Subgroups — C₂²×C₅₃×#498 #27268164

Order 13568.

2398 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₅₃	C₂	1	2	yes	maximal
2	C₂×C₅₃×#498	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#106×C₅₃	C₂	1	4	yes	maximal
4	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₅₃	—	2	2	no	—
6	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
7	C₂×C₄⋊Q₈×C₅₃	—	2	12	no	—
8	C₂×C₄⋊Q₈×C₅₃	C₂²	1	24	yes	—
9	C₂²×C₄⋊C₄×C₅₃	—	2	2	no	—
10	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#498	C₂²	1	64	yes	direct product factor
13	C₂×#106×C₅₃	C₂²	1	48	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	12	yes	—
15	C₂²×Q₈×C₅₃	—	2	24	no	—
16	C₂²×Q₈×C₅₃	—	4	16	no	—
17	C₂³×C₄×C₅₃	—	2	4	no	—
18	C₂³×C₄×C₅₃	D₄	1	2	yes	—
19	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	Q₈	1	16	yes	complemented
21	C₂²×C₈×C₅₃	C₂³	1	12	yes	—
22	C₄⋊Q₈×C₅₃	—	2	48	no	—
23	C₄⋊Q₈×C₅₃	C₂³	1	32	yes	—
24	C₂×C₄⋊C₄×C₅₃	—	2	24	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	12	no	—
26	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
27	C₂×C₄²×C₅₃	D₄	1	6	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
29	#106×C₅₃	C₂³	1	64	yes	—
30	C₄×C₈×C₅₃	C₂³	1	16	yes	—
31	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×Q₈×C₅₃	—	2	32	no	—
33	C₂×Q₈×C₅₃	—	4	96	no	—
34	C₂²×C₄×C₅₃	—	2	24	no	—
35	C₂²×C₄×C₅₃	—	4	16	no	—
36	C₂²×C₄×C₅₃	SD₁₆	1	16	yes	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
39	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
40	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
41	C₂×C₈×C₅₃	C₂×Q₈	1	48	yes	complemented
42	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
43	C₄⋊C₄×C₅₃	—	2	32	no	—
44	C₄⋊C₄×C₅₃	—	2	24	no	—
45	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
46	C₄²×C₅₃	C₂×D₄	1	12	yes	—
47	C₄²×C₅₃	C₂⁴	1	4	yes	—
48	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
49	C₂³×C₅₃	C₂×SD₁₆	1	8	yes	—
50	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
51	Q₈×C₅₃	—	4	64	no	—
52	C₂×C₄×C₅₃	—	2	16	no	—
53	C₂×C₄×C₅₃	—	4	48	no	—
54	C₂×C₄×C₅₃	C₄⋊Q₈	1	12	yes	—
55	C₂×C₄×C₅₃	C₂×SD₁₆	1	48	yes	—
56	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
57	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
58	C₂×C₄×C₅₃	C₂²×Q₈	1	12	yes	—
59	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
60	C₈×C₅₃	C₂²×Q₈	1	16	yes	complemented
61	C₂²×#498	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂²×C₅₃	#498	1	16	yes	direct product factor
63	C₂²×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
64	C₂²×C₅₃	C₂²×SD₁₆	1	12	yes	—
65	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
66	C₄×C₅₃	—	4	16	no	—
67	C₄×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
68	C₄×C₅₃	C₂²×SD₁₆	1	16	yes	—
69	C₄×C₅₃	C₂³×Q₈	1	2	yes	—
70	C₂²×C₄⋊Q₈	C₂×C₅₃	1	2	yes	—
71	C₂×#498	C₂×C₅₃	1	24	yes	direct product factor
72	C₂²×#106	C₂×C₅₃	1	4	yes	—
73	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
74	C₂×C₅₃	C₂×#498	1	12	yes	direct product factor
75	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₅₃	C₂³×SD₁₆	1	2	yes	—
77	C₂³×Q₈	—	2	2	no	—
78	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
79	C₂×C₄⋊Q₈	—	2	12	no	—
80	C₂×C₄⋊Q₈	C₂²×C₅₃	1	24	yes	—
