Subgroups — C₂×#27134380 #37589941

Order 24608.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27134380	C₂	1	8	yes	maximal, direct product factor
2	C₂×Dic₁₅₃₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊D₇₆₉	C₂	1	2	yes	maximal, complemented
4	C₂²×Dic₇₆₉	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₇₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×C₇₆₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₇₆₉	C₂²	1	1	yes	characteristic
8	Dic₁₅₃₈	C₂²	1	4	yes	complemented
9	C₂²⋊D₇₆₉	C₂²	1	8	yes	complemented
10	C₂×Dic₇₆₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₇₆₉	C₂²	1	2	yes	—
12	C₂×Dic₇₆₉	C₂²	1	8	yes	complemented
13	C₄×D₇₆₉	C₂²	1	4	yes	complemented
14	C₂³×C₇₆₉	C₂²	1	2	yes	—
15	D₄×C₇₆₉	C₂²	1	4	yes	complemented
16	C₂×C₄×C₇₆₉	C₂²	1	1	yes	characteristic
17	C₂²×C₇₆₉	—	2	4	no	—
18	C₂²×C₇₆₉	C₂³	1	4	yes	—
19	C₂²×C₇₆₉	C₂³	1	1	yes	characteristic
20	C₂×D₇₆₉	—	2	2	no	—
21	C₂×D₇₆₉	C₂³	1	2	yes	—
22	C₄×C₇₆₉	C₂³	1	2	yes	—
23	Dic₇₆₉	C₂³	1	2	yes	—
24	Dic₇₆₉	C₂³	1	4	yes	—
25	C₂×C₇₆₉	—	2	4	no	—
26	C₂×C₇₆₉	#41	1	2	yes	complemented
27	C₂×C₇₆₉	C₂⁴	1	1	yes	characteristic
28	D₇₆₉	—	2	2	no	—
29	C₇₆₉	C₂×#41	1	1	yes	characteristic, complemented, Sylow
30	C₂×#41	—	769	1	no	maximal, Sylow
31	#41	—	769	8	no	—
32	C₂×Q₈	—	769	1	no	—
33	C₂×D₄	—	769	2	no	—
34	C₂×D₄	D₇₆₉	1	1	yes	characteristic, complemented
35	C₂²×C₄	—	769	2	no	—
36	C₂²×C₄	—	769	1	no	—
37	C₂³	—	769	1	no	—
38	C₂³	C₂×D₇₆₉	1	2	yes	—
39	Q₈	—	769	4	no	—
40	D₄	—	769	8	no	—
41	D₄	C₂×D₇₆₉	1	4	yes	complemented
42	C₂×C₄	—	769	8	no	—
43	C₂×C₄	—	769	1	no	—
44	C₂×C₄	—	769	2	no	—
45	C₂×C₄	—	769	4	no	—
46	C₂×C₄	C₂×D₇₆₉	1	1	yes	characteristic
47	C₂²	—	769	2	no	—
48	C₂²	—	2	4	no	—
49	C₂²	—	1538	2	no	—
50	C₂²	C₂²×D₇₆₉	1	4	yes	—
51	C₂²	C₂²×D₇₆₉	1	1	yes	characteristic
52	C₄	—	769	2	no	—
53	C₄	—	769	4	no	—
54	C₄	C₂²×D₇₆₉	1	2	yes	—
55	C₂	—	2	4	no	—
56	C₂	—	1538	2	no	—
57	C₂	#27134380	1	2	yes	direct product factor
58	C₂	C₂³×D₇₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₆₉
commutator, cosocle	C₂×C₇₆₉	C₂⁴
Frattini	C₂	C₂³×D₇₆₉
Fitting	C₂×D₄×C₇₆₉	C₂
socle	C₂²×C₇₆₉	C₂³
hypercenter	C₂×D₄	D₇₆₉
hypocenter	C₇₆₉	C₂×#41
LCS	G ▷ C₂×C₇₆₉ ▷ C₇₆₉
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #27134380	C₂	direct (N × H)
2. C₂×Dic₁₅₃₈	C₂	semidirect
3. C₂×C₂²⋊D₇₆₉	C₂	semidirect
4. C₂²×Dic₇₆₉	C₂	semidirect
5. C₂×C₄×D₇₆₉	C₂	semidirect
6. C₂×D₄×C₇₆₉	C₂	semidirect
8. Dic₁₅₃₈	C₂²	semidirect
9. C₂²⋊D₇₆₉	C₂²	semidirect
12. C₂×Dic₇₆₉	C₂²	semidirect
13. C₄×D₇₆₉	C₂²	semidirect
15. D₄×C₇₆₉	C₂²	semidirect
26. C₂×C₇₆₉	#41	semidirect
29. C₇₆₉	C₂×#41	semidirect
34. C₂×D₄	D₇₆₉	semidirect
41. D₄	C₂×D₇₆₉	semidirect
57. C₂	#27134380	direct (N × H)

Maximal subgroups

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

subgroup	index
#27134380	2
C₂×Dic₁₅₃₈	2
C₂×C₂²⋊D₇₆₉	2
C₂²×Dic₇₆₉	2
C₂×C₄×D₇₆₉	2
C₂×D₄×C₇₆₉	2
C₂×#41	769

Subgroups — C2×#27134380 #37589941

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27134380 #37589941