Subgroups — #29504814

Order 15296.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21439919	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₂₃₉	C₂	1	1	yes	maximal, characteristic
3	#21439894	C₂	1	2	yes	maximal, complemented
4	#21439892	C₂	1	2	yes	maximal, complemented
5	C₂×M₁₆×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₃₉	—	2	1	no	—
7	C₄⋊Dic₂₃₉	—	2	1	no	—
8	C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
10	C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₂₃₉	—	2	1	no	—
12	C₄×Dic₂₃₉	—	2	1	no	—
13	C₂²×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
14	M₁₆×C₂₃₉	C₄	1	4	yes	complemented
15	C₂×C₈×C₂₃₉	C₂²	1	2	yes	—
16	C₂×Dic₂₃₉	—	2	1	no	—
17	C₂×Dic₂₃₉	—	2	1	no	—
18	C₂×Dic₂₃₉	—	2	2	no	—
19	C₂×Dic₂₃₉	—	4	2	no	—
20	C₂³×C₂₃₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₃₉	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₂₃₉	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₃₉	Q₈	1	2	yes	—
24	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
25	C₈×C₂₃₉	C₂×C₄	1	4	yes	complemented
26	C₂²×C₂₃₉	—	2	1	no	—
27	C₂²×C₂₃₉	—	2	1	no	—
28	C₂²×C₂₃₉	C₄⋊C₄	1	2	yes	—
29	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
30	C₄×C₂₃₉	C₄⋊C₄	1	2	yes	—
31	C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
33	Dic₂₃₉	—	4	1	no	—
34	Dic₂₃₉	—	4	2	no	—
35	Dic₂₃₉	—	4	1	no	—
36	C₂×C₂₃₉	—	2	2	no	—
37	C₂×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₂₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
39	C₂×C₂₃₉	#137	1	1	yes	characteristic
40	C₂₃₉	#428	1	1	yes	characteristic, complemented, Sylow
41	#428	—	239	1	no	maximal, Sylow
42	C₂×M₁₆	D₂₃₉	1	1	yes	characteristic
43	#117	—	239	1	no	—
44	C₂×C₄⋊C₄	—	239	1	no	—
45	#107	—	239	2	no	—
46	#106	—	239	2	no	—
47	C₂²×C₄	—	478	1	no	—
48	C₂²×C₄	C₂×D₂₃₉	1	1	yes	characteristic
49	M₁₆	Dic₂₃₉	1	4	yes	complemented
50	C₂×C₈	C₂×D₂₃₉	1	2	yes	—
51	C₄⋊C₄	—	478	1	no	—
52	C₄⋊C₄	—	239	1	no	—
53	C₄⋊C₄	—	239	2	no	—
54	C₄⋊C₄	—	239	1	no	—
55	C₂²⋊C₄	—	478	1	no	—
56	C₄²	—	478	1	no	—
57	C₂³	D₉₅₆	1	1	yes	characteristic
58	C₂×C₄	—	956	2	no	—
59	C₂×C₄	—	478	1	no	—
60	C₂×C₄	—	478	1	no	—
61	C₂×C₄	—	478	2	no	—
62	C₂×C₄	C₂×Dic₂₃₉	1	2	yes	—
63	C₂×C₄	D₉₅₆	1	1	yes	characteristic
64	C₂×C₄	Dic₄₇₈	1	2	yes	—
65	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
66	C₈	C₂×Dic₂₃₉	1	4	yes	complemented
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	C₄⋊Dic₂₃₉	1	2	yes	—
70	C₂²	C₂×D₉₅₆	1	1	yes	characteristic
71	C₄	—	956	2	no	—
72	C₄	—	956	1	no	—
73	C₄	—	956	1	no	—
74	C₄	C₄⋊Dic₂₃₉	1	2	yes	—
75	C₄	C₂²×Dic₂₃₉	1	1	yes	characteristic
76	C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	C₂×C₄⋊Dic₂₃₉	1	1	yes	characteristic
79	C₂	#21439988	1	1	yes	characteristic
80	C₂	#21439993	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₉₅₆
commutator	C₄×C₂₃₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₃₉
Fitting	C₂×M₁₆×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₂×D₄
prime radical	C₂³×C₂₃₉	D₄
cosocle	C₂×C₄×C₂₃₉	C₂³
hypercenter	C₂×M₁₆	D₂₃₉
hypocenter	C₂₃₉	#428
LCS	G ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #21439894	C₂	semidirect
4. #21439892	C₂	semidirect
14. M₁₆×C₂₃₉	C₄	semidirect
25. C₈×C₂₃₉	C₂×C₄	semidirect
40. C₂₃₉	#428	semidirect
49. M₁₆	Dic₂₃₉	semidirect
66. C₈	C₂×Dic₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#21439919	2
C₂×C₄⋊Dic₂₃₉	2
#21439894	2
#21439892	2
C₂×M₁₆×C₂₃₉	2
#428	239