Subgroups — C₂×C₅₃×#1842 #27228301

Order 13568.

1470 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₈×C₅₃	C₂	1	6	yes	maximal
2	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1842	C₂	1	8	yes	maximal, direct product factor
4	C₂³×C₈×C₅₃	C₂²	1	4	yes	—
5	C₂²×C₄²×C₅₃	C₄	1	3	yes	—
6	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
8	C₂×C₄⋊C₈×C₅₃	—	2	24	no	—
9	C₂×C₄⋊C₈×C₅₃	C₂²	1	24	yes	—
10	C₄³×C₅₃	C₄	1	4	yes	—
11	C₄³×C₅₃	C₂²	1	4	yes	—
12	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
13	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
14	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	—	2	24	no	—
16	C₂²×C₈×C₅₃	C₂³	1	8	yes	—
17	C₂×C₄²×C₅₃	—	2	12	no	—
18	C₂×C₄²×C₅₃	C₈	1	8	yes	complemented
19	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
20	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₅₃	D₄	1	12	yes	—
23	C₂×C₄²×C₅₃	Q₈	1	12	yes	—
24	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
25	C₄⋊C₈×C₅₃	—	2	48	no	—
26	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	—	2	12	no	—
28	C₂²×C₄×C₅₃	C₄⋊C₄	1	12	yes	—
29	C₂²×C₄×C₅₃	C₄⋊C₄	1	12	yes	—
30	C₂²×C₄×C₅₃	C₂×C₈	1	12	yes	—
31	C₂²×C₄×C₅₃	M₁₆	1	12	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
36	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
37	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
38	C₂×C₈×C₅₃	—	2	24	no	—
39	C₂×C₈×C₅₃	—	4	16	no	—
40	C₄²×C₅₃	—	2	24	no	—
41	C₄²×C₅₃	C₄⋊C₄	1	24	yes	—
42	C₄²×C₅₃	C₂×C₈	1	16	yes	complemented
43	C₄²×C₅₃	C₂×D₄	1	12	yes	—
44	C₄²×C₅₃	C₂×Q₈	1	12	yes	—
45	C₂³×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
46	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
47	C₂³×C₅₃	C₂×M₁₆	1	6	yes	—
48	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	—	2	12	no	—
50	C₂×C₄×C₅₃	—	2	24	no	—
51	C₂×C₄×C₅₃	C₄⋊C₈	1	48	yes	complemented
52	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
53	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
54	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
55	C₂×C₄×C₅₃	#127	1	4	yes	—
56	C₂×C₄×C₅₃	C₄⋊Q₈	1	12	yes	—
57	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
58	C₂×C₄×C₅₃	C₂×M₁₆	1	12	yes	—
59	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
60	C₂×C₄×C₅₃	C₂²×Q₈	1	3	yes	—
61	C₈×C₅₃	—	4	8	no	—
62	C₂×#1842	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂²×C₅₃	#384	1	4	yes	—
64	C₂²×C₅₃	C₂×C₄⋊C₈	1	12	yes	—
65	C₂²×C₅₃	#423	1	12	yes	—
66	C₂²×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
67	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
68	C₂²×C₅₃	C₂²×M₁₆	1	3	yes	—
69	C₄×C₅₃	—	2	12	no	—
70	C₄×C₅₃	#384	1	4	yes	—
71	C₄×C₅₃	C₂×C₄⋊C₈	1	24	yes	complemented
72	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
73	C₄×C₅₃	C₂×C₄⋊Q₈	1	3	yes	—
74	C₂²×C₄⋊C₈	C₂×C₅₃	1	6	yes	—
75	C₂×C₄³	C₂×C₅₃	1	1	yes	characteristic
76	#1842	C₂×C₅₃	1	8	yes	direct product factor
77	C₂×C₅₃	#1842	1	8	yes	direct product factor
78	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
79	C₂×C₅₃	C₂²×C₄⋊C₈	1	3	yes	—
80	C₂×C₅₃	C₂×#423	1	3	yes	—
81	C₂³×C₈	C₂²×C₅₃	1	4	yes	—
82	C₂²×C₄²	C₄×C₅₃	1	3	yes	—
83	C₂²×C₄²	C₄×C₅₃	1	1	yes	characteristic
84	C₂²×C₄²	C₂²×C₅₃	1	3	yes	—
85	C₂×C₄⋊C₈	—	2	24	no	—
86	C₂×C₄⋊C₈	C₂²×C₅₃	1	24	yes	—
87	C₄³	C₄×C₅₃	1	4	yes	—
