Subgroups — C₂²×#10587488 #14537356

Order 2176.

600 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₁₇	C₂	1	2	yes	maximal
2	C₂×#10587488	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
5	C₂³×Dic₁₇	—	2	2	no	—
6	C₂²×C₈×C₁₇	C₄	1	8	yes	complemented
7	C₂²×C₈×C₁₇	C₂²	1	6	yes	—
8	C₂×C₄⋊Dic₁₇	—	2	6	no	—
9	C₂×C₄⋊Dic₁₇	C₂²	1	12	yes	—
10	#10587488	C₂²	1	16	yes	direct product factor
11	C₂⁴×C₁₇	D₄	1	1	yes	characteristic
12	C₂²×C₄×C₁₇	C₂×C₄	1	4	yes	—
13	C₂²×C₄×C₁₇	D₄	1	3	yes	—
14	C₂²×C₄×C₁₇	Q₈	1	4	yes	—
15	C₂²×C₄×C₁₇	C₂³	1	3	yes	—
16	C₂²×Dic₁₇	—	2	12	no	—
17	C₂²×Dic₁₇	—	4	8	no	—
18	C₂×C₈×C₁₇	C₂×C₄	1	24	yes	complemented
19	C₂×C₈×C₁₇	C₂³	1	4	yes	—
20	C₄⋊Dic₁₇	—	2	12	no	—
21	C₄⋊Dic₁₇	C₂³	1	8	yes	—
22	C₂³×C₁₇	C₄⋊C₄	1	4	yes	—
23	C₂³×C₁₇	SD₁₆	1	8	yes	—
24	C₂³×C₁₇	C₂×D₄	1	3	yes	—
25	C₂×C₄×C₁₇	C₄⋊C₄	1	12	yes	—
26	C₂×C₄×C₁₇	C₂²×C₄	1	6	yes	—
27	C₂×C₄×C₁₇	C₂×D₄	1	3	yes	—
28	C₂×C₄×C₁₇	C₂×Q₈	1	6	yes	—
29	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
30	C₂×Dic₁₇	—	2	8	no	—
31	C₂×Dic₁₇	—	4	24	no	—
32	C₈×C₁₇	C₂²×C₄	1	8	yes	complemented
33	C₂²×#106	—	17	1	no	maximal, Sylow
34	C₂²×C₁₇	#106	1	16	yes	complemented
35	C₂²×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
36	C₂²×C₁₇	C₂×SD₁₆	1	12	yes	—
37	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
38	C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
39	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
40	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
41	Dic₁₇	—	4	8	no	—
42	C₂³×C₈	D₁₇	1	1	yes	characteristic
43	C₂²×C₄⋊C₄	—	17	2	no	—
44	C₂×#106	—	17	12	no	—
45	C₂×C₁₇	C₂×#106	1	12	yes	complemented
46	C₂×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₁₇	C₂²×SD₁₆	1	2	yes	—
48	C₂³×C₄	—	34	2	no	—
49	C₂³×C₄	C₂×D₁₇	1	1	yes	characteristic
50	C₂²×C₈	Dic₁₇	1	8	yes	complemented
51	C₂²×C₈	C₂×D₁₇	1	6	yes	—
52	C₂×C₄⋊C₄	—	17	12	no	—
53	C₂×C₄⋊C₄	—	34	6	no	—
54	#106	—	17	16	no	—
55	C₁₇	C₂²×#106	1	1	yes	characteristic, complemented, Sylow
56	C₂⁴	D₆₈	1	1	yes	characteristic
57	C₂²×C₄	—	34	12	no	—
58	C₂²×C₄	—	68	8	no	—
59	C₂²×C₄	Dic₃₄	1	4	yes	—
60	C₂²×C₄	D₆₈	1	3	yes	—
61	C₂²×C₄	C₂×Dic₁₇	1	4	yes	—
62	C₂²×C₄	C₂²×D₁₇	1	3	yes	—
63	C₂×C₈	C₂×Dic₁₇	1	24	yes	complemented
64	C₂×C₈	C₂²×D₁₇	1	4	yes	—
65	C₄⋊C₄	—	17	8	no	—
66	C₄⋊C₄	—	34	12	no	—
67	C₂³	#63219	1	8	yes	—
68	C₂³	C₄⋊Dic₁₇	1	4	yes	—
69	C₂³	C₂×D₆₈	1	3	yes	—
70	C₂×C₄	—	34	8	no	—
71	C₂×C₄	—	68	24	no	—
72	C₂×C₄	C₄⋊Dic₁₇	1	12	yes	—
73	C₂×C₄	C₂×Dic₃₄	1	6	yes	—
74	C₂×C₄	C₂×D₆₈	1	3	yes	—
75	C₂×C₄	C₂²×Dic₁₇	1	6	yes	—
76	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
77	C₈	C₂²×Dic₁₇	1	8	yes	complemented
78	C₂²	#10587488	1	16	yes	direct product factor
79	C₂²	C₂×#63219	1	12	yes	—
80	C₂²	C₂×C₄⋊Dic₁₇	1	6	yes	—
81	C₂²	C₂²×D₆₈	1	1	yes	characteristic
82	C₄	—	68	8	no	—
83	C₄	C₂×C₄⋊Dic₁₇	1	6	yes	—
84	C₄	C₂²×Dic₃₄	1	1	yes	characteristic
85	C₄	C₂³×Dic₁₇	1	1	yes	characteristic
86	C₂	C₂×#10587488	1	12	yes	direct product factor
87	C₂	C₂²×C₄⋊Dic₁₇	1	1	yes	characteristic
88	C₂	C₂²×#63219	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₆₈
commutator	C₄×C₁₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₂³×C₈×C₁₇	C₂
socle, prime radical	C₂⁴×C₁₇	D₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	C₂³×C₈	D₁₇
hypocenter	C₁₇	C₂²×#106
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#10587488	C₂	direct (N × H)
6. C₂²×C₈×C₁₇	C₄	semidirect
10. #10587488	C₂²	direct (N × H)
18. C₂×C₈×C₁₇	C₂×C₄	semidirect
32. C₈×C₁₇	C₂²×C₄	semidirect
34. C₂²×C₁₇	#106	semidirect
45. C₂×C₁₇	C₂×#106	semidirect
50. C₂²×C₈	Dic₁₇	semidirect
55. C₁₇	C₂²×#106	semidirect
63. C₂×C₈	C₂×Dic₁₇	semidirect
77. C₈	C₂²×Dic₁₇	semidirect
78. C₂²	#10587488	direct (N × H)
86. C₂	C₂×#10587488	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×C₄⋊Dic₁₇	2
C₂×#10587488	2
C₂³×C₈×C₁₇	2
C₂²×#106	17

Subgroups — C22×#10587488 #14537356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10587488 #14537356