Subgroups — C₂×C₅₃×#557 #20272413

Order 6784.

1048 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#557	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₄⋊Q₈×C₅₃	C₂	1	3	yes	maximal
3	C₂×#125×C₅₃	C₂	1	6	yes	maximal
4	C₂×C₄×Q₈×C₅₃	C₂	1	2	yes	maximal
5	C₂×C₄×Q₈×C₅₃	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₅₃	C₂²	1	3	yes	—
8	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₄⋊Q₈×C₅₃	C₂²	1	24	yes	—
10	#125×C₅₃	C₂²	1	48	yes	—
11	C₄×Q₈×C₅₃	C₂²	1	24	yes	—
12	C₄×Q₈×C₅₃	C₂²	1	16	yes	—
13	C₄×Q₈×C₅₃	C₂²	1	8	yes	—
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
19	C₂×C₄²×C₅₃	C₂²	1	6	yes	—
20	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
21	C₂×Q₈×C₅₃	—	2	12	no	—
22	C₂×Q₈×C₅₃	Q₈	1	8	yes	complemented
23	C₂×Q₈×C₅₃	C₂³	1	12	yes	—
24	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
27	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
28	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
29	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
31	C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
32	C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
33	C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
34	C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
35	C₄²×C₅₃	C₂³	1	24	yes	—
36	C₄²×C₅₃	C₂³	1	12	yes	—
37	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
38	Q₈×C₅₃	—	2	24	no	—
39	Q₈×C₅₃	C₂×Q₈	1	16	yes	complemented
40	C₂×C₄×C₅₃	—	2	12	no	—
41	C₂×C₄×C₅₃	—	2	4	no	—
42	C₂×C₄×C₅₃	—	2	6	no	—
43	C₂×C₄×C₅₃	C₂×Q₈	1	12	yes	—
44	C₂×C₄×C₅₃	#41	1	4	yes	—
45	C₂×C₄×C₅₃	C₂⁴	1	12	yes	—
46	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
47	C₂×C₄×C₅₃	C₂⁴	1	6	yes	—
48	C₂×C₄×C₅₃	C₂⁴	1	6	yes	—
49	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
50	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
51	C₂²×C₅₃	C₂×#41	1	2	yes	—
52	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
53	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
54	C₄×C₅₃	—	2	12	no	—
55	C₄×C₅₃	—	2	4	no	—
56	C₄×C₅₃	—	2	6	no	—
57	C₄×C₅₃	C₂²×Q₈	1	12	yes	—
58	C₄×C₅₃	C₂×#41	1	4	yes	—
59	C₂×#557	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×C₅₃	#557	1	4	yes	direct product factor
61	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
62	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
63	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
64	#557	C₂×C₅₃	1	16	yes	direct product factor
65	C₂×C₄⋊Q₈	C₂×C₅₃	1	3	yes	—
66	C₂×#125	C₂×C₅₃	1	6	yes	—
67	C₂×C₄×Q₈	C₂×C₅₃	1	2	yes	—
68	C₂×C₄×Q₈	C₂×C₅₃	1	3	yes	—
69	C₂×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic
70	C₅₃	C₂×#557	1	1	yes	characteristic, direct product factor, Sylow
71	C₂²×Q₈	C₂²×C₅₃	1	3	yes	—
72	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
73	C₄⋊Q₈	C₂²×C₅₃	1	24	yes	—
74	#125	C₂²×C₅₃	1	48	yes	—
75	C₄×Q₈	C₂²×C₅₃	1	16	yes	—
76	C₄×Q₈	C₂²×C₅₃	1	24	yes	—
77	C₄×Q₈	C₂²×C₅₃	1	8	yes	—
78	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
79	C₂×C₄⋊C₄	C₂²×C₅₃	1	6	yes	—
80	C₂×C₄⋊C₄	C₂²×C₅₃	1	3	yes	—
81	C₂×C₄⋊C₄	C₂²×C₅₃	1	6	yes	—
82	C₂×C₄⋊C₄	C₂²×C₅₃	1	6	yes	—
83	C₂×C₄²	C₂²×C₅₃	1	6	yes	—
84	C₂×C₄²	C₂²×C₅₃	1	3	yes	—
85	C₂×Q₈	—	2	12	no	—
86	C₂×Q₈	Q₈×C₅₃	1	8	yes	complemented
87	C₂×Q₈	C₂³×C₅₃	1	12	yes	—
88	C₂×Q₈	C₂³×C₅₃	1	4	yes	—
89	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
90	C₂²×C₄	C₂³×C₅₃	1	3	yes	—
91	C₂²×C₄	C₂³×C₅₃	1	6	yes	—
92	C₂²×C₄	C₂³×C₅₃	1	3	yes	—
93	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
94	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
95	C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
96	C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
97	C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
98	C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
99	C₄²	C₂³×C₅₃	1	24	yes	—
100	C₄²	C₂³×C₅₃	1	12	yes	—
101	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
102	Q₈	—	2	24	no	—
103	Q₈	C₂×Q₈×C₅₃	1	16	yes	complemented
104	C₂×C₄	—	2	12	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	6	no	—
107	C₂×C₄	C₂×Q₈×C₅₃	1	12	yes	—
108	C₂×C₄	#41×C₅₃	1	4	yes	—
109	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
110	C₂×C₄	C₂⁴×C₅₃	1	6	yes	—
111	C₂×C₄	C₂⁴×C₅₃	1	12	yes	—
112	C₂×C₄	C₂⁴×C₅₃	1	6	yes	—
113	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
114	C₂²	C₂²×Q₈×C₅₃	1	2	yes	—
115	C₂²	C₂×#41×C₅₃	1	2	yes	—
116	C₂²	2₋¹⁺⁴×C₅₃	1	2	yes	—
117	C₂²	C₂⁵×C₅₃	1	1	yes	characteristic
118	C₄	—	2	12	no	—
119	C₄	—	2	4	no	—
120	C₄	—	2	6	no	—
121	C₄	C₂²×Q₈×C₅₃	1	12	yes	—
122	C₄	C₂×#41×C₅₃	1	4	yes	—
123	C₂	C₅₃×#557	1	4	yes	direct product factor
124	C₂	C₂³×Q₈×C₅₃	1	1	yes	characteristic
125	C₂	C₂²×#41×C₅₃	1	1	yes	characteristic
126	C₂	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₅₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#557	C₂	direct (N × H)
22. C₂×Q₈×C₅₃	Q₈	semidirect
39. Q₈×C₅₃	C₂×Q₈	semidirect
59. C₂×#557	C₅₃	direct (N × H)
60. C₂×C₅₃	#557	direct (N × H)
64. #557	C₂×C₅₃	direct (N × H)
70. C₅₃	C₂×#557	direct (N × H)
86. C₂×Q₈	Q₈×C₅₃	semidirect
103. Q₈	C₂×Q₈×C₅₃	semidirect
123. C₂	C₅₃×#557	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#557	2
C₂×C₄⋊Q₈×C₅₃	2
C₂×#125×C₅₃	2
C₂×C₄×Q₈×C₅₃	2
C₂×#557	53

Subgroups — C2×C53×#557 #20272413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#557 #20272413