81	C₂²×C₄⋊C₄	—	2	2	no	—
82	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
83	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
84	#498	C₂²×C₅₃	1	64	yes	direct product factor
85	C₂×#106	C₂²×C₅₃	1	48	yes	—
86	C₂×C₄×C₈	C₂²×C₅₃	1	12	yes	—
87	C₅₃	C₂²×#498	1	1	yes	characteristic, direct product factor, Sylow
88	C₂²×Q₈	—	2	24	no	—
89	C₂²×Q₈	—	4	16	no	—
90	C₂³×C₄	—	2	4	no	—
91	C₂³×C₄	D₄×C₅₃	1	2	yes	—
92	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
93	C₂²×C₈	Q₈×C₅₃	1	16	yes	complemented
94	C₂²×C₈	C₂³×C₅₃	1	12	yes	—
95	C₄⋊Q₈	—	2	48	no	—
96	C₄⋊Q₈	C₂³×C₅₃	1	32	yes	—
97	C₂×C₄⋊C₄	—	2	24	no	—
98	C₂×C₄⋊C₄	—	2	12	no	—
99	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
100	C₂×C₄²	D₄×C₅₃	1	6	yes	—
101	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
102	#106	C₂³×C₅₃	1	64	yes	—
103	C₄×C₈	C₂³×C₅₃	1	16	yes	—
104	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
105	C₂×Q₈	—	2	32	no	—
106	C₂×Q₈	—	4	96	no	—
107	C₂²×C₄	—	2	24	no	—
108	C₂²×C₄	—	4	16	no	—
109	C₂²×C₄	SD₁₆×C₅₃	1	16	yes	—
110	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
111	C₂²×C₄	C₂×D₄×C₅₃	1	12	yes	—
112	C₂²×C₄	C₂×Q₈×C₅₃	1	8	yes	—
113	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
114	C₂×C₈	C₂×Q₈×C₅₃	1	48	yes	complemented
115	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
116	C₄⋊C₄	—	2	32	no	—
117	C₄⋊C₄	—	2	24	no	—
118	C₄⋊C₄	C₂⁴×C₅₃	1	16	yes	—
119	C₄²	C₂×D₄×C₅₃	1	12	yes	—
120	C₄²	C₂⁴×C₅₃	1	4	yes	—
121	C₂³	C₄⋊Q₈×C₅₃	1	4	yes	—
122	C₂³	C₂×SD₁₆×C₅₃	1	8	yes	—
123	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
124	Q₈	—	4	64	no	—
125	C₂×C₄	—	2	16	no	—
126	C₂×C₄	—	4	48	no	—
127	C₂×C₄	C₄⋊Q₈×C₅₃	1	12	yes	—
128	C₂×C₄	C₂×SD₁₆×C₅₃	1	48	yes	—
129	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
130	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
131	C₂×C₄	C₂²×Q₈×C₅₃	1	12	yes	—
132	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
133	C₈	C₂²×Q₈×C₅₃	1	16	yes	complemented
134	C₂²	C₅₃×#498	1	16	yes	direct product factor
135	C₂²	C₂×C₄⋊Q₈×C₅₃	1	6	yes	—
136	C₂²	C₂²×SD₁₆×C₅₃	1	12	yes	—
137	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
138	C₄	—	4	16	no	—
139	C₄	C₂×C₄⋊Q₈×C₅₃	1	6	yes	—
140	C₄	C₂²×SD₁₆×C₅₃	1	16	yes	—
141	C₄	C₂³×Q₈×C₅₃	1	2	yes	—
142	C₂	C₂×C₅₃×#498	1	12	yes	direct product factor
143	C₂	C₂²×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
144	C₂	C₂³×SD₁₆×C₅₃	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#498	C₂	direct (N × H)
12. C₅₃×#498	C₂²	direct (N × H)
20. C₂²×C₈×C₅₃	Q₈	semidirect
41. C₂×C₈×C₅₃	C₂×Q₈	semidirect
60. C₈×C₅₃	C₂²×Q₈	semidirect
61. C₂²×#498	C₅₃	direct (N × H)
62. C₂²×C₅₃	#498	direct (N × H)
71. C₂×#498	C₂×C₅₃	direct (N × H)
74. C₂×C₅₃	C₂×#498	direct (N × H)
84. #498	C₂²×C₅₃	direct (N × H)
87. C₅₃	C₂²×#498	direct (N × H)
93. C₂²×C₈	Q₈×C₅₃	semidirect
114. C₂×C₈	C₂×Q₈×C₅₃	semidirect
133. C₈	C₂²×Q₈×C₅₃	semidirect
134. C₂²	C₅₃×#498	direct (N × H)
142. C₂	C₂×C₅₃×#498	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⋊Q₈×C₅₃	2
C₂×C₅₃×#498	2
C₂²×#106×C₅₃	2
C₂²×C₄×C₈×C₅₃	2
C₂²×#498	53

Subgroups — C22×C53×#498 #27268164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#498 #27268164