Subgroups — #14544441

Order 2176.

328 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10587542	C₂	1	3	yes	maximal
2	C₂×C₄×#3624	C₂	1	3	yes	maximal
3	#11762654	C₂	1	1	yes	maximal, characteristic
4	C₂²×#3624	C₂²	1	1	yes	characteristic
5	C₂²×#3624	C₂²	1	3	yes	—
6	C₂×C₄×Dic₁₇	C₂²	1	3	yes	—
7	#10587542	C₄	1	12	yes	complemented
8	C₄×#3624	C₄	1	12	yes	complemented
9	C₁₇×#95	C₄	1	1	yes	characteristic
10	#10587502	C₄	1	3	yes	—
11	C₂²×C₄×C₁₇	C₂×C₄	1	3	yes	—
12	C₂²×Dic₁₇	C₂×C₄	1	3	yes	—
13	C₂²×Dic₁₇	C₂³	1	1	yes	characteristic
14	C₂×#3624	C₂×C₄	1	6	yes	—
15	C₂×#3624	C₂×C₄	1	6	yes	—
16	C₂×#3624	C₂×C₄	1	12	yes	—
17	C₄×Dic₁₇	C₂×C₄	1	6	yes	—
18	C₄×Dic₁₇	C₂×C₄	1	6	yes	—
19	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₇	—	2	6	no	—
21	C₂×C₄×C₁₇	C₄²	1	6	yes	—
22	C₂×Dic₁₇	—	2	3	no	—
23	C₂×Dic₁₇	—	2	3	no	—
24	C₂×Dic₁₇	C₄²	1	6	yes	—
25	C₂×Dic₁₇	C₂²×C₄	1	3	yes	—
26	C₂×Dic₁₇	C₂²×C₄	1	3	yes	—
27	#3624	C₄²	1	4	yes	—
28	#3624	C₄²	1	12	yes	—
29	#1726	—	17	1	no	maximal, Sylow
30	C₂²×C₁₇	C₂×C₄²	1	3	yes	—
31	C₂²×C₁₇	#131	1	3	yes	—
32	C₂²×C₁₇	#131	1	1	yes	characteristic
33	C₄×C₁₇	—	2	6	no	—
34	Dic₁₇	—	2	6	no	—
35	Dic₁₇	C₂×C₄²	1	1	yes	characteristic
36	Dic₁₇	C₂×C₄²	1	3	yes	—
37	C₂×#97	—	17	3	no	—
38	C₂×C₄×C₈	—	17	3	no	—
39	#376	—	17	1	no	—
40	C₂×C₁₇	C₄³	1	1	yes	characteristic
41	C₂×C₁₇	#405	1	3	yes	—
42	C₂×C₁₇	#405	1	3	yes	—
43	C₂²×C₈	—	17	1	no	—
44	C₂²×C₈	—	17	3	no	—
45	C₂×C₄²	—	17	3	no	—
46	#97	—	17	12	no	—
47	C₄×C₈	—	17	12	no	—
48	#95	—	17	3	no	—
49	#95	#595	1	1	yes	characteristic
50	C₁₇	#1726	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₄	—	17	1	no	—
52	C₂²×C₄	—	17	3	no	—
53	C₂²×C₄	C₂×#595	1	3	yes	—
54	C₂×C₈	—	17	6	no	—
55	C₂×C₈	—	17	6	no	—
56	C₂×C₈	—	17	12	no	—
57	C₄²	—	17	6	no	—
58	C₄²	—	17	6	no	—
59	C₂³	C₂²×#595	1	1	yes	characteristic
60	C₂×C₄	—	2	6	no	—
61	C₂×C₄	—	17	3	no	—
62	C₂×C₄	—	17	3	no	—
63	C₂×C₄	—	17	6	no	—
64	C₂×C₄	—	34	3	no	—
65	C₂×C₄	—	34	3	no	—
66	C₂×C₄	C₄×#595	1	6	yes	—
67	C₈	—	17	4	no	—
68	C₈	—	17	12	no	—
69	C₂²	C₂×C₄×#595	1	3	yes	—
70	C₂²	#10587678	1	3	yes	—
71	C₂²	#10587680	1	1	yes	characteristic
72	C₄	—	2	6	no	—
73	C₄	—	17	1	no	—
74	C₄	—	17	3	no	—
75	C₄	—	34	6	no	—
76	C₂	C₄²×#595	1	1	yes	characteristic
77	C₂	#11763773	1	3	yes	—
78	C₂	#11763774	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#595
commutator	C₂×C₁₇	C₄³
Fitting	C₁₇×#95	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₄
cosocle	C₂²×Dic₁₇	C₂³
hypercenter	#95	#595
hypocenter	C₁₇	#1726
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
7. #10587542	C₄	semidirect
8. C₄×#3624	C₄	semidirect
50. C₁₇	#1726	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10587542	2
C₂×C₄×#3624	2
#11762654	2
#1726	17