Subgroups — C₂²×#63230 #11763581

Order 1088.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₆₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#63230	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#3622	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₁₇	—	2	1	no	—
6	C₂×Q₈×C₁₇	—	2	3	no	—
7	C₂×Q₈×C₁₇	C₂²	1	6	yes	complemented
8	C₂²×C₄×C₁₇	—	2	1	no	—
9	C₂²×C₄×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×D₆₈	—	2	3	no	—
11	C₂×D₆₈	C₂²	1	6	yes	—
12	#63230	C₂²	1	16	yes	direct product factor
13	C₂×#3622	C₂²	1	6	yes	complemented
14	C₂³×C₁₇	D₄	1	1	yes	characteristic
15	C₂²×D₁₇	—	2	6	no	—
16	C₂²×D₁₇	—	4	4	no	—
17	Q₈×C₁₇	—	2	6	no	—
18	Q₈×C₁₇	C₂³	1	4	yes	complemented
19	C₂×C₄×C₁₇	—	2	6	no	—
20	C₂×C₄×C₁₇	D₄	1	3	yes	—
21	C₂×C₄×C₁₇	C₂³	1	3	yes	—
22	D₆₈	—	2	6	no	—
23	D₆₈	C₂³	1	4	yes	—
24	#3622	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	—	2	4	no	—
30	C₄×C₁₇	C₂×D₄	1	3	yes	—
31	C₄×C₁₇	C₂⁴	1	1	yes	characteristic
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	D₁₇	1	1	yes	characteristic, complemented
37	C₂²×D₄	—	17	1	no	—
38	C₂×SD₁₆	—	17	12	no	—
39	C₂²×C₈	—	17	1	no	—
40	C₁₇	C₂²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
41	C₂⁴	—	34	1	no	—
42	C₂×Q₈	—	2	3	no	—
43	C₂×Q₈	C₂×D₁₇	1	6	yes	complemented
44	C₂×D₄	—	17	6	no	—
45	C₂×D₄	—	34	3	no	—
46	C₂²×C₄	—	2	1	no	—
47	C₂²×C₄	C₂×D₁₇	1	1	yes	characteristic
48	SD₁₆	—	17	16	no	—
49	C₂×C₈	—	17	6	no	—
50	C₂³	—	34	6	no	—
51	C₂³	—	68	4	no	—
52	C₂³	C₂²⋊D₁₇	1	1	yes	characteristic
53	Q₈	—	2	6	no	—
54	Q₈	C₂²×D₁₇	1	4	yes	complemented
55	D₄	—	17	4	no	—
56	D₄	—	34	6	no	—
57	C₂×C₄	—	2	6	no	—
58	C₂×C₄	C₂²⋊D₁₇	1	3	yes	—
59	C₂×C₄	C₂²×D₁₇	1	3	yes	—
60	C₈	—	17	4	no	—
61	C₂²	—	34	4	no	—
62	C₂²	—	68	12	no	—
63	C₂²	#63230	1	4	yes	direct product factor
64	C₂²	C₂×C₂²⋊D₁₇	1	3	yes	—
65	C₄	—	2	4	no	—
66	C₄	C₂×C₂²⋊D₁₇	1	3	yes	—
67	C₄	C₂³×D₁₇	1	1	yes	characteristic
68	C₂	—	68	4	no	—
69	C₂	C₂×#63230	1	6	yes	direct product factor
70	C₂	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₁₇
commutator, cosocle	C₄×C₁₇	C₂⁴
Frattini	C₄	C₂³×D₁₇
Fitting	C₂²×Q₈×C₁₇	C₂
socle	C₂³×C₁₇	D₄
prime radical	C₂²×D₆₈	C₂
hypercenter	C₂²×Q₈	D₁₇
hypocenter	C₁₇	C₂²×SD₁₆
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×C₁₇	C₂	semidirect
3. C₂×#63230	C₂	direct (N × H)
4. C₂²×#3622	C₂	semidirect
7. C₂×Q₈×C₁₇	C₂²	semidirect
12. #63230	C₂²	direct (N × H)
13. C₂×#3622	C₂²	semidirect
18. Q₈×C₁₇	C₂³	semidirect
24. #3622	C₂³	semidirect
25. C₂²×C₁₇	SD₁₆	semidirect
33. C₂×C₁₇	C₂×SD₁₆	semidirect
36. C₂²×Q₈	D₁₇	semidirect
40. C₁₇	C₂²×SD₁₆	semidirect
43. C₂×Q₈	C₂×D₁₇	semidirect
54. Q₈	C₂²×D₁₇	semidirect
63. C₂²	#63230	direct (N × H)
69. C₂	C₂×#63230	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×Q₈×C₁₇	2
C₂²×D₆₈	2
C₂×#63230	2
C₂²×#3622	2
C₂²×SD₁₆	17

Subgroups — C22×#63230 #11763581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#63230 #11763581