Subgroups — #14526959

Order 2176.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11762606	C₂	1	1	yes	maximal, characteristic
2	#11762604	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#488	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#125	C₄	1	1	yes	characteristic, complemented
5	C₁₇×#124	C₂²	1	1	yes	characteristic
6	C₁₇×#103	—	2	1	no	—
7	C₁₇×#102	—	2	1	no	—
8	C₄×C₈×C₁₇	C₄	1	1	yes	characteristic, complemented
9	#10587507	—	2	1	no	—
10	#10587505	—	2	1	no	—
11	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
12	C₂×D₄×C₁₇	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₇	—	2	1	no	—
14	C₄⋊C₄×C₁₇	—	2	1	no	—
15	C₄⋊C₄×C₁₇	—	4	1	no	—
16	C₂²⋊C₄×C₁₇	—	4	1	no	—
17	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂²⋊Dic₁₇	—	4	1	no	—
19	#63223	—	4	1	no	—
20	C₂³×C₁₇	—	2	1	no	—
21	Q₈×C₁₇	—	4	1	no	—
22	D₄×C₁₇	—	4	1	no	—
23	C₂×C₄×C₁₇	—	2	1	no	—
24	C₂×C₄×C₁₇	—	2	1	no	—
25	C₂×C₄×C₁₇	—	4	1	no	—
26	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×Dic₁₇	—	8	1	no	—
28	C₈×C₁₇	—	4	1	no	—
29	#3622	—	8	1	no	—
30	#1414	—	17	1	no	maximal, Sylow
31	C₂²×C₁₇	—	4	1	no	—
32	C₂²×C₁₇	—	8	1	no	—
33	C₂²×C₁₇	#99	1	1	yes	characteristic
34	C₄×C₁₇	—	2	1	no	—
35	C₄×C₁₇	—	4	1	no	—
36	C₄×C₁₇	—	4	1	no	—
37	C₄×C₁₇	—	8	1	no	—
38	Dic₁₇	—	16	1	no	—
39	#488	D₁₇	1	1	yes	characteristic
40	#355	—	17	1	no	—
41	#354	—	17	1	no	—
42	C₂×C₁₇	—	2	1	no	—
43	C₂×C₁₇	—	8	1	no	—
44	C₂×C₁₇	#354	1	1	yes	characteristic
45	#125	Dic₁₇	1	1	yes	characteristic, complemented
46	#124	C₂×D₁₇	1	1	yes	characteristic
47	#103	—	2	1	no	—
48	#102	—	2	1	no	—
49	#101	—	34	1	no	—
50	#99	—	34	1	no	—
51	C₄×C₈	Dic₁₇	1	1	yes	characteristic, complemented
52	C₁₇	#1414	1	1	yes	characteristic, complemented, Sylow
53	C₂×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
54	C₂×D₄	C₂²⋊D₁₇	1	1	yes	characteristic
55	M₁₆	—	68	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	—	4	1	no	—
59	C₂²⋊C₄	—	4	1	no	—
60	C₂²⋊C₄	—	68	1	no	—
61	C₄²	C₂×Dic₁₇	1	1	yes	characteristic
62	C₂³	—	2	1	no	—
63	Q₈	—	4	1	no	—
64	D₄	—	4	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	4	1	no	—
68	C₂×C₄	—	136	1	no	—
69	C₂×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
70	C₈	—	4	1	no	—
71	C₈	—	136	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	8	1	no	—
74	C₂²	#10587505	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	8	1	no	—
79	C₄	—	272	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	#11762604	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11762604
commutator	C₄²×C₁₇	C₂×C₄
Frattini	#124	C₂×D₁₇
Fitting	C₁₇×#488	C₂
socle	C₂×C₁₇	#354
prime radical	C₂×D₄×C₁₇	D₄
cosocle	C₁₇×#124	C₂²
hypercenter	#488	D₁₇
hypocenter	C₁₇	#1414
LCS	G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #488

Complements

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

N	H ≅ G/N	type
4. C₁₇×#125	C₄	semidirect
8. C₄×C₈×C₁₇	C₄	semidirect
45. #125	Dic₁₇	semidirect
51. C₄×C₈	Dic₁₇	semidirect
52. C₁₇	#1414	semidirect

Maximal subgroups

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

subgroup	index
#11762606	2
#11762604	2
C₁₇×#488	2
#1414	17