Subgroups — #16811228

Order 3392.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133214	C₂	1	1	yes	maximal, characteristic, complemented
2	#13133209	C₂	1	1	yes	maximal, characteristic, complemented
3	#13133202	C₂	1	4	yes	maximal, complemented
4	#13133191	C₂	1	2	yes	maximal, complemented
5	#13133190	C₂	1	4	yes	maximal, complemented
6	C₄×D₂₁₂	C₂	1	2	yes	maximal, complemented
7	#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊D₅₃	C₂²	1	4	yes	—
10	C₂×D₂₁₂	C₂²	1	2	yes	—
11	C₂×D₂₁₂	C₂²	1	4	yes	—
12	C₂×C₄×D₅₃	C₂²	1	4	yes	—
13	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
14	C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	#11719921	C₂²	1	4	yes	—
18	#11719921	C₂²	1	4	yes	—
19	C₄⋊Dic₅₃	C₂²	1	2	yes	—
20	#11719919	C₂²	1	4	yes	—
21	C₂³×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×D₅₃	C₂³	1	4	yes	—
23	C₂×C₄×C₅₃	—	2	2	no	—
24	C₂×C₄×C₅₃	C₂³	1	2	yes	—
25	C₂×C₄×C₅₃	C₂³	1	2	yes	—
26	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²⋊D₅₃	—	4	4	no	—
29	C₂×Dic₅₃	C₂³	1	4	yes	—
30	D₂₁₂	—	2	4	no	—
31	D₂₁₂	—	4	4	no	—
32	C₄×D₅₃	—	4	4	no	—
33	C₂²×C₅₃	—	2	1	no	—
34	C₂²×C₅₃	—	2	2	no	—
35	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
36	C₂×D₅₃	—	2	4	no	—
37	C₂×D₅₃	—	2	4	no	—
38	C₂×D₅₃	—	2	4	no	—
39	C₄×C₅₃	—	2	1	no	—
40	C₄×C₅₃	—	2	2	no	—
41	C₄×C₅₃	—	2	2	no	—
42	C₄×C₅₃	#41	1	2	yes	—
43	Dic₅₃	—	2	4	no	—
44	C₂×C₅₃	—	4	1	no	—
45	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
46	C₂×C₅₃	H(5,𝔽₂)	1	2	yes	—
47	D₅₃	—	4	4	no	—
48	#540	—	53	1	no	maximal, Sylow
49	C₅₃	#540	1	1	yes	characteristic, complemented, Sylow
50	#127	—	53	1	no	—
51	#125	—	53	1	no	—
52	#123	—	53	4	no	—
53	#121	—	53	2	no	—
54	#121	—	53	4	no	—
55	C₄×D₄	—	53	2	no	—
56	#117	D₅₃	1	1	yes	characteristic, complemented
57	C₂×D₄	—	53	4	no	—
58	C₂×D₄	—	53	2	no	—
59	C₂×D₄	—	53	4	no	—
60	C₂²×C₄	—	53	4	no	—
61	C₂²×C₄	C₂×D₅₃	1	1	yes	characteristic
62	C₄⋊C₄	—	53	2	no	—
63	C₄⋊C₄	—	53	4	no	—
64	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
65	C₂²⋊C₄	—	53	4	no	—
66	C₂²⋊C₄	—	53	4	no	—
67	C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
68	C₄²	C₂×D₅₃	1	1	yes	characteristic
69	C₄²	C₂×D₅₃	1	1	yes	characteristic
70	C₂³	—	53	4	no	—
71	C₂³	C₂²×D₅₃	1	1	yes	characteristic
72	D₄	—	106	4	no	—
73	D₄	—	212	4	no	—
74	D₄	—	212	4	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	53	4	no	—
77	C₂×C₄	—	212	4	no	—
78	C₂×C₄	C₂²×D₅₃	1	2	yes	—
79	C₂×C₄	C₂²×D₅₃	1	2	yes	—
80	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
81	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
82	C₂²	—	2	1	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	106	4	no	—
85	C₂²	—	106	4	no	—
86	C₂²	—	106	4	no	—
87	C₂²	C₂³×D₅₃	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	2	no	—
90	C₄	—	2	2	no	—
91	C₄	—	106	4	no	—
92	C₄	#11719945	1	2	yes	—
93	C₂	—	4	1	no	—
94	C₂	—	212	4	no	—
95	C₂	C₂×#11719945	1	1	yes	characteristic
96	C₂	#13133263	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	#117×C₅₃	C₂
hypercenter	#117	D₅₃
hypocenter	C₅₃	#540
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #13133214	C₂	semidirect
2. #13133209	C₂	semidirect
3. #13133202	C₂	semidirect
4. #13133191	C₂	semidirect
5. #13133190	C₂	semidirect
6. C₄×D₂₁₂	C₂	semidirect
7. #117×C₅₃	C₂	semidirect
49. C₅₃	#540	semidirect
56. #117	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#13133214	2
#13133209	2
#13133202	2
#13133191	2
#13133190	2
C₄×D₂₁₂	2
#117×C₅₃	2
#540	53