Subgroups — #88152

Order 416.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#6103	C₂	1	1	yes	maximal, characteristic
3	#6103	C₂	1	1	yes	maximal, characteristic
4	#6103	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#6101	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃	C₂³	1	1	yes	characteristic
16	C₂×D₁₃	—	2	1	no	—
17	C₂×D₁₃	—	2	1	no	—
18	C₂×D₁₃	—	2	1	no	—
19	C₄×C₁₃	—	2	1	no	—
20	C₄×C₁₃	—	2	1	no	—
21	C₄×C₁₃	—	2	1	no	—
22	Dic₁₃	—	2	1	no	—
23	Dic₁₃	—	2	1	no	—
24	Dic₁₃	—	2	1	no	—
25	#126	—	13	1	no	maximal, Sylow
26	C₂×C₁₃	#41	1	1	yes	characteristic
27	C₂×C₁₃	#41	1	1	yes	characteristic
28	C₂×C₁₃	#41	1	1	yes	characteristic
29	D₁₃	—	4	1	no	—
30	C₄⋊C₄	—	13	1	no	—
31	C₄⋊C₄	—	13	1	no	—
32	C₄⋊C₄	D₁₃	1	1	yes	characteristic, complemented
33	C₂²⋊C₄	—	13	1	no	—
34	C₂²⋊C₄	—	13	1	no	—
35	C₂²⋊C₄	—	13	1	no	—
36	C₄²	—	13	1	no	—
37	C₁₃	#126	1	1	yes	characteristic, complemented, Sylow
38	C₂³	—	13	1	no	—
39	C₂×C₄	—	13	1	no	—
40	C₂×C₄	—	13	1	no	—
41	C₂×C₄	—	13	1	no	—
42	C₂×C₄	C₂×D₁₃	1	1	yes	characteristic
43	C₂×C₄	C₂×D₁₃	1	1	yes	characteristic
44	C₂×C₄	C₂×D₁₃	1	1	yes	characteristic
45	C₂²	—	26	1	no	—
46	C₂²	—	26	1	no	—
47	C₂²	—	26	1	no	—
48	C₂²	C₂²×D₁₃	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	—	2	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	26	1	no	—
53	C₄	—	26	1	no	—
54	C₄	—	26	1	no	—
55	C₂	—	52	1	no	—
56	C₂	#6127	1	1	yes	characteristic
57	C₂	#6129	1	1	yes	characteristic
58	C₂	#6131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂³
Fitting	C₁₃×C₄⋊C₄	C₂
prime radical	C₂²×D₁₃	C₂²
hypercenter	C₄⋊C₄	D₁₃
hypocenter	C₁₃	#126
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₁₃×C₄⋊C₄	C₂	semidirect
5. C₄⋊Dic₁₃	C₂	semidirect
6. #6101	C₂	semidirect
7. C₄×Dic₁₃	C₂	semidirect
32. C₄⋊C₄	D₁₃	semidirect
37. C₁₃	#126	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×C₄⋊C₄	2
#6103	2
C₄⋊Dic₁₃	2
#6101	2
C₄×Dic₁₃	2
#126	13