Subgroups — #14526697

Order 2176.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11762833	C₂	1	2	yes	maximal
2	C₁₇×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×C₁₇	C₄	1	2	yes	—
4	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
5	C₂×#63223	C₄	1	4	yes	—
6	C₁₇×C₄⋊C₈	—	2	2	no	—
7	C₄⋊#3622	—	2	2	no	—
8	C₄⋊#3622	—	2	2	no	—
9	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
10	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
11	M₁₆×C₁₇	—	4	2	no	—
12	C₂×C₈×C₁₇	—	2	2	no	—
13	C₄²×C₁₇	D₄	1	1	yes	characteristic
14	C₄²×C₁₇	D₄	1	2	yes	—
15	C₄²×C₁₇	Q₈	1	1	yes	characteristic
16	#63223	—	4	4	no	—
17	C₂×#3622	—	2	4	no	—
18	C₂³×C₁₇	C₄²	1	1	yes	characteristic
19	C₂×C₄×C₁₇	—	2	1	no	—
20	C₂×C₄×C₁₇	—	2	2	no	—
21	C₂×C₄×C₁₇	—	2	2	no	—
22	C₂×C₄×C₁₇	—	2	1	no	—
23	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
25	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
27	C₈×C₁₇	—	4	2	no	—
28	#3622	—	4	4	no	—
29	#1309	—	17	1	no	maximal, Sylow
30	C₂²×C₁₇	—	2	2	no	—
31	C₂²×C₁₇	—	2	1	no	—
32	C₂²×C₁₇	#95	1	1	yes	characteristic
33	C₄×C₁₇	—	2	2	no	—
34	C₄×C₁₇	—	2	1	no	—
35	C₄×C₁₇	#100	1	2	yes	—
36	C₄×C₁₇	#100	1	1	yes	characteristic
37	C₄×C₁₇	#101	1	2	yes	—
38	C₄×C₁₇	#101	1	1	yes	characteristic
39	#423	—	17	2	no	—
40	#423	D₁₇	1	1	yes	characteristic
41	C₂×C₁₇	—	4	1	no	—
42	C₂×C₁₇	#343	1	2	yes	—
43	C₂×C₁₇	#343	1	1	yes	characteristic
44	C₂×M₁₆	—	17	4	no	—
45	C₂×M₁₆	Dic₁₇	1	2	yes	—
46	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
47	C₄⋊C₈	—	2	2	no	—
48	C₄⋊C₈	—	34	2	no	—
49	C₄⋊C₈	—	34	2	no	—
50	C₁₇	#1309	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₄	C₄×D₁₇	1	2	yes	—
52	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
53	M₁₆	—	4	2	no	—
54	M₁₆	—	68	4	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	34	4	no	—
57	C₄²	Dic₃₄	1	1	yes	characteristic
58	C₄²	D₆₈	1	1	yes	characteristic
59	C₄²	C₂²⋊D₁₇	1	2	yes	—
60	C₂³	C₄×Dic₁₇	1	1	yes	characteristic
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	#63225	1	2	yes	—
66	C₂×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
67	C₂×C₄	#63227	1	2	yes	—
68	C₂×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
69	C₈	—	4	2	no	—
70	C₈	—	68	4	no	—
71	C₂²	—	2	2	no	—
72	C₂²	—	2	1	no	—
73	C₂²	#10587502	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	2	1	no	—
76	C₄	#10587494	1	2	yes	—
77	C₄	#10587495	1	2	yes	—
78	C₄	#10587504	1	1	yes	characteristic
79	C₄	#10587507	1	1	yes	characteristic
80	C₂	—	4	1	no	—
81	C₂	#11762577	1	2	yes	—
82	C₂	#11762578	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10587502
commutator, prime radical	C₂³×C₁₇	C₄²
Frattini	C₂×C₄²	C₂×D₁₇
Fitting	C₁₇×#423	C₂
socle	C₂²×C₁₇	#95
cosocle	C₂×C₄²×C₁₇	C₂²
hypercenter	#423	D₁₇
hypocenter	C₁₇	#1309
LCS	G ▷ C₂³×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
50. C₁₇	#1309	semidirect

Maximal subgroups

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

subgroup	index
#11762833	2
C₁₇×#423	2
#1309	17