Subgroups — #14544716

Order 2176.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#3624	C₂	1	6	yes	maximal
2	C₄²×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3624	C₂²	1	4	yes	—
4	C₂×C₄²×C₁₇	C₄	1	1	yes	characteristic
5	C₂×C₄×Dic₁₇	C₄	1	3	yes	—
6	C₂×C₄×Dic₁₇	C₂²	1	3	yes	—
7	C₄⋊#3624	—	2	12	no	—
8	C₂²×C₄×C₁₇	C₂×C₄	1	3	yes	—
9	C₂²×Dic₁₇	C₂×C₄	1	3	yes	—
10	C₂²×Dic₁₇	C₂³	1	1	yes	characteristic
11	C₂×#3624	—	2	12	no	—
12	C₄²×C₁₇	C₈	1	4	yes	complemented
13	C₄×Dic₁₇	—	2	6	no	—
14	C₄×Dic₁₇	D₄	1	6	yes	—
15	C₄×Dic₁₇	Q₈	1	6	yes	—
16	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
17	C₂×C₄×C₁₇	C₄⋊C₄	1	6	yes	—
18	C₂×C₄×C₁₇	C₂×C₈	1	6	yes	—
19	C₂×C₄×C₁₇	M₁₆	1	6	yes	—
20	C₂×Dic₁₇	—	2	6	no	—
21	C₂×Dic₁₇	C₄⋊C₄	1	6	yes	—
22	C₂×Dic₁₇	C₂×D₄	1	3	yes	—
23	C₂×Dic₁₇	C₂×Q₈	1	3	yes	—
24	#3624	—	4	4	no	—
25	#1842	—	17	1	no	maximal, Sylow
26	C₂²×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
27	C₂²×C₁₇	C₂²×C₈	1	1	yes	characteristic
28	C₂²×C₁₇	C₂×M₁₆	1	3	yes	—
29	C₄×C₁₇	C₄⋊C₈	1	12	yes	complemented
30	Dic₁₇	—	2	6	no	—
31	Dic₁₇	#127	1	1	yes	characteristic
32	Dic₁₇	C₄⋊Q₈	1	3	yes	—
33	C₂×C₄⋊C₈	—	17	6	no	—
34	C₄³	—	17	1	no	—
35	C₂×C₁₇	#384	1	1	yes	characteristic
36	C₂×C₁₇	C₂×C₄⋊C₈	1	3	yes	—
37	C₂×C₁₇	#423	1	3	yes	—
38	C₂²×C₈	—	17	4	no	—
39	C₂×C₄²	—	17	3	no	—
40	C₂×C₄²	—	17	3	no	—
41	C₂×C₄²	#595	1	1	yes	characteristic
42	C₄⋊C₈	—	34	12	no	—
43	C₁₇	#1842	1	1	yes	characteristic, complemented, Sylow
44	C₂²×C₄	—	17	1	no	—
45	C₂²×C₄	—	17	3	no	—
46	C₂²×C₄	C₂×#595	1	3	yes	—
47	C₂×C₈	—	34	12	no	—
48	C₄²	—	17	6	no	—
49	C₄²	—	17	6	no	—
50	C₄²	—	34	6	no	—
51	C₄²	#3624	1	4	yes	complemented
52	C₂³	C₂²×#595	1	1	yes	characteristic
53	C₂×C₄	—	17	3	no	—
54	C₂×C₄	—	17	3	no	—
55	C₂×C₄	—	17	6	no	—
56	C₂×C₄	—	34	6	no	—
57	C₂×C₄	#63243	1	6	yes	—
58	C₂×C₄	C₄⋊#595	1	6	yes	—
59	C₂×C₄	C₂×#3624	1	6	yes	—
60	C₈	—	68	4	no	—
61	C₂²	C₂×#63243	1	3	yes	—
62	C₂²	C₂×C₄⋊#595	1	3	yes	—
63	C₂²	C₂²×#3624	1	1	yes	characteristic
64	C₄	—	17	1	no	—
65	C₄	—	17	3	no	—
66	C₄	—	34	6	no	—
67	C₄	C₄⋊#3624	1	12	yes	complemented
68	C₂	#11763755	1	1	yes	characteristic
69	C₂	C₂×C₄⋊#3624	1	3	yes	—
70	C₂	#11763813	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#595
commutator	C₂²×C₁₇	C₂²×C₈
Fitting	C₂×C₄²×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₄
cosocle	C₂²×Dic₁₇	C₂³
hypercenter	C₂×C₄²	#595
hypocenter	C₁₇	#1842
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
12. C₄²×C₁₇	C₈	semidirect
29. C₄×C₁₇	C₄⋊C₈	semidirect
43. C₁₇	#1842	semidirect
51. C₄²	#3624	semidirect
67. C₄	C₄⋊#3624	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#3624	2
C₄²×Dic₁₇	2
#1842	17