Subgroups — #11763825

Order 1088.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊#3624	C₂	1	2	yes	maximal
3	C₄⋊#3624	C₂	1	2	yes	maximal, complemented
4	C₄×#3624	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₁₇	C₄	1	1	yes	characteristic
6	C₂²×Dic₁₇	C₄	1	1	yes	characteristic
7	C₂²×Dic₁₇	C₂²	1	1	yes	characteristic
8	C₂×#3624	C₂²	1	4	yes	—
9	C₄×Dic₁₇	C₄	1	1	yes	characteristic
10	C₄×Dic₁₇	C₄	1	1	yes	characteristic
11	C₄×Dic₁₇	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₇	C₂²	1	1	yes	characteristic
13	C₂³×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₇	C₈	1	4	yes	complemented
15	C₂×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₁₇	—	2	2	no	—
18	C₂×Dic₁₇	—	2	2	no	—
19	C₂×Dic₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₁₇	C₂³	1	1	yes	characteristic
23	#3624	—	2	4	no	—
24	C₂²×C₁₇	—	2	1	no	—
25	C₂²×C₁₇	—	2	1	no	—
26	C₂²×C₁₇	C₂×C₈	1	2	yes	—
27	C₂²×C₁₇	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₁₇	C₂×C₈	1	2	yes	—
29	C₄×C₁₇	C₂×C₈	1	2	yes	—
30	Dic₁₇	—	2	1	no	—
31	Dic₁₇	—	2	1	no	—
32	Dic₁₇	M₁₆	1	2	yes	—
33	Dic₁₇	#41	1	2	yes	—
34	#431	—	17	1	no	maximal, Sylow
35	C₂×C₁₇	—	2	2	no	—
36	C₂×C₁₇	#117	1	1	yes	characteristic
37	C₂×C₁₇	C₂²×C₈	1	1	yes	characteristic
38	C₂×C₁₇	C₂×M₁₆	1	1	yes	characteristic
39	C₂×C₄²	—	17	1	no	—
40	C₄⋊C₈	—	17	2	no	—
41	C₂²⋊C₈	—	17	2	no	—
42	C₄×C₈	—	17	2	no	—
43	C₁₇	#431	1	1	yes	characteristic, complemented, Sylow
44	C₂²×C₄	—	17	1	no	—
45	C₂²×C₄	—	17	1	no	—
46	C₂²×C₄	#595	1	1	yes	characteristic
47	C₂×C₈	—	17	4	no	—
48	C₄²	—	17	1	no	—
49	C₄²	—	17	1	no	—
50	C₄²	—	17	1	no	—
51	C₄²	—	17	1	no	—
52	C₂³	C₂×#595	1	1	yes	characteristic
53	C₂×C₄	—	17	1	no	—
54	C₂×C₄	—	17	1	no	—
55	C₂×C₄	—	17	1	no	—
56	C₂×C₄	—	17	1	no	—
57	C₂×C₄	—	34	2	no	—
58	C₂×C₄	—	34	2	no	—
59	C₂×C₄	#3624	1	4	yes	complemented
60	C₂×C₄	C₂×#595	1	1	yes	characteristic
61	C₂×C₄	C₂×#595	1	1	yes	characteristic
62	C₈	—	34	4	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	C₂×#3624	1	2	yes	—
66	C₂²	C₂²×#595	1	1	yes	characteristic
67	C₄	—	17	2	no	—
68	C₄	—	17	2	no	—
69	C₄	—	34	1	no	—
70	C₄	—	34	1	no	—
71	C₄	C₂×#3624	1	2	yes	—
72	C₄	C₂×#3624	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	D₁₇⋊M₁₆	1	1	yes	characteristic
75	C₂	#10587677	1	1	yes	characteristic
76	C₂	C₂²×#3624	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#595
commutator	C₂×C₁₇	C₂²×C₈
Frattini	C₂²	C₂²×#595
Fitting	C₂²×C₄×C₁₇	C₄
socle	C₂²×C₁₇	C₂²×C₄
prime radical	C₂³×C₁₇	C₂×C₄
cosocle	C₂×Dic₁₇	C₂³
hypercenter	C₂²×C₄	#595
hypocenter	C₁₇	#431
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. C₄⋊#3624	C₂	semidirect
4. C₄×#3624	C₂	semidirect
14. C₂×C₄×C₁₇	C₈	semidirect
43. C₁₇	#431	semidirect
59. C₂×C₄	#3624	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₇	2
C₂²⋊#3624	2
C₄⋊#3624	2
C₄×#3624	2
#431	17