Subgroups — #14543806

Order 2176.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#63216	C₂	1	2	yes	maximal
2	C₈×#3622	C₂	1	1	yes	maximal, characteristic
3	C₂×#63216	—	2	2	no	—
4	C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
5	C₄×#3622	C₄	1	1	yes	characteristic
6	C₄×#3622	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₁₇	C₈	1	1	yes	characteristic
8	C₂×C₈×C₁₇	C₈	1	1	yes	characteristic
9	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
10	C₂×#3622	—	2	1	no	—
11	C₂×#3622	D₄	1	1	yes	characteristic
12	C₂×#3622	Q₈	1	1	yes	characteristic
13	#63216	—	4	2	no	—
14	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
16	C₂×C₄×C₁₇	M₁₆	1	1	yes	characteristic
17	C₈×C₁₇	—	2	1	no	—
18	C₈×C₁₇	C₁₆	1	2	yes	complemented
19	#3622	—	2	2	no	—
20	#3622	—	2	1	no	—
21	#3622	SD₁₆	1	2	yes	—
22	#1367	—	17	1	no	maximal, Sylow
23	C₂²×C₁₇	C₄⋊C₈	1	1	yes	characteristic
24	C₄×C₁₇	—	2	1	no	—
25	C₄×C₁₇	#106	1	1	yes	characteristic
26	C₄×C₁₇	#108	1	1	yes	characteristic
27	C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
28	C₄×C₁₇	M₃₂	1	1	yes	characteristic
29	C₄⋊C₁₆	—	17	2	no	—
30	C₈²	—	17	1	no	—
31	C₂×C₁₇	#334	1	1	yes	characteristic
32	C₂×C₁₇	C₄⋊C₁₆	1	1	yes	characteristic
33	C₂×C₁₇	#364	1	1	yes	characteristic
34	C₂×C₁₆	—	34	2	no	—
35	C₄×C₈	—	17	1	no	—
36	C₄×C₈	—	17	1	no	—
37	C₄×C₈	#595	1	1	yes	characteristic
38	C₁₇	#1367	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₈	—	17	1	no	—
40	C₂×C₈	—	17	1	no	—
41	C₂×C₈	—	34	1	no	—
42	C₂×C₈	#3624	1	1	yes	characteristic
43	C₂×C₈	#3624	1	1	yes	characteristic
44	C₄²	C₂×#595	1	1	yes	characteristic
45	C₁₆	—	68	2	no	—
46	C₂×C₄	#63243	1	1	yes	characteristic
47	C₂×C₄	C₄⋊#595	1	1	yes	characteristic
48	C₂×C₄	C₂×#3624	1	1	yes	characteristic
49	C₈	—	2	1	no	—
50	C₈	—	17	2	no	—
51	C₈	—	34	2	no	—
52	C₈	—	34	1	no	—
53	C₈	#63216	1	2	yes	complemented
54	C₂²	C₄⋊#3624	1	1	yes	characteristic
55	C₄	—	2	1	no	—
56	C₄	#10587533	1	1	yes	characteristic
57	C₄	#10587535	1	1	yes	characteristic
58	C₄	C₂×#63216	1	1	yes	characteristic
59	C₄	#10587538	1	1	yes	characteristic
60	C₂	#11763670	1	1	yes	characteristic
61	C₂	C₄⋊#63216	1	1	yes	characteristic
62	C₂	#11763731	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#595
commutator	C₄×C₁₇	C₂×C₁₆
Frattini	C₄²	C₂×#595
Fitting	C₄×C₈×C₁₇	C₄
socle, prime radical	C₂²×C₁₇	C₄⋊C₈
cosocle	C₄×#3622	C₂²
hypercenter	C₄×C₈	#595
hypocenter	C₁₇	#1367
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
18. C₈×C₁₇	C₁₆	semidirect
38. C₁₇	#1367	semidirect
53. C₈	#63216	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#63216	2
C₈×#3622	2
#1367	17