Subgroups — #29504903

Order 15296.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₃₉⋊C₈)	C₂	1	2	yes	maximal
3	#21439871	C₂	1	2	yes	maximal
4	C₄×C₂₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₂₃₉⋊C₈	C₂²	1	2	yes	—
8	C₂×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×Q₈×C₂₃₉	C₄	1	1	yes	characteristic
10	C₄⋊C₄×C₂₃₉	C₄	1	2	yes	—
11	C₄⋊C₄×C₂₃₉	C₄	1	1	yes	characteristic
12	C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
13	C₄²×C₂₃₉	C₂²	1	2	yes	—
14	C₂₃₉⋊C₈	—	2	1	no	—
15	C₂₃₉⋊C₈	—	2	2	no	—
16	C₂₃₉⋊C₈	Q₈	1	2	yes	complemented
17	Q₈×C₂₃₉	—	2	2	no	—
18	C₂×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₃₉	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₂₃₉	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
23	C₂²×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₂₃₉	—	2	2	no	—
25	C₄×C₂₃₉	—	2	1	no	—
26	C₄×C₂₃₉	—	2	2	no	—
27	C₄×C₂₃₉	M₁₆	1	2	yes	—
28	C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₂₃₉	#41	1	1	yes	characteristic
30	C₂×C₂₃₉	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₂₃₉	C₂×M₁₆	1	1	yes	characteristic
32	C₂×C₂₃₉	#131	1	1	yes	characteristic
33	C₂₃₉	#446	1	1	yes	characteristic, complemented, Sylow
34	#446	—	239	1	no	maximal, Sylow
35	C₄×Q₈	D₂₃₉	1	1	yes	characteristic
36	C₄⋊C₈	—	239	1	no	—
37	C₄⋊C₈	—	239	2	no	—
38	#97	—	239	2	no	—
39	C₄×C₈	—	239	1	no	—
40	C₂×Q₈	Dic₂₃₉	1	1	yes	characteristic
41	C₂×C₈	—	239	1	no	—
42	C₂×C₈	—	239	2	no	—
43	C₂×C₈	—	239	1	no	—
44	C₄⋊C₄	Dic₂₃₉	1	2	yes	—
45	C₄⋊C₄	Dic₂₃₉	1	1	yes	characteristic
46	C₄²	C₂×D₂₃₉	1	1	yes	characteristic
47	C₄²	C₂×D₂₃₉	1	2	yes	—
48	Q₈	—	2	2	no	—
49	C₂×C₄	C₂×Dic₂₃₉	1	1	yes	characteristic
50	C₂×C₄	C₂×Dic₂₃₉	1	2	yes	—
51	C₂×C₄	C₂×Dic₂₃₉	1	2	yes	—
52	C₂×C₄	C₂×Dic₂₃₉	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
54	C₈	—	478	1	no	—
55	C₈	—	478	2	no	—
56	C₈	—	239	2	no	—
57	C₂²	C₂²×Dic₂₃₉	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	2	1	no	—
60	C₄	—	2	2	no	—
61	C₄	#16838532	1	2	yes	—
62	C₄	Q₈×D₂₃₉	1	1	yes	characteristic
63	C₄	#16838549	1	1	yes	characteristic
64	C₂	Q₈×Dic₂₃₉	1	1	yes	characteristic
65	C₂	C₂×#16838532	1	1	yes	characteristic
66	C₂	#21439967	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃₉
commutator, socle, prime radical	C₂²×C₂₃₉	C₂²×C₄
Fitting	C₄×Q₈×C₂₃₉	C₂
cosocle	C₂×C₄×C₂₃₉	C₂³
hypercenter	C₄×Q₈	D₂₃₉
hypocenter	C₂₃₉	#446
LCS	G ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
16. C₂₃₉⋊C₈	Q₈	semidirect
33. C₂₃₉	#446	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₂₃₉⋊C₈)	2
#21439871	2
C₄×C₂₃₉⋊C₈	2
C₄×Q₈×C₂₃₉	2
#446	239