Subgroups — #88139

Order 416.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊Dic₁₃	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	C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₁₃	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	C₂²⋊D₁₃	—	2	2	no	—
13	C₂×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₃	C₂²	1	1	yes	characteristic
16	Dic₂₆	—	2	2	no	—
17	C₂²×C₁₃	—	2	1	no	—
18	C₂²×C₁₃	—	2	1	no	—
19	C₂²×C₁₃	—	2	1	no	—
20	C₂²×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×D₁₃	—	2	1	no	—
22	C₂×D₁₃	—	2	1	no	—
23	C₂×D₁₃	—	2	1	no	—
24	C₄×C₁₃	—	2	1	no	—
25	C₄×C₁₃	—	2	1	no	—
26	Dic₁₃	—	2	1	no	—
27	Dic₁₃	—	2	1	no	—
28	Dic₁₃	D₄	1	2	yes	—
29	#124	—	13	1	no	maximal, Sylow
30	C₂×C₁₃	—	4	1	no	—
31	C₂×C₁₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₁₃	#41	1	1	yes	characteristic
33	C₂×C₁₃	#41	1	1	yes	characteristic
34	D₁₃	—	4	1	no	—
35	C₂×Q₈	—	13	1	no	—
36	C₂×D₄	—	13	1	no	—
37	C₂²⋊C₄	—	13	1	no	—
38	C₂²⋊C₄	—	13	1	no	—
39	C₂²⋊C₄	—	13	1	no	—
40	C₂²⋊C₄	D₁₃	1	1	yes	characteristic, complemented
41	C₄²	—	13	1	no	—
42	C₁₃	#124	1	1	yes	characteristic, complemented, Sylow
43	C₂³	—	13	1	no	—
44	C₂³	C₂×D₁₃	1	1	yes	characteristic
45	Q₈	—	26	2	no	—
46	D₄	—	26	2	no	—
47	C₂×C₄	—	13	1	no	—
48	C₂×C₄	—	13	1	no	—
49	C₂×C₄	—	13	1	no	—
50	C₂×C₄	C₂×D₁₃	1	1	yes	characteristic
51	C₂×C₄	C₂×D₁₃	1	1	yes	characteristic
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	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	—	13	2	no	—
62	C₄	—	26	1	no	—
63	C₄	—	26	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	52	1	no	—
66	C₂	#6127	1	1	yes	characteristic
67	C₂	D₄×D₁₃	1	1	yes	characteristic
68	C₂	#6129	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂³
Fitting	C₁₃×C₂²⋊C₄	C₂
prime radical	C₂×C₂²⋊D₁₃	C₂
hypercenter	C₂²⋊C₄	D₁₃
hypocenter	C₁₃	#124
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₂₆	C₂	semidirect
3. C₁₃×C₂²⋊C₄	C₂	semidirect
4. C₂²⋊Dic₁₃	C₂	semidirect
5. #6103	C₂	semidirect
6. #6103	C₂	semidirect
7. C₄×Dic₁₃	C₂	semidirect
40. C₂²⋊C₄	D₁₃	semidirect
42. C₁₃	#124	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊D₁₃	2
C₂×Dic₂₆	2
C₁₃×C₂²⋊C₄	2
C₂²⋊Dic₁₃	2
#6103	2
C₄×Dic₁₃	2
#124	13