Subgroups — D₄×#10587739 #17984168

Order 4384.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587739	C₂	1	2	yes	maximal, complemented
2	C₄⋊#10587739	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#10587739	C₂	1	2	yes	maximal, complemented
4	C₄×#10587739	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×D₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587739	—	2	4	no	—
7	C₂×#10587739	C₂²	1	1	yes	characteristic
8	C₂×#10587739	C₂²	1	2	yes	—
9	C₂×#10587739	C₂²	1	1	yes	characteristic
10	C₂²×D₁₃₇	C₂²	1	2	yes	—
11	D₅₄₈	C₄	1	1	yes	characteristic, complemented
12	C₂²⋊D₁₃₇	C₄	1	2	yes	complemented
13	C₄×D₁₃₇	C₂²	1	1	yes	characteristic
14	D₄×C₁₃₇	C₄	1	1	yes	characteristic, complemented
15	C₂²×C₁₃₇	C₂×C₄	1	2	yes	complemented
16	C₂×D₁₃₇	—	2	2	no	—
17	C₂×D₁₃₇	—	2	2	no	—
18	C₂×D₁₃₇	C₂×C₄	1	2	yes	complemented
19	C₂×D₁₃₇	C₂³	1	1	yes	characteristic
20	#10587739	—	2	2	no	—
21	#10587739	—	2	1	no	—
22	#10587739	D₄	1	2	yes	direct product factor
23	C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
24	Dic₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
25	C₂×C₁₃₇	—	2	2	no	—
26	C₂×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
27	D₁₃₇	—	2	2	no	—
28	D₁₃₇	C₂×D₄	1	1	yes	characteristic
29	D₁₃₇	#41	1	1	yes	characteristic
30	C₁₃₇	C₄×D₄	1	1	yes	characteristic, complemented, Sylow
31	C₄×D₄	—	137	1	no	maximal, Sylow
32	C₂×D₄	—	137	1	no	—
33	C₂²×C₄	—	137	2	no	—
34	C₄⋊C₄	—	137	1	no	—
35	C₂²⋊C₄	—	137	2	no	—
36	C₄²	—	137	1	no	—
37	C₂³	—	137	2	no	—
38	D₄	—	137	2	no	—
39	D₄	—	137	1	no	—
40	D₄	#10587739	1	1	yes	characteristic, direct product factor
41	C₂×C₄	—	274	4	no	—
42	C₂×C₄	—	137	1	no	—
43	C₂×C₄	—	137	2	no	—
44	C₂×C₄	—	137	1	no	—
45	C₂×C₄	—	137	1	no	—
46	C₂²	—	274	2	no	—
47	C₂²	—	274	2	no	—
48	C₂²	—	137	2	no	—
49	C₂²	—	137	1	no	—
50	C₂²	C₂×#10587739	1	2	yes	complemented
51	C₄	—	274	2	no	—
52	C₄	—	274	1	no	—
53	C₄	—	137	2	no	—
54	C₄	—	137	1	no	—
55	C₄	C₂×#10587739	1	1	yes	characteristic, complemented
56	C₂	—	2	2	no	—
57	C₂	—	274	2	no	—
58	C₂	—	137	1	no	—
59	C₂	—	137	1	no	—
60	C₂	C₂²×#10587739	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×#10587739
commutator, socle	C₂×C₁₃₇	C₂²×C₄
Fitting	D₄×C₁₃₇	C₄
prime radical	D₄×D₁₃₇	C₂
cosocle	C₂×D₁₃₇	C₂³
hypercenter	D₄	#10587739
hypocenter	C₁₃₇	C₄×D₄
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#10587739	C₂	semidirect
2. C₄⋊#10587739	C₂	semidirect
3. C₂²⋊#10587739	C₂	semidirect
4. C₄×#10587739	C₂	semidirect
11. D₅₄₈	C₄	semidirect
12. C₂²⋊D₁₃₇	C₄	semidirect
14. D₄×C₁₃₇	C₄	semidirect
15. C₂²×C₁₃₇	C₂×C₄	semidirect
18. C₂×D₁₃₇	C₂×C₄	semidirect
22. #10587739	D₄	direct (N × H)
23. C₄×C₁₃₇	C₂×C₄	semidirect
24. Dic₁₃₇	C₂×C₄	semidirect
30. C₁₃₇	C₄×D₄	semidirect
40. D₄	#10587739	direct (N × H)
50. C₂²	C₂×#10587739	semidirect
55. C₄	C₂×#10587739	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10587739	2
C₄⋊#10587739	2
C₂²⋊#10587739	2
C₄×#10587739	2
D₄×D₁₃₇	2
C₄×D₄	137

Subgroups — D4×#10587739 #17984168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#10587739 #17984168