Subgroups — C₂²×Dic₁₃₆ #14541897

Order 2176.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₈	C₂	1	2	yes	maximal
2	C₂×Dic₁₃₆	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₃₄	—	2	2	no	—
5	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
6	C₂×Dic₆₈	—	2	6	no	—
7	C₂×Dic₆₈	C₂²	1	12	yes	—
8	C₂×C₁₆×C₁₇	C₂²	1	6	yes	—
9	Dic₁₃₆	C₂²	1	16	yes	direct product factor
10	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
11	C₂²×Dic₁₇	—	4	2	no	—
12	C₂×Dic₃₄	—	2	12	no	—
13	C₂×Dic₃₄	—	4	6	no	—
14	C₂×C₈×C₁₇	D₄	1	3	yes	—
15	C₂×C₈×C₁₇	C₂³	1	3	yes	—
16	Dic₆₈	—	2	12	no	—
17	Dic₆₈	C₂³	1	8	yes	—
18	C₁₆×C₁₇	C₂³	1	4	yes	—
19	C₂³×C₁₇	D₈	1	1	yes	characteristic
20	C₂×C₄×C₁₇	D₈	1	3	yes	—
21	C₂×C₄×C₁₇	C₂×D₄	1	3	yes	—
22	C₂×Dic₁₇	—	4	12	no	—
23	Dic₃₄	—	2	8	no	—
24	Dic₃₄	—	4	12	no	—
25	C₈×C₁₇	C₂×D₄	1	3	yes	—
26	C₈×C₁₇	C₂⁴	1	1	yes	characteristic
27	C₂²×Q₃₂	—	17	1	no	maximal, Sylow
28	C₂²×C₁₇	Q₃₂	1	4	yes	complemented
29	C₂²×C₁₇	C₂×D₈	1	3	yes	—
30	C₄×C₁₇	C₂×D₈	1	3	yes	—
31	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
32	Dic₁₇	—	4	8	no	—
33	C₂²×Q₁₆	—	17	2	no	—
34	C₂×Q₃₂	—	17	12	no	—
35	C₂²×C₁₆	D₁₇	1	1	yes	characteristic
36	C₂×C₁₇	C₂×Q₃₂	1	6	yes	complemented
37	C₂×C₁₇	C₂²×D₈	1	1	yes	characteristic
38	C₂²×Q₈	—	34	2	no	—
39	C₂×Q₁₆	—	17	12	no	—
40	C₂×Q₁₆	—	34	6	no	—
41	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
42	Q₃₂	—	17	16	no	—
43	C₂×C₁₆	C₂×D₁₇	1	6	yes	—
44	C₁₇	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
45	C₂×Q₈	—	34	12	no	—
46	C₂×Q₈	—	68	6	no	—
47	C₂²×C₄	—	68	2	no	—
48	C₂²×C₄	D₆₈	1	1	yes	characteristic
49	Q₁₆	—	17	8	no	—
50	Q₁₆	—	34	12	no	—
51	C₂×C₈	D₆₈	1	3	yes	—
52	C₂×C₈	C₂²×D₁₇	1	3	yes	—
53	C₁₆	C₂²×D₁₇	1	4	yes	—
54	C₂³	D₁₃₆	1	1	yes	characteristic
55	Q₈	—	34	8	no	—
56	Q₈	—	68	12	no	—
57	C₂×C₄	—	68	12	no	—
58	C₂×C₄	D₁₃₆	1	3	yes	—
59	C₂×C₄	C₂×D₆₈	1	3	yes	—
60	C₈	C₂×D₆₈	1	3	yes	—
61	C₈	C₂³×D₁₇	1	1	yes	characteristic
62	C₂²	Dic₁₃₆	1	4	yes	direct product factor
63	C₂²	C₂×D₁₃₆	1	3	yes	—
64	C₄	—	68	8	no	—
65	C₄	C₂×D₁₃₆	1	3	yes	—
66	C₄	C₂²×D₆₈	1	1	yes	characteristic
67	C₂	C₂×Dic₁₃₆	1	6	yes	direct product factor
68	C₂	C₂²×D₁₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₁₃₆
commutator, cosocle	C₈×C₁₇	C₂⁴
Frattini	C₈	C₂³×D₁₇
Fitting	C₂²×C₁₆×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	D₈
hypercenter	C₂²×C₁₆	D₁₇
hypocenter	C₁₇	C₂²×Q₃₂
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₃₆	C₂	direct (N × H)
9. Dic₁₃₆	C₂²	direct (N × H)
28. C₂²×C₁₇	Q₃₂	semidirect
36. C₂×C₁₇	C₂×Q₃₂	semidirect
44. C₁₇	C₂²×Q₃₂	semidirect
62. C₂²	Dic₁₃₆	direct (N × H)
67. C₂	C₂×Dic₁₃₆	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×Dic₆₈	2
C₂×Dic₁₃₆	2
C₂²×C₁₆×C₁₇	2
C₂²×Q₃₂	17

Subgroups — C22×Dic136 #14541897

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₁₃₆ #14541897