Subgroups — #17981335

Order 4352.

126 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	C₄	1	4	yes	—
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	C₈	1	2	yes	—
8	C₂×#63216	C₂×C₄	1	2	yes	—
9	#10587528	C₈	1	8	yes	complemented
10	C₄×C₈×C₁₇	C₈	1	1	yes	characteristic
11	C₄×#3622	C₈	1	1	yes	characteristic
12	C₄×#3622	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₁₇	C₁₆	1	2	yes	—
14	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
15	C₂×#3622	C₁₆	1	2	yes	—
16	C₂×#3622	C₄²	1	1	yes	characteristic
17	C₂×#3622	C₂×C₈	1	1	yes	characteristic
18	#63216	C₁₆	1	4	yes	—
19	#63216	C₂×C₈	1	4	yes	—
20	#7329	—	17	1	no	maximal, Sylow
21	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
22	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
23	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
24	C₈×C₁₇	—	2	2	no	—
25	#3622	—	2	2	no	—
26	#3622	C₄×C₈	1	2	yes	—
27	#3622	C₂×C₁₆	1	2	yes	—
28	C₄×C₃₂	—	17	2	no	—
29	C₈×C₁₆	—	17	1	no	—
30	C₂²×C₁₇	C₄×C₁₆	1	1	yes	characteristic
31	C₄×C₁₇	C₈²	1	1	yes	characteristic
32	C₄×C₁₇	C₄×C₁₆	1	1	yes	characteristic
33	C₄×C₁₇	M₆₄	1	2	yes	—
34	C₄×C₁₇	M₆₄	1	2	yes	—
35	C₂×C₃₂	—	17	4	no	—
36	C₄×C₁₆	—	17	1	no	—
37	C₄×C₁₆	—	17	1	no	—
38	C₈²	—	17	1	no	—
39	C₂×C₁₇	C₈×C₁₆	1	1	yes	characteristic
40	C₂×C₁₇	#1397	1	1	yes	characteristic
41	C₂×C₁₇	#1397	1	1	yes	characteristic
42	C₂×C₁₆	—	17	2	no	—
43	C₂×C₁₆	—	17	2	no	—
44	C₄×C₈	—	17	1	no	—
45	C₄×C₈	—	17	1	no	—
46	C₄×C₈	#3633	1	1	yes	characteristic
47	C₃₂	—	17	8	no	—
48	C₁₇	#7329	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₈	—	17	1	no	—
50	C₂×C₈	—	17	1	no	—
51	C₂×C₈	—	17	2	no	—
52	C₂×C₈	#63241	1	2	yes	—
53	C₄²	C₂×#3633	1	1	yes	characteristic
54	C₁₆	—	17	4	no	—
55	C₁₆	—	17	4	no	—
56	C₂×C₄	#10587665	1	1	yes	characteristic
57	C₂×C₄	C₄×#3633	1	1	yes	characteristic
58	C₂×C₄	C₂×#63241	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	17	2	no	—
61	C₈	—	17	2	no	—
62	C₈	—	34	2	no	—
63	C₂²	C₄×#63241	1	1	yes	characteristic
64	C₄	C₈×#3633	1	1	yes	characteristic
65	C₄	#11763902	1	1	yes	characteristic
66	C₄	#11763917	1	2	yes	—
67	C₄	#11763918	1	2	yes	—
68	C₂	C₈×#63241	1	1	yes	characteristic
69	C₂	#14545869	1	1	yes	characteristic
70	C₂	#14545870	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂×#3633
commutator	C₂×C₁₇	C₈×C₁₆
Fitting	C₄×C₈×C₁₇	C₈
socle, prime radical	C₂²×C₁₇	C₄×C₁₆
cosocle	C₄×#63216	C₂²
hypercenter	C₄×C₈	#3633
hypocenter	C₁₇	#7329
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
9. #10587528	C₈	semidirect
48. C₁₇	#7329	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
#7329	17