Subgroups — C₂×C₄×Dic₇₆₆ #37588816

Order 24512.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₇₆₆	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₇₆₆	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₃₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₈₃⋊C₄	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₃₈₃	C₂	1	2	yes	maximal
6	C₂×C₄²×C₃₈₃	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₇₆₆	C₄	1	8	yes	direct product factor
8	C₂×Dic₇₆₆	C₂²	1	4	yes	—
9	C₂²×Dic₃₈₃	C₂²	1	2	yes	—
10	C₂²×Dic₃₈₃	C₂²	1	2	yes	—
11	C₄⋊Dic₃₈₃	C₂²	1	4	yes	—
12	Dic₃₈₃⋊C₄	C₂²	1	8	yes	—
13	C₄×Dic₃₈₃	C₂²	1	8	yes	—
14	C₂²×C₄×C₃₈₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₃₈₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₃₈₃	C₂²	1	1	yes	characteristic
17	C₄²×C₃₈₃	C₂²	1	4	yes	—
18	Dic₇₆₆	C₂×C₄	1	16	yes	direct product factor
19	C₂×Dic₃₈₃	—	2	4	no	—
20	C₂×Dic₃₈₃	C₂×C₄	1	8	yes	—
21	C₂×Dic₃₈₃	C₂³	1	4	yes	—
22	C₂×Dic₃₈₃	C₂³	1	4	yes	—
23	C₂³×C₃₈₃	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₃₈₃	—	2	2	no	—
25	C₂×C₄×C₃₈₃	C₂×C₄	1	4	yes	—
26	C₂×C₄×C₃₈₃	Q₈	1	4	yes	complemented
27	C₂×C₄×C₃₈₃	C₂³	1	2	yes	—
28	C₂×C₄×C₃₈₃	C₂³	1	2	yes	—
29	C₂×C₄×C₃₈₃	C₂³	1	2	yes	—
30	C₂²×C₃₈₃	C₂²×C₄	1	2	yes	—
31	C₂²×C₃₈₃	C₂×Q₈	1	2	yes	—
32	C₂²×C₃₈₃	#41	1	2	yes	—
33	C₂²×C₃₈₃	C₂⁴	1	1	yes	characteristic
34	C₄×C₃₈₃	—	2	2	no	—
35	C₄×C₃₈₃	C₂²×C₄	1	4	yes	—
36	C₄×C₃₈₃	C₂×Q₈	1	4	yes	complemented
37	Dic₃₈₃	—	2	4	no	—
38	Dic₃₈₃	C₂²×C₄	1	8	yes	—
39	C₂×C₃₈₃	C₄×Q₈	1	4	yes	complemented
40	C₂×C₃₈₃	C₂³×C₄	1	1	yes	characteristic
41	C₂×C₃₈₃	C₂²×Q₈	1	1	yes	characteristic
42	C₂×C₃₈₃	C₂×#41	1	1	yes	characteristic
43	C₃₈₃	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₄×Q₈	—	383	1	no	maximal, Sylow
45	C₂²×Q₈	—	383	1	no	—
46	C₄×Q₈	—	383	8	no	—
47	C₂×C₄⋊C₄	—	383	2	no	—
48	C₂×C₄⋊C₄	—	383	1	no	—
49	C₂×C₄²	—	383	2	no	—
50	C₂×C₄²	D₃₈₃	1	1	yes	characteristic
51	C₂×Q₈	—	383	8	no	—
52	C₂×Q₈	—	383	4	no	—
53	C₂²×C₄	—	383	2	no	—
54	C₂²×C₄	—	383	2	no	—
55	C₂²×C₄	C₂×D₃₈₃	1	1	yes	characteristic
56	C₂²×C₄	C₂×D₃₈₃	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₃₈₃	1	1	yes	characteristic
58	C₄⋊C₄	—	383	8	no	—
59	C₄⋊C₄	—	383	4	no	—
60	C₄²	—	383	8	no	—
61	C₄²	C₂×D₃₈₃	1	4	yes	—
62	C₂³	C₂²×D₃₈₃	1	1	yes	characteristic
63	Q₈	—	383	16	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	383	8	no	—
66	C₂×C₄	—	383	4	no	—
67	C₂×C₄	—	383	4	no	—
68	C₂×C₄	—	766	4	no	—
69	C₂×C₄	C₄×D₃₈₃	1	4	yes	—
70	C₂×C₄	Dic₇₆₆	1	4	yes	direct product factor
71	C₂×C₄	C₂²×D₃₈₃	1	2	yes	—
72	C₂×C₄	C₂²×D₃₈₃	1	2	yes	—
73	C₂×C₄	C₂²×D₃₈₃	1	2	yes	—
74	C₂²	C₂×C₄×D₃₈₃	1	2	yes	—
75	C₂²	C₂×Dic₇₆₆	1	2	yes	—
76	C₂²	#20259457	1	2	yes	—
77	C₂²	C₂³×D₃₈₃	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	383	8	no	—
80	C₄	—	766	4	no	—
81	C₄	C₂×C₄×D₃₈₃	1	4	yes	—
82	C₄	C₂×Dic₇₆₆	1	4	yes	direct product factor
83	C₂	C₄×Dic₇₆₆	1	4	yes	direct product factor
84	C₂	C₂²×C₄×D₃₈₃	1	1	yes	characteristic
85	C₂	C₂²×Dic₇₆₆	1	1	yes	characteristic
86	C₂	C₂×#20259457	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₈₃
commutator	C₂×C₃₈₃	C₂³×C₄
Frattini	C₂²	C₂³×D₃₈₃
Fitting	C₂×C₄²×C₃₈₃	C₂
socle, prime radical	C₂³×C₃₈₃	C₂³
cosocle	C₂²×C₃₈₃	C₂⁴
hypercenter	C₂×C₄²	D₃₈₃
hypocenter	C₃₈₃	C₂×C₄×Q₈
LCS	G ▷ C₂×C₃₈₃ ▷ C₃₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×Dic₇₆₆	C₂	direct (N × H)
7. C₂×Dic₇₆₆	C₄	direct (N × H)
18. Dic₇₆₆	C₂×C₄	direct (N × H)
26. C₂×C₄×C₃₈₃	Q₈	semidirect
36. C₄×C₃₈₃	C₂×Q₈	semidirect
39. C₂×C₃₈₃	C₄×Q₈	semidirect
43. C₃₈₃	C₂×C₄×Q₈	semidirect
70. C₂×C₄	Dic₇₆₆	direct (N × H)
82. C₄	C₂×Dic₇₆₆	direct (N × H)
83. C₂	C₄×Dic₇₆₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₇₆₆	2
C₄×Dic₇₆₆	2
C₂×C₄⋊Dic₃₈₃	2
C₂×Dic₃₈₃⋊C₄	2
C₂×C₄×Dic₃₈₃	2
C₂×C₄²×C₃₈₃	2
C₂×C₄×Q₈	383

Subgroups — C2×C4×Dic766 #37588816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Dic₇₆₆ #37588816