Subgroups — C₂×C₅₃×C₄⋊C₃₂ #27221138

Order 13568.

230 conjugacy classes of subgroups.

The subgroup lattice has 130 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₂×C₅₃	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×C₄⋊C₃₂	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₅₃	—	2	4	no	—
8	C₂×C₃₂×C₅₃	C₂²	1	4	yes	—
9	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
10	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
12	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	—	2	2	no	—
15	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
16	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
18	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
19	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₅₃	C₈	1	2	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₃₂×C₅₃	—	2	4	no	—
23	C₂²×C₄×C₅₃	C₁₆	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₁₆	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
26	C₂×C₈×C₅₃	—	2	2	no	—
27	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
28	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
31	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₄²×C₅₃	C₁₆	1	2	yes	—
33	C₄²×C₅₃	C₂×C₈	1	2	yes	—
34	C₁₆×C₅₃	—	2	2	no	—
35	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	C₃₂	1	4	yes	complemented
41	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
42	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
45	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
46	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
47	C₈×C₅₃	—	2	2	no	—
48	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
49	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
51	C₂×C₄⋊C₃₂	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
53	C₂²×C₅₃	C₂×C₃₂	1	2	yes	—
54	C₂²×C₅₃	M₆₄	1	2	yes	—
55	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
58	C₄×C₅₃	C₂×C₃₂	1	4	yes	complemented
59	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
60	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
61	C₂²×C₃₂	C₂×C₅₃	1	2	yes	—
62	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic
63	C₄⋊C₃₂	C₂×C₅₃	1	4	yes	direct product factor
64	C₂×C₅₃	C₄⋊C₃₂	1	4	yes	direct product factor
65	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
66	C₂×C₅₃	C₂²×C₃₂	1	1	yes	characteristic
67	C₂×C₅₃	C₂×M₆₄	1	1	yes	characteristic
68	C₂²×C₁₆	C₄×C₅₃	1	1	yes	characteristic
69	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
70	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
71	C₂×C₃₂	—	2	4	no	—
72	C₂×C₃₂	C₂²×C₅₃	1	4	yes	—
73	C₄×C₁₆	C₄×C₅₃	1	2	yes	—
74	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
75	C₅₃	C₂×C₄⋊C₃₂	1	1	yes	characteristic, direct product factor, Sylow
76	C₂²×C₈	C₈×C₅₃	1	1	yes	characteristic
77	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
78	C₂×C₄²	C₈×C₅₃	1	1	yes	characteristic
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
81	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
82	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
83	C₂×C₁₆	Q₈×C₅₃	1	2	yes	—
84	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
85	C₄×C₈	C₈×C₅₃	1	2	yes	—
86	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
87	C₃₂	—	2	4	no	—
88	C₂²×C₄	C₁₆×C₅₃	1	1	yes	characteristic
89	C₂²×C₄	C₁₆×C₅₃	1	1	yes	characteristic
90	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
93	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
94	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
95	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
96	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
97	C₄²	C₁₆×C₅₃	1	2	yes	—
98	C₄²	C₂×C₈×C₅₃	1	2	yes	—
99	C₁₆	—	2	2	no	—
100	C₁₆	C₄⋊C₄×C₅₃	1	2	yes	—
101	C₁₆	C₂×D₄×C₅₃	1	1	yes	characteristic
102	C₁₆	C₂×Q₈×C₅₃	1	1	yes	characteristic
103	C₂³	C₂×C₁₆×C₅₃	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₃₂×C₅₃	1	4	yes	complemented
106	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
107	C₂×C₄	C₂×C₁₆×C₅₃	1	2	yes	—
108	C₂×C₄	C₂×C₁₆×C₅₃	1	1	yes	characteristic
109	C₂×C₄	C₂×C₁₆×C₅₃	1	2	yes	—
110	C₂×C₄	M₃₂×C₅₃	1	2	yes	—
111	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	C₄⋊C₈×C₅₃	1	2	yes	—
114	C₈	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
115	C₈	C₂×M₁₆×C₅₃	1	1	yes	characteristic
116	C₂²	C₅₃×C₄⋊C₁₆	1	2	yes	—
117	C₂²	C₂×C₃₂×C₅₃	1	2	yes	—
118	C₂²	C₅₃×M₆₄	1	2	yes	—
119	C₂²	C₂²×C₁₆×C₅₃	1	1	yes	characteristic
120	C₄	—	2	2	no	—
121	C₄	C₅₃×C₄⋊C₁₆	1	2	yes	—
122	C₄	C₂×C₃₂×C₅₃	1	4	yes	complemented
123	C₄	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
124	C₄	C₂×M₃₂×C₅₃	1	1	yes	characteristic
125	C₂	C₅₃×C₄⋊C₃₂	1	4	yes	direct product factor
126	C₂	C₂×C₅₃×C₄⋊C₁₆	1	1	yes	characteristic
127	C₂	C₂²×C₃₂×C₅₃	1	1	yes	characteristic
128	C₂	C₂×C₅₃×M₆₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆×C₅₃	C₂²
commutator	C₂	C₂²×C₃₂×C₅₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₂×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×C₄⋊C₃₂	C₂	direct (N × H)
40. C₂×C₄×C₅₃	C₃₂	semidirect
51. C₂×C₄⋊C₃₂	C₅₃	direct (N × H)
58. C₄×C₅₃	C₂×C₃₂	semidirect
63. C₄⋊C₃₂	C₂×C₅₃	direct (N × H)
64. C₂×C₅₃	C₄⋊C₃₂	direct (N × H)
75. C₅₃	C₂×C₄⋊C₃₂	direct (N × H)
105. C₂×C₄	C₃₂×C₅₃	semidirect
122. C₄	C₂×C₃₂×C₅₃	semidirect
125. C₂	C₅₃×C₄⋊C₃₂	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×C₃₂×C₅₃	2
C₂×C₄×C₁₆×C₅₃	2
C₅₃×C₄⋊C₃₂	2
C₂×C₄⋊C₃₂	53

Subgroups — C2×C53×C4⋊C32 #27221138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×C₄⋊C₃₂ #27221138