Subgroups — C₂×#14528399 #17022015

Order 4352.

1342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763186	C₂	1	6	yes	maximal, complemented
2	#14528399	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₈²×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₁₃₆	—	2	12	no	—
5	C₂×#10587560	C₂²	1	4	yes	—
6	#11763186	—	2	12	no	—
7	#11763186	C₂²	1	24	yes	complemented
8	C₂×C₄×C₈×C₁₇	C₂²	1	3	yes	—
9	C₈²×C₁₇	C₂²	1	4	yes	complemented
10	C₂²×D₆₈	—	2	12	no	—
11	C₂²×C₈×C₁₇	D₄	1	6	yes	—
12	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
13	C₂×D₁₃₆	—	2	48	no	—
14	C₂×D₁₃₆	—	4	48	no	—
15	#10587560	—	2	12	no	—
16	#10587560	C₂³	1	8	yes	—
17	C₄×C₈×C₁₇	D₄	1	6	yes	—
18	C₄×C₈×C₁₇	C₂³	1	6	yes	—
19	C₂³×D₁₇	—	4	4	no	—
20	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
21	C₂×D₆₈	—	2	24	no	—
22	C₂×D₆₈	—	4	48	no	—
23	C₂×D₆₈	—	4	12	no	—
24	C₂×C₈×C₁₇	D₈	1	24	yes	complemented
25	C₂×C₈×C₁₇	C₂×D₄	1	12	yes	—
26	C₄²×C₁₇	C₂×D₄	1	3	yes	—
27	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
28	D₁₃₆	—	4	96	no	—
29	C₂×#1712	—	17	1	no	maximal, Sylow
30	C₂³×C₁₇	#127	1	1	yes	characteristic
31	C₂²×D₁₇	—	4	8	no	—
32	C₂²×D₁₇	—	8	24	no	—
33	C₂×C₄×C₁₇	#127	1	3	yes	—
34	C₂×C₄×C₁₇	C₂×D₈	1	12	yes	—
35	C₂×C₄×C₁₇	C₂²×D₄	1	3	yes	—
36	D₆₈	—	4	48	no	—
37	D₆₈	—	8	24	no	—
38	C₈×C₁₇	C₂×D₈	1	24	yes	complemented
39	C₂×#493	—	17	6	no	—
40	#1712	—	17	8	no	—
41	C₂×C₈²	D₁₇	1	1	yes	characteristic, complemented
42	C₂²×C₁₇	#493	1	6	yes	—
43	C₂²×C₁₇	C₂×#127	1	1	yes	characteristic
44	C₂×D₁₇	—	8	24	no	—
45	C₂×D₁₇	—	16	16	no	—
46	C₄×C₁₇	#493	1	6	yes	—
47	C₄×C₁₇	C₂²×D₈	1	6	yes	—
48	C₂²×D₈	—	34	12	no	—
49	C₂×#127	—	17	4	no	—
50	#493	—	17	24	no	—
51	#493	—	34	12	no	—
52	C₂×C₄×C₈	C₂×D₁₇	1	3	yes	—
53	C₈²	C₂×D₁₇	1	4	yes	complemented
54	C₂×C₁₇	#1712	1	4	yes	complemented
55	C₂×C₁₇	C₂×#493	1	3	yes	—
56	D₁₇	—	16	8	no	—
57	C₂²×D₄	—	34	12	no	—
58	C₂×D₈	—	34	48	no	—
59	C₂×D₈	—	68	48	no	—
60	C₂²×C₈	D₆₈	1	6	yes	—
61	#127	—	17	8	no	—
62	#127	—	34	12	no	—
63	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
64	C₄×C₈	D₆₈	1	6	yes	—
65	C₄×C₈	C₂²×D₁₇	1	6	yes	—
66	C₁₇	C₂×#1712	1	1	yes	characteristic, complemented, Sylow
67	C₂⁴	—	68	4	no	—
68	C₂×D₄	—	34	24	no	—
69	C₂×D₄	—	68	12	no	—
70	C₂×D₄	—	68	48	no	—
71	C₂²×C₄	C₂×D₆₈	1	3	yes	—
72	D₈	—	68	96	no	—
73	C₂×C₈	D₁₃₆	1	24	yes	complemented
74	C₂×C₈	C₂×D₆₈	1	12	yes	—
75	C₄²	C₂×D₆₈	1	3	yes	—
76	C₄²	C₂³×D₁₇	1	1	yes	characteristic
77	C₂³	—	68	8	no	—
78	C₂³	—	136	24	no	—
79	C₂³	#10587560	1	1	yes	characteristic
80	D₄	—	68	48	no	—
81	D₄	—	136	24	no	—
82	C₂×C₄	#10587560	1	3	yes	—
83	C₂×C₄	C₂×D₁₃₆	1	12	yes	—
84	C₂×C₄	C₂²×D₆₈	1	3	yes	—
85	C₈	C₂×D₁₃₆	1	24	yes	complemented
86	C₂²	—	272	16	no	—
87	C₂²	—	136	24	no	—
88	C₂²	#11763186	1	6	yes	—
89	C₂²	C₂×#10587560	1	1	yes	characteristic
90	C₄	#11763186	1	6	yes	—
91	C₄	C₂²×D₁₃₆	1	6	yes	—
92	C₂	—	272	8	no	—
93	C₂	#14528399	1	4	yes	direct product factor
94	C₂	C₂×#11763186	1	3	yes	—

	subgroup	G/H
center	C₂³	#10587560
commutator, cosocle	C₄²×C₁₇	C₂⁴
Frattini	C₄²	C₂³×D₁₇
Fitting	C₂×C₈²×C₁₇	C₂
socle	C₂³×C₁₇	#127
hypercenter	C₂×C₈²	D₁₇
hypocenter	C₁₇	C₂×#1712
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. C₂×#11763186	C₂	semidirect
2. #14528399	C₂	direct (N × H)
3. C₂×C₈²×C₁₇	C₂	semidirect
7. #11763186	C₂²	semidirect
9. C₈²×C₁₇	C₂²	semidirect
24. C₂×C₈×C₁₇	D₈	semidirect
38. C₈×C₁₇	C₂×D₈	semidirect
41. C₂×C₈²	D₁₇	semidirect
53. C₈²	C₂×D₁₇	semidirect
54. C₂×C₁₇	#1712	semidirect
66. C₁₇	C₂×#1712	semidirect
73. C₂×C₈	D₁₃₆	semidirect
85. C₈	C₂×D₁₃₆	semidirect
93. C₂	#14528399	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11763186	2
#14528399	2
C₂×C₈²×C₁₇	2
C₂×#1712	17

Subgroups — C2×#14528399 #17022015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14528399 #17022015