Subgroups — #14528372

Order 2176.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763205	C₂	1	2	yes	maximal, complemented
2	#11763183	C₂	1	2	yes	maximal
3	#11763182	C₂	1	1	yes	maximal, characteristic
4	C₈×#3622	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×#63230	—	2	4	no	—
8	C₂×#63219	—	2	2	no	—
9	#10587560	C₂²	1	1	yes	characteristic
10	#10587555	C₂²	1	1	yes	characteristic
11	C₁₇×#106	—	2	2	no	—
12	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	#10587479	—	2	4	no	—
14	C₄×#3622	C₂²	1	2	yes	—
15	C₂×Q₈×C₁₇	—	2	2	no	—
16	C₂×D₆₈	—	2	1	no	—
17	C₂×D₆₈	—	2	2	no	—
18	C₂×Dic₃₄	—	2	1	no	—
19	C₂×C₈×C₁₇	D₄	1	2	yes	—
20	C₄⋊C₄×C₁₇	—	2	2	no	—
21	C₄⋊C₄×C₁₇	—	2	2	no	—
22	C₄²×C₁₇	C₂³	1	1	yes	characteristic
23	#63230	—	4	8	no	—
24	C₄⋊Dic₁₇	—	2	2	no	—
25	C₂×#3622	D₄	1	4	yes	—
26	#63219	—	4	4	no	—
27	C₂²×D₁₇	—	4	1	no	—
28	Q₈×C₁₇	—	4	4	no	—
29	C₂×C₄×C₁₇	—	4	2	no	—
30	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
31	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂×Dic₁₇	—	4	1	no	—
33	D₆₈	—	4	2	no	—
34	D₆₈	—	4	4	no	—
35	Dic₃₄	—	4	2	no	—
36	C₈×C₁₇	SD₁₆	1	4	yes	complemented
37	#3622	SD₁₆	1	8	yes	complemented
38	#1705	—	17	1	no	maximal, Sylow
39	C₂²×C₁₇	#127	1	1	yes	characteristic
40	C₂×D₁₇	—	8	2	no	—
41	C₂×D₁₇	—	8	1	no	—
42	C₄×C₁₇	—	8	2	no	—
43	C₄×C₁₇	C₂×SD₁₆	1	4	yes	—
44	C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
45	Dic₁₇	—	8	1	no	—
46	#498	—	17	2	no	—
47	#498	D₁₇	1	1	yes	characteristic, complemented
48	#492	—	17	2	no	—
49	#492	—	17	1	no	—
50	C₈²	—	17	1	no	—
51	C₂×C₁₇	#492	1	2	yes	—
52	C₂×C₁₇	#492	1	1	yes	characteristic
53	D₁₇	—	16	1	no	—
54	C₂×SD₁₆	—	34	4	no	—
55	C₂×SD₁₆	—	34	2	no	—
56	C₄⋊Q₈	—	17	1	no	—
57	C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
58	#127	—	17	1	no	—
59	#106	—	2	2	no	—
60	#106	—	34	4	no	—
61	C₄×C₈	—	17	2	no	—
62	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
63	C₁₇	#1705	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₈	—	2	2	no	—
65	C₂×Q₈	—	34	1	no	—
66	C₂×D₄	—	34	2	no	—
67	C₂×D₄	—	34	1	no	—
68	SD₁₆	—	68	8	no	—
69	SD₁₆	—	68	4	no	—
70	C₂×C₈	—	17	4	no	—
71	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	34	2	no	—
75	C₄²	C₂²×D₁₇	1	1	yes	characteristic
76	C₂³	—	68	1	no	—
77	Q₈	—	4	4	no	—
78	Q₈	—	68	2	no	—
79	D₄	—	68	4	no	—
80	D₄	—	68	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	68	1	no	—
83	C₂×C₄	D₄×D₁₇	1	2	yes	—
84	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
85	C₈	—	17	8	no	—
86	C₈	#63230	1	4	yes	complemented
87	C₂²	—	136	2	no	—
88	C₂²	—	136	1	no	—
89	C₂²	#10587626	1	1	yes	characteristic
90	C₄	—	8	2	no	—
91	C₄	—	136	1	no	—
92	C₄	SD₁₆×D₁₇	1	4	yes	—
93	C₄	C₂×#63230	1	2	yes	—
94	C₂	—	272	1	no	—
95	C₂	#11763180	1	2	yes	—
96	C₂	#11763183	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10587626
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#498	C₂
socle	C₂²×C₁₇	#127
prime radical	#10587560	C₂²
hypercenter	#498	D₁₇
hypocenter	C₁₇	#1705
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

Complements

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

N	H ≅ G/N	type
1. #11763205	C₂	semidirect
4. C₈×#3622	C₂	semidirect
5. C₁₇×#498	C₂	semidirect
36. C₈×C₁₇	SD₁₆	semidirect
37. #3622	SD₁₆	semidirect
47. #498	D₁₇	semidirect
63. C₁₇	#1705	semidirect
86. C₈	#63230	semidirect

Maximal subgroups

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

subgroup	index
#11763205	2
#11763183	2
#11763182	2
C₈×#3622	2
C₁₇×#498	2
#1705	17