Subgroups — C₂×#28380702 #43244060

Order 28096.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28380702	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊Dic₄₃₉	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊Dic₄₃₉	C₂	1	2	yes	maximal
4	C₂×C₄×Dic₄₃₉	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×C₄₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₃₉	C₂²	1	4	yes	—
7	C₄⋊Dic₄₃₉	C₂²	1	8	yes	complemented
8	C₂²⋊Dic₄₃₉	C₂²	1	8	yes	—
9	C₄×Dic₄₃₉	C₂²	1	8	yes	complemented
10	C₂⁴×C₄₃₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₄₃₉	C₄	1	8	yes	complemented
12	C₂²×C₄×C₄₃₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₄₃₉	C₂²	1	4	yes	—
14	C₂²×C₄×C₄₃₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₃₉	—	2	8	no	—
16	C₂×Dic₄₃₉	C₂³	1	8	yes	—
17	C₂³×C₄₃₉	—	2	4	no	—
18	C₂³×C₄₃₉	C₂×C₄	1	4	yes	—
19	C₂³×C₄₃₉	C₂³	1	2	yes	—
20	C₂³×C₄₃₉	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₄₃₉	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₄₃₉	C₂×C₄	1	4	yes	—
23	C₂×C₄×C₄₃₉	C₂×C₄	1	16	yes	complemented
24	C₂×C₄×C₄₃₉	C₂³	1	2	yes	—
25	C₂×C₄×C₄₃₉	C₂³	1	2	yes	—
26	C₂²×C₄₃₉	—	2	8	no	—
27	C₂²×C₄₃₉	—	2	4	no	—
28	C₂²×C₄₃₉	C₂²×C₄	1	4	yes	—
29	C₂²×C₄₃₉	C₂²×C₄	1	2	yes	—
30	C₂²×C₄₃₉	#41	1	4	yes	—
31	C₂²×C₄₃₉	C₂⁴	1	1	yes	characteristic
32	C₄×C₄₃₉	C₂²×C₄	1	4	yes	—
33	C₄×C₄₃₉	C₂²×C₄	1	4	yes	—
34	Dic₄₃₉	—	2	8	no	—
35	C₂×C₄₃₉	—	2	4	no	—
36	C₂×C₄₃₉	#117	1	4	yes	complemented
37	C₂×C₄₃₉	C₂³×C₄	1	1	yes	characteristic
38	C₂×C₄₃₉	C₂×#41	1	2	yes	—
39	C₄₃₉	C₂×#117	1	1	yes	characteristic, complemented, Sylow
40	C₂×#117	—	439	1	no	maximal, Sylow
41	C₂³×C₄	D₄₃₉	1	1	yes	characteristic
42	#117	—	439	8	no	—
43	C₂×C₄⋊C₄	—	439	2	no	—
44	C₂×C₂²⋊C₄	—	439	2	no	—
45	C₂×C₄²	—	439	2	no	—
46	C₂⁴	C₂×D₄₃₉	1	1	yes	characteristic
47	C₂²×C₄	—	439	4	no	—
48	C₂²×C₄	Dic₄₃₉	1	8	yes	complemented
49	C₂²×C₄	C₂×D₄₃₉	1	1	yes	characteristic
50	C₂²×C₄	C₂×D₄₃₉	1	4	yes	—
51	C₂²×C₄	C₂×D₄₃₉	1	1	yes	characteristic
52	C₄⋊C₄	—	439	8	no	—
53	C₂²⋊C₄	—	439	8	no	—
54	C₄²	—	439	8	no	—
55	C₂³	—	2	4	no	—
56	C₂³	C₂×Dic₄₃₉	1	4	yes	—
57	C₂³	C₂²×D₄₃₉	1	2	yes	—
58	C₂³	C₂²×D₄₃₉	1	1	yes	characteristic
59	C₂×C₄	—	878	8	no	—
60	C₂×C₄	—	439	8	no	—
61	C₂×C₄	C₂×Dic₄₃₉	1	4	yes	—
62	C₂×C₄	C₂×Dic₄₃₉	1	4	yes	—
63	C₂×C₄	C₂×Dic₄₃₉	1	16	yes	complemented
64	C₂×C₄	C₂²×D₄₃₉	1	2	yes	—
65	C₂×C₄	C₂²×D₄₃₉	1	2	yes	—
66	C₂²	—	2	8	no	—
67	C₂²	—	2	4	no	—
68	C₂²	C₂²×Dic₄₃₉	1	4	yes	—
69	C₂²	C₂²×Dic₄₃₉	1	2	yes	—
70	C₂²	#20296329	1	4	yes	—
71	C₂²	C₂³×D₄₃₉	1	1	yes	characteristic
72	C₄	—	878	8	no	—
73	C₄	C₂²×Dic₄₃₉	1	4	yes	—
74	C₄	C₂²×Dic₄₃₉	1	4	yes	—
75	C₂	—	2	4	no	—
76	C₂	#28380702	1	4	yes	direct product factor
77	C₂	C₂³×Dic₄₃₉	1	1	yes	characteristic
78	C₂	C₂×#20296329	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₄₃₉
commutator	C₂×C₄₃₉	C₂³×C₄
Frattini	C₂²	C₂³×D₄₃₉
Fitting	C₂³×C₄×C₄₃₉	C₂
socle	C₂³×C₄₃₉	C₂³
prime radical	C₂⁴×C₄₃₉	C₂²
cosocle	C₂²×C₄₃₉	C₂⁴
hypercenter	C₂³×C₄	D₄₃₉
hypocenter	C₄₃₉	C₂×#117
LCS	G ▷ C₂×C₄₃₉ ▷ C₄₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. #28380702	C₂	direct (N × H)
2. C₂×C₄⋊Dic₄₃₉	C₂	semidirect
4. C₂×C₄×Dic₄₃₉	C₂	semidirect
7. C₄⋊Dic₄₃₉	C₂²	semidirect
9. C₄×Dic₄₃₉	C₂²	semidirect
11. C₂²×C₄×C₄₃₉	C₄	semidirect
23. C₂×C₄×C₄₃₉	C₂×C₄	semidirect
36. C₂×C₄₃₉	#117	semidirect
39. C₄₃₉	C₂×#117	semidirect
48. C₂²×C₄	Dic₄₃₉	semidirect
63. C₂×C₄	C₂×Dic₄₃₉	semidirect
76. C₂	#28380702	direct (N × H)

Maximal subgroups

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

subgroup	index
#28380702	2
C₂×C₄⋊Dic₄₃₉	2
C₂×C₂²⋊Dic₄₃₉	2
C₂×C₄×Dic₄₃₉	2
C₂³×C₄×C₄₃₉	2
C₂×#117	439

Subgroups — C2×#28380702 #43244060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#28380702 #43244060