Subgroups — #14527055

Order 2176.

736 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10587502	C₂	1	6	yes	maximal
2	C₂³×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×Q₈×C₁₇	—	2	12	no	—
4	C₂²×Q₈×C₁₇	C₄	1	4	yes	complemented
5	C₂³×C₄×C₁₇	C₂²	1	3	yes	—
6	C₂³×Dic₁₇	C₂²	1	4	yes	—
7	#10587502	—	2	12	no	—
8	C₂⁴×C₁₇	C₂³	1	1	yes	characteristic
9	C₂×Q₈×C₁₇	—	2	32	no	—
10	C₂×Q₈×C₁₇	—	2	24	no	—
11	C₂²×C₄×C₁₇	—	2	12	no	—
12	C₂²×C₄×C₁₇	C₂×C₄	1	6	yes	—
13	C₂²×C₄×C₁₇	D₄	1	12	yes	—
14	C₂²×Dic₁₇	—	2	12	no	—
15	C₂²×Dic₁₇	—	2	12	no	—
16	C₂²×Dic₁₇	Q₈	1	8	yes	complemented
17	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
18	C₂³×C₁₇	C₂×D₄	1	6	yes	—
19	C₂³×C₁₇	C₂×Q₈	1	4	yes	—
20	C₂³×C₁₇	#41	1	4	yes	—
21	Q₈×C₁₇	—	2	32	no	—
22	C₂×C₄×C₁₇	—	2	12	no	—
23	C₂×C₄×C₁₇	—	2	24	no	—
24	C₂×C₄×C₁₇	C₂²⋊C₄	1	12	yes	—
25	C₂×Dic₁₇	—	2	24	no	—
26	C₂×Dic₁₇	—	4	4	no	—
27	C₂×Dic₁₇	—	4	12	no	—
28	#1446	—	17	1	no	maximal, Sylow
29	C₂²×C₁₇	C₂×C₂²⋊C₄	1	3	yes	—
30	C₂²×C₁₇	C₄×Q₈	1	4	yes	—
31	C₂²×C₁₇	#120	1	4	yes	—
32	C₂²×C₁₇	C₂²⋊Q₈	1	12	yes	—
33	C₂²×C₁₇	#124	1	6	yes	—
34	C₂²×C₁₇	C₄⋊Q₈	1	6	yes	—
35	C₄×C₁₇	—	2	12	no	—
36	Dic₁₇	—	4	8	no	—
37	C₂³×Q₈	D₁₇	1	1	yes	characteristic
38	C₂×#95	—	17	6	no	—
39	C₂×C₁₇	#379	1	1	yes	characteristic
40	C₂×C₁₇	#391	1	6	yes	—
41	C₂×C₁₇	#393	1	4	yes	—
42	C₂×C₁₇	#395	1	4	yes	—
43	C₂²×Q₈	—	2	12	no	—
44	C₂²×Q₈	Dic₁₇	1	4	yes	complemented
45	C₂³×C₄	—	17	4	no	—
46	C₂³×C₄	C₂×D₁₇	1	3	yes	—
47	#95	—	34	12	no	—
48	C₁₇	#1446	1	1	yes	characteristic, complemented, Sylow
49	C₂⁴	C₂²×D₁₇	1	1	yes	characteristic
50	C₂×Q₈	—	2	32	no	—
51	C₂×Q₈	—	2	24	no	—
52	C₂²×C₄	—	2	12	no	—
53	C₂²×C₄	—	17	8	no	—
54	C₂²×C₄	—	34	12	no	—
55	C₂²×C₄	—	34	12	no	—
56	C₂²×C₄	C₂×Dic₁₇	1	6	yes	—
57	C₂²×C₄	C₂²⋊D₁₇	1	12	yes	—
58	C₂³	Q₈×D₁₇	1	4	yes	—
59	C₂³	#63256	1	4	yes	—
60	C₂³	C₂²×Dic₁₇	1	1	yes	characteristic
61	C₂³	C₂×C₂²⋊D₁₇	1	6	yes	—
62	Q₈	—	2	32	no	—
63	C₂×C₄	—	2	12	no	—
64	C₂×C₄	—	2	24	no	—
65	C₂×C₄	—	34	24	no	—
66	C₂×C₄	—	68	12	no	—
67	C₂×C₄	—	68	4	no	—
68	C₂×C₄	C₂²⋊Dic₁₇	1	12	yes	—
69	C₂²	#10587630	1	6	yes	—
70	C₂²	Q₈×Dic₁₇	1	4	yes	—
71	C₂²	#10587632	1	12	yes	—
72	C₂²	#10587633	1	6	yes	—
73	C₂²	C₂×C₂²⋊Dic₁₇	1	3	yes	—
74	C₂²	#10587639	1	4	yes	—
75	C₄	—	2	12	no	—
76	C₄	—	68	8	no	—
77	C₂	#11762665	1	1	yes	characteristic
78	C₂	#11762716	1	6	yes	—
79	C₂	#11762720	1	4	yes	—
80	C₂	#11762728	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₁₇
commutator	C₂³×C₁₇	C₂²×C₄
Fitting	C₂³×Q₈×C₁₇	C₂
socle, prime radical, cosocle	C₂⁴×C₁₇	C₂³
hypercenter	C₂³×Q₈	D₁₇
hypocenter	C₁₇	#1446
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
4. C₂²×Q₈×C₁₇	C₄	semidirect
16. C₂²×Dic₁₇	Q₈	semidirect
44. C₂²×Q₈	Dic₁₇	semidirect
48. C₁₇	#1446	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10587502	2
C₂³×Q₈×C₁₇	2
#1446	17