Subgroups — #14543916

Order 2176.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763714	C₂	1	2	yes	maximal, complemented
2	#11763186	C₂	1	1	yes	maximal, characteristic
3	C₂×D₁₃₆	—	2	2	no	—
4	#10587560	C₄	1	1	yes	characteristic, complemented
5	#10587560	C₂²	1	1	yes	characteristic
6	#10587539	—	2	2	no	—
7	C₄×C₈×C₁₇	C₄	1	1	yes	characteristic, complemented
8	C₂×D₆₈	—	2	1	no	—
9	C₂×D₆₈	—	4	1	no	—
10	C₂×D₆₈	—	4	1	no	—
11	C₂×D₆₈	D₄	1	2	yes	—
12	C₂²⋊#595	—	4	2	no	—
13	C₂×C₈×C₁₇	—	2	1	no	—
14	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
15	D₁₃₆	—	4	4	no	—
16	C₂²×D₁₇	—	2	2	no	—
17	C₂²×D₁₇	—	4	1	no	—
18	C₂×#595	—	8	2	no	—
19	C₂×C₄×C₁₇	—	2	1	no	—
20	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
21	D₆₈	—	4	2	no	—
22	D₆₈	—	4	2	no	—
23	D₆₈	—	8	2	no	—
24	D₆₈	—	8	2	no	—
25	C₈×C₁₇	—	2	2	no	—
26	#1408	—	17	1	no	maximal, Sylow
27	C₂²×C₁₇	#99	1	1	yes	characteristic
28	C₂×D₁₇	—	4	2	no	—
29	C₂×D₁₇	—	8	2	no	—
30	C₂×D₁₇	—	8	2	no	—
31	C₂×D₁₇	—	8	1	no	—
32	#595	—	16	2	no	—
33	C₄×C₁₇	—	2	2	no	—
34	C₄×C₁₇	—	2	1	no	—
35	#493	—	17	1	no	—
36	#353	—	17	2	no	—
37	C₂×C₁₇	—	2	1	no	—
38	C₂×C₁₇	#353	1	1	yes	characteristic
39	D₁₇	—	8	2	no	—
40	D₁₇	—	16	1	no	—
41	C₂×D₈	—	34	2	no	—
42	#127	—	17	1	no	—
43	#127	—	17	1	no	—
44	#99	—	34	2	no	—
45	C₄×C₈	#595	1	1	yes	characteristic, complemented
46	C₁₇	#1408	1	1	yes	characteristic, complemented, Sylow
47	C₂×D₄	—	17	2	no	—
48	C₂×D₄	—	34	1	no	—
49	C₂×D₄	—	68	1	no	—
50	C₂×D₄	—	68	1	no	—
51	D₈	—	68	4	no	—
52	C₂×C₈	—	2	1	no	—
53	C₂²⋊C₄	—	68	2	no	—
54	C₄²	C₂×#595	1	1	yes	characteristic
55	C₂³	—	34	2	no	—
56	C₂³	—	68	1	no	—
57	D₄	—	68	2	no	—
58	D₄	—	68	2	no	—
59	D₄	—	136	2	no	—
60	D₄	—	136	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	136	2	no	—
63	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₂²	—	68	2	no	—
66	C₂²	—	136	2	no	—
67	C₂²	—	136	2	no	—
68	C₂²	—	136	1	no	—
69	C₂²	#10587539	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	—	272	2	no	—
73	C₂	—	2	1	no	—
74	C₂	—	272	1	no	—
75	C₂	—	136	2	no	—
76	C₂	#11763714	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11763714
commutator	C₄²×C₁₇	C₂×C₄
Frattini	C₄²	C₂×#595
Fitting	C₄×C₈×C₁₇	C₄
socle	C₂×C₁₇	#353
prime radical	#11763186	C₂
cosocle	#10587560	C₂²
hypercenter	C₄×C₈	#595
hypocenter	C₁₇	#1408
LCS	G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #11763714	C₂	semidirect
4. #10587560	C₄	semidirect
7. C₄×C₈×C₁₇	C₄	semidirect
45. C₄×C₈	#595	semidirect
46. C₁₇	#1408	semidirect

Maximal subgroups

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

subgroup	index
#11763714	2
#11763186	2
#1408	17