Subgroups — #16811229

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	#13133202	C₂	1	4	yes	maximal, complemented
3	#13133192	C₂	1	2	yes	maximal, complemented
4	#13133192	C₂	1	4	yes	maximal, complemented
5	#13133179	C₂	1	2	yes	maximal, complemented
6	#13133174	C₂	1	1	yes	maximal, characteristic, complemented
7	#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₂₁₂	C₂²	1	2	yes	—
9	C₂×D₂₁₂	C₂²	1	4	yes	—
10	C₂×D₂₁₂	C₂²	1	4	yes	—
11	C₂×C₄×D₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₅₃	C₂²	1	4	yes	—
13	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
14	C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
15	C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	#11719921	C₂²	1	2	yes	—
17	#11719921	C₂²	1	4	yes	—
18	#11719921	C₂²	1	4	yes	—
19	#11719919	C₂²	1	2	yes	—
20	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
21	C₂²×D₅₃	C₂³	1	1	yes	characteristic
22	C₂²×D₅₃	C₂³	1	4	yes	—
23	C₂×C₄×C₅₃	C₂³	1	4	yes	—
24	C₂×C₄×C₅₃	C₂³	1	2	yes	—
25	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₅₃	C₂³	1	2	yes	—
27	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
28	D₂₁₂	—	2	4	no	—
29	D₂₁₂	—	4	4	no	—
30	D₂₁₂	—	4	4	no	—
31	C₄×D₅₃	—	2	2	no	—
32	C₄×D₅₃	—	4	4	no	—
33	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
34	C₂×D₅₃	—	2	1	no	—
35	C₂×D₅₃	—	2	4	no	—
36	C₂×D₅₃	—	2	4	no	—
37	C₂×D₅₃	—	2	2	no	—
38	C₂×D₅₃	—	2	4	no	—
39	C₄×C₅₃	—	2	4	no	—
40	C₄×C₅₃	—	2	2	no	—
41	C₄×C₅₃	#41	1	2	yes	—
42	Dic₅₃	—	2	1	no	—
43	Dic₅₃	—	2	2	no	—
44	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
45	C₂×C₅₃	H(5,𝔽₂)	1	2	yes	—
46	D₅₃	—	4	1	no	—
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	D₅₃	1	1	yes	characteristic, complemented
52	#123	—	53	4	no	—
53	#121	—	53	4	no	—
54	#121	—	53	2	no	—
55	C₄×D₄	—	53	2	no	—
56	#117	—	53	1	no	—
57	C₂×D₄	—	53	2	no	—
58	C₂×D₄	—	53	4	no	—
59	C₂×D₄	—	53	4	no	—
60	C₂²×C₄	—	53	1	no	—
61	C₂²×C₄	—	53	4	no	—
62	C₄⋊C₄	—	53	2	no	—
63	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
64	C₄⋊C₄	C₂×D₅₃	1	4	yes	—
65	C₂²⋊C₄	—	53	2	no	—
66	C₂²⋊C₄	—	53	4	no	—
67	C₂²⋊C₄	—	53	4	no	—
68	C₄²	—	53	1	no	—
69	C₄²	C₂×D₅₃	1	1	yes	characteristic
70	C₂³	—	53	1	no	—
71	C₂³	—	53	4	no	—
72	D₄	—	106	4	no	—
73	D₄	—	212	4	no	—
74	D₄	—	212	4	no	—
75	C₂×C₄	—	53	2	no	—
76	C₂×C₄	—	53	1	no	—
77	C₂×C₄	—	106	2	no	—
78	C₂×C₄	—	212	4	no	—
79	C₂×C₄	C₂²×D₅₃	1	2	yes	—
80	C₂×C₄	C₂²×D₅₃	1	4	yes	—
81	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
82	C₂²	—	106	1	no	—
83	C₂²	—	106	4	no	—
84	C₂²	—	106	4	no	—
85	C₂²	—	106	2	no	—
86	C₂²	—	106	4	no	—
87	C₂²	C₂³×D₅₃	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	—	2	2	no	—
90	C₄	—	106	1	no	—
91	C₄	—	106	2	no	—
92	C₄	#11719949	1	2	yes	—
93	C₂	—	212	1	no	—
94	C₂	—	212	4	no	—
95	C₂	C₂×#11719949	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	#125×C₅₃	C₂
hypercenter	#125	D₅₃
hypocenter	C₅₃	#540
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #13133214	C₂	semidirect
2. #13133202	C₂	semidirect
3. #13133192	C₂	semidirect
4. #13133192	C₂	semidirect
5. #13133179	C₂	semidirect
6. #13133174	C₂	semidirect
7. #125×C₅₃	C₂	semidirect
49. C₅₃	#540	semidirect
51. #125	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
#13133202	2
#13133192	2
#13133179	2
#13133174	2
#125×C₅₃	2
#540	53