Subgroups — C₂³×#16864390 #46825604

Order 34432.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#16864390	C₂	1	28	yes	maximal, direct product factor
3	C₂³×#14524130	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₃₈	—	2	7	no	—
6	C₂²×Dic₅₃₈	C₂²	1	14	yes	complemented
7	C₂³×Dic₂₆₉	—	2	1	no	—
8	C₂×#16864390	C₂²	1	112	yes	direct product factor
9	C₂²×#14524130	C₂²	1	14	yes	complemented
10	C₂⁵×C₂₆₉	—	2	1	no	—
11	C₂²×D₄×C₂₆₉	—	2	7	no	—
12	C₂²×D₄×C₂₆₉	C₂²	1	14	yes	—
13	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₅₃₈	—	2	42	no	—
15	C₂×Dic₅₃₈	C₂³	1	28	yes	complemented
16	C₂²×Dic₂₆₉	—	2	14	no	—
17	#16864390	C₂³	1	64	yes	direct product factor
18	C₂×#14524130	C₂³	1	28	yes	complemented
19	C₂⁴×C₂₆₉	—	2	14	no	—
20	C₂⁴×C₂₆₉	—	4	8	no	—
21	C₂⁴×C₂₆₉	D₄	1	1	yes	characteristic
22	C₂×D₄×C₂₆₉	—	2	42	no	—
23	C₂×D₄×C₂₆₉	C₂³	1	28	yes	—
24	C₂²×C₄×C₂₆₉	D₄	1	7	yes	—
25	C₂²×C₄×C₂₆₉	C₂³	1	7	yes	—
26	Dic₅₃₈	—	2	28	no	—
27	Dic₅₃₈	C₂⁴	1	8	yes	complemented
28	C₂×Dic₂₆₉	—	2	28	no	—
29	#14524130	C₂⁴	1	8	yes	complemented
30	C₂³×C₂₆₉	—	2	28	no	—
31	C₂³×C₂₆₉	—	4	56	no	—
32	C₂³×C₂₆₉	SD₁₆	1	8	yes	complemented
33	C₂³×C₂₆₉	C₂×D₄	1	7	yes	—
34	D₄×C₂₆₉	—	2	28	no	—
35	D₄×C₂₆₉	C₂⁴	1	8	yes	—
36	C₂×C₄×C₂₆₉	C₂×D₄	1	21	yes	—
37	C₂×C₄×C₂₆₉	C₂⁴	1	7	yes	—
38	C₂²×C₂₆₉	—	2	8	no	—
39	C₂²×C₂₆₉	—	4	56	no	—
40	C₂²×C₂₆₉	C₂×SD₁₆	1	28	yes	complemented
41	C₂²×C₂₆₉	C₂²×D₄	1	7	yes	—
42	C₄×C₂₆₉	C₂²×D₄	1	7	yes	—
43	C₄×C₂₆₉	C₂⁵	1	1	yes	characteristic
44	Dic₂₆₉	—	2	8	no	—
45	C₂×C₂₆₉	—	4	8	no	—
46	C₂×C₂₆₉	C₂²×SD₁₆	1	14	yes	complemented
47	C₂×C₂₆₉	C₂³×D₄	1	1	yes	characteristic
48	C₂₆₉	C₂³×SD₁₆	1	1	yes	characteristic, complemented, Sylow
49	C₂³×SD₁₆	—	269	1	no	maximal, Sylow
50	C₂³×Q₈	—	269	1	no	—
51	C₂³×D₄	D₂₆₉	1	1	yes	characteristic
52	C₂²×SD₁₆	—	269	28	no	—
53	C₂³×C₈	—	269	1	no	—
54	C₂⁵	—	2	1	no	—
55	C₂²×Q₈	—	269	14	no	—
56	C₂²×Q₈	—	538	7	no	—
57	C₂²×D₄	—	2	7	no	—
58	C₂²×D₄	C₂×D₂₆₉	1	14	yes	—
59	C₂³×C₄	—	538	1	no	—
60	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
61	C₂×SD₁₆	—	269	112	no	—
