Subgroups — C₂×C₄×Dic₂₂₆ #20299063

Order 7232.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₂₂₆	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14268626	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₁₁₃	C₂	1	2	yes	maximal
6	C₂×C₄²×C₁₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₂₂₆	C₄	1	8	yes	direct product factor
8	C₂×Dic₂₂₆	C₂²	1	4	yes	—
9	C₂²×Dic₁₁₃	C₂²	1	2	yes	—
10	C₂²×Dic₁₁₃	C₂²	1	2	yes	—
11	C₄⋊Dic₁₁₃	C₂²	1	4	yes	—
12	#14268626	C₂²	1	8	yes	—
13	C₄×Dic₁₁₃	C₂²	1	8	yes	—
14	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
17	C₄²×C₁₁₃	C₂²	1	4	yes	—
18	C₂³×C₁₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₁₁₃	—	2	2	no	—
20	C₂×C₄×C₁₁₃	C₂×C₄	1	4	yes	—
21	C₂×C₄×C₁₁₃	Q₈	1	4	yes	complemented
22	C₂×C₄×C₁₁₃	C₂³	1	2	yes	—
23	C₂×C₄×C₁₁₃	C₂³	1	2	yes	—
24	C₂×C₄×C₁₁₃	C₂³	1	2	yes	—
25	C₂×Dic₁₁₃	—	2	4	no	—
26	C₂×Dic₁₁₃	C₂×C₄	1	8	yes	—
27	C₂×Dic₁₁₃	C₂³	1	4	yes	—
28	C₂×Dic₁₁₃	C₂³	1	4	yes	—
29	Dic₂₂₆	C₂×C₄	1	16	yes	direct product factor
30	C₂²×C₁₁₃	C₂²×C₄	1	2	yes	—
31	C₂²×C₁₁₃	C₂×Q₈	1	2	yes	—
32	C₂²×C₁₁₃	#41	1	2	yes	—
33	C₂²×C₁₁₃	C₂⁴	1	1	yes	characteristic
34	C₄×C₁₁₃	—	2	2	no	—
35	C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
36	C₄×C₁₁₃	C₂×Q₈	1	4	yes	complemented
37	Dic₁₁₃	—	2	4	no	—
38	Dic₁₁₃	C₂²×C₄	1	8	yes	—
39	C₂×C₁₁₃	C₄×Q₈	1	4	yes	complemented
40	C₂×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
41	C₂×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
42	C₂×C₁₁₃	C₂×#41	1	1	yes	characteristic
43	C₁₁₃	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₄×Q₈	—	113	1	no	maximal, Sylow
45	C₂²×Q₈	—	113	1	no	—
46	C₄×Q₈	—	113	8	no	—
47	C₂×C₄⋊C₄	—	113	2	no	—
48	C₂×C₄⋊C₄	—	113	1	no	—
49	C₂×C₄²	—	113	2	no	—
50	C₂×C₄²	D₁₁₃	1	1	yes	characteristic
51	C₂×Q₈	—	113	8	no	—
52	C₂×Q₈	—	113	4	no	—
53	C₂²×C₄	—	113	2	no	—
54	C₂²×C₄	—	113	2	no	—
55	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
56	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
58	C₄⋊C₄	—	113	8	no	—
59	C₄⋊C₄	—	113	4	no	—
60	C₄²	—	113	8	no	—
61	C₄²	C₂×D₁₁₃	1	4	yes	—
62	C₂³	C₂²×D₁₁₃	1	1	yes	characteristic
63	Q₈	—	113	16	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	113	8	no	—
66	C₂×C₄	—	113	4	no	—
67	C₂×C₄	—	113	4	no	—
68	C₂×C₄	—	226	4	no	—
69	C₂×C₄	Dic₂₂₆	1	4	yes	direct product factor
70	C₂×C₄	C₄×D₁₁₃	1	4	yes	—
71	C₂×C₄	C₂²×D₁₁₃	1	2	yes	—
72	C₂×C₄	C₂²×D₁₁₃	1	2	yes	—
73	C₂×C₄	C₂²×D₁₁₃	1	2	yes	—
74	C₂²	C₂×C₄×D₁₁₃	1	2	yes	—
75	C₂²	C₂×Dic₂₂₆	1	2	yes	—
76	C₂²	#14268647	1	2	yes	—
77	C₂²	C₂³×D₁₁₃	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	113	8	no	—
80	C₄	—	226	4	no	—
81	C₄	C₂×C₄×D₁₁₃	1	4	yes	—
82	C₄	C₂×Dic₂₂₆	1	4	yes	direct product factor
83	C₂	C₄×Dic₂₂₆	1	4	yes	direct product factor
84	C₂	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
85	C₂	C₂²×Dic₂₂₆	1	1	yes	characteristic
86	C₂	C₂×#14268647	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₁₃
commutator	C₂×C₁₁₃	C₂³×C₄
Frattini	C₂²	C₂³×D₁₁₃
Fitting	C₂×C₄²×C₁₁₃	C₂
socle	C₂³×C₁₁₃	C₂³
hypercenter	C₂×C₄²	D₁₁₃
hypocenter	C₁₁₃	C₂×C₄×Q₈
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×Dic₂₂₆	C₂	direct (N × H)
7. C₂×Dic₂₂₆	C₄	direct (N × H)
21. C₂×C₄×C₁₁₃	Q₈	semidirect
29. Dic₂₂₆	C₂×C₄	direct (N × H)
36. C₄×C₁₁₃	C₂×Q₈	semidirect
39. C₂×C₁₁₃	C₄×Q₈	semidirect
43. C₁₁₃	C₂×C₄×Q₈	semidirect
69. C₂×C₄	Dic₂₂₆	direct (N × H)
82. C₄	C₂×Dic₂₂₆	direct (N × H)
83. C₂	C₄×Dic₂₂₆	direct (N × H)

Subgroups — C2×C4×Dic226 #20299063

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₄×Dic₂₂₆ #20299063