Subgroups — Dic₄₃₆₈ #72215811

Order 17472.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₂₁₈₄	C₂	1	2	yes	maximal
2	Dic₁₄₅₆	—	3	1	no	maximal, Hall
3	Dic₁₀₉₂	—	2	2	no	—
4	Dic₇₂₈	—	3	2	no	—
5	Dic₆₂₄	—	7	1	no	maximal, Hall
6	Dic₅₄₆	—	4	2	no	—
7	C₃×C₇×C₁₃×C₃₂	C₂	1	1	yes	maximal, characteristic
8	C₃×C₇×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₇×C₁₃×C₃₂	D₃	1	1	yes	characteristic
10	C₃×C₇×C₈×C₁₃	D₄	1	1	yes	characteristic
11	Dic₃₆₄	—	6	2	no	—
12	C₇×C₁₃×C₁₆	C₂×D₃	1	1	yes	characteristic
13	Dic₃₃₆	—	13	1	no	maximal, Hall
14	Dic₃₁₂	—	7	2	no	—
15	C₃×C₁₃×C₃₂	D₇	1	1	yes	characteristic
16	C₃×C₄×C₇×C₁₃	D₈	1	1	yes	characteristic
17	Dic₂₇₃	—	8	2	no	—
18	Dic₂₀₈	—	21	1	no	Hall
19	Dic₁₈₂	—	12	2	no	—
20	C₇×C₈×C₁₃	D₁₂	1	1	yes	characteristic
21	Dic₁₆₈	—	13	2	no	—
22	C₃×C₇×C₃₂	D₁₃	1	1	yes	characteristic
23	Dic₁₅₆	—	14	2	no	—
24	C₃×C₁₃×C₁₆	C₂×D₇	1	1	yes	characteristic
25	C₂×C₃×C₇×C₁₃	D₁₆	1	1	yes	characteristic
26	Dic₁₁₂	—	39	1	no	Hall
27	Dic₁₀₄	—	21	2	no	—
28	C₁₃×C₃₂	D₂₁	1	1	yes	characteristic
29	C₄×C₇×C₁₃	D₂₄	1	1	yes	characteristic
30	Dic₉₁	—	24	2	no	—
31	Dic₈₄	—	26	2	no	—
32	C₃×C₇×C₁₆	C₂×D₁₃	1	1	yes	characteristic
33	Dic₇₈	—	28	2	no	—
34	C₃×C₈×C₁₃	D₂₈	1	1	yes	characteristic
35	C₃×C₇×C₁₃	Q₆₄	1	1	yes	characteristic, complemented, Hall
36	Dic₅₆	—	39	2	no	—
37	C₇×C₃₂	D₃₉	1	1	yes	characteristic
38	Dic₅₂	—	42	2	no	—
39	C₁₃×C₁₆	C₂×D₂₁	1	1	yes	characteristic
40	Dic₄₈	—	91	1	no	Hall
41	C₂×C₇×C₁₃	D₄₈	1	1	yes	characteristic
42	Dic₄₂	—	52	2	no	—
43	C₃×C₇×C₈	D₅₂	1	1	yes	characteristic
44	C₃×C₄×C₁₃	D₅₆	1	1	yes	characteristic
45	Dic₃₉	—	56	2	no	—
46	Dic₂₈	—	78	2	no	—
47	C₇×C₁₆	C₂×D₃₉	1	1	yes	characteristic
48	Dic₂₆	—	84	2	no	—
49	C₈×C₁₃	D₈₄	1	1	yes	characteristic
50	Dic₂₄	—	91	2	no	—
51	C₃×C₃₂	D₉₁	1	1	yes	characteristic
52	C₇×C₁₃	Dic₄₈	1	1	yes	characteristic, complemented, Hall
53	C₃×C₄×C₇	D₁₀₄	1	1	yes	characteristic
54	Dic₂₁	—	104	2	no	—
55	C₂×C₃×C₁₃	D₁₁₂	1	1	yes	characteristic
56	Q₆₄	—	273	1	no	Sylow
57	Dic₁₄	—	156	2	no	—
58	C₇×C₈	D₁₅₆	1	1	yes	characteristic
59	C₄×C₁₃	D₁₆₈	1	1	yes	characteristic
60	Dic₁₃	—	168	2	no	—
61	Dic₁₂	—	182	2	no	—
62	C₃×C₁₆	C₂×D₉₁	1	1	yes	characteristic
63	C₂×C₃×C₇	D₂₀₈	1	1	yes	characteristic
64	C₃×C₁₃	Dic₁₁₂	1	1	yes	characteristic, complemented, Hall
65	Q₃₂	—	273	2	no	—
66	C₃₂	D₂₇₃	1	1	yes	characteristic
67	C₄×C₇	D₃₁₂	1	1	yes	characteristic
68	Dic₇	—	312	2	no	—
69	C₂×C₁₃	D₃₃₆	1	1	yes	characteristic
70	Dic₆	—	364	2	no	—
71	C₃×C₈	D₃₆₄	1	1	yes	characteristic
72	C₃×C₇	Dic₂₀₈	1	1	yes	characteristic, complemented, Hall
73	Q₁₆	—	546	2	no	—
74	C₁₆	C₂×D₂₇₃	1	1	yes	characteristic
75	C₂×C₇	D₆₂₄	1	1	yes	characteristic
76	C₁₃	Dic₃₃₆	1	1	yes	characteristic, complemented, Sylow
77	C₃×C₄	D₇₂₈	1	1	yes	characteristic
78	Dic₃	—	728	2	no	—
79	Q₈	—	1092	2	no	—
80	C₈	D₁₀₉₂	1	1	yes	characteristic
81	C₇	Dic₆₂₄	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₃	D₁₄₅₆	1	1	yes	characteristic
83	C₄	—	2184	2	no	—
84	C₄	D₂₁₈₄	1	1	yes	characteristic
85	C₃	Dic₁₄₅₆	1	1	yes	characteristic, complemented, Sylow
86	C₂	D₄₃₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄₃₆₈
commutator, cosocle	C₃×C₇×C₁₃×C₁₆	C₂²
Frattini	C₁₆	C₂×D₂₇₃
Fitting	C₃×C₇×C₁₃×C₃₂	C₂
socle, prime radical	C₂×C₃×C₇×C₁₃	D₁₆
hypercenter	C₃₂	D₂₇₃
hypocenter	C₃×C₇×C₁₃	Q₆₄
LCS	G ▷ C₃×C₇×C₁₃×C₁₆ ▷ C₃×C₇×C₈×C₁₃ ▷ C₃×C₄×C₇×C₁₃ ▷ C₂×C₃×C₇×C₁₃ ▷ C₃×C₇×C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

Complements

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

N	H ≅ G/N	type
35. C₃×C₇×C₁₃	Q₆₄	semidirect
52. C₇×C₁₃	Dic₄₈	semidirect
64. C₃×C₁₃	Dic₁₁₂	semidirect
72. C₃×C₇	Dic₂₀₈	semidirect
76. C₁₃	Dic₃₃₆	semidirect
81. C₇	Dic₆₂₄	semidirect
85. C₃	Dic₁₄₅₆	semidirect

Maximal subgroups

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

subgroup	index
Dic₂₁₈₄	2
Dic₁₄₅₆	3
Dic₆₂₄	7
C₃×C₇×C₁₃×C₃₂	2
Dic₃₃₆	13

Subgroups — Dic4368 #72215811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₄₃₆₈ #72215811