Subgroups — #17981456

Order 4352.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#10587528	C₂	1	2	yes	maximal
2	C₈×#63216	C₂	1	1	yes	maximal, characteristic
3	C₂×#10587528	—	2	2	no	—
4	C₄×#63216	C₄	1	1	yes	characteristic
5	C₄×#63216	C₂²	1	1	yes	characteristic
6	C₈×#3622	C₄	1	1	yes	characteristic
7	C₂×#63216	—	2	1	no	—
8	C₂×#63216	D₄	1	1	yes	characteristic
9	C₂×#63216	Q₈	1	1	yes	characteristic
10	#10587528	—	4	2	no	—
11	C₄×C₈×C₁₇	C₈	1	1	yes	characteristic
12	C₄×#3622	C₈	1	1	yes	characteristic
13	C₄×#3622	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₇	—	2	1	no	—
15	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
16	C₂×#3622	C₁₆	1	1	yes	characteristic
17	C₂×#3622	C₁₆	1	1	yes	characteristic
18	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
19	C₂×#3622	M₁₆	1	1	yes	characteristic
20	#63216	—	2	2	no	—
21	#63216	—	2	1	no	—
22	#63216	D₈	1	1	yes	characteristic
23	#63216	Q₁₆	1	1	yes	characteristic
24	#7458	—	17	1	no	maximal, Sylow
25	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
27	C₂×C₄×C₁₇	M₃₂	1	1	yes	characteristic
28	C₈×C₁₇	—	2	2	no	—
29	#3622	—	2	1	no	—
30	#3622	—	2	1	no	—
31	#3622	C₃₂	1	2	yes	complemented
32	#3622	#107	1	1	yes	characteristic
33	#3622	#108	1	1	yes	characteristic
34	C₄⋊C₃₂	—	17	2	no	—
35	C₈×C₁₆	—	17	1	no	—
36	C₂²×C₁₇	C₄⋊C₁₆	1	1	yes	characteristic
37	C₄×C₁₇	—	2	1	no	—
38	C₄×C₁₇	#335	1	1	yes	characteristic
39	C₄×C₁₇	#364	1	1	yes	characteristic
40	C₄×C₁₇	C₂×C₃₂	1	1	yes	characteristic
41	C₄×C₁₇	M₆₄	1	1	yes	characteristic
42	C₂×C₃₂	—	34	2	no	—
43	C₄×C₁₆	—	17	1	no	—
44	C₄×C₁₆	—	17	1	no	—
45	C₈²	—	17	1	no	—
46	C₂×C₁₇	#1370	1	1	yes	characteristic
47	C₂×C₁₇	C₄⋊C₃₂	1	1	yes	characteristic
48	C₂×C₁₇	#1422	1	1	yes	characteristic
49	C₂×C₁₆	—	17	1	no	—
50	C₂×C₁₆	—	17	1	no	—
51	C₂×C₁₆	—	34	1	no	—
52	C₄×C₈	—	17	1	no	—
53	C₄×C₈	—	17	1	no	—
54	C₄×C₈	#3633	1	1	yes	characteristic
55	C₃₂	—	68	2	no	—
56	C₁₇	#7458	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	17	1	no	—
59	C₂×C₈	—	17	1	no	—
60	C₂×C₈	—	17	1	no	—
61	C₂×C₈	—	17	1	no	—
62	C₄²	C₂×#3633	1	1	yes	characteristic
63	C₁₆	—	17	1	no	—
64	C₁₆	—	17	1	no	—
65	C₁₆	—	34	1	no	—
66	C₁₆	—	34	2	no	—
67	C₂×C₄	#10587665	1	1	yes	characteristic
68	C₂×C₄	C₄⋊#3633	1	1	yes	characteristic
69	C₂×C₄	#10587670	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	17	1	no	—
72	C₈	—	17	1	no	—
73	C₈	—	17	2	no	—
74	C₈	—	34	1	no	—
75	C₈	—	34	1	no	—
76	C₂²	#11763912	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	#11763899	1	1	yes	characteristic
79	C₄	#11763913	1	1	yes	characteristic
80	C₄	#11763916	1	1	yes	characteristic
81	C₄	#11763918	1	1	yes	characteristic
82	C₂	#14545849	1	1	yes	characteristic
83	C₂	#14545883	1	1	yes	characteristic
84	C₂	#14545885	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#3633
commutator	C₄×C₁₇	C₂×C₃₂
Frattini	C₄²	C₂×#3633
Fitting	C₄×C₈×C₁₇	C₈
socle, prime radical	C₂²×C₁₇	C₄⋊C₁₆
cosocle	C₄×#63216	C₂²
hypercenter	C₄×C₈	#3633
hypocenter	C₁₇	#7458
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
31. #3622	C₃₂	semidirect
56. C₁₇	#7458	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#10587528	2
C₈×#63216	2
#7458	17