Subgroups — #14543722

Order 2176.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#63216	C₂	1	2	yes	maximal
2	#11762836	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#117	C₄	1	1	yes	characteristic
4	C₂×#63223	C₄	1	1	yes	characteristic
5	C₂²×#3622	C₂²	1	1	yes	characteristic
6	C₂×#63216	—	2	2	no	—
7	C₄⋊#3622	—	2	2	no	—
8	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
9	C₄⋊C₄×C₁₇	C₈	1	2	yes	—
10	C₂²⋊C₄×C₁₇	—	2	1	no	—
11	C₄²×C₁₇	—	2	1	no	—
12	#63223	—	4	1	no	—
13	C₂×#3622	—	2	1	no	—
14	C₂×#3622	—	4	1	no	—
15	C₂×#3622	D₄	1	2	yes	—
16	#63216	—	4	2	no	—
17	C₂³×C₁₇	M₁₆	1	1	yes	characteristic
18	C₂×C₄×C₁₇	—	2	1	no	—
19	C₂×C₄×C₁₇	—	2	1	no	—
20	C₂×C₄×C₁₇	—	2	2	no	—
21	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
23	#3622	—	2	2	no	—
24	#3622	—	4	1	no	—
25	#1323	—	17	1	no	maximal, Sylow
26	C₂²×C₁₇	—	2	1	no	—
27	C₂²×C₁₇	—	2	2	no	—
28	C₂²×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
29	C₄×C₁₇	—	2	1	no	—
30	C₄×C₁₇	—	4	2	no	—
31	C₄×C₁₇	#99	1	1	yes	characteristic
32	C₄×C₁₇	#101	1	1	yes	characteristic
33	#424	—	17	1	no	—
34	C₂²⋊C₁₆	—	17	2	no	—
35	C₂×C₁₇	—	4	1	no	—
36	C₂×C₁₇	#324	1	1	yes	characteristic
37	C₂×C₁₇	#350	1	2	yes	—
38	C₂×M₁₆	—	17	1	no	—
39	C₂²×C₈	—	17	1	no	—
40	#117	#595	1	1	yes	characteristic
41	C₂×C₁₆	—	34	2	no	—
42	C₄⋊C₈	—	34	2	no	—
43	C₁₇	#1323	1	1	yes	characteristic, complemented, Sylow
44	C₂²×C₄	C₂×#595	1	1	yes	characteristic
45	M₁₆	—	68	1	no	—
46	C₂×C₈	—	17	2	no	—
47	C₂×C₈	—	34	1	no	—
48	C₂×C₈	—	68	1	no	—
49	C₄⋊C₄	#3624	1	2	yes	—
50	C₂²⋊C₄	—	2	1	no	—
51	C₄²	—	2	1	no	—
52	C₁₆	—	68	2	no	—
53	C₂³	#63247	1	1	yes	characteristic
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	C₂×#3624	1	1	yes	characteristic
58	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
59	C₈	—	34	2	no	—
60	C₈	—	68	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	2	2	no	—
63	C₂²	C₂²⋊#3624	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	4	2	no	—
66	C₄	#10587539	1	1	yes	characteristic
67	C₄	#10587545	1	1	yes	characteristic
68	C₂	—	4	1	no	—
69	C₂	#11763654	1	1	yes	characteristic
70	C₂	#11763707	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊#595
commutator	C₂×C₄×C₁₇	C₂×C₈
Frattini	C₂²×C₄	C₂×#595
Fitting	C₁₇×#117	C₄
socle	C₂²×C₁₇	C₂²⋊C₈
prime radical	C₂³×C₁₇	M₁₆
cosocle	C₂²×#3622	C₂²
hypercenter	#117	#595
hypocenter	C₁₇	#1323
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
43. C₁₇	#1323	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#63216	2
#11762836	2
#1323	17