Subgroups — C₅₃×#2961 #20271898

Order 6784.

348 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×#533	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#446	C₂	1	8	yes	maximal, complemented
3	C₅₃×#424	C₂	1	2	yes	maximal
4	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#405	C₂	1	2	yes	maximal
6	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈×C₅₃	C₄	1	2	yes	—
11	#125×C₅₃	C₄	1	2	yes	—
12	C₂²⋊Q₈×C₅₃	C₄	1	4	yes	—
13	C₄×Q₈×C₅₃	C₂²	1	4	yes	—
14	#117×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
17	C₄⋊C₈×C₅₃	C₂²	1	8	yes	—
18	#97×C₅₃	C₂²	1	4	yes	—
19	#97×C₅₃	C₂²	1	4	yes	—
20	C₄×C₈×C₅₃	C₂²	1	4	yes	—
21	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	M₁₆×C₅₃	—	2	4	no	—
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₂×C₈×C₅₃	Q₈	1	4	yes	complemented
27	C₂×C₈×C₅₃	C₂³	1	4	yes	—
28	C₂×C₈×C₅₃	C₂³	1	2	yes	—
29	C₂×C₈×C₅₃	C₂³	1	2	yes	—
30	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
31	C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
32	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
34	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₅₃	C₂³	1	4	yes	—
37	C₄²×C₅₃	C₂³	1	2	yes	—
38	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	Q₈×C₅₃	—	4	4	no	—
40	C₂×C₄×C₅₃	—	4	2	no	—
41	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₅₃	#41	1	2	yes	—
47	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₈×C₅₃	—	2	2	no	—
49	C₈×C₅₃	—	2	4	no	—
50	C₈×C₅₃	C₂×Q₈	1	4	yes	—
51	C₂²×C₅₃	—	2	2	no	—
52	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
53	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₅₃	—	2	2	no	—
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	—	2	8	no	—
57	C₄×C₅₃	C₄×Q₈	1	2	yes	—
58	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
59	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
60	#2961	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×C₅₃	—	2	2	no	—
62	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
63	C₂×C₅₃	#568	1	2	yes	—
64	#533	C₂×C₅₃	1	1	yes	characteristic
65	#446	C₂×C₅₃	1	8	yes	complemented
66	#424	C₂×C₅₃	1	2	yes	—
67	#423	C₂×C₅₃	1	1	yes	characteristic
68	#405	C₂×C₅₃	1	2	yes	—
69	C₂×#97	C₂×C₅₃	1	1	yes	characteristic
70	C₅₃	#2961	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×M₁₆	C₂²×C₅₃	1	2	yes	—
72	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
73	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
74	C₄⋊Q₈	C₄×C₅₃	1	2	yes	—
75	#125	C₄×C₅₃	1	2	yes	—
76	C₂²⋊Q₈	C₄×C₅₃	1	4	yes	—
77	C₄×Q₈	C₂²×C₅₃	1	4	yes	—
78	#117	C₂²×C₅₃	1	2	yes	—
79	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
80	C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
81	C₄⋊C₈	C₂²×C₅₃	1	8	yes	—
82	#97	C₂²×C₅₃	1	4	yes	—
83	#97	C₂²×C₅₃	1	4	yes	—
84	C₄×C₈	C₂²×C₅₃	1	4	yes	—
85	C₂×Q₈	C₂×C₄×C₅₃	1	4	yes	—
86	C₂²×C₄	C₂×C₄×C₅₃	1	2	yes	—
87	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
88	M₁₆	—	2	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	Q₈×C₅₃	1	4	yes	complemented
91	C₂×C₈	C₂³×C₅₃	1	4	yes	—
92	C₂×C₈	C₂³×C₅₃	1	2	yes	—
93	C₂×C₈	C₂³×C₅₃	1	2	yes	—
94	C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
95	C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
96	C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
97	C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
98	C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
99	C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
100	C₄²	C₂³×C₅₃	1	4	yes	—
101	C₄²	C₂³×C₅₃	1	2	yes	—
102	C₂³	C₂²×C₄×C₅₃	1	1	yes	characteristic
103	Q₈	—	4	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
106	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
107	C₂×C₄	C₂²×C₄×C₅₃	1	8	yes	—
108	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
109	C₂×C₄	C₂×Q₈×C₅₃	1	2	yes	—
110	C₂×C₄	#41×C₅₃	1	2	yes	—
111	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	2	4	no	—
114	C₈	C₂×Q₈×C₅₃	1	4	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	C₄×Q₈×C₅₃	1	2	yes	—
117	C₂²	C₂³×C₄×C₅₃	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	2	8	no	—
121	C₄	C₄×Q₈×C₅₃	1	2	yes	—
122	C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
123	C₄	C₂×#41×C₅₃	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
126	C₂	C₅₃×#568	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂³×C₄
prime radical	C₂³×C₅₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#446	C₂	semidirect
26. C₂×C₈×C₅₃	Q₈	semidirect
60. #2961	C₅₃	direct (N × H)
65. #446	C₂×C₅₃	semidirect
70. C₅₃	#2961	direct (N × H)
90. C₂×C₈	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#533	2
C₅₃×#446	2
C₅₃×#424	2
C₅₃×#423	2
C₅₃×#405	2
C₂×#97×C₅₃	2
#2961	53

Subgroups — C53×#2961 #20271898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2961 #20271898