Subgroups — C₂²×Dic₇₈ #11929300

Order 1248.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₃₉	C₂	1	2	yes	maximal
3	C₂×Dic₇₈	C₂	1	12	yes	maximal, direct product factor
4	C₂²×Dic₂₆	—	3	1	no	maximal, Hall
5	C₂³×C₃×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₄×C₁₃	C₂²	1	6	yes	—
7	C₂×Dic₃₉	C₂²	1	12	yes	—
8	Dic₇₈	C₂²	1	16	yes	direct product factor
9	C₂²×C₄×C₁₃	D₃	1	1	yes	characteristic
10	C₂²×Dic₁₃	—	3	2	no	—
11	C₂×Dic₂₆	—	3	12	no	—
12	C₂²×C₃×C₁₃	Q₈	1	4	yes	complemented
13	C₂²×C₃×C₁₃	C₂³	1	3	yes	—
14	C₃×C₄×C₁₃	C₂³	1	4	yes	—
15	Dic₃₉	C₂³	1	8	yes	—
16	C₂³×C₁₃	C₂×D₃	1	1	yes	characteristic
17	C₂×C₄×C₁₃	C₂×D₃	1	6	yes	—
18	C₂×Dic₁₃	—	3	12	no	—
19	Dic₂₆	—	3	16	no	—
20	C₂²×Dic₆	—	13	1	no	maximal, Hall
21	C₂×C₃×C₁₃	C₂×Q₈	1	6	yes	complemented
22	C₂×C₃×C₁₃	C₂⁴	1	1	yes	characteristic
23	C₂²×C₁₃	Dic₆	1	4	yes	complemented
24	C₂²×C₁₃	C₂²×D₃	1	3	yes	—
25	C₄×C₁₃	C₂²×D₃	1	4	yes	—
26	Dic₁₃	—	3	8	no	—
27	C₂²×C₃×C₄	D₁₃	1	1	yes	characteristic
28	C₂²×Dic₃	—	13	2	no	—
29	C₂×Dic₆	—	13	12	no	—
30	C₃×C₁₃	C₂²×Q₈	1	1	yes	characteristic, complemented, Hall
31	C₂²×Q₈	—	39	1	no	Sylow
32	C₂×C₁₃	C₂×Dic₆	1	6	yes	complemented
33	C₂×C₁₃	C₂³×D₃	1	1	yes	characteristic
34	C₂³×C₃	C₂×D₁₃	1	1	yes	characteristic
35	C₂×C₃×C₄	C₂×D₁₃	1	6	yes	—
36	C₂×Dic₃	—	13	12	no	—
37	Dic₆	—	13	16	no	—
38	C₂×Q₈	—	39	12	no	—
39	C₂²×C₄	—	39	2	no	—
40	C₂²×C₄	D₃₉	1	1	yes	characteristic
41	C₁₃	C₂²×Dic₆	1	1	yes	characteristic, complemented, Sylow
42	C₂²×C₃	Dic₂₆	1	4	yes	complemented
43	C₂²×C₃	C₂²×D₁₃	1	3	yes	—
44	C₃×C₄	C₂²×D₁₃	1	4	yes	—
45	Dic₃	—	13	8	no	—
46	C₂³	C₂×D₃₉	1	1	yes	characteristic
47	Q₈	—	39	16	no	—
48	C₂×C₄	—	39	12	no	—
49	C₂×C₄	C₂×D₃₉	1	6	yes	—
50	C₂×C₃	C₂×Dic₂₆	1	6	yes	complemented
51	C₂×C₃	C₂³×D₁₃	1	1	yes	characteristic
52	C₂²	Dic₇₈	1	4	yes	direct product factor
53	C₂²	C₂²×D₃₉	1	3	yes	—
54	C₄	—	39	8	no	—
55	C₄	C₂²×D₃₉	1	4	yes	—
56	C₃	C₂²×Dic₂₆	1	1	yes	characteristic, complemented, Sylow
57	C₂	C₂×Dic₇₈	1	6	yes	direct product factor
58	C₂	C₂³×D₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₉
commutator, cosocle	C₂×C₃×C₁₃	C₂⁴
Frattini	C₂	C₂³×D₃₉
Fitting	C₂²×C₃×C₄×C₁₃	C₂
socle, prime radical	C₂³×C₃×C₁₃	C₂²
hypercenter	C₂²×C₄	D₃₉
hypocenter	C₃×C₁₃	C₂²×Q₈
LCS	G ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×Dic₇₈	C₂	direct (N × H)
8. Dic₇₈	C₂²	direct (N × H)
12. C₂²×C₃×C₁₃	Q₈	semidirect
21. C₂×C₃×C₁₃	C₂×Q₈	semidirect
23. C₂²×C₁₃	Dic₆	semidirect
30. C₃×C₁₃	C₂²×Q₈	semidirect
32. C₂×C₁₃	C₂×Dic₆	semidirect
41. C₁₃	C₂²×Dic₆	semidirect
42. C₂²×C₃	Dic₂₆	semidirect
50. C₂×C₃	C₂×Dic₂₆	semidirect
52. C₂²	Dic₇₈	direct (N × H)
56. C₃	C₂²×Dic₂₆	semidirect
57. C₂	C₂×Dic₇₈	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×C₃×C₄×C₁₃	2
C₂²×Dic₃₉	2
C₂×Dic₇₈	2
C₂²×Dic₂₆	3
C₂²×Dic₆	13

Subgroups — C22×Dic78 #11929300

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₇₈ #11929300