Subgroups — #14543692

Order 2176.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763812	C₂	1	2	yes	maximal
2	#11762832	C₂	1	1	yes	maximal, characteristic
3	C₂×#63247	C₄	1	4	yes	—
4	C₂×M₁₆×C₁₇	C₄	1	1	yes	characteristic
5	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
6	C₂×#63223	C₄	1	1	yes	characteristic
7	#10587544	—	2	2	no	—
8	C₄⋊#3624	—	2	2	no	—
9	#10587485	—	2	2	no	—
10	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
11	C₂²×Dic₁₇	C₂×C₄	1	2	yes	—
12	#63247	—	4	4	no	—
13	C₂×#3624	—	2	4	no	—
14	M₁₆×C₁₇	—	4	1	no	—
15	C₂×C₈×C₁₇	—	2	1	no	—
16	C₄×Dic₁₇	D₄	1	1	yes	characteristic
17	C₄×Dic₁₇	D₄	1	2	yes	—
18	C₄×Dic₁₇	Q₈	1	1	yes	characteristic
19	#63223	—	4	1	no	—
20	C₂×#3622	—	2	1	no	—
21	C₂³×C₁₇	C₄²	1	1	yes	characteristic
22	C₂×C₄×C₁₇	—	2	1	no	—
23	C₂×C₄×C₁₇	—	2	1	no	—
24	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂×Dic₁₇	—	2	2	no	—
27	C₂×Dic₁₇	—	2	2	no	—
28	C₂×Dic₁₇	C₂²⋊C₄	1	2	yes	—
29	C₂×Dic₁₇	C₄⋊C₄	1	2	yes	—
30	#3624	—	4	4	no	—
31	C₈×C₁₇	—	4	1	no	—
32	#3622	—	4	1	no	—
33	#1309	—	17	1	no	maximal, Sylow
34	C₂²×C₁₇	—	2	2	no	—
35	C₂²×C₁₇	—	2	1	no	—
36	C₂²×C₁₇	#95	1	1	yes	characteristic
37	C₄×C₁₇	—	2	1	no	—
38	C₄×C₁₇	#100	1	1	yes	characteristic
39	C₄×C₁₇	#101	1	1	yes	characteristic
40	Dic₁₇	—	2	2	no	—
41	Dic₁₇	#100	1	2	yes	—
42	Dic₁₇	#101	1	2	yes	—
43	#423	—	17	2	no	—
44	#423	—	17	1	no	—
45	C₂×C₁₇	—	4	1	no	—
46	C₂×C₁₇	#343	1	2	yes	—
47	C₂×C₁₇	#343	1	1	yes	characteristic
48	C₂×M₁₆	—	17	4	no	—
49	C₂×M₁₆	—	17	1	no	—
50	C₂×M₁₆	#595	1	1	yes	characteristic
51	C₂×C₄²	—	17	1	no	—
52	C₄⋊C₈	—	34	2	no	—
53	C₄⋊C₈	—	34	2	no	—
54	C₄⋊C₈	—	34	2	no	—
55	C₁₇	#1309	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₄	—	17	2	no	—
57	C₂²×C₄	C₂×#595	1	1	yes	characteristic
58	M₁₆	—	4	1	no	—
59	M₁₆	—	68	1	no	—
60	M₁₆	—	68	4	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	34	4	no	—
63	C₂×C₈	—	34	1	no	—
64	C₄²	—	17	1	no	—
65	C₄²	—	17	1	no	—
66	C₄²	—	17	2	no	—
67	C₂³	C₄×#595	1	1	yes	characteristic
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	17	2	no	—
71	C₂×C₄	—	17	2	no	—
72	C₂×C₄	—	34	2	no	—
73	C₂×C₄	—	34	2	no	—
74	C₂×C₄	C₄⋊#595	1	1	yes	characteristic
75	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
76	C₈	—	4	1	no	—
77	C₈	—	68	4	no	—
78	C₈	—	68	1	no	—
79	C₂²	—	2	2	no	—
80	C₂²	—	2	1	no	—
81	C₂²	(C₂×#595)⋊C₄	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	17	2	no	—
84	C₄	—	17	2	no	—
85	C₄	—	34	2	no	—
86	C₄	#10587545	1	1	yes	characteristic
87	C₄	#10587553	1	1	yes	characteristic
88	C₂	—	4	1	no	—
89	C₂	#11763688	1	2	yes	—
90	C₂	#11763689	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×#595)⋊C₄
commutator, prime radical	C₂³×C₁₇	C₄²
Frattini	C₂²×C₄	C₂×#595
Fitting	C₂×M₁₆×C₁₇	C₄
socle	C₂²×C₁₇	#95
cosocle	C₂×C₄×Dic₁₇	C₂²
hypercenter	C₂×M₁₆	#595
hypocenter	C₁₇	#1309
LCS	G ▷ C₂³×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

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

Maximal subgroups

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

subgroup	index
#11763812	2
#11762832	2
#1309	17