Subgroups — C₂³×#107×C₅₃ #27267614

Order 13568.

3614 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	C₂⁴×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal
3	C₂²×#107×C₅₃	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₄×C₅₃	—	2	2	no	—
5	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₅₃	C₄	1	16	yes	complemented
7	C₂³×C₈×C₅₃	C₂²	1	14	yes	—
8	C₂²×C₄⋊C₄×C₅₃	—	2	14	no	—
9	C₂²×C₄⋊C₄×C₅₃	C₂²	1	28	yes	—
10	C₂×#107×C₅₃	C₂²	1	112	yes	direct product factor
11	C₂⁵×C₅₃	D₄	1	1	yes	characteristic
12	C₂³×C₄×C₅₃	—	2	28	no	—
13	C₂³×C₄×C₅₃	—	4	16	no	—
14	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
15	C₂³×C₄×C₅₃	D₄	1	7	yes	—
16	C₂³×C₄×C₅₃	Q₈	1	8	yes	—
17	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	112	yes	complemented
19	C₂²×C₈×C₅₃	C₂³	1	28	yes	—
20	C₂×C₄⋊C₄×C₅₃	—	2	84	no	—
21	C₂×C₄⋊C₄×C₅₃	C₂³	1	56	yes	—
22	#107×C₅₃	C₂³	1	64	yes	direct product factor
23	C₂⁴×C₅₃	C₄⋊C₄	1	8	yes	—
24	C₂⁴×C₅₃	D₈	1	8	yes	—
25	C₂⁴×C₅₃	Q₁₆	1	8	yes	—
26	C₂⁴×C₅₃	C₂×D₄	1	7	yes	—
27	C₂²×C₄×C₅₃	—	2	56	no	—
28	C₂²×C₄×C₅₃	—	4	112	no	—
29	C₂²×C₄×C₅₃	C₄⋊C₄	1	56	yes	—
30	C₂²×C₄×C₅₃	C₂²×C₄	1	28	yes	—
31	C₂²×C₄×C₅₃	C₂×D₄	1	21	yes	—
32	C₂²×C₄×C₅₃	C₂×Q₈	1	28	yes	—
33	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
34	C₂×C₈×C₅₃	C₂²×C₄	1	112	yes	complemented
35	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
36	C₄⋊C₄×C₅₃	—	2	56	no	—
37	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
38	C₂³×C₅₃	#107	1	64	yes	direct product factor
39	C₂³×C₅₃	C₂×C₄⋊C₄	1	28	yes	—
40	C₂³×C₅₃	C₂×D₈	1	28	yes	—
41	C₂³×C₅₃	C₂×Q₁₆	1	28	yes	—
42	C₂³×C₅₃	C₂²×D₄	1	7	yes	—
43	C₂×C₄×C₅₃	—	2	16	no	—
44	C₂×C₄×C₅₃	—	4	112	no	—
45	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	84	yes	—
46	C₂×C₄×C₅₃	C₂³×C₄	1	14	yes	—
47	C₂×C₄×C₅₃	C₂²×D₄	1	7	yes	—
48	C₂×C₄×C₅₃	C₂²×Q₈	1	14	yes	—
49	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
50	C₈×C₅₃	C₂³×C₄	1	16	yes	complemented
51	C₂³×#107	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×C₅₃	C₂×#107	1	112	yes	direct product factor
53	C₂²×C₅₃	C₂²×C₄⋊C₄	1	14	yes	—
54	C₂²×C₅₃	C₂²×D₈	1	14	yes	—
55	C₂²×C₅₃	C₂²×Q₁₆	1	14	yes	—
56	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
57	C₄×C₅₃	—	4	16	no	—
58	C₄×C₅₃	C₂²×C₄⋊C₄	1	14	yes	—
59	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
60	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
61	C₂⁴×C₈	C₂×C₅₃	1	1	yes	characteristic
62	C₂³×C₄⋊C₄	C₂×C₅₃	1	2	yes	—
63	C₂²×#107	C₂×C₅₃	1	28	yes	direct product factor
64	C₂×C₅₃	C₂²×#107	1	28	yes	direct product factor
65	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₅₃	C₂³×D₈	1	1	yes	characteristic
67	C₂×C₅₃	C₂³×Q₁₆	1	1	yes	characteristic
68	C₂⁴×C₄	—	2	2	no	—
69	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
70	C₂³×C₈	C₄×C₅₃	1	16	yes	complemented
71	C₂³×C₈	C₂²×C₅₃	1	14	yes	—
72	C₂²×C₄⋊C₄	—	2	14	no	—
73	C₂²×C₄⋊C₄	C₂²×C₅₃	1	28	yes	—
74	C₂×#107	C₂²×C₅₃	1	112	yes	direct product factor
