Subgroups — #14526752

Order 2176.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₃₆	C₂	1	1	yes	maximal, characteristic
2	#11762560	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₃₆	C₄	1	1	yes	characteristic
5	C₄×D₆₈	—	2	1	no	—
6	C₂×C₁₆×C₁₇	—	2	1	no	—
7	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	#10587492	—	2	1	no	—
9	#10587489	C₄	1	1	yes	characteristic
10	C₄⋊#3622	—	2	1	no	—
11	C₂×D₆₈	—	2	1	no	—
12	C₂×C₄×D₁₇	—	4	1	no	—
13	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
15	C₄²×C₁₇	D₄	1	1	yes	characteristic
16	#63227	—	4	1	no	—
17	C₄⋊Dic₁₇	—	2	1	no	—
18	C₂×#3622	—	4	1	no	—
19	D₁₃₆	—	2	1	no	—
20	D₁₃₆	C₈	1	2	yes	complemented
21	C₁₆×C₁₇	—	2	2	no	—
22	C₂²×D₁₇	—	4	1	no	—
23	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₇	D₈	1	1	yes	characteristic
25	C₂×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
26	C₂×Dic₁₇	—	4	1	no	—
27	D₆₈	—	2	2	no	—
28	D₆₈	—	4	1	no	—
29	C₄×D₁₇	—	8	2	no	—
30	C₈×C₁₇	—	2	1	no	—
31	C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₇	M₁₆	1	1	yes	characteristic
33	#3622	—	8	1	no	—
34	#1333	—	17	1	no	maximal, Sylow
35	C₂²×C₁₇	#102	1	1	yes	characteristic
36	C₂×D₁₇	—	4	2	no	—
37	C₂×D₁₇	—	8	1	no	—
38	C₂×D₁₇	—	8	1	no	—
39	C₄×C₁₇	—	2	1	no	—
40	C₄×C₁₇	—	2	1	no	—
41	C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
42	C₄×C₁₇	#104	1	1	yes	characteristic
43	Dic₁₇	—	8	1	no	—
44	C₄×D₈	—	17	1	no	—
45	#346	D₁₇	1	1	yes	characteristic, complemented
46	#334	—	17	1	no	—
47	C₂×C₁₇	D₄⋊C₈	1	1	yes	characteristic
48	C₂×C₁₇	#360	1	1	yes	characteristic
49	C₂×C₁₇	#361	1	1	yes	characteristic
50	D₁₇	—	8	2	no	—
51	C₂×D₈	—	17	1	no	—
52	C₄×D₄	—	34	1	no	—
53	C₂×C₁₆	—	2	1	no	—
54	#107	—	17	1	no	—
55	C₄⋊C₈	—	34	1	no	—
56	#102	—	34	1	no	—
57	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
58	C₁₇	#1333	1	1	yes	characteristic, complemented, Sylow
59	C₂×D₄	—	34	1	no	—
60	C₂²×C₄	—	68	1	no	—
61	D₈	—	17	2	no	—
62	D₈	—	34	1	no	—
63	C₂×C₈	—	68	1	no	—
64	C₂×C₈	C₄×D₁₇	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
66	C₄⋊C₄	—	34	1	no	—
67	C₂²⋊C₄	—	68	1	no	—
68	C₄²	D₆₈	1	1	yes	characteristic
69	C₁₆	—	2	2	no	—
70	C₂³	—	68	1	no	—
71	D₄	—	34	2	no	—
72	D₄	—	68	1	no	—
73	C₂×C₄	—	68	1	no	—
74	C₂×C₄	—	136	2	no	—
75	C₂×C₄	#63219	1	1	yes	characteristic
76	C₂×C₄	D₁₃₆	1	1	yes	characteristic
77	C₂×C₄	#63227	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	136	1	no	—
80	C₈	C₈×D₁₇	1	1	yes	characteristic
81	C₈	#63218	1	1	yes	characteristic
82	C₂²	—	68	2	no	—
83	C₂²	—	136	1	no	—
84	C₂²	—	136	1	no	—
85	C₂²	#10587492	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	136	1	no	—
89	C₄	#10587476	1	1	yes	characteristic
90	C₄	#10587491	1	1	yes	characteristic
91	C₂	—	136	2	no	—
92	C₂	#11762538	1	1	yes	characteristic
93	C₂	#11762620	1	1	yes	characteristic
94	C₂	#11762623	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10587492
commutator	C₈×C₁₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₇
Fitting	C₁₇×#346	C₂
socle	C₂²×C₁₇	#102
prime radical	C₂×D₁₃₆	C₄
cosocle	C₄×C₈×C₁₇	C₂²
hypercenter	#346	D₁₇
hypocenter	C₁₇	#1333
LCS	G ▷ 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
2. #11762560	C₂	semidirect
3. C₁₇×#346	C₂	semidirect
20. D₁₃₆	C₈	semidirect
45. #346	D₁₇	semidirect
58. C₁₇	#1333	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₃₆	2
#11762560	2
C₁₇×#346	2
#1333	17