Subgroups — #88148

Order 416.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#6103	C₂	1	1	yes	maximal, characteristic, complemented
5	#6103	C₂	1	1	yes	maximal, characteristic, complemented
6	#6103	C₂	1	1	yes	maximal, characteristic, complemented
7	#6101	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₃	C₂²	1	1	yes	characteristic
9	C₂²×D₁₃	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×C₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₃	C₂²	1	1	yes	characteristic
15	D₅₂	—	2	2	no	—
16	C₄×D₁₃	—	2	2	no	—
17	C₂²×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×D₁₃	—	2	1	no	—
19	C₂×D₁₃	—	2	1	no	—
20	C₂×D₁₃	—	2	1	no	—
21	C₂×D₁₃	—	2	1	no	—
22	C₂×D₁₃	—	2	1	no	—
23	C₂×D₁₃	D₄	1	2	yes	—
24	C₄×C₁₃	—	2	1	no	—
25	C₄×C₁₃	—	2	1	no	—
26	C₄×C₁₃	—	2	1	no	—
27	Dic₁₃	—	2	1	no	—
28	Dic₁₃	—	2	1	no	—
29	#123	—	13	1	no	maximal, Sylow
30	C₂×C₁₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₁₃	#41	1	1	yes	characteristic
32	C₂×C₁₃	#41	1	1	yes	characteristic
33	D₁₃	—	2	2	no	—
34	D₁₃	—	4	1	no	—
35	C₂×D₄	—	13	1	no	—
36	C₂²×C₄	—	13	1	no	—
37	C₄⋊C₄	—	13	1	no	—
38	C₄⋊C₄	D₁₃	1	1	yes	characteristic, complemented
39	C₂²⋊C₄	—	13	1	no	—
40	C₂²⋊C₄	—	13	1	no	—
41	C₂²⋊C₄	—	13	1	no	—
42	C₁₃	#123	1	1	yes	characteristic, complemented, Sylow
43	C₂³	—	13	1	no	—
44	C₂³	—	13	1	no	—
45	D₄	—	26	2	no	—
46	C₂×C₄	—	13	1	no	—
47	C₂×C₄	—	13	1	no	—
48	C₂×C₄	—	26	2	no	—
49	C₂×C₄	C₂×D₁₃	1	1	yes	characteristic
50	C₂×C₄	C₂×D₁₃	1	1	yes	characteristic
51	C₂×C₄	C₂×D₁₃	1	1	yes	characteristic
52	C₂²	—	13	2	no	—
53	C₂²	—	26	1	no	—
54	C₂²	—	26	1	no	—
55	C₂²	—	26	1	no	—
56	C₂²	—	26	1	no	—
57	C₂²	—	26	1	no	—
58	C₂²	C₂²×D₁₃	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	2	1	no	—
61	C₄	—	2	1	no	—
62	C₄	—	26	1	no	—
63	C₄	—	26	1	no	—
64	C₂	—	26	2	no	—
65	C₂	—	52	1	no	—
66	C₂	#6127	1	1	yes	characteristic
67	C₂	D₄×D₁₃	1	1	yes	characteristic
68	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	C₂×D₅₂	C₂
hypercenter	C₄⋊C₄	D₁₃
hypocenter	C₁₃	#123
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
2. C₂×C₄×D₁₃	C₂	semidirect
3. C₁₃×C₄⋊C₄	C₂	semidirect
4. #6103	C₂	semidirect
5. #6103	C₂	semidirect
6. #6103	C₂	semidirect
7. #6101	C₂	semidirect
38. C₄⋊C₄	D₁₃	semidirect
42. C₁₃	#123	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅₂	2
C₂×C₄×D₁₃	2
C₁₃×C₄⋊C₄	2
#6103	2
#6101	2
#123	13