75	C₅₃	C₂³×#107	1	1	yes	characteristic, direct product factor, Sylow
76	C₂⁵	D₄×C₅₃	1	1	yes	characteristic
77	C₂³×C₄	—	2	28	no	—
78	C₂³×C₄	—	4	16	no	—
79	C₂³×C₄	C₂×C₄×C₅₃	1	8	yes	—
80	C₂³×C₄	D₄×C₅₃	1	7	yes	—
81	C₂³×C₄	Q₈×C₅₃	1	8	yes	—
82	C₂³×C₄	C₂³×C₅₃	1	7	yes	—
83	C₂²×C₈	C₂×C₄×C₅₃	1	112	yes	complemented
84	C₂²×C₈	C₂³×C₅₃	1	28	yes	—
85	C₂×C₄⋊C₄	—	2	84	no	—
86	C₂×C₄⋊C₄	C₂³×C₅₃	1	56	yes	—
87	#107	C₂³×C₅₃	1	64	yes	direct product factor
88	C₂⁴	C₄⋊C₄×C₅₃	1	8	yes	—
89	C₂⁴	D₈×C₅₃	1	8	yes	—
90	C₂⁴	Q₁₆×C₅₃	1	8	yes	—
91	C₂⁴	C₂×D₄×C₅₃	1	7	yes	—
92	C₂²×C₄	—	2	56	no	—
93	C₂²×C₄	—	4	112	no	—
94	C₂²×C₄	C₄⋊C₄×C₅₃	1	56	yes	—
95	C₂²×C₄	C₂²×C₄×C₅₃	1	28	yes	—
96	C₂²×C₄	C₂×D₄×C₅₃	1	21	yes	—
97	C₂²×C₄	C₂×Q₈×C₅₃	1	28	yes	—
98	C₂²×C₄	C₂⁴×C₅₃	1	7	yes	—
99	C₂×C₈	C₂²×C₄×C₅₃	1	112	yes	complemented
100	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
101	C₄⋊C₄	—	2	56	no	—
102	C₄⋊C₄	C₂⁴×C₅₃	1	16	yes	—
103	C₂³	#107×C₅₃	1	64	yes	direct product factor
104	C₂³	C₂×C₄⋊C₄×C₅₃	1	28	yes	—
105	C₂³	C₂×D₈×C₅₃	1	28	yes	—
106	C₂³	C₂×Q₁₆×C₅₃	1	28	yes	—
107	C₂³	C₂²×D₄×C₅₃	1	7	yes	—
108	C₂×C₄	—	2	16	no	—
109	C₂×C₄	—	4	112	no	—
110	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	84	yes	—
111	C₂×C₄	C₂³×C₄×C₅₃	1	14	yes	—
112	C₂×C₄	C₂²×D₄×C₅₃	1	7	yes	—
113	C₂×C₄	C₂²×Q₈×C₅₃	1	14	yes	—
114	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
115	C₈	C₂³×C₄×C₅₃	1	16	yes	complemented
116	C₂²	C₂×#107×C₅₃	1	112	yes	direct product factor
117	C₂²	C₂²×C₄⋊C₄×C₅₃	1	14	yes	—
118	C₂²	C₂²×D₈×C₅₃	1	14	yes	—
119	C₂²	C₂²×Q₁₆×C₅₃	1	14	yes	—
120	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
121	C₄	—	4	16	no	—
122	C₄	C₂²×C₄⋊C₄×C₅₃	1	14	yes	—
123	C₄	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
124	C₄	C₂³×Q₈×C₅₃	1	1	yes	characteristic
125	C₂	C₂²×#107×C₅₃	1	28	yes	direct product factor
126	C₂	C₂³×C₄⋊C₄×C₅₃	1	1	yes	characteristic
127	C₂	C₂³×D₈×C₅₃	1	1	yes	characteristic
128	C₂	C₂³×Q₁₆×C₅₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂²×#107×C₅₃	C₂	direct (N × H)
6. C₂³×C₈×C₅₃	C₄	semidirect
10. C₂×#107×C₅₃	C₂²	direct (N × H)
18. C₂²×C₈×C₅₃	C₂×C₄	semidirect
22. #107×C₅₃	C₂³	direct (N × H)
34. C₂×C₈×C₅₃	C₂²×C₄	semidirect
38. C₂³×C₅₃	#107	direct (N × H)
50. C₈×C₅₃	C₂³×C₄	semidirect
51. C₂³×#107	C₅₃	direct (N × H)
52. C₂²×C₅₃	C₂×#107	direct (N × H)
63. C₂²×#107	C₂×C₅₃	direct (N × H)
64. C₂×C₅₃	C₂²×#107	direct (N × H)
70. C₂³×C₈	C₄×C₅₃	semidirect
74. C₂×#107	C₂²×C₅₃	direct (N × H)
75. C₅₃	C₂³×#107	direct (N × H)
83. C₂²×C₈	C₂×C₄×C₅₃	semidirect
87. #107	C₂³×C₅₃	direct (N × H)
99. C₂×C₈	C₂²×C₄×C₅₃	semidirect
103. C₂³	#107×C₅₃	direct (N × H)
115. C₈	C₂³×C₄×C₅₃	semidirect
116. C₂²	C₂×#107×C₅₃	direct (N × H)
125. C₂	C₂²×#107×C₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₅₃	2
C₂³×C₄⋊C₄×C₅₃	2
C₂²×#107×C₅₃	2
C₂³×#107	53

Subgroups — C23×#107×C53 #27267614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#107×C₅₃ #27267614