124	C₄²	—	2	4	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	2	2	no	—
128	C₄²	C₁₆×C₅₃	1	4	yes	complemented
129	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
130	C₄²	M₁₆×C₅₃	1	2	yes	—
131	C₁₆	—	4	4	no	—
132	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
139	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
140	C₂×C₄	C₂×C₁₆×C₅₃	1	4	yes	—
141	C₂×C₄	C₂×C₁₆×C₅₃	1	2	yes	—
142	C₂×C₄	M₃₂×C₅₃	1	4	yes	—
143	C₂×C₄	M₃₂×C₅₃	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
145	C₂×C₄	#117×C₅₃	1	2	yes	—
146	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
147	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	2	4	no	—
150	C₈	#124×C₅₃	1	2	yes	—
151	C₈	#125×C₅₃	1	2	yes	—
152	C₂²	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
153	C₂²	C₅₃×#431	1	2	yes	—
154	C₂²	C₂²×C₁₆×C₅₃	1	1	yes	characteristic
155	C₂²	C₂×M₃₂×C₅₃	1	1	yes	characteristic
156	C₂²	C₂×M₃₂×C₅₃	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	4	no	—
159	C₄	—	2	4	no	—
160	C₄	C₅₃×C₄⋊C₁₆	1	4	yes	—
161	C₄	C₅₃×#382	1	1	yes	characteristic
162	C₄	C₅₃×#423	1	1	yes	characteristic
163	C₄	C₅₃×#432	1	2	yes	—
164	C₂	C₅₃×#1831	1	1	yes	characteristic
165	C₂	C₂×C₅₃×C₄⋊C₁₆	1	1	yes	characteristic
166	C₂	C₅₃×#2150	1	1	yes	characteristic
167	C₂	C₅₃×#2162	1	2	yes	—
168	C₂	C₅₃×#2163	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₅₃	C₂³
commutator	C₂²	C₂²×C₁₆×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
43. C₄²×C₅₃	C₁₆	semidirect
67. #12879	C₅₃	direct (N × H)
95. C₅₃	#12879	direct (N × H)
128. C₄²	C₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×C₄⋊C₁₆	2
C₄²×C₈×C₅₃	2
C₅₃×#1374	2
#12879	53

Subgroups — C53×#12879 #27220391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12879 #27220391