Subgroups — C₂²×Dic₂₇₂ #17192447

Order 4352.

462 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₃₆	C₂	1	2	yes	maximal
2	C₂×Dic₂₇₂	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₇×C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₆₈	—	2	2	no	—
5	C₂×Dic₁₃₆	—	2	6	no	—
6	C₂×Dic₁₃₆	C₂²	1	12	yes	—
7	Dic₂₇₂	C₂²	1	16	yes	direct product factor
8	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₇×C₃₂	C₂²	1	6	yes	—
10	C₂²×Dic₃₄	—	4	2	no	—
11	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
12	C₂×Dic₆₈	—	2	12	no	—
13	C₂×Dic₆₈	—	4	6	no	—
14	C₂×C₁₆×C₁₇	D₄	1	3	yes	—
15	C₂×C₁₆×C₁₇	C₂³	1	3	yes	—
16	Dic₁₃₆	—	2	12	no	—
17	Dic₁₃₆	C₂³	1	8	yes	—
18	C₁₇×C₃₂	C₂³	1	4	yes	—
19	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
20	C₂²×Dic₁₇	—	8	2	no	—
21	C₂×Dic₃₄	—	4	12	no	—
22	C₂×Dic₃₄	—	8	6	no	—
23	C₂×C₈×C₁₇	D₈	1	3	yes	—
24	C₂×C₈×C₁₇	C₂×D₄	1	3	yes	—
25	Dic₆₈	—	2	8	no	—
26	Dic₆₈	—	4	12	no	—
27	C₁₆×C₁₇	C₂×D₄	1	3	yes	—
28	C₁₆×C₁₇	C₂⁴	1	1	yes	characteristic
29	C₂²×Q₆₄	—	17	1	no	maximal, Sylow
30	C₂³×C₁₇	D₁₆	1	1	yes	characteristic
31	C₂×C₄×C₁₇	D₁₆	1	3	yes	—
32	C₂×C₄×C₁₇	C₂×D₈	1	3	yes	—
33	C₂×Dic₁₇	—	8	12	no	—
34	Dic₃₄	—	4	8	no	—
35	Dic₃₄	—	8	12	no	—
36	C₈×C₁₇	C₂×D₈	1	3	yes	—
37	C₈×C₁₇	C₂²×D₄	1	1	yes	characteristic
38	C₂²×Q₃₂	—	17	2	no	—
39	C₂×Q₆₄	—	17	12	no	—
40	C₂²×C₃₂	D₁₇	1	1	yes	characteristic
41	C₂²×C₁₇	Q₆₄	1	4	yes	complemented
42	C₂²×C₁₇	C₂×D₁₆	1	3	yes	—
43	C₄×C₁₇	C₂×D₁₆	1	3	yes	—
44	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
45	Dic₁₇	—	8	8	no	—
46	C₂²×Q₁₆	—	34	2	no	—
47	C₂×Q₃₂	—	17	12	no	—
48	C₂×Q₃₂	—	34	6	no	—
49	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
50	Q₆₄	—	17	16	no	—
51	C₂×C₃₂	C₂×D₁₇	1	6	yes	—
52	C₂×C₁₇	C₂×Q₆₄	1	6	yes	complemented
53	C₂×C₁₇	C₂²×D₁₆	1	1	yes	characteristic
54	C₂²×Q₈	—	68	2	no	—
55	C₂×Q₁₆	—	34	12	no	—
56	C₂×Q₁₆	—	68	6	no	—
57	C₂²×C₈	D₆₈	1	1	yes	characteristic
58	Q₃₂	—	17	8	no	—
59	Q₃₂	—	34	12	no	—
60	C₂×C₁₆	D₆₈	1	3	yes	—
61	C₂×C₁₆	C₂²×D₁₇	1	3	yes	—
62	C₃₂	C₂²×D₁₇	1	4	yes	—
63	C₁₇	C₂²×Q₆₄	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₈	—	68	12	no	—
65	C₂×Q₈	—	136	6	no	—
66	C₂²×C₄	—	136	2	no	—
67	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
68	Q₁₆	—	34	8	no	—
69	Q₁₆	—	68	12	no	—
70	C₂×C₈	D₁₃₆	1	3	yes	—
71	C₂×C₈	C₂×D₆₈	1	3	yes	—
72	C₁₆	C₂×D₆₈	1	3	yes	—
73	C₁₆	C₂³×D₁₇	1	1	yes	characteristic
74	C₂³	D₂₇₂	1	1	yes	characteristic
75	Q₈	—	68	8	no	—
76	Q₈	—	136	12	no	—
77	C₂×C₄	—	136	12	no	—
78	C₂×C₄	D₂₇₂	1	3	yes	—
79	C₂×C₄	C₂×D₁₃₆	1	3	yes	—
80	C₈	C₂×D₁₃₆	1	3	yes	—
81	C₈	C₂²×D₆₈	1	1	yes	characteristic
82	C₂²	Dic₂₇₂	1	4	yes	direct product factor
83	C₂²	C₂×D₂₇₂	1	3	yes	—
84	C₄	—	136	8	no	—
85	C₄	C₂×D₂₇₂	1	3	yes	—
86	C₄	C₂²×D₁₃₆	1	1	yes	characteristic
87	C₂	C₂×Dic₂₇₂	1	6	yes	direct product factor
88	C₂	C₂²×D₂₇₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₂₇₂
commutator, cosocle	C₁₆×C₁₇	C₂⁴
Frattini	C₁₆	C₂³×D₁₇
Fitting	C₂²×C₁₇×C₃₂	C₂
socle, prime radical	C₂³×C₁₇	D₁₆
hypercenter	C₂²×C₃₂	D₁₇
hypocenter	C₁₇	C₂²×Q₆₄
LCS	G ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₂₇₂	C₂	direct (N × H)
7. Dic₂₇₂	C₂²	direct (N × H)
41. C₂²×C₁₇	Q₆₄	semidirect
52. C₂×C₁₇	C₂×Q₆₄	semidirect
63. C₁₇	C₂²×Q₆₄	semidirect
82. C₂²	Dic₂₇₂	direct (N × H)
87. C₂	C₂×Dic₂₇₂	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×Dic₁₃₆	2
C₂×Dic₂₇₂	2
C₂²×C₁₇×C₃₂	2
C₂²×Q₆₄	17

Subgroups — C22×Dic272 #17192447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₂₇₂ #17192447