Subgroups — #14543917

Order 2176.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763714	C₂	1	2	yes	maximal
2	#11763183	C₂	1	1	yes	maximal, characteristic
3	C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
4	C₂×#63230	—	2	2	no	—
5	#10587560	C₂²	1	1	yes	characteristic
6	#10587539	—	2	2	no	—
7	C₄×#3622	C₄	1	1	yes	characteristic, complemented
8	C₂×Q₈×C₁₇	—	2	1	no	—
9	C₂×D₆₈	—	4	1	no	—
10	C₂×D₆₈	D₄	1	2	yes	—
11	C₂²⋊#595	—	4	2	no	—
12	C₄⋊C₄×C₁₇	—	4	1	no	—
13	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
14	#63230	—	4	4	no	—
15	C₂×#3622	—	2	1	no	—
16	C₂²×D₁₇	—	2	2	no	—
17	C₂×#595	—	8	2	no	—
18	Q₈×C₁₇	—	4	2	no	—
19	C₂×C₄×C₁₇	—	2	1	no	—
20	C₂×C₄×C₁₇	—	4	1	no	—
21	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
22	D₆₈	—	4	2	no	—
23	D₆₈	—	8	2	no	—
24	#3622	—	2	2	no	—
25	#1409	—	17	1	no	maximal, Sylow
26	C₂²×C₁₇	#99	1	1	yes	characteristic
27	C₂×D₁₇	—	4	2	no	—
28	C₂×D₁₇	—	8	2	no	—
29	#595	—	16	2	no	—
30	C₄×C₁₇	—	2	2	no	—
31	C₄×C₁₇	—	2	1	no	—
32	C₄×C₁₇	—	8	1	no	—
33	#492	—	17	1	no	—
34	#353	—	17	2	no	—
35	C₂×C₁₇	—	2	1	no	—
36	C₂×C₁₇	#353	1	1	yes	characteristic
37	D₁₇	—	8	2	no	—
38	C₂×SD₁₆	—	34	2	no	—
39	C₄⋊Q₈	#595	1	1	yes	characteristic, complemented
40	#127	—	17	1	no	—
41	#99	—	34	2	no	—
42	C₄×C₈	—	17	1	no	—
43	C₁₇	#1409	1	1	yes	characteristic, complemented, Sylow
44	C₂×Q₈	—	2	1	no	—
45	C₂×D₄	—	17	2	no	—
46	C₂×D₄	—	68	1	no	—
47	SD₁₆	—	68	4	no	—
48	C₂×C₈	—	34	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₂²⋊C₄	—	68	2	no	—
51	C₄²	C₂×#595	1	1	yes	characteristic
52	C₂³	—	34	2	no	—
53	Q₈	—	4	2	no	—
54	D₄	—	68	2	no	—
55	D₄	—	136	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	—	136	2	no	—
59	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
60	C₈	—	34	2	no	—
61	C₂²	—	68	2	no	—
62	C₂²	—	136	2	no	—
63	C₂²	#10587551	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	2	1	no	—
66	C₄	—	8	1	no	—
67	C₄	—	272	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	136	2	no	—
70	C₂	#11763713	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11763713
commutator	C₄²×C₁₇	C₂×C₄
Frattini	C₄²	C₂×#595
Fitting	C₁₇×C₄⋊Q₈	C₄
socle	C₂×C₁₇	#353
prime radical, cosocle	#10587560	C₂²
hypercenter	C₄⋊Q₈	#595
hypocenter	C₁₇	#1409
LCS	G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. C₁₇×C₄⋊Q₈	C₄	semidirect
7. C₄×#3622	C₄	semidirect
39. C₄⋊Q₈	#595	semidirect
43. C₁₇	#1409	semidirect

Maximal subgroups

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

subgroup	index
#11763714	2
#11763183	2
#1409	17