Subgroups — C₂²×C₅₃×#494 #27268142

Order 13568.

3262 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₅₃	C₂	1	4	yes	maximal
2	C₂²×C₄⋊Q₈×C₅₃	C₂	1	2	yes	maximal
3	C₂×C₅₃×#494	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₅₃	C₂²	1	4	yes	—
6	C₂²×Q₁₆×C₅₃	—	2	32	no	—
7	C₂²×Q₁₆×C₅₃	C₂²	1	48	yes	—
8	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₅₃	—	2	12	no	—
10	C₂×C₄⋊Q₈×C₅₃	C₂²	1	24	yes	—
11	C₂²×C₄⋊C₄×C₅₃	—	2	4	no	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#494	C₂²	1	64	yes	direct product factor
14	C₂×C₄×C₈×C₅₃	C₂²	1	12	yes	—
15	C₂²×Q₈×C₅₃	—	2	12	no	—
16	C₂²×Q₈×C₅₃	—	2	32	no	—
17	C₂²×Q₈×C₅₃	C₂³	1	24	yes	—
18	C₂³×C₄×C₅₃	—	2	4	no	—
19	C₂³×C₄×C₅₃	D₄	1	2	yes	—
20	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×Q₁₆×C₅₃	—	2	192	no	—
22	C₂×Q₁₆×C₅₃	C₂³	1	64	yes	—
23	C₂²×C₈×C₅₃	D₄	1	16	yes	—
24	C₂²×C₈×C₅₃	C₂³	1	12	yes	—
25	C₄⋊Q₈×C₅₃	—	2	48	no	—
26	C₄⋊Q₈×C₅₃	C₂³	1	32	yes	—
27	C₂×C₄⋊C₄×C₅₃	—	2	48	no	—
28	C₂×C₄²×C₅₃	D₄	1	6	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
30	C₄×C₈×C₅₃	C₂³	1	16	yes	—
31	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×Q₈×C₅₃	—	2	24	no	—
33	C₂×Q₈×C₅₃	—	2	96	no	—
34	C₂×Q₈×C₅₃	—	4	48	no	—
35	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	—
36	C₂²×C₄×C₅₃	—	2	24	no	—
37	C₂²×C₄×C₅₃	—	4	16	no	—
38	C₂²×C₄×C₅₃	Q₁₆	1	16	yes	complemented
39	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
42	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
43	Q₁₆×C₅₃	—	2	128	no	—
44	C₂×C₈×C₅₃	C₂×D₄	1	48	yes	—
45	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
46	C₄⋊C₄×C₅₃	—	2	64	no	—
47	C₄²×C₅₃	C₂×D₄	1	12	yes	—
48	C₄²×C₅₃	C₂⁴	1	4	yes	—
49	C₂³×C₅₃	#127	1	4	yes	—
50	C₂³×C₅₃	C₂×Q₁₆	1	8	yes	—
51	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
52	Q₈×C₅₃	—	2	32	no	—
53	Q₈×C₅₃	—	4	48	no	—
54	C₂×C₄×C₅₃	—	2	16	no	—
55	C₂×C₄×C₅₃	—	4	48	no	—
56	C₂×C₄×C₅₃	#127	1	12	yes	—
57	C₂×C₄×C₅₃	C₂×Q₁₆	1	48	yes	complemented
58	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
59	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
60	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
61	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
62	C₈×C₅₃	C₂²×D₄	1	16	yes	—
63	C₂²×#494	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂²×C₅₃	#494	1	16	yes	direct product factor
65	C₂²×C₅₃	C₂×#127	1	6	yes	—
66	C₂²×C₅₃	C₂²×Q₁₆	1	12	yes	—
67	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
68	C₄×C₅₃	—	4	16	no	—
69	C₄×C₅₃	C₂×#127	1	6	yes	—
70	C₄×C₅₃	C₂²×Q₁₆	1	16	yes	complemented
71	C₄×C₅₃	C₂³×D₄	1	2	yes	—
72	C₂³×Q₁₆	C₂×C₅₃	1	4	yes	—
73	C₂²×C₄⋊Q₈	C₂×C₅₃	1	2	yes	—
74	C₂×#494	C₂×C₅₃	1	24	yes	direct product factor
75	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
76	C₂×C₅₃	C₂×#494	1	12	yes	direct product factor
77	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
78	C₂×C₅₃	C₂³×Q₁₆	1	2	yes	—
79	C₂³×Q₈	C₂²×C₅₃	1	4	yes	—
80	C₂²×Q₁₆	—	2	32	no	—
81	C₂²×Q₁₆	C₂²×C₅₃	1	48	yes	—
82	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
83	C₂×C₄⋊Q₈	—	2	12	no	—
84	C₂×C₄⋊Q₈	C₂²×C₅₃	1	24	yes	—
85	C₂²×C₄⋊C₄	—	2	4	no	—