62	C₂²×C₈	—	269	14	no	—
63	C₂⁴	—	2	14	no	—
64	C₂⁴	—	4	8	no	—
65	C₂⁴	C₂²⋊D₂₆₉	1	1	yes	characteristic
66	C₂×Q₈	—	269	28	no	—
67	C₂×Q₈	—	538	42	no	—
68	C₂×D₄	—	2	42	no	—
69	C₂×D₄	C₂²×D₂₆₉	1	28	yes	—
70	C₂²×C₄	—	538	14	no	—
71	C₂²×C₄	C₂²⋊D₂₆₉	1	7	yes	—
72	C₂²×C₄	C₂²×D₂₆₉	1	7	yes	—
73	SD₁₆	—	269	64	no	—
74	C₂×C₈	—	269	28	no	—
75	C₂³	—	2	28	no	—
76	C₂³	—	4	56	no	—
77	C₂³	#16864390	1	8	yes	direct product factor
78	C₂³	C₂×C₂²⋊D₂₆₉	1	7	yes	—
79	Q₈	—	269	8	no	—
80	Q₈	—	538	28	no	—
81	D₄	—	2	28	no	—
82	D₄	C₂³×D₂₆₉	1	8	yes	—
83	C₂×C₄	—	538	28	no	—
84	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	21	yes	—
85	C₂×C₄	C₂³×D₂₆₉	1	7	yes	—
86	C₈	—	269	8	no	—
87	C₂²	—	2	8	no	—
88	C₂²	—	4	56	no	—
89	C₂²	C₂×#16864390	1	28	yes	direct product factor
90	C₂²	C₂²×C₂²⋊D₂₆₉	1	7	yes	—
91	C₄	—	538	8	no	—
92	C₄	C₂²×C₂²⋊D₂₆₉	1	7	yes	—
93	C₄	C₂⁴×D₂₆₉	1	1	yes	characteristic
94	C₂	—	4	8	no	—
95	C₂	C₂²×#16864390	1	14	yes	direct product factor
96	C₂	C₂³×C₂²⋊D₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊D₂₆₉
commutator, cosocle	C₄×C₂₆₉	C₂⁵
Frattini	C₄	C₂⁴×D₂₆₉
Fitting, prime radical	C₂³×D₄×C₂₆₉	C₂
socle	C₂⁴×C₂₆₉	D₄
hypercenter	C₂³×D₄	D₂₆₉
hypocenter	C₂₆₉	C₂³×SD₁₆
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₅₃₈	C₂	semidirect
2. C₂²×#16864390	C₂	direct (N × H)
3. C₂³×#14524130	C₂	semidirect
6. C₂²×Dic₅₃₈	C₂²	semidirect
8. C₂×#16864390	C₂²	direct (N × H)
9. C₂²×#14524130	C₂²	semidirect
15. C₂×Dic₅₃₈	C₂³	semidirect
17. #16864390	C₂³	direct (N × H)
18. C₂×#14524130	C₂³	semidirect
27. Dic₅₃₈	C₂⁴	semidirect
29. #14524130	C₂⁴	semidirect
32. C₂³×C₂₆₉	SD₁₆	semidirect
40. C₂²×C₂₆₉	C₂×SD₁₆	semidirect
46. C₂×C₂₆₉	C₂²×SD₁₆	semidirect
48. C₂₆₉	C₂³×SD₁₆	semidirect
77. C₂³	#16864390	direct (N × H)
89. C₂²	C₂×#16864390	direct (N × H)
95. C₂	C₂²×#16864390	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₅₃₈	2
C₂²×#16864390	2
C₂³×#14524130	2
C₂³×D₄×C₂₆₉	2
C₂³×SD₁₆	269

Subgroups — C23×#16864390 #46825604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#16864390 #46825604