Subgroups — #11928470

Order 1248.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×Dic₂₆	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₃×C₁₃	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊Dic₁₃	C₂	1	4	yes	maximal
5	#88123	—	3	1	no	maximal, Hall
6	C₂×C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×Dic₃₉	C₂²	1	2	yes	—
8	Dic₇₈	—	2	2	no	—
9	C₂×Dic₃×C₁₃	C₂²	1	2	yes	—
10	C₂×C₃×Dic₁₃	C₂²	1	2	yes	—
11	C₃×Dic₂₆	—	2	2	no	—
12	C₂×Dic₂₆	—	3	1	no	—
13	C₂×Dic₂₆	D₃	1	1	yes	characteristic
14	C₄²×C₁₃	—	3	1	no	—
15	C₄⋊Dic₁₃	—	3	4	no	—
16	C₂²×C₃×C₁₃	C₂³	1	1	yes	characteristic
17	C₃×C₄×C₁₃	D₄	1	2	yes	—
18	Dic₃₉	—	2	2	no	—
19	C₃×Dic₁₃	—	2	2	no	—
20	Dic₃×C₁₃	Q₈	1	4	yes	complemented
21	C₂×C₄×C₁₃	—	3	2	no	—
22	C₂×C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
23	C₂×Dic₁₃	—	3	2	no	—
24	C₂×Dic₁₃	C₂×D₃	1	2	yes	—
25	Dic₂₆	—	2	2	no	—
26	Dic₂₆	—	6	2	no	—
27	#944	—	13	1	no	maximal, Hall
28	C₂×C₃×C₁₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₃×C₁₃	C₂×Q₈	1	2	yes	—
30	C₂²×C₁₃	C₂²×D₃	1	1	yes	characteristic
31	C₄×C₁₃	—	3	4	no	—
32	C₄×C₁₃	#67	1	2	yes	—
33	Dic₁₃	—	2	2	no	—
34	Dic₁₃	—	6	2	no	—
35	C₂×C₃×Q₈	—	13	1	no	—
36	C₂×Dic₆	—	13	1	no	—
37	Dic₃⋊C₄	—	13	4	no	—
38	C₄×Dic₃	D₁₃	1	1	yes	characteristic
39	C₃×C₁₃	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
40	C₄⋊Q₈	—	39	1	no	Sylow
41	C₂×C₁₃	D₃×Q₈	1	2	yes	—
42	C₂×C₁₃	C₂×#67	1	1	yes	characteristic
43	C₃×Q₈	—	26	2	no	—
44	C₂×C₃×C₄	—	13	2	no	—
45	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
46	C₂×Dic₃	—	13	2	no	—
47	C₂×Dic₃	C₂×D₁₃	1	2	yes	—
48	Dic₆	—	26	2	no	—
49	C₂×Q₈	—	13	1	no	—
50	C₂×Q₈	—	39	1	no	—
51	C₄⋊C₄	—	39	4	no	—
52	C₄²	—	3	1	no	—
53	C₁₃	#944	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₃	C₂²×D₁₃	1	1	yes	characteristic
55	C₃×C₄	—	26	2	no	—
56	C₃×C₄	D₅₂	1	2	yes	—
57	Dic₃	—	26	2	no	—
58	Dic₃	Dic₂₆	1	4	yes	complemented
59	Q₈	—	26	2	no	—
60	Q₈	—	78	2	no	—
61	C₂×C₄	—	3	2	no	—
62	C₂×C₄	—	13	2	no	—
63	C₂×C₄	—	39	2	no	—
64	C₂×C₄	D₃×D₁₃	1	1	yes	characteristic
65	C₂×C₃	C₂×Dic₂₆	1	2	yes	—
66	C₂×C₃	C₂×D₅₂	1	1	yes	characteristic
67	C₂²	C₂×D₃×D₁₃	1	1	yes	characteristic
68	C₄	—	3	4	no	—
69	C₄	—	26	2	no	—
70	C₄	—	78	2	no	—
71	C₄	#64603	1	2	yes	—
72	C₃	#88123	1	1	yes	characteristic, complemented, Sylow
73	C₂	D₃×Dic₂₆	1	2	yes	—
74	C₂	C₂×#64603	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
20. Dic₃×C₁₃	Q₈	semidirect
39. C₃×C₁₃	C₄⋊Q₈	semidirect
53. C₁₃	#944	semidirect
58. Dic₃	Dic₂₆	semidirect
72. 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×Dic₇₈	2
C₂×C₃×Dic₂₆	2
C₄×Dic₃×C₁₃	2
Dic₃⋊Dic₁₃	2
#88123	3
#944	13