86	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
87	#494	C₂²×C₅₃	1	64	yes	direct product factor
88	C₂×C₄×C₈	C₂²×C₅₃	1	12	yes	—
89	C₅₃	C₂²×#494	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²×Q₈	—	2	12	no	—
91	C₂²×Q₈	—	2	32	no	—
92	C₂²×Q₈	C₂³×C₅₃	1	24	yes	—
93	C₂³×C₄	—	2	4	no	—
94	C₂³×C₄	D₄×C₅₃	1	2	yes	—
95	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
96	C₂×Q₁₆	—	2	192	no	—
97	C₂×Q₁₆	C₂³×C₅₃	1	64	yes	—
98	C₂²×C₈	D₄×C₅₃	1	16	yes	—
99	C₂²×C₈	C₂³×C₅₃	1	12	yes	—
100	C₄⋊Q₈	—	2	48	no	—
101	C₄⋊Q₈	C₂³×C₅₃	1	32	yes	—
102	C₂×C₄⋊C₄	—	2	48	no	—
103	C₂×C₄²	D₄×C₅₃	1	6	yes	—
104	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
105	C₄×C₈	C₂³×C₅₃	1	16	yes	—
106	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
107	C₂×Q₈	—	2	24	no	—
108	C₂×Q₈	—	2	96	no	—
109	C₂×Q₈	—	4	48	no	—
110	C₂×Q₈	C₂⁴×C₅₃	1	16	yes	—
111	C₂²×C₄	—	2	24	no	—
112	C₂²×C₄	—	4	16	no	—
113	C₂²×C₄	Q₁₆×C₅₃	1	16	yes	complemented
114	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
115	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
116	C₂²×C₄	C₂×D₄×C₅₃	1	12	yes	—
117	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
118	Q₁₆	—	2	128	no	—
119	C₂×C₈	C₂×D₄×C₅₃	1	48	yes	—
120	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
121	C₄⋊C₄	—	2	64	no	—
122	C₄²	C₂×D₄×C₅₃	1	12	yes	—
123	C₄²	C₂⁴×C₅₃	1	4	yes	—
124	C₂³	#127×C₅₃	1	4	yes	—
125	C₂³	C₂×Q₁₆×C₅₃	1	8	yes	—
126	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
127	Q₈	—	2	32	no	—
128	Q₈	—	4	48	no	—
129	C₂×C₄	—	2	16	no	—
130	C₂×C₄	—	4	48	no	—
131	C₂×C₄	#127×C₅₃	1	12	yes	—
132	C₂×C₄	C₂×Q₁₆×C₅₃	1	48	yes	complemented
133	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
134	C₂×C₄	C₂²×D₄×C₅₃	1	12	yes	—
135	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
136	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
137	C₈	C₂²×D₄×C₅₃	1	16	yes	—
138	C₂²	C₅₃×#494	1	16	yes	direct product factor
139	C₂²	C₂×#127×C₅₃	1	6	yes	—
140	C₂²	C₂²×Q₁₆×C₅₃	1	12	yes	—
141	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
142	C₄	—	4	16	no	—
143	C₄	C₂×#127×C₅₃	1	6	yes	—
144	C₄	C₂²×Q₁₆×C₅₃	1	16	yes	complemented
145	C₄	C₂³×D₄×C₅₃	1	2	yes	—
146	C₂	C₂×C₅₃×#494	1	12	yes	direct product factor
147	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
148	C₂	C₂³×Q₁₆×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
3. C₂×C₅₃×#494	C₂	direct (N × H)
13. C₅₃×#494	C₂²	direct (N × H)
38. C₂²×C₄×C₅₃	Q₁₆	semidirect
57. C₂×C₄×C₅₃	C₂×Q₁₆	semidirect
63. C₂²×#494	C₅₃	direct (N × H)
64. C₂²×C₅₃	#494	direct (N × H)
70. C₄×C₅₃	C₂²×Q₁₆	semidirect
74. C₂×#494	C₂×C₅₃	direct (N × H)
76. C₂×C₅₃	C₂×#494	direct (N × H)
87. #494	C₂²×C₅₃	direct (N × H)
89. C₅₃	C₂²×#494	direct (N × H)
113. C₂²×C₄	Q₁₆×C₅₃	semidirect
132. C₂×C₄	C₂×Q₁₆×C₅₃	semidirect
138. C₂²	C₅₃×#494	direct (N × H)
144. C₄	C₂²×Q₁₆×C₅₃	semidirect
146. C₂	C₂×C₅₃×#494	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₅₃	2
C₂²×C₄⋊Q₈×C₅₃	2
C₂×C₅₃×#494	2
C₂²×C₄×C₈×C₅₃	2
C₂²×#494	53

Subgroups — C22×C53×#494 #27268142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#494 #27268142