Subgroups — #14545928

Order 2176.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#3633	C₂	1	6	yes	maximal
2	C₄²×#595	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3633	C₂²	1	4	yes	—
4	C₂×C₄×#595	C₄	1	3	yes	—
5	C₂×C₄×#595	C₂²	1	3	yes	—
6	C₄⋊#3633	—	2	12	no	—
7	C₄²×D₁₇	C₄	1	1	yes	characteristic
8	C₂²×#595	C₂×C₄	1	3	yes	—
9	C₂²×#595	C₂³	1	1	yes	characteristic
10	C₂×#3633	—	2	12	no	—
11	C₂×C₄×D₁₇	C₂×C₄	1	3	yes	—
12	C₄×#595	—	2	6	no	—
13	C₄×#595	D₄	1	6	yes	—
14	C₄×#595	Q₈	1	6	yes	—
15	C₄²×C₁₇	C₈	1	1	yes	characteristic, complemented
16	C₄×Dic₁₇	C₈	1	3	yes	complemented
17	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
18	C₂×#595	—	2	6	no	—
19	C₂×#595	C₄⋊C₄	1	6	yes	—
20	C₂×#595	C₂×D₄	1	3	yes	—
21	C₂×#595	C₂×Q₈	1	3	yes	—
22	#3633	—	4	4	no	—
23	C₂×C₄×C₁₇	C₂×C₈	1	3	yes	—
24	C₂×Dic₁₇	C₂×C₈	1	3	yes	—
25	C₄×D₁₇	C₄⋊C₄	1	6	yes	—
26	C₄×D₁₇	M₁₆	1	6	yes	—
27	#1842	—	17	1	no	maximal, Sylow
28	C₂²×C₁₇	C₂²×C₈	1	1	yes	characteristic
29	C₂×D₁₇	C₂×C₄⋊C₄	1	3	yes	—
30	C₂×D₁₇	C₂×M₁₆	1	3	yes	—
31	#595	—	2	6	no	—
32	#595	#127	1	1	yes	characteristic
33	#595	C₄⋊Q₈	1	3	yes	—
34	C₄×C₁₇	C₄⋊C₈	1	6	yes	complemented
35	Dic₁₇	C₄⋊C₈	1	6	yes	complemented
36	C₂×C₄⋊C₈	—	17	6	no	—
37	C₄³	—	17	1	no	—
38	C₂×C₁₇	C₂×C₄⋊C₈	1	3	yes	—
39	D₁₇	#384	1	1	yes	characteristic
40	D₁₇	#423	1	3	yes	—
41	C₂²×C₈	—	17	4	no	—
42	C₂×C₄²	—	17	3	no	—
43	C₂×C₄²	—	17	1	no	—
44	C₂×C₄²	—	17	3	no	—
45	C₄⋊C₈	—	34	12	no	—
46	C₁₇	#1842	1	1	yes	characteristic, complemented, Sylow
47	C₂²×C₄	—	17	1	no	—
48	C₂²×C₄	—	17	3	no	—
49	C₂²×C₄	—	17	3	no	—
50	C₂×C₈	—	34	12	no	—
51	C₄²	—	17	6	no	—
52	C₄²	—	17	6	no	—
53	C₄²	—	17	3	no	—
54	C₄²	—	34	6	no	—
55	C₄²	#3633	1	1	yes	characteristic, complemented
56	C₂³	—	17	1	no	—
57	C₂×C₄	—	17	3	no	—
58	C₂×C₄	—	17	3	no	—
59	C₂×C₄	—	17	3	no	—
60	C₂×C₄	—	17	6	no	—
61	C₂×C₄	—	17	6	no	—
62	C₂×C₄	—	17	6	no	—
63	C₂×C₄	—	34	6	no	—
64	C₂×C₄	C₂×#3633	1	3	yes	—
65	C₈	—	68	4	no	—
66	C₂²	—	17	3	no	—
67	C₂²	—	17	3	no	—
68	C₂²	C₂²×#3633	1	1	yes	characteristic
69	C₄	—	17	1	no	—
70	C₄	—	17	3	no	—
71	C₄	—	17	6	no	—
72	C₄	—	34	6	no	—
73	C₄	C₄⋊#3633	1	6	yes	complemented
74	C₂	—	17	1	no	—
75	C₂	—	17	3	no	—
76	C₂	C₂×C₄⋊#3633	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#3633
commutator, socle	C₂²×C₁₇	C₂²×C₈
Fitting	C₄²×C₁₇	C₈
prime radical	C₂²×D₁₇	C₂²×C₄
cosocle	C₂²×#595	C₂³
hypercenter	C₄²	#3633
hypocenter	C₁₇	#1842
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
15. C₄²×C₁₇	C₈	semidirect
16. C₄×Dic₁₇	C₈	semidirect
34. C₄×C₁₇	C₄⋊C₈	semidirect
35. Dic₁₇	C₄⋊C₈	semidirect
46. C₁₇	#1842	semidirect
55. C₄²	#3633	semidirect
73. C₄	C₄⋊#3633	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#3633	2
C₄²×#595	2
#1842	17