Subgroups — #17981499

Order 4352.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545888	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11763891	C₂	1	1	yes	maximal, characteristic, complemented
3	#14545207	C₂	1	1	yes	maximal, characteristic
4	#11763891	—	2	1	no	—
5	#11763891	—	2	1	no	—
6	#11763741	—	2	1	no	—
7	#11763741	C₄	1	1	yes	characteristic
8	C₂×#10587467	—	2	1	no	—
9	C₂×#10587467	C₂²	1	1	yes	characteristic
10	#11763226	C₄	1	1	yes	characteristic
11	C₁₇×#131	C₈	1	1	yes	characteristic
12	#10587483	—	2	1	no	—
13	#10587483	C₈	1	1	yes	characteristic
14	C₂×#63214	—	2	1	no	—
15	C₂×#63214	C₂×C₄	1	1	yes	characteristic
16	#10587467	—	2	1	no	—
17	#10587467	D₄	1	1	yes	characteristic
18	#10587467	D₄	1	1	yes	characteristic
19	#41×C₁₇	C₁₆	1	1	yes	characteristic
20	M₁₆×C₁₇	—	2	1	no	—
21	M₁₆×C₁₇	C₁₆	1	1	yes	characteristic
22	C₂×C₈×C₁₇	—	2	1	no	—
23	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
24	#63214	—	2	1	no	—
25	#63214	C₂²⋊C₄	1	1	yes	characteristic
26	#63214	M₁₆	1	1	yes	characteristic
27	#7514	—	17	1	no	maximal, Sylow
28	Q₈×C₁₇	C₃₂	1	1	yes	characteristic
29	D₄×C₁₇	—	2	1	no	—
30	D₄×C₁₇	C₃₂	1	1	yes	characteristic
31	C₂×C₄×C₁₇	—	2	1	no	—
32	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
33	C₈×C₁₇	—	2	1	no	—
34	C₈×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
35	C₈×C₁₇	M₃₂	1	1	yes	characteristic
36	#2258	—	17	1	no	—
37	M₁₂₈	—	17	1	no	—
38	C₂×C₆₄	—	17	1	no	—
39	C₂²×C₁₇	—	2	1	no	—
40	C₂²×C₁₇	M₆₄	1	1	yes	characteristic
41	C₄×C₁₇	—	2	1	no	—
42	C₄×C₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
43	C₄×C₁₇	C₂×C₃₂	1	1	yes	characteristic
44	#504	—	17	1	no	—
45	M₆₄	—	17	1	no	—
46	M₆₄	—	34	1	no	—
47	C₂×C₃₂	—	17	1	no	—
48	C₂×C₃₂	—	34	1	no	—
49	C₆₄	—	34	1	no	—
50	C₆₄	—	34	1	no	—
51	C₂×C₁₇	—	2	1	no	—
52	C₂×C₁₇	—	4	1	no	—
53	C₂×C₁₇	C₂²⋊C₃₂	1	1	yes	characteristic
54	#131	#3633	1	1	yes	characteristic
55	M₃₂	—	17	1	no	—
56	M₃₂	—	34	1	no	—
57	C₂×C₁₆	—	17	1	no	—
58	C₂×C₁₆	—	34	1	no	—
59	C₃₂	—	17	1	no	—
60	C₃₂	—	17	1	no	—
61	C₃₂	—	34	1	no	—
62	C₁₇	#7514	1	1	yes	characteristic, complemented, Sylow
63	#41	#63241	1	1	yes	characteristic
64	M₁₆	—	2	1	no	—
65	M₁₆	#63241	1	1	yes	characteristic
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	C₂×#3633	1	1	yes	characteristic
68	C₁₆	—	17	1	no	—
69	C₁₆	—	17	1	no	—
70	C₁₆	—	34	1	no	—
71	Q₈	#10587528	1	1	yes	characteristic
72	D₄	—	2	1	no	—
73	D₄	#10587528	1	1	yes	characteristic
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	C₂×#63241	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	#10587670	1	1	yes	characteristic
78	C₈	C₂²⋊#3633	1	1	yes	characteristic
79	C₂²	—	2	1	no	—
80	C₂²	#11763917	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	C₂²⋊#63241	1	1	yes	characteristic
83	C₄	C₂×#10587528	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	4	1	no	—
86	C₂	C₂²⋊#10587528	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊#3633
commutator	C₄×C₁₇	C₂×C₃₂
Frattini	C₂×C₈	C₂×#3633
Fitting	C₁₇×#131	C₈
socle	C₂×C₁₇	C₂²⋊C₃₂
prime radical	#41×C₁₇	C₁₆
cosocle	C₂×#10587467	C₂²
hypercenter	#131	#3633
hypocenter	C₁₇	#7514
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
1. #14545888	C₂	semidirect
2. C₂×#11763891	C₂	semidirect
62. C₁₇	#7514	semidirect

Maximal subgroups

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

subgroup	index
#14545888	2
C₂×#11763891	2
#14545207	2
#7514	17