Subgroups — #11928951

Order 1248.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₈	C₂	1	2	yes	maximal
2	C₃×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₃₉	C₂	1	1	yes	maximal, characteristic
4	#10597664	C₂	1	2	yes	maximal
5	C₄×Dic₃₉	C₂	1	1	yes	maximal, characteristic
6	#88142	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄×C₁₃	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₃₉	C₂²	1	2	yes	—
10	C₂×Dic₃₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₉	C₂²	1	1	yes	characteristic
12	Dic₇₈	—	2	4	no	—
13	C₂×Dic₂₆	—	3	2	no	—
14	C₁₃×C₄⋊C₄	D₃	1	1	yes	characteristic
15	C₄⋊Dic₁₃	—	3	1	no	—
16	#6101	—	3	2	no	—
17	C₄×Dic₁₃	—	3	1	no	—
18	C₂²×C₃×C₁₃	C₂³	1	1	yes	characteristic
19	C₃×C₄×C₁₃	—	2	2	no	—
20	C₃×C₄×C₁₃	Q₈	1	2	yes	complemented
21	Dic₃₉	—	2	2	no	—
22	Dic₃₉	D₄	1	2	yes	—
23	Dic₃₉	Q₈	1	2	yes	complemented
24	C₂×C₄×C₁₃	C₂×D₃	1	2	yes	—
25	C₂×C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
26	C₂×Dic₁₃	—	3	1	no	—
27	C₂×Dic₁₃	—	3	1	no	—
28	C₂×Dic₁₃	—	3	2	no	—
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	—	2	2	no	—
36	C₄×C₁₃	Dic₆	1	2	yes	complemented
37	Dic₁₃	—	3	2	no	—
38	Dic₁₃	—	3	2	no	—
39	Dic₁₃	—	6	2	no	—
40	C₂×Dic₆	—	13	2	no	—
41	C₃×C₄⋊C₄	D₁₃	1	1	yes	characteristic
42	C₄⋊Dic₃	—	13	1	no	—
43	Dic₃⋊C₄	—	13	2	no	—
44	C₄×Dic₃	—	13	1	no	—
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	C₂×D₁₃	1	2	yes	—
51	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
52	C₂×Dic₃	—	13	1	no	—
53	C₂×Dic₃	—	13	1	no	—
54	C₂×Dic₃	—	13	2	no	—
55	Dic₆	—	26	4	no	—
56	C₂×Q₈	—	39	2	no	—
57	C₄⋊C₄	—	39	2	no	—
58	C₄⋊C₄	—	39	1	no	—
59	C₄⋊C₄	D₃₉	1	1	yes	characteristic
60	C₄²	—	39	1	no	—
61	C₁₃	#888	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₃	C₂²×D₁₃	1	1	yes	characteristic
63	C₃×C₄	—	2	2	no	—
64	C₃×C₄	Dic₂₆	1	2	yes	complemented
65	Dic₃	—	13	2	no	—
66	Dic₃	—	13	2	no	—
67	Dic₃	—	26	2	no	—
68	Q₈	—	78	4	no	—
69	C₂×C₄	—	39	1	no	—
70	C₂×C₄	—	39	1	no	—
71	C₂×C₄	—	39	2	no	—
72	C₂×C₄	C₂×D₃₉	1	2	yes	—
73	C₂×C₄	C₂×D₃₉	1	1	yes	characteristic
74	C₂×C₃	C₂×Dic₂₆	1	1	yes	characteristic
75	C₂×C₃	D₄×D₁₃	1	1	yes	characteristic
76	C₂×C₃	Q₈×D₁₃	1	1	yes	characteristic
77	C₂²	C₂²×D₃₉	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	39	2	no	—
80	C₄	—	39	2	no	—
81	C₄	—	78	2	no	—
82	C₄	Dic₇₈	1	2	yes	complemented
83	C₃	#88142	1	1	yes	characteristic, complemented, Sylow
84	C₂	C₂×Dic₇₈	1	1	yes	characteristic
85	C₂	D₄×D₃₉	1	1	yes	characteristic
86	C₂	Q₈×D₃₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₉
commutator, socle, prime radical, cosocle	C₂²×C₃×C₁₃	C₂³
Fitting	C₃×C₁₃×C₄⋊C₄	C₂
hypercenter	C₄⋊C₄	D₃₉
hypocenter	C₃×C₁₃	C₄⋊Q₈
LCS	G ▷ C₂²×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
20. C₃×C₄×C₁₃	Q₈	semidirect
23. Dic₃₉	Q₈	semidirect
36. C₄×C₁₃	Dic₆	semidirect
45. C₃×C₁₃	C₄⋊Q₈	semidirect
61. C₁₃	#888	semidirect
64. C₃×C₄	Dic₂₆	semidirect
82. C₄	Dic₇₈	semidirect
83. C₃	#88142	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₈	2
C₃×C₁₃×C₄⋊C₄	2
C₄⋊Dic₃₉	2
#10597664	2
C₄×Dic₃₉	2
#88142	3
#888	13