Subgroups — #14528533

Order 2176.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10587474	C₂	1	2	yes	maximal
2	#11762564	C₂	1	4	yes	maximal
3	C₄³×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄²×C₁₇	C₄	1	4	yes	—
5	C₂×C₄²×C₁₇	C₂²	1	2	yes	—
6	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
7	C₂²×#3622	C₂²	1	4	yes	—
8	#10587474	C₄	1	8	yes	complemented
9	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
10	C₂²×C₄×C₁₇	C₂×C₄	1	4	yes	—
11	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
12	C₄²×C₁₇	—	2	8	no	—
13	C₄²×C₁₇	—	2	4	no	—
14	C₄²×C₁₇	C₂×C₄	1	4	yes	—
15	C₂×#3622	—	2	8	no	—
16	C₂×#3622	C₂×C₄	1	8	yes	—
17	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₇	—	2	4	no	—
19	C₂×C₄×C₁₇	—	2	4	no	—
20	C₂×C₄×C₁₇	—	2	4	no	—
21	C₂×C₄×C₁₇	C₄²	1	4	yes	—
22	C₂×C₄×C₁₇	M₁₆	1	8	yes	—
23	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
24	C₂×C₄×C₁₇	#41	1	4	yes	—
25	#3622	—	2	8	no	—
26	#1745	—	17	1	no	maximal, Sylow
27	C₂²×C₁₇	C₂×C₄²	1	1	yes	characteristic
28	C₂²×C₁₇	#117	1	2	yes	—
29	C₂²×C₁₇	C₂×M₁₆	1	4	yes	—
30	C₄×C₁₇	—	2	4	no	—
31	C₄×C₁₇	—	2	8	no	—
32	C₄×C₁₇	#117	1	4	yes	—
33	C₂×#97	—	17	2	no	—
34	C₄³	D₁₇	1	1	yes	characteristic
35	#336	—	17	4	no	—
36	C₂×C₁₇	#376	1	1	yes	characteristic
37	C₂×C₁₇	C₄×M₁₆	1	2	yes	—
38	C₂×C₁₇	#432	1	4	yes	—
39	C₂²×C₈	—	17	4	no	—
40	C₂×C₄²	Dic₁₇	1	4	yes	—
41	C₂×C₄²	C₂×D₁₇	1	2	yes	—
42	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
43	#97	—	17	8	no	—
44	C₁₇	#1745	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₄	C₂×Dic₁₇	1	4	yes	—
46	C₂²×C₄	C₂×Dic₁₇	1	2	yes	—
47	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
48	C₂×C₈	—	17	8	no	—
49	C₂×C₈	—	34	8	no	—
50	C₄²	—	2	4	no	—
51	C₄²	—	2	8	no	—
52	C₄²	C₄×D₁₇	1	4	yes	—
53	C₂³	C₂²×Dic₁₇	1	1	yes	characteristic
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	#63223	1	8	yes	—
58	C₂×C₄	C₄×Dic₁₇	1	4	yes	—
59	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
60	C₂×C₄	#63252	1	4	yes	—
61	C₈	—	34	8	no	—
62	C₂²	C₂×#63223	1	4	yes	—
63	C₂²	C₂×C₄×Dic₁₇	1	1	yes	characteristic
64	C₂²	#10587612	1	2	yes	—
65	C₄	—	2	4	no	—
66	C₄	—	2	8	no	—
67	C₄	#10587558	1	4	yes	—
68	C₂	#11762655	1	1	yes	characteristic
69	C₂	C₄×#63223	1	2	yes	—
70	C₂	#11762866	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₇
commutator	C₂²×C₁₇	C₂×C₄²
Fitting	C₄³×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₄³	D₁₇
hypocenter	C₁₇	#1745
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
8. #10587474	C₄	semidirect
44. C₁₇	#1745	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10587474	2
#11762564	2
C₄³×C₁₇	2
#1745	17