Subgroups — #29505065

Order 15296.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21439959	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₂₃₉	C₂	1	1	yes	maximal, characteristic
3	#21439894	C₂	1	1	yes	maximal, characteristic
4	#21439893	C₂	1	1	yes	maximal, characteristic
5	Dic₂₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#21439884	C₂	1	1	yes	maximal, characteristic
7	#103×C₂₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₇₈	—	2	1	no	—
9	C₂×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₃₉	—	2	1	no	—
11	C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
13	Dic₂₃₉⋊C₄	—	2	1	no	—
14	C₄×Dic₂₃₉	—	2	1	no	—
15	C₄×Dic₂₃₉	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
19	Dic₄₇₈	—	4	2	no	—
20	C₂×Dic₂₃₉	—	2	1	no	—
21	C₂×Dic₂₃₉	—	2	1	no	—
22	C₂×Dic₂₃₉	D₄	1	1	yes	characteristic
23	C₂×Dic₂₃₉	D₄	1	1	yes	characteristic
24	C₂₃₉⋊C₈	—	2	1	no	—
25	Q₈×C₂₃₉	—	2	1	no	—
26	Q₈×C₂₃₉	Q₈	1	2	yes	complemented
27	C₂×C₄×C₂₃₉	—	2	1	no	—
28	C₂×C₄×C₂₃₉	—	2	1	no	—
29	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
30	C₈×C₂₃₉	—	2	1	no	—
31	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
32	C₄×C₂₃₉	—	2	2	no	—
33	C₄×C₂₃₉	—	4	1	no	—
34	C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₂₃₉	#41	1	1	yes	characteristic
36	Dic₂₃₉	—	2	1	no	—
37	Dic₂₃₉	—	4	1	no	—
38	Dic₂₃₉	—	4	1	no	—
39	Dic₂₃₉	Q₁₆	1	2	yes	—
40	C₂×C₂₃₉	C₂²⋊Q₈	1	1	yes	characteristic
41	C₂×C₂₃₉	C₂×Q₁₆	1	1	yes	characteristic
42	C₂×C₂₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
43	C₂₃₉	#477	1	1	yes	characteristic, complemented, Sylow
44	#477	—	239	1	no	maximal, Sylow
45	C₄⋊Q₈	—	239	1	no	—
46	C₄×Q₈	—	239	1	no	—
47	#107	—	239	1	no	—
48	#107	—	239	1	no	—
49	C₄⋊C₈	—	239	1	no	—
50	#103	—	239	1	no	—
51	#103	D₂₃₉	1	1	yes	characteristic
52	C₂×Q₈	—	478	1	no	—
53	C₂×Q₈	C₂×D₂₃₉	1	1	yes	characteristic
54	C₂×C₈	—	239	1	no	—
55	C₂×C₈	C₂×D₂₃₉	1	1	yes	characteristic
56	C₄⋊C₄	—	478	1	no	—
57	C₄⋊C₄	—	478	1	no	—
58	C₄⋊C₄	—	239	1	no	—
59	C₄⋊C₄	—	239	1	no	—
60	C₄⋊C₄	C₂×D₂₃₉	1	1	yes	characteristic
61	C₄²	—	478	1	no	—
62	C₄²	—	239	1	no	—
63	Q₈	—	2	1	no	—
64	Q₈	—	956	2	no	—
65	Q₈	Dic₄₇₈	1	2	yes	complemented
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	478	1	no	—
69	C₂×C₄	—	478	1	no	—
70	C₂×C₄	—	239	1	no	—
71	C₂×C₄	—	239	1	no	—
72	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	478	1	no	—
75	C₂²	D₄×D₂₃₉	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	956	1	no	—
79	C₄	—	956	1	no	—
80	C₄	—	478	1	no	—
81	C₄	—	239	2	no	—
82	C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
83	C₄	#16838547	1	1	yes	characteristic
84	C₂	#21439937	1	1	yes	characteristic
85	C₂	Q₁₆×D₂₃₉	1	1	yes	characteristic
86	C₂	#21439987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₃₉
commutator, cosocle	C₂×C₄×C₂₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₃₉
Fitting	#103×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	C₂×D₄
hypercenter	#103	D₂₃₉
hypocenter	C₂₃₉	#477
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
26. Q₈×C₂₃₉	Q₈	semidirect
43. C₂₃₉	#477	semidirect
65. Q₈	Dic₄₇₈	semidirect

Maximal subgroups

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

subgroup	index
#21439959	2
Q₈×Dic₂₃₉	2
#21439894	2
#21439893	2
Dic₂₃₉⋊C₈	2
#21439884	2
#103×C₂₃₉	2
#477	239