Subgroups — #14530183

Order 2176.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11762857	C₂	1	3	yes	maximal, complemented
2	C₂×#10587494	C₂	1	3	yes	maximal
3	C₁₇×#338	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₆₈	C₂²	1	1	yes	characteristic
5	C₂×M₁₆×C₁₇	C₂²	1	3	yes	—
6	C₂×#63223	C₂²	1	3	yes	—
7	#10587494	—	2	6	no	—
8	#10587493	—	2	6	no	—
9	C₂³×D₁₇	Q₈	1	2	yes	—
10	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
11	C₂×D₆₈	—	2	3	no	—
12	C₂×D₆₈	—	2	3	no	—
13	M₁₆×C₁₇	—	2	3	no	—
14	M₁₆×C₁₇	—	2	3	no	—
15	C₂×C₈×C₁₇	—	2	3	no	—
16	#63223	—	2	3	no	—
17	#63223	D₄	1	6	yes	—
18	C₂×#3622	—	2	3	no	—
19	C₂³×C₁₇	C₂×Q₈	1	1	yes	characteristic
20	C₂²×D₁₇	—	2	6	no	—
21	C₂²×D₁₇	—	8	2	no	—
22	C₂×C₄×C₁₇	C₂×D₄	1	3	yes	—
23	C₂×C₄×C₁₇	#41	1	3	yes	—
24	D₆₈	—	4	3	no	—
25	D₆₈	—	4	1	no	—
26	C₈×C₁₇	—	4	3	no	—
27	#3622	—	2	6	no	—
28	#2069	—	17	1	no	maximal, Sylow
29	C₂²×C₁₇	—	4	1	no	—
30	C₂²×C₁₇	C₂²⋊Q₈	1	3	yes	—
31	C₂×D₁₇	—	4	2	no	—
32	C₂×D₁₇	—	8	6	no	—
33	C₄×C₁₇	#121	1	3	yes	—
34	C₄×C₁₇	#126	1	1	yes	characteristic
35	#430	—	17	3	no	—
36	C₂×#100	—	17	3	no	—
37	#338	D₁₇	1	1	yes	characteristic, complemented
38	C₂×C₁₇	—	2	3	no	—
39	C₂×C₁₇	#396	1	1	yes	characteristic
40	D₁₇	—	8	2	no	—
41	C₂²×D₄	—	17	1	no	—
42	C₂×M₁₆	—	17	3	no	—
43	C₂×M₁₆	C₂×D₁₇	1	3	yes	—
44	#108	—	34	6	no	—
45	#100	—	34	6	no	—
46	C₁₇	#2069	1	1	yes	characteristic, complemented, Sylow
47	C₂⁴	—	17	2	no	—
48	C₂×D₄	—	34	3	no	—
49	C₂×D₄	—	34	3	no	—
50	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
51	M₁₆	—	2	3	no	—
52	M₁₆	—	2	3	no	—
53	M₁₆	—	17	6	no	—
54	M₁₆	—	34	3	no	—
55	C₂×C₈	—	2	3	no	—
56	C₂×C₈	—	34	3	no	—
57	C₂³	—	34	6	no	—
58	C₂³	—	136	2	no	—
59	C₂³	Q₈×D₁₇	1	1	yes	characteristic
60	D₄	—	68	1	no	—
61	D₄	—	68	3	no	—
62	C₂×C₄	#63252	1	3	yes	—
63	C₂×C₄	D₄×D₁₇	1	3	yes	—
64	C₈	—	4	3	no	—
65	C₈	—	34	6	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	68	2	no	—
68	C₂²	—	136	6	no	—
69	C₂²	#10587582	1	3	yes	—
70	C₄	#10587562	1	1	yes	characteristic
71	C₄	#10587570	1	3	yes	—
72	C₂	—	2	3	no	—
73	C₂	—	136	2	no	—
74	C₂	#11762733	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11762733
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#338	C₂
socle	C₂×C₁₇	#396
prime radical	C₂²×D₆₈	C₂²
hypercenter	#338	D₁₇
hypocenter	C₁₇	#2069
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #11762857	C₂	semidirect
3. C₁₇×#338	C₂	semidirect
37. #338	D₁₇	semidirect
46. C₁₇	#2069	semidirect

Maximal subgroups

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

subgroup	index
#11762857	2
C₂×#10587494	2
C₁₇×#338	2
#2069	17