Subgroups — #14531026

Order 2176.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763211	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763186	C₂	1	1	yes	maximal, characteristic
3	#11762611	C₂	1	4	yes	maximal, complemented
4	C₁₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₃₆	—	2	1	no	—
6	C₂×D₁₃₆	C₂²	1	2	yes	—
7	#10587560	—	2	1	no	—
8	#10587556	—	2	1	no	—
9	C₂×C₁₆×C₁₇	C₂²	1	2	yes	—
10	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
11	#10587489	C₂²	1	2	yes	—
12	#10587488	—	2	1	no	—
13	C₂×D₆₈	—	2	2	no	—
14	C₂×D₆₈	—	4	2	no	—
15	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
17	C₄²×C₁₇	D₄	1	1	yes	characteristic
18	C₄⋊Dic₁₇	—	2	2	no	—
19	#63225	—	4	2	no	—
20	D₁₃₆	—	2	4	no	—
21	D₁₃₆	—	4	2	no	—
22	C₁₆×C₁₇	—	2	2	no	—
23	C₂²×D₁₇	—	4	2	no	—
24	C₂×C₄×C₁₇	D₈	1	2	yes	—
25	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
26	C₂×Dic₁₇	—	4	2	no	—
27	D₆₈	—	4	2	no	—
28	D₆₈	—	4	2	no	—
29	D₆₈	—	8	4	no	—
30	C₈×C₁₇	SD₁₆	1	2	yes	—
31	C₈×C₁₇	#41	1	2	yes	—
32	#2244	—	17	1	no	maximal, Sylow
33	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
34	C₂×D₁₇	—	8	2	no	—
35	C₂×D₁₇	—	8	4	no	—
36	C₄×C₁₇	—	2	2	no	—
37	C₄×C₁₇	#124	1	1	yes	characteristic
38	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
39	Dic₁₇	—	8	2	no	—
40	#499	—	17	1	no	—
41	#493	—	17	1	no	—
42	#357	—	17	4	no	—
43	#346	D₁₇	1	1	yes	characteristic, complemented
44	C₂×C₁₇	#486	1	1	yes	characteristic
45	C₂×C₁₇	#509	1	2	yes	—
46	D₁₇	—	16	2	no	—
47	C₂×D₈	—	17	2	no	—
48	C₂×D₈	—	34	1	no	—
49	#127	—	34	1	no	—
50	#125	—	34	1	no	—
51	C₂×C₁₆	C₂×D₁₇	1	2	yes	—
52	#107	—	17	2	no	—
53	#106	—	34	1	no	—
54	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
55	C₁₇	#2244	1	1	yes	characteristic, complemented, Sylow
56	C₂×D₄	—	34	2	no	—
57	C₂×D₄	—	68	2	no	—
58	D₈	—	34	4	no	—
59	D₈	—	68	2	no	—
60	C₂×C₈	D₆₈	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
62	C₄⋊C₄	—	34	2	no	—
63	C₄⋊C₄	—	68	2	no	—
64	C₄²	D₆₈	1	1	yes	characteristic
65	C₁₆	—	2	2	no	—
66	C₂³	—	68	2	no	—
67	D₄	—	68	2	no	—
68	D₄	—	68	2	no	—
69	D₄	—	136	4	no	—
70	C₂×C₄	—	68	2	no	—
71	C₂×C₄	D₁₃₆	1	2	yes	—
72	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
73	C₈	#63219	1	2	yes	—
74	C₈	#63252	1	2	yes	—
75	C₂²	—	136	4	no	—
76	C₂²	—	136	2	no	—
77	C₂²	C₂×D₁₃₆	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	136	2	no	—
80	C₄	#10587561	1	1	yes	characteristic
81	C₄	C₂×#63219	1	1	yes	characteristic
82	C₂	—	272	2	no	—
83	C₂	#11763156	1	1	yes	characteristic
84	C₂	#11763247	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₃₆
commutator, cosocle	C₂×C₈×C₁₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₁₇×#346	C₂
socle	C₂²×C₁₇	C₂×D₈
prime radical	#11763186	C₂
hypercenter	#346	D₁₇
hypocenter	C₁₇	#2244
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
1. #11763211	C₂	semidirect
3. #11762611	C₂	semidirect
4. C₁₇×#346	C₂	semidirect
43. #346	D₁₇	semidirect
55. C₁₇	#2244	semidirect

Maximal subgroups

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

subgroup	index
#11763211	2
#11763186	2
#11762611	2
C₁₇×#346	2
#2244	17