88	C₄³	C₂²×C₅₃	1	4	yes	—
89	C₅₃	C₂×#1842	1	1	yes	characteristic, direct product factor, Sylow
90	C₂³×C₄	C₂×C₄×C₅₃	1	3	yes	—
91	C₂³×C₄	C₂×C₄×C₅₃	1	3	yes	—
92	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
93	C₂²×C₈	—	2	24	no	—
94	C₂²×C₈	C₂³×C₅₃	1	8	yes	—
95	C₂×C₄²	—	2	12	no	—
96	C₂×C₄²	C₈×C₅₃	1	8	yes	complemented
97	C₂×C₄²	C₂×C₄×C₅₃	1	12	yes	—
98	C₂×C₄²	C₂×C₄×C₅₃	1	6	yes	—
99	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
100	C₂×C₄²	D₄×C₅₃	1	12	yes	—
101	C₂×C₄²	Q₈×C₅₃	1	12	yes	—
102	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
103	C₄⋊C₈	—	2	48	no	—
104	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
105	C₂²×C₄	—	2	12	no	—
106	C₂²×C₄	C₄⋊C₄×C₅₃	1	12	yes	—
107	C₂²×C₄	C₄⋊C₄×C₅₃	1	12	yes	—
108	C₂²×C₄	C₂×C₈×C₅₃	1	12	yes	—
109	C₂²×C₄	M₁₆×C₅₃	1	12	yes	—
110	C₂²×C₄	C₂²×C₄×C₅₃	1	6	yes	—
111	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
112	C₂²×C₄	C₂²×C₄×C₅₃	1	6	yes	—
113	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
114	C₂²×C₄	C₂×Q₈×C₅₃	1	6	yes	—
115	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
116	C₂×C₈	—	2	24	no	—
117	C₂×C₈	—	4	16	no	—
118	C₄²	—	2	24	no	—
119	C₄²	C₄⋊C₄×C₅₃	1	24	yes	—
120	C₄²	C₂×C₈×C₅₃	1	16	yes	complemented
121	C₄²	C₂×D₄×C₅₃	1	12	yes	—
122	C₄²	C₂×Q₈×C₅₃	1	12	yes	—
123	C₂³	C₂×C₄⋊C₄×C₅₃	1	6	yes	—
124	C₂³	C₂²×C₈×C₅₃	1	2	yes	—
125	C₂³	C₂×M₁₆×C₅₃	1	6	yes	—
126	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
127	C₂×C₄	—	2	12	no	—
128	C₂×C₄	—	2	24	no	—
129	C₂×C₄	C₄⋊C₈×C₅₃	1	48	yes	complemented
130	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	12	yes	—
131	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	6	yes	—
132	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	12	yes	—
133	C₂×C₄	#127×C₅₃	1	4	yes	—
134	C₂×C₄	C₄⋊Q₈×C₅₃	1	12	yes	—
135	C₂×C₄	C₂²×C₈×C₅₃	1	12	yes	—
136	C₂×C₄	C₂×M₁₆×C₅₃	1	12	yes	—
137	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
138	C₂×C₄	C₂²×Q₈×C₅₃	1	3	yes	—
139	C₈	—	4	8	no	—
140	C₂²	C₅₃×#384	1	4	yes	—
141	C₂²	C₂×C₄⋊C₈×C₅₃	1	12	yes	—
142	C₂²	C₅₃×#423	1	12	yes	—
143	C₂²	C₂²×C₄⋊C₄×C₅₃	1	3	yes	—
144	C₂²	C₂³×C₈×C₅₃	1	1	yes	characteristic
145	C₂²	C₂²×M₁₆×C₅₃	1	3	yes	—
146	C₄	—	2	12	no	—
147	C₄	C₅₃×#384	1	4	yes	—
148	C₄	C₂×C₄⋊C₈×C₅₃	1	24	yes	complemented
149	C₄	C₂×#127×C₅₃	1	1	yes	characteristic
150	C₄	C₂×C₄⋊Q₈×C₅₃	1	3	yes	—
151	C₂	C₅₃×#1842	1	8	yes	direct product factor
152	C₂	C₂×C₅₃×#384	1	1	yes	characteristic
153	C₂	C₂²×C₄⋊C₈×C₅₃	1	3	yes	—
154	C₂	C₂×C₅₃×#423	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄×C₅₃	C₂³
commutator	C₂²	C₂³×C₈×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1842	C₂	direct (N × H)
18. C₂×C₄²×C₅₃	C₈	semidirect
42. C₄²×C₅₃	C₂×C₈	semidirect
51. C₂×C₄×C₅₃	C₄⋊C₈	semidirect
62. C₂×#1842	C₅₃	direct (N × H)
71. C₄×C₅₃	C₂×C₄⋊C₈	semidirect
76. #1842	C₂×C₅₃	direct (N × H)
77. C₂×C₅₃	#1842	direct (N × H)
89. C₅₃	C₂×#1842	direct (N × H)
96. C₂×C₄²	C₈×C₅₃	semidirect
120. C₄²	C₂×C₈×C₅₃	semidirect
129. C₂×C₄	C₄⋊C₈×C₅₃	semidirect
148. C₄	C₂×C₄⋊C₈×C₅₃	semidirect
151. C₂	C₅₃×#1842	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₈×C₅₃	2
C₂×C₄³×C₅₃	2
C₅₃×#1842	2
C₂×#1842	53

Subgroups — C2×C53×#1842 #27228301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1842 #27228301