Subgroups — C₂²×#63219 #11763583

Order 1088.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₃₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#63219	C₂	1	12	yes	maximal, direct product factor
5	C₂³×D₁₇	—	2	1	no	—
6	C₂²×C₄×C₁₇	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₇	—	2	1	no	—
8	C₂×D₆₈	—	2	3	no	—
9	C₂×D₆₈	C₂²	1	6	yes	—
10	C₂×Dic₃₄	—	2	3	no	—
11	C₂×Dic₃₄	C₂²	1	6	yes	complemented
12	C₂×C₈×C₁₇	C₂²	1	6	yes	complemented
13	#63219	C₂²	1	16	yes	direct product factor
14	C₂³×C₁₇	D₄	1	1	yes	characteristic
15	C₂²×D₁₇	—	2	6	no	—
16	C₂²×D₁₇	—	4	4	no	—
17	C₂×C₄×C₁₇	D₄	1	3	yes	—
18	C₂×C₄×C₁₇	C₂³	1	3	yes	—
19	C₂×Dic₁₇	—	2	6	no	—
20	D₆₈	—	2	6	no	—
21	D₆₈	C₂³	1	4	yes	—
22	Dic₃₄	—	2	6	no	—
23	Dic₃₄	C₂³	1	4	yes	complemented
24	C₈×C₁₇	C₂³	1	4	yes	complemented
25	C₂²×C₁₇	SD₁₆	1	4	yes	complemented
26	C₂²×C₁₇	C₂×D₄	1	3	yes	—
27	C₂×D₁₇	—	2	4	no	—
28	C₂×D₁₇	—	4	12	no	—
29	C₄×C₁₇	C₂×D₄	1	3	yes	—
30	C₄×C₁₇	C₂⁴	1	1	yes	characteristic
31	Dic₁₇	—	2	4	no	—
32	C₂²×SD₁₆	—	17	1	no	maximal, Sylow
33	C₂×C₁₇	C₂×SD₁₆	1	6	yes	complemented
34	C₂×C₁₇	C₂²×D₄	1	1	yes	characteristic
35	D₁₇	—	4	4	no	—
36	C₂²×Q₈	—	17	1	no	—
37	C₂²×D₄	—	17	1	no	—
38	C₂×SD₁₆	—	17	12	no	—
39	C₂²×C₈	D₁₇	1	1	yes	characteristic, complemented
40	C₁₇	C₂²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
41	C₂⁴	—	34	1	no	—
42	C₂×Q₈	—	17	6	no	—
43	C₂×Q₈	—	34	3	no	—
44	C₂×D₄	—	17	6	no	—
45	C₂×D₄	—	34	3	no	—
46	C₂²×C₄	—	34	1	no	—
47	C₂²×C₄	C₂×D₁₇	1	1	yes	characteristic
48	SD₁₆	—	17	16	no	—
49	C₂×C₈	C₂×D₁₇	1	6	yes	complemented
50	C₂³	—	34	6	no	—
51	C₂³	—	68	4	no	—
52	C₂³	D₆₈	1	1	yes	characteristic
53	Q₈	—	17	4	no	—
54	Q₈	—	34	6	no	—
55	D₄	—	17	4	no	—
56	D₄	—	34	6	no	—
57	C₂×C₄	—	34	6	no	—
58	C₂×C₄	D₆₈	1	3	yes	—
59	C₂×C₄	C₂²×D₁₇	1	3	yes	—
60	C₈	C₂²×D₁₇	1	4	yes	complemented
61	C₂²	—	34	4	no	—
62	C₂²	—	68	12	no	—
63	C₂²	#63219	1	4	yes	direct product factor
64	C₂²	C₂×D₆₈	1	3	yes	—
65	C₄	—	34	4	no	—
66	C₄	C₂×D₆₈	1	3	yes	—
67	C₄	C₂³×D₁₇	1	1	yes	characteristic
68	C₂	—	68	4	no	—
69	C₂	C₂×#63219	1	6	yes	direct product factor
70	C₂	C₂²×D₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₆₈
commutator, cosocle	C₄×C₁₇	C₂⁴
Frattini	C₄	C₂³×D₁₇
Fitting	C₂²×C₈×C₁₇	C₂
socle	C₂³×C₁₇	D₄
prime radical	C₂²×D₆₈	C₂
hypercenter	C₂²×C₈	D₁₇
hypocenter	C₁₇	C₂²×SD₁₆
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×Dic₃₄	C₂	semidirect
3. C₂²×C₈×C₁₇	C₂	semidirect
4. C₂×#63219	C₂	direct (N × H)
11. C₂×Dic₃₄	C₂²	semidirect
12. C₂×C₈×C₁₇	C₂²	semidirect
13. #63219	C₂²	direct (N × H)
23. Dic₃₄	C₂³	semidirect
24. C₈×C₁₇	C₂³	semidirect
25. C₂²×C₁₇	SD₁₆	semidirect
33. C₂×C₁₇	C₂×SD₁₆	semidirect
39. C₂²×C₈	D₁₇	semidirect
40. C₁₇	C₂²×SD₁₆	semidirect
49. C₂×C₈	C₂×D₁₇	semidirect
60. C₈	C₂²×D₁₇	semidirect
63. C₂²	#63219	direct (N × H)
69. C₂	C₂×#63219	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₆₈	2
C₂²×Dic₃₄	2
C₂²×C₈×C₁₇	2
C₂×#63219	2
C₂²×SD₁₆	17

Subgroups — C22×#63219 #11763583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#63219 #11763583