Subgroups — #16810684

Order 3392.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#88322	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#88322	C₂	1	2	yes	maximal
3	#13133128	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	#117×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×C₅₃	C₂²	1	1	yes	characteristic
7	C₄⋊C₄×C₅₃	C₄	1	2	yes	—
8	C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
9	C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#88322	C₂²	1	2	yes	—
12	C₂×#88322	C₂²	1	2	yes	—
13	C₂³×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₅₃	—	2	2	no	—
15	C₂×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	#88322	—	2	2	no	—
20	#88322	—	2	2	no	—
21	C₂²×C₅₃	—	2	1	no	—
22	C₂²×C₅₃	—	2	2	no	—
23	C₂²×C₅₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₅₃	—	2	1	no	—
25	C₄×C₅₃	—	2	2	no	—
26	C₄×C₅₃	—	2	2	no	—
27	C₄×C₅₃	#41	1	2	yes	—
28	C₂×C₅₃	—	4	1	no	—
29	C₂×C₅₃	#117	1	1	yes	characteristic
30	C₂×C₅₃	#131	1	2	yes	—
31	#433	—	53	1	no	maximal, Sylow
32	C₅₃	#433	1	1	yes	characteristic, complemented, Sylow
33	#117	D₅₃	1	1	yes	characteristic
34	C₄⋊C₈	—	53	2	no	—
35	C₂²⋊C₈	—	53	2	no	—
36	#97	—	53	1	no	—
37	C₄×C₈	—	53	1	no	—
38	C₂²×C₄	C₂×D₅₃	1	1	yes	characteristic
39	C₂×C₈	—	53	2	no	—
40	C₂×C₈	—	53	2	no	—
41	C₄⋊C₄	Dic₅₃	1	2	yes	—
42	C₂²⋊C₄	Dic₅₃	1	2	yes	—
43	C₄²	C₂×D₅₃	1	1	yes	characteristic
44	C₄²	C₂×D₅₃	1	1	yes	characteristic
45	C₂³	C₂×Dic₅₃	1	1	yes	characteristic
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	C₂×Dic₅₃	1	2	yes	—
48	C₂×C₄	C₂×Dic₅₃	1	2	yes	—
49	C₂×C₄	C₂×Dic₅₃	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
51	C₈	—	106	2	no	—
52	C₈	—	106	2	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	2	no	—
55	C₂²	C₂²×Dic₅₃	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	2	2	no	—
58	C₄	—	2	2	no	—
59	C₄	#11719945	1	2	yes	—
60	C₂	—	4	1	no	—
61	C₂	#13133176	1	1	yes	characteristic
62	C₂	#13133224	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄
Fitting	#117×C₅₃	C₂
prime radical	C₂³×C₅₃	C₂×C₄
cosocle	C₂×C₄×C₅₃	C₂³
hypercenter	#117	D₅₃
hypocenter	C₅₃	#433
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
1. C₄⋊#88322	C₂	semidirect
3. #13133128	C₂	semidirect
4. C₄×#88322	C₂	semidirect
32. C₅₃	#433	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#88322	2
C₂²⋊#88322	2
#13133128	2
C₄×#88322	2
#117×C₅₃	2
#433	53