Subgroups — D₄×#157 #64288

Order 288.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#157	C₂	1	2	yes	maximal, complemented
2	D₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#157	C₂	1	2	yes	maximal, complemented
4	C₄⋊#157	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#157	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃⋊D₃	C₂²	1	2	yes	—
7	C₂×#157	—	2	4	no	—
8	C₂×#157	C₂²	1	2	yes	—
9	C₂×#157	C₂²	1	1	yes	characteristic
10	C₂×#157	C₂²	1	1	yes	characteristic
11	C₃²×D₄	C₄	1	1	yes	characteristic, complemented
12	#638	C₄	1	2	yes	complemented
13	C₄⋊(C₃⋊D₃)	C₄	1	1	yes	characteristic, complemented
14	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
15	D₃×D₄	—	6	2	no	—
16	C₂²×C₃²	C₂×C₄	1	2	yes	complemented
17	C₂×C₃⋊D₃	—	2	2	no	—
18	C₂×C₃⋊D₃	—	2	2	no	—
19	C₂×C₃⋊D₃	C₂×C₄	1	2	yes	complemented
20	C₂×C₃⋊D₃	C₂³	1	1	yes	characteristic
21	#157	—	2	2	no	—
22	#157	—	2	1	no	—
23	#157	D₄	1	2	yes	direct product factor
24	C₃²×C₄	C₂×C₄	1	1	yes	characteristic, complemented
25	C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic, complemented
26	C₄×D₄	—	9	1	no	maximal, Sylow
27	C₂²×D₃	—	6	4	no	—
28	C₃×D₄	—	2	2	no	—
29	#67	—	6	4	no	—
30	D₁₂	—	6	2	no	—
31	C₄×D₃	—	6	2	no	—
32	C₂×C₃²	—	2	2	no	—
33	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
34	C₃⋊D₃	—	2	2	no	—
35	C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
36	C₃⋊D₃	#41	1	1	yes	characteristic
37	C₂×D₄	—	9	1	no	—
38	C₂²×C₄	—	9	2	no	—
39	C₄⋊C₄	—	9	1	no	—
40	C₂²⋊C₄	—	9	2	no	—
41	C₄²	—	9	1	no	—
42	C₂²×C₃	—	2	4	no	—
43	C₂×D₃	—	6	2	no	—
44	C₂×D₃	—	6	4	no	—
45	C₂×D₃	—	12	4	no	—
46	C₂×D₃	—	12	4	no	—
47	C₃×C₄	—	2	2	no	—
48	Dic₃	—	6	2	no	—
49	C₃²	C₄×D₄	1	1	yes	characteristic, complemented, Sylow
50	C₂³	—	9	2	no	—
51	D₄	—	9	2	no	—
52	D₄	—	9	1	no	—
53	D₄	#157	1	1	yes	characteristic, direct product factor
54	C₂×C₄	—	9	1	no	—
55	C₂×C₄	—	9	1	no	—
56	C₂×C₄	—	9	1	no	—
57	C₂×C₄	—	9	2	no	—
58	C₂×C₄	—	18	4	no	—
59	C₂×C₃	—	2	2	no	—
60	C₂×C₃	—	4	4	no	—
61	D₃	—	6	2	no	—
62	D₃	—	6	2	no	—
63	D₃	—	12	4	no	—
64	C₂²	—	9	1	no	—
65	C₂²	—	9	2	no	—
66	C₂²	—	18	2	no	—
67	C₂²	—	18	2	no	—
68	C₂²	C₂×#157	1	2	yes	complemented
69	C₄	—	9	2	no	—
70	C₄	—	9	1	no	—
71	C₄	—	18	2	no	—
72	C₄	—	18	1	no	—
73	C₄	C₂×#157	1	1	yes	characteristic, complemented
74	C₃	—	2	2	no	—
75	C₂	—	2	2	no	—
76	C₂	—	9	1	no	—
77	C₂	—	9	1	no	—
78	C₂	—	18	2	no	—
79	C₂	C₂²×#157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×#157
commutator, socle	C₂×C₃²	C₂²×C₄
Fitting	C₃²×D₄	C₄
prime radical	D₄×C₃⋊D₃	C₂
cosocle	C₂×C₃⋊D₃	C₂³
hypercenter	D₄	#157
hypocenter	C₃²	C₄×D₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#157	C₂	semidirect
3. C₂²⋊#157	C₂	semidirect
4. C₄⋊#157	C₂	semidirect
5. C₄×#157	C₂	semidirect
11. C₃²×D₄	C₄	semidirect
12. #638	C₄	semidirect
13. C₄⋊(C₃⋊D₃)	C₄	semidirect
16. C₂²×C₃²	C₂×C₄	semidirect
19. C₂×C₃⋊D₃	C₂×C₄	semidirect
23. #157	D₄	direct (N × H)
24. C₃²×C₄	C₂×C₄	semidirect
25. C₃⋊Dic₃	C₂×C₄	semidirect
49. C₃²	C₄×D₄	semidirect
53. D₄	#157	direct (N × H)
68. C₂²	C₂×#157	semidirect
73. C₄	C₂×#157	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#157	2
D₄×C₃⋊D₃	2
C₂²⋊#157	2
C₄⋊#157	2
C₄×#157	2
C₄×D₄	9

Subgroups — D4×#157 #64288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#157 #64288