Subgroups — #29504902

Order 15296.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₄₇₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	Dic₂₃₉⋊C₈	C₂	1	2	yes	maximal
4	#21439870	C₂	1	2	yes	maximal
5	C₄×C₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₇₈	C₄	1	1	yes	characteristic
7	C₂×C₂₃₉⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₂₃₉	C₄	1	1	yes	characteristic
9	Dic₂₃₉⋊C₄	C₄	1	2	yes	—
10	C₄×Dic₂₃₉	C₂²	1	2	yes	—
11	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
13	C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
14	Dic₄₇₈	—	2	2	no	—
15	C₂×Dic₂₃₉	C₂×C₄	1	2	yes	—
16	C₂×Dic₂₃