Subgroups — #14528455

Order 2176.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763214	C₂	1	2	yes	maximal, complemented
2	#11763189	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763182	C₂	1	1	yes	maximal, characteristic
4	#11763156	C₂	1	2	yes	maximal
5	C₁₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₆₈	—	2	2	no	—
7	C₂×#63219	—	2	2	no	—
8	#10587560	C₂²	1	1	yes	characteristic
9	#10587555	C₂²	1	1	yes	characteristic
10	#10587555	C₂²	1	2	yes	—
11	C₄×C₈×C₁₇	C₂²	1	2	yes	—
12	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	#10587492	—	2	4	no	—
14	#10587489	—	2	4	no	—
15	C₂×D₆₈	—	2	2	no	—
16	C₂×D₆₈	—	2	1	no	—
17	C₂×Dic₃₄	—	2	1	no	—
18	C₂×Dic₃₄	—	2	2	no	—
19	C₂×C₈×C₁₇	D₄	1	4	yes	—
20	C₂×C₈×C₁₇	D₄	1	2	yes	—
21	C₄²×C₁₇	C₂³	1	1	yes	characteristic
22	C₄⋊Dic₁₇	—	2	2	no	—
23	C₄⋊Dic₁₇	—	2	2	no	—
24	C₄⋊Dic₁₇	—	2	2	no	—
25	Dic₆₈	—	4	4	no	—
26	#63219	—	4	4	no	—
27	C₂²×D₁₇	—	4	1	no	—
28	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
30	C₂×Dic₁₇	—	4	1	no	—
31	C₂×Dic₁₇	—	4	2	no	—
32	D₆₈	—	4	2	no	—
33	D₆₈	—	4	4	no	—
34	Dic₃₄	—	4	2	no	—
35	Dic₃₄	—	4	4	no	—
36	C₈×C₁₇	—	2	4	no	—
37	C₈×C₁₇	D₈	1	4	yes	—
38	#1722	—	17	1	no	maximal, Sylow
39	C₂²×C₁₇	#127	1	1	yes	characteristic
40	C₂×D₁₇	—	8	1	no	—
41	C₂×D₁₇	—	8	2	no	—
42	C₄×C₁₇	C₂×D₈	1	2	yes	—
43	C₄×C₁₇	#137	1	4	yes	—
44	Dic₁₇	—	8	1	no	—
45	Dic₁₇	—	8	2	no	—
46	#500	—	17	2	no	—
47	#494	—	17	1	no	—
48	#492	—	17	1	no	—
49	#486	—	17	2	no	—
50	#322	D₁₇	1	1	yes	characteristic, complemented
51	C₂×C₁₇	#493	1	1	yes	characteristic
52	C₂×C₁₇	#497	1	2	yes	—
53	D₁₇	—	16	1	no	—
54	C₂×Q₁₆	—	34	2	no	—
55	C₂×SD₁₆	—	34	2	no	—
56	C₄⋊Q₈	—	17	2	no	—
57	C₄⋊Q₈	—	17	1	no	—
58	#127	—	17	1	no	—
59	#107	—	34	4	no	—
60	#102	—	34	4	no	—
61	C₄×C₈	C₂×D₁₇	1	2	yes	—
62	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
63	C₁₇	#1722	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₈	—	34	1	no	—
65	C₂×Q₈	—	34	2	no	—
66	C₂×D₄	—	34	2	no	—
67	C₂×D₄	—	34	1	no	—
68	Q₁₆	—	68	4	no	—
69	SD₁₆	—	68	4	no	—
70	C₂×C₈	D₆₈	1	4	yes	—
71	C₂×C₈	D₆₈	1	2	yes	—
72	C₄⋊C₄	—	34	2	no	—
73	C₄⋊C₄	—	34	2	no	—
74	C₄⋊C₄	—	34	2	no	—
75	C₄²	C₂²×D₁₇	1	1	yes	characteristic
76	C₂³	—	68	1	no	—
77	Q₈	—	68	2	no	—
78	Q₈	—	68	4	no	—
79	D₄	—	68	2	no	—
80	D₄	—	68	4	no	—
81	C₂×C₄	—	68	2	no	—
82	C₂×C₄	—	68	1	no	—
83	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
84	C₂×C₄	C₂×D₆₈	1	2	yes	—
85	C₈	—	2	4	no	—
86	C₈	D₁₃₆	1	4	yes	—
87	C₂²	—	136	2	no	—
88	C₂²	—	136	1	no	—
89	C₂²	#10587560	1	1	yes	characteristic
90	C₄	—	136	1	no	—
91	C₄	—	136	2	no	—
92	C₄	C₂×D₁₃₆	1	2	yes	—
93	C₄	#10587595	1	4	yes	—
94	C₂	—	272	1	no	—
95	C₂	#11763186	1	1	yes	characteristic
96	C₂	#11763203	1	2	yes	—

	subgroup	G/H
center	C₂²	#10587560
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#322	C₂
socle	C₂²×C₁₇	#127
prime radical	#10587560	C₂²
hypercenter	#322	D₁₇
hypocenter	C₁₇	#1722
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
1. #11763214	C₂	semidirect
2. #11763189	C₂	semidirect
5. C₁₇×#322	C₂	semidirect
50. #322	D₁₇	semidirect
63. C₁₇	#1722	semidirect

Maximal subgroups

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

subgroup	index
#11763214	2
#11763189	2
#11763182	2
#11763156	2
C₁₇×#322	2
#1722	17