Subgroups — #14274136

Order 1904.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₂₃₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₇×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745752	C₂	1	2	yes	maximal, complemented
5	D₇×Dic₁₇	C₂	1	2	yes	maximal, complemented
6	C₂×D₇×C₁₇	C₂²	1	2	yes	—
7	C₂×C₇×D₁₇	C₂²	1	1	yes	characteristic
8	C₄×C₇×C₁₇	C₂²	1	1	yes	characteristic
9	Dic₁₁₉	C₂²	1	2	yes	—
10	C₇×Dic₁₇	C₂²	1	1	yes	characteristic
11	#63254	—	7	1	no	maximal, Hall
12	C₂×C₇×C₁₇	C₂³	1	1	yes	characteristic
13	C₇×D₁₇	—	2	1	no	—
14	D₇×C₁₇	—	2	2	no	—
15	D₄×C₁₇	—	7	1	no	—
16	C₂²⋊D₁₇	—	7	2	no	—
17	C₂×Dic₁₇	—	7	2	no	—
18	C₄×D₁₇	D₇	1	1	yes	characteristic, complemented
19	Dic₃₄	—	7	1	no	—
20	C₇×C₁₇	#41	1	1	yes	characteristic, complemented, Hall
21	#1157	—	17	1	no	maximal, Hall
22	C₂²×C₁₇	—	7	2	no	—
23	C₂×D₁₇	C₂×D₇	1	1	yes	characteristic
24	C₄×C₁₇	C₂×D₇	1	1	yes	characteristic
25	Dic₁₇	—	7	2	no	—
26	Dic₁₇	C₂×D₇	1	1	yes	characteristic
27	C₂×C₄×C₇	—	17	1	no	—
28	C₂²⋊D₇	—	17	2	no	—
29	D₂₈	D₁₇	1	1	yes	characteristic, complemented
30	C₄×D₇	—	17	2	no	—
31	Dic₁₄	—	17	1	no	—
32	C₂×C₁₇	—	14	2	no	—
33	C₂×C₁₇	C₂²×D₇	1	1	yes	characteristic
34	D₁₇	—	2	1	no	—
35	C₂²×C₇	—	17	1	no	—
36	C₂×D₇	C₂×D₁₇	1	2	yes	—
37	C₄×C₇	—	17	1	no	—
38	C₄×C₇	C₂×D₁₇	1	1	yes	characteristic
39	Dic₇	—	17	2	no	—
40	C₁₇	#1157	1	1	yes	characteristic, complemented, Sylow
41	#41	—	119	1	no	Sylow
42	C₂×C₇	—	34	1	no	—
43	C₂×C₇	C₂²×D₁₇	1	1	yes	characteristic
44	D₇	—	2	2	no	—
45	Q₈	—	119	1	no	—
46	D₄	—	7	1	no	—
47	D₄	—	119	2	no	—
48	C₂×C₄	—	17	1	no	—
49	C₂×C₄	—	119	2	no	—
50	C₇	#63254	1	1	yes	characteristic, complemented, Sylow
51	C₂²	—	7	2	no	—
52	C₂²	—	17	1	no	—
53	C₄	—	17	1	no	—
54	C₄	—	119	2	no	—
55	C₄	D₇×D₁₇	1	1	yes	characteristic
56	C₂	—	14	2	no	—
57	C₂	—	34	1	no	—
58	C₂	C₂×D₇×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₇×D₁₇
commutator, socle, cosocle	C₂×C₇×C₁₇	C₂³
Fitting	C₄×C₇×C₁₇	C₂²
hypercenter	C₄	D₇×D₁₇
hypocenter	C₇×C₁₇	#41
LCS	G ▷ C₂×C₇×C₁₇ ▷ C₇×C₁₇
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. Dic₂₃₈	C₂	semidirect
2. C₁₇×D₂₈	C₂	semidirect
3. C₄×C₇×D₁₇	C₂	semidirect
4. #11745752	C₂	semidirect
5. D₇×Dic₁₇	C₂	semidirect
18. C₄×D₁₇	D₇	semidirect
20. C₇×C₁₇	#41	semidirect
29. D₂₈	D₁₇	semidirect
40. C₁₇	#1157	semidirect
50. C₇	#63254	semidirect

Maximal subgroups

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

subgroup	index
Dic₂₃₈	2
C₁₇×D₂₈	2
C₄×C₇×D₁₇	2
#11745752	2
D₇×Dic₁₇	2
#63254	7
#1157	17