Subgroups — C₂×C₅₃×#2104 #27231358

Order 13568.

822 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2104	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₅₃×#342	C₂	1	7	yes	maximal
3	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	7	yes	—
4	C₅₃×#342	C₂²	1	28	yes	—
5	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
6	C₂³×C₄×C₅₃	—	2	7	no	—
7	C₂×C₂²⋊C₄×C₅₃	—	2	7	no	—
8	C₂×C₂²⋊C₄×C₅₃	—	2	14	no	—
9	C₂×C₂²⋊C₄×C₅₃	—	2	14	no	—
10	C₂×C₂²⋊C₄×C₅₃	C₂³	1	14	yes	—
11	C₂⁴×C₅₃	—	8	2	no	—
12	C₂⁴×C₅₃	#41	1	7	yes	—
13	C₂⁴×C₅₃	#41	1	7	yes	—
14	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	—	2	14	no	—
16	C₂²×C₄×C₅₃	—	4	7	no	—
17	C₂²×C₄×C₅₃	—	4	7	no	—
18	C₂²×C₄×C₅₃	—	8	14	no	—
19	C₂²⋊C₄×C₅₃	—	2	28	no	—
20	C₂²⋊C₄×C₅₃	—	4	14	no	—
21	C₂³×C₅₃	—	4	14	no	—
22	C₂³×C₅₃	—	8	7	no	—
23	C₂³×C₅₃	—	8	1	no	—
24	C₂³×C₅₃	#126	1	7	yes	—
25	C₂³×C₅₃	#126	1	7	yes	—
26	C₂³×C₅₃	#126	1	14	yes	—
27	C₂³×C₅₃	C₂×#41	1	7	yes	—
28	C₂×C₄×C₅₃	—	4	14	no	—
29	C₂×C₄×C₅₃	—	8	14	no	—
30	C₂×C₄×C₅₃	—	8	14	no	—
31	C₂×C₄×C₅₃	—	8	14	no	—
32	C₂×#2104	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂²×C₅₃	—	2	14	no	—
34	C₂²×C₅₃	—	4	7	no	—
35	C₂²×C₅₃	—	4	7	no	—
36	C₂²×C₅₃	—	4	14	no	—
37	C₂²×C₅₃	#401	1	7	yes	—
38	C₂²×C₅₃	#401	1	1	yes	characteristic
39	C₂²×C₅₃	C₂×#126	1	7	yes	—
40	C₄×C₅₃	—	8	14	no	—
41	#2104	C₂×C₅₃	1	8	yes	direct product factor
42	C₂×#342	C₂×C₅₃	1	7	yes	—
43	C₂×C₅₃	—	2	7	no	—
44	C₂×C₅₃	—	2	7	no	—
45	C₂×C₅₃	#2104	1	2	yes	direct product factor
46	C₂×C₅₃	C₂×#401	1	1	yes	characteristic
47	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	7	yes	—
48	#342	C₂²×C₅₃	1	28	yes	—
49	C₅₃	C₂×#2104	1	1	yes	characteristic, direct product factor, Sylow
50	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
51	C₂³×C₄	—	2	7	no	—
52	C₂×C₂²⋊C₄	—	2	14	no	—
53	C₂×C₂²⋊C₄	—	2	14	no	—
54	C₂×C₂²⋊C₄	—	2	7	no	—
55	C₂×C₂²⋊C₄	C₂³×C₅₃	1	14	yes	—
56	C₂⁴	—	8	2	no	—
57	C₂⁴	#41×C₅₃	1	7	yes	—
58	C₂⁴	#41×C₅₃	1	7	yes	—
59	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
60	C₂²×C₄	—	2	14	no	—
61	C₂²×C₄	—	4	7	no	—
62	C₂²×C₄	—	4	7	no	—
63	C₂²×C₄	—	8	14	no	—
64	C₂²⋊C₄	—	2	28	no	—
65	C₂²⋊C₄	—	4	14	no	—
66	C₂³	—	4	14	no	—
67	C₂³	—	8	7	no	—
68	C₂³	—	8	1	no	—
69	C₂³	#126×C₅₃	1	7	yes	—
70	C₂³	#126×C₅₃	1	14	yes	—
71	C₂³	#126×C₅₃	1	7	yes	—
72	C₂³	C₂×#41×C₅₃	1	7	yes	—
73	C₂×C₄	—	4	14	no	—
74	C₂×C₄	—	8	14	no	—
75	C₂×C₄	—	8	14	no	—
76	C₂×C₄	—	8	14	no	—
77	C₂²	—	2	14	no	—
78	C₂²	—	4	7	no	—
79	C₂²	—	4	7	no	—
80	C₂²	—	4	14	no	—
81	C₂²	C₅₃×#401	1	7	yes	—
82	C₂²	C₅₃×#401	1	1	yes	characteristic
83	C₂²	C₂×#126×C₅₃	1	7	yes	—
84	C₄	—	8	14	no	—
85	C₂	—	2	7	no	—
86	C₂	—	2	7	no	—
87	C₂	C₅₃×#2104	1	2	yes	direct product factor
88	C₂	C₂×C₅₃×#401	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#401
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₅₃
prime radical	C₂⁵×C₅₃	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂⁵×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#2104	C₂	direct (N × H)
32. C₂×#2104	C₅₃	direct (N × H)
41. #2104	C₂×C₅₃	direct (N × H)
45. C₂×C₅₃	#2104	direct (N × H)
49. C₅₃	C₂×#2104	direct (N × H)
87. C₂	C₅₃×#2104	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×#2104	2
C₂×C₅₃×#342	2
C₂×#2104	53

Subgroups — C2×C53×#2104 #27231358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2104 #27231358