Subgroups — Dic₁₅₄₉₈ #72293673

Order 61992.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₇₇₄₉	C₂	1	2	yes	maximal
2	Dic₅₁₆₆	—	3	1	no	maximal
3	Dic₂₂₁₄	—	7	1	no	maximal, Hall
4	Dic₁₇₂₂	—	9	1	no	—
5	Dic₁₁₀₇	—	7	2	no	—
6	Dic₇₃₈	—	21	1	no	—
7	Dic₅₇₄	—	27	1	no	Hall
8	C₄×C₇×C₂₇×C₄₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₇×C₂₇×C₄₁	C₂²	1	1	yes	characteristic
10	C₇×C₂₇×C₄₁	Q₈	1	1	yes	characteristic, complemented, Hall
11	C₄×C₂₇×C₄₁	D₇	1	1	yes	characteristic
12	C₂×C₂₇×C₄₁	C₂×D₇	1	1	yes	characteristic
13	Dic₂₅₈₃	—	3	2	no	—
14	C₄×C₇×C₉×C₄₁	D₃	1	1	yes	characteristic
15	C₂×C₇×C₉×C₄₁	C₂×D₃	1	1	yes	characteristic
16	Dic₈₆₁	—	9	2	no	—
17	C₃×C₄×C₇×C₄₁	D₉	1	1	yes	characteristic
18	C₇×C₉×C₄₁	Dic₆	1	1	yes	characteristic
19	C₂×C₃×C₇×C₄₁	C₂×D₉	1	1	yes	characteristic
20	Dic₃₇₈	—	41	1	no	maximal, Hall
21	C₄×C₉×C₄₁	D₂₁	1	1	yes	characteristic
22	Dic₃₆₉	—	21	2	no	—
23	C₄×C₇×C₄₁	D₂₇	1	1	yes	characteristic
24	Dic₂₈₇	—	27	2	no	—
25	C₂₇×C₄₁	Dic₁₄	1	1	yes	characteristic, complemented, Hall
26	Dic₂₄₆	—	63	1	no	—
27	C₃×C₇×C₄₁	Dic₁₈	1	1	yes	characteristic
28	C₄×C₇×C₂₇	D₄₁	1	1	yes	characteristic
29	Dic₁₈₉	—	41	2	no	—
30	C₂×C₉×C₄₁	C₂×D₂₁	1	1	yes	characteristic
31	C₂×C₇×C₄₁	C₂×D₂₇	1	1	yes	characteristic
32	Dic₁₂₆	—	123	1	no	—
33	C₃×C₄×C₄₁	D₆₃	1	1	yes	characteristic
34	Dic₁₂₃	—	63	2	no	—
35	C₂×C₇×C₂₇	C₂×D₄₁	1	1	yes	characteristic
36	C₉×C₄₁	Dic₄₂	1	1	yes	characteristic
37	Dic₈₂	—	189	1	no	Hall
38	C₇×C₄₁	Dic₅₄	1	1	yes	characteristic, complemented, Hall
39	C₄×C₇×C₉	D₁₂₃	1	1	yes	characteristic
40	Dic₆₃	—	123	2	no	—
41	C₂×C₃×C₄₁	C₂×D₆₃	1	1	yes	characteristic
42	Dic₅₄	—	287	1	no	Hall
43	C₇×C₂₇	Dic₈₂	1	1	yes	characteristic, complemented, Hall
44	Dic₄₂	—	369	1	no	—
45	C₄×C₄₁	D₁₈₉	1	1	yes	characteristic
46	Dic₄₁	—	189	2	no	—
47	C₂×C₇×C₉	C₂×D₁₂₃	1	1	yes	characteristic
48	C₃×C₄₁	Dic₁₂₆	1	1	yes	characteristic
49	C₄×C₂₇	D₂₈₇	1	1	yes	characteristic
50	Dic₂₇	—	287	2	no	—
51	C₃×C₄×C₇	D₃₆₉	1	1	yes	characteristic
52	Dic₂₁	—	369	2	no	—
53	C₂×C₄₁	C₂×D₁₈₉	1	1	yes	characteristic
54	Dic₁₈	—	861	1	no	—
55	C₇×C₉	Dic₂₄₆	1	1	yes	characteristic
56	Dic₁₄	—	1107	1	no	Hall
57	C₂×C₂₇	C₂×D₂₈₇	1	1	yes	characteristic
58	C₂×C₃×C₇	C₂×D₃₆₉	1	1	yes	characteristic
59	C₄₁	Dic₃₇₈	1	1	yes	characteristic, complemented, Sylow
60	C₄×C₉	D₈₆₁	1	1	yes	characteristic
61	Dic₉	—	861	2	no	—
62	C₄×C₇	D₁₁₀₇	1	1	yes	characteristic
63	Dic₇	—	1107	2	no	—
64	C₂₇	Dic₅₇₄	1	1	yes	characteristic, complemented, Sylow
65	Dic₆	—	2583	1	no	—
66	C₃×C₇	Dic₇₃₈	1	1	yes	characteristic
67	C₂×C₉	C₂×D₈₆₁	1	1	yes	characteristic
68	C₂×C₇	C₂×D₁₁₀₇	1	1	yes	characteristic
69	C₃×C₄	D₂₅₈₃	1	1	yes	characteristic
70	Dic₃	—	2583	2	no	—
71	C₉	Dic₁₇₂₂	1	1	yes	characteristic
72	Q₈	—	7749	1	no	Sylow
73	C₇	Dic₂₂₁₄	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₃	C₂×D₂₅₈₃	1	1	yes	characteristic
75	C₄	—	7749	2	no	—
76	C₄	D₇₇₄₉	1	1	yes	characteristic
77	C₃	Dic₅₁₆₆	1	1	yes	characteristic
78	C₂	C₂×D₇₇₄₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₇₇₄₉
commutator, cosocle	C₂×C₇×C₂₇×C₄₁	C₂²
Frattini	C₂×C₉	C₂×D₈₆₁
Fitting	C₄×C₇×C₂₇×C₄₁	C₂
socle, prime radical	C₂×C₃×C₇×C₄₁	C₂×D₉
hypercenter	C₄	D₇₇₄₉
hypocenter	C₇×C₂₇×C₄₁	Q₈
LCS	G ▷ C₂×C₇×C₂₇×C₄₁ ▷ C₇×C₂₇×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
10. C₇×C₂₇×C₄₁	Q₈	semidirect
25. C₂₇×C₄₁	Dic₁₄	semidirect
38. C₇×C₄₁	Dic₅₄	semidirect
43. C₇×C₂₇	Dic₈₂	semidirect
59. C₄₁	Dic₃₇₈	semidirect
64. C₂₇	Dic₅₇₄	semidirect
73. C₇	Dic₂₂₁₄	semidirect

Maximal subgroups

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

subgroup	index
Dic₇₇₄₉	2
Dic₅₁₆₆	3
Dic₂₂₁₄	7
C₄×C₇×C₂₇×C₄₁	2
Dic₃₇₈	41

Subgroups — Dic15498 #72293673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₁₅₄₉₈ #72293673