Subgroups — #11928608

Order 1248.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64605	C₂	1	2	yes	maximal
2	C₄⋊Dic₃₉	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₃×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	Dic₃⋊Dic₁₃	C₂	1	2	yes	maximal
6	#88123	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×Dic₃₉	C₂²	1	2	yes	—
9	C₂×Dic₃×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×Dic₃×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₃×Dic₁₃	C₂²	1	2	yes	—
12	#64605	—	2	4	no	—
13	C₂×Dic₂₆	—	3	2	no	—
14	C₄²×C₁₃	—	3	1	no	—
15	C₄⋊Dic₁₃	—	3	1	no	—
16	C₄⋊Dic₁₃	—	3	2	no	—
17	C₄⋊Dic₁₃	D₃	1	1	yes	characteristic
18	C₂²×C₃×C₁₃	C₂³	1	1	yes	characteristic
19	C₃×C₄×C₁₃	Q₈	1	2	yes	complemented
20	Dic₃₉	—	2	2	no	—
21	C₃×Dic₁₃	—	2	2	no	—
22	Dic₃×C₁₃	D₄	1	2	yes	—
23	Dic₃×C₁₃	Q₈	1	2	yes	complemented
24	C₂×C₄×C₁₃	—	3	1	no	—
25	C₂×C₄×C₁₃	—	3	1	no	—
26	C₂×C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
27	C₂×Dic₁₃	—	3	2	no	—
28	C₂×Dic₁₃	C₂×D₃	1	2	yes	—
29	Dic₂₆	—	6	4	no	—
30	#888	—	13	1	no	maximal, Hall
31	C₂×C₃×C₁₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₃×C₁₃	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₃×C₁₃	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₁₃	C₂²×D₃	1	1	yes	characteristic
35	C₄×C₁₃	—	3	2	no	—
36	C₄×C₁₃	—	3	2	no	—
37	C₄×C₁₃	Dic₆	1	2	yes	complemented
38	Dic₁₃	—	2	2	no	—
39	Dic₁₃	—	6	2	no	—
40	C₂×Dic₆	—	13	2	no	—
41	C₃×C₄⋊C₄	—	13	1	no	—
42	C₄⋊Dic₃	—	13	1	no	—
43	Dic₃⋊C₄	—	13	2	no	—
44	C₄×Dic₃	D₁₃	1	1	yes	characteristic
45	C₃×C₁₃	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
46	C₄⋊Q₈	—	39	1	no	Sylow
47	C₂×C₁₃	C₂×Dic₆	1	1	yes	characteristic
48	C₂×C₁₃	D₃×D₄	1	1	yes	characteristic
49	C₂×C₁₃	D₃×Q₈	1	1	yes	characteristic
50	C₂×C₃×C₄	—	13	2	no	—
51	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
52	C₂×Dic₃	—	13	2	no	—
53	C₂×Dic₃	C₂×D₁₃	1	1	yes	characteristic
54	C₂×Dic₃	C₂×D₁₃	1	1	yes	characteristic
55	Dic₆	—	26	4	no	—
56	C₂×Q₈	—	39	2	no	—
57	C₄⋊C₄	—	13	1	no	—
58	C₄⋊C₄	—	39	2	no	—
59	C₄⋊C₄	—	39	1	no	—
60	C₄²	—	3	1	no	—
61	C₁₃	#888	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₃	C₂²×D₁₃	1	1	yes	characteristic
63	C₃×C₄	—	26	2	no	—
64	C₃×C₄	Dic₂₆	1	2	yes	complemented
65	Dic₃	—	26	2	no	—
66	Dic₃	Dic₂₆	1	2	yes	complemented
67	Dic₃	D₅₂	1	2	yes	—
68	Q₈	—	78	4	no	—
69	C₂×C₄	—	3	1	no	—
70	C₂×C₄	—	3	1	no	—
71	C₂×C₄	—	13	2	no	—
72	C₂×C₄	—	39	2	no	—
73	C₂×C₄	D₃×D₁₃	1	1	yes	characteristic
74	C₂×C₃	C₂×Dic₂₆	1	1	yes	characteristic
75	C₂×C₃	C₂×Dic₂₆	1	1	yes	characteristic
76	C₂×C₃	C₂×D₅₂	1	1	yes	characteristic
77	C₂²	C₂×D₃×D₁₃	1	1	yes	characteristic
78	C₄	—	3	2	no	—
79	C₄	—	3	2	no	—
80	C₄	—	26	2	no	—
81	C₄	—	78	2	no	—
82	C₄	#64605	1	2	yes	complemented
83	C₃	#88123	1	1	yes	characteristic, complemented, Sylow
84	C₂	D₃×Dic₂₆	1	1	yes	characteristic
85	C₂	D₃×D₅₂	1	1	yes	characteristic
86	C₂	C₂×#64605	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃×D₁₃
commutator, socle, prime radical, cosocle	C₂²×C₃×C₁₃	C₂³
Fitting	C₂×C₃×C₄×C₁₃	C₂²
hypercenter	C₂×C₄	D₃×D₁₃
hypocenter	C₃×C₁₃	C₄⋊Q₈
LCS	G ▷ C₂²×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
19. C₃×C₄×C₁₃	Q₈	semidirect
23. Dic₃×C₁₃	Q₈	semidirect
37. C₄×C₁₃	Dic₆	semidirect
45. C₃×C₁₃	C₄⋊Q₈	semidirect
61. C₁₃	#888	semidirect
64. C₃×C₄	Dic₂₆	semidirect
66. Dic₃	Dic₂₆	semidirect
82. C₄	#64605	semidirect
83. C₃	#88123	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64605	2
C₄⋊Dic₃₉	2
C₄×Dic₃×C₁₃	2
C₃×C₄⋊Dic₁₃	2
Dic₃⋊Dic₁₃	2
#88123	3
#888	13