Subgroups — #16954838

Order 4352.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762639	C₂	1	2	yes	maximal
2	#14526988	C₂	1	4	yes	maximal, complemented
3	C₂×C₁₇×M₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587490	—	2	2	no	—
5	#11762639	—	2	2	no	—
6	#11762639	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
8	C₁₇×M₆₄	C₄	1	4	yes	—
9	C₂×C₁₇×C₃₂	C₂²	1	2	yes	—
10	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
11	C₂×#63223	—	4	2	no	—
12	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
13	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₁₇	Q₈	1	2	yes	—
15	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
16	#10587490	—	2	4	no	—
17	#10587490	—	4	2	no	—
18	C₁₇×C₃₂	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
20	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
21	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
22	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
23	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
24	#63223	—	4	4	no	—
25	#63223	—	8	2	no	—
26	C₂×#3622	—	8	2	no	—
27	C₁₆×C₁₇	C₄⋊C₄	1	2	yes	—
28	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
29	C₁₆×C₁₇	C₂×Q₈	1	1	yes	characteristic
30	#13636	—	17	1	no	maximal, Sylow
31	C₂³×C₁₇	Q₃₂	1	1	yes	characteristic
32	C₂×C₄×C₁₇	#107	1	2	yes	—
33	C₂×C₄×C₁₇	D₁₆	1	2	yes	—
34	C₂×C₄×C₁₇	Q₃₂	1	1	yes	characteristic
35	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
36	C₈×C₁₇	#107	1	2	yes	—
37	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₈×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
39	#3622	—	8	4	no	—
40	C₂×M₆₄	D₁₇	1	1	yes	characteristic
41	C₂×#368	—	17	2	no	—
42	#1425	—	17	4	no	—
43	C₂²×C₁₇	—	4	1	no	—
44	C₂²×C₁₇	#366	1	2	yes	—
45	C₂²×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
46	C₄×C₁₇	#366	1	2	yes	—
47	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
48	C₄×C₁₇	C₂×D₁₆	1	1	yes	characteristic
49	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
50	C₂×#108	—	34	2	no	—
51	M₆₄	Dic₁₇	1	4	yes	—
52	C₂×C₃₂	C₂×D₁₇	1	2	yes	—
53	#368	—	17	4	no	—
54	#368	—	34	2	no	—
55	C₂×C₁₇	—	2	2	no	—
56	C₂×C₁₇	—	2	1	no	—
57	C₂×C₁₇	C₂×#366	1	1	yes	characteristic
58	C₂×M₁₆	—	68	2	no	—
59	C₂²×C₈	D₆₈	1	1	yes	characteristic
60	C₂×C₁₆	Dic₃₄	1	2	yes	—
61	C₂×C₁₆	D₆₈	1	1	yes	characteristic
62	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
63	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
64	#108	—	34	4	no	—
65	#108	—	68	2	no	—
66	C₃₂	C₂×Dic₁₇	1	4	yes	—
67	C₁₇	#13636	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
69	M₁₆	—	68	4	no	—
70	M₁₆	—	136	2	no	—
71	C₂×C₈	—	136	2	no	—
72	C₂×C₈	D₁₃₆	1	1	yes	characteristic
73	C₂×C₈	Dic₆₈	1	2	yes	—
74	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
75	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
76	C₁₆	C₄⋊Dic₁₇	1	2	yes	—
77	C₁₆	C₂×Dic₃₄	1	1	yes	characteristic
78	C₁₆	C₂²×Dic₁₇	1	1	yes	characteristic
79	C₂³	Dic₁₃₆	1	1	yes	characteristic
80	C₂×C₄	D₂₇₂	1	2	yes	—
81	C₂×C₄	Dic₁₃₆	1	1	yes	characteristic
82	C₂×C₄	#10587489	1	2	yes	—
83	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
84	C₈	—	136	4	no	—
85	C₈	#10587489	1	2	yes	—
86	C₈	C₂×Dic₆₈	1	1	yes	characteristic
87	C₈	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
88	C₂²	—	4	1	no	—
89	C₂²	#11762635	1	2	yes	—
90	C₂²	C₂×Dic₁₃₆	1	1	yes	characteristic
91	C₄	#11762635	1	2	yes	—
92	C₄	C₂×#10587489	1	1	yes	characteristic
93	C₄	C₂×D₂₇₂	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	2	2	no	—
96	C₂	C₂×#11762635	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₂₇₂
commutator	C₁₆×C₁₇	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₁₇
Fitting	C₂×C₁₇×M₆₄	C₂
socle	C₂×C₁₇	C₂×#366
prime radical	C₂³×C₁₇	Q₃₂
cosocle	C₂×C₁₆×C₁₇	C₂³
hypercenter	C₂×M₆₄	D₁₇
hypocenter	C₁₇	#13636
LCS	G ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
2. #14526988	C₂	semidirect
67. C₁₇	#13636	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762639	2
#14526988	2
C₂×C₁₇×M₆₄	2
#13636	17