Subgroups — #14531099

Order 2176.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763241	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762648	C₂	1	2	yes	maximal, complemented
4	D₅₄₄	C₂	1	2	yes	maximal, complemented
5	C₁₇×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#10587590	—	2	1	no	—
7	C₂×D₁₃₆	—	2	1	no	—
8	C₂×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
9	Dic₁₃₆	C₂²	1	1	yes	characteristic, complemented
10	#10587471	—	2	1	no	—
11	D₂₇₂	—	2	1	no	—
12	D₂₇₂	C₂²	1	2	yes	—
13	D₂₇₂	C₂²	1	1	yes	characteristic, complemented
14	C₁₇×C₃₂	C₂²	1	2	yes	complemented
15	#63252	—	4	1	no	—
16	C₂×D₆₈	—	4	1	no	—
17	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
18	Dic₆₈	—	2	1	no	—
19	D₁₃₆	—	2	2	no	—
20	D₁₃₆	—	2	1	no	—
21	D₁₃₆	—	4	1	no	—
22	#63219	—	4	1	no	—
23	C₁₆×C₁₇	D₄	1	1	yes	characteristic
24	C₁₆×C₁₇	C₂³	1	1	yes	characteristic
25	C₂²×D₁₇	—	8	1	no	—
26	C₂×C₄×C₁₇	D₈	1	1	yes	characteristic
27	C₂²⋊D₁₇	—	8	1	no	—
28	D₆₈	—	4	2	no	—
29	D₆₈	—	4	1	no	—
30	D₆₈	—	8	1	no	—
31	C₄×D₁₇	—	8	1	no	—
32	Dic₃₄	—	4	1	no	—
33	C₈×C₁₇	D₈	1	1	yes	characteristic
34	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
35	#2263	—	17	1	no	maximal, Sylow
36	C₂²×C₁₇	D₁₆	1	1	yes	characteristic
37	C₂×D₁₇	—	8	1	no	—
38	C₂×D₁₇	—	8	2	no	—
39	C₂×D₁₇	—	16	2	no	—
40	C₄×C₁₇	D₁₆	1	1	yes	characteristic
41	C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
42	Dic₁₇	—	8	1	no	—
43	#508	—	17	1	no	—
44	C₂×D₁₆	—	17	1	no	—
45	SD₆₄	—	17	2	no	—
46	D₃₂	—	17	2	no	—
47	M₆₄	D₁₇	1	1	yes	characteristic, complemented
48	C₂×C₁₇	—	2	1	no	—
49	C₂×C₁₇	C₂×D₁₆	1	1	yes	characteristic
50	D₁₇	—	16	1	no	—
51	D₁₇	—	16	2	no	—
52	#135	—	34	1	no	—
53	C₂×D₈	—	34	1	no	—
54	Q₃₂	—	17	1	no	—
55	SD₃₂	—	34	1	no	—
56	D₁₆	—	17	2	no	—
57	D₁₆	—	17	1	no	—
58	D₁₆	—	34	1	no	—
59	C₂×C₁₆	C₂×D₁₇	1	1	yes	characteristic
60	C₃₂	C₂×D₁₇	1	2	yes	complemented
61	C₁₇	#2263	1	1	yes	characteristic, complemented, Sylow
62	#41	—	68	1	no	—
63	C₂×D₄	—	68	1	no	—
64	Q₁₆	—	34	1	no	—
65	SD₁₆	—	68	1	no	—
66	D₈	—	34	2	no	—
67	D₈	—	34	1	no	—
68	D₈	—	68	1	no	—
69	C₂×C₈	D₆₈	1	1	yes	characteristic
70	C₁₆	D₆₈	1	1	yes	characteristic
71	C₁₆	C₂²×D₁₇	1	1	yes	characteristic
72	C₂³	—	136	1	no	—
73	Q₈	—	68	1	no	—
74	D₄	—	68	2	no	—
75	D₄	—	68	1	no	—
76	D₄	—	136	1	no	—
77	D₄	—	136	1	no	—
78	C₂×C₄	—	136	1	no	—
79	C₂×C₄	D₁₃₆	1	1	yes	characteristic
80	C₈	D₁₃₆	1	1	yes	characteristic
81	C₈	C₂×D₆₈	1	1	yes	characteristic
82	C₂²	—	272	2	no	—
83	C₂²	—	136	1	no	—
84	C₂²	—	136	2	no	—
85	C₂²	D₂₇₂	1	1	yes	characteristic
86	C₄	—	136	1	no	—
87	C₄	D₂₇₂	1	1	yes	characteristic
88	C₄	C₂×D₁₃₆	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	272	2	no	—
91	C₂	—	272	1	no	—
92	C₂	C₂×D₂₇₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₇₂
commutator, cosocle	C₁₆×C₁₇	C₂³
Frattini	C₁₆	C₂²×D₁₇
Fitting	C₁₇×M₆₄	C₂
socle	C₂×C₁₇	C₂×D₁₆
hypercenter	M₆₄	D₁₇
hypocenter	C₁₇	#2263
LCS	G ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
1. #11763241	C₂	semidirect
2. C₂×D₂₇₂	C₂	semidirect
3. #11762648	C₂	semidirect
4. D₅₄₄	C₂	semidirect
5. C₁₇×M₆₄	C₂	semidirect
9. Dic₁₃₆	C₂²	semidirect
13. D₂₇₂	C₂²	semidirect
14. C₁₇×C₃₂	C₂²	semidirect
47. M₆₄	D₁₇	semidirect
60. C₃₂	C₂×D₁₇	semidirect
61. C₁₇	#2263	semidirect

Maximal subgroups

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

subgroup	index
#11763241	2
C₂×D₂₇₂	2
#11762648	2
D₅₄₄	2
C₁₇×M₆₄	2
#2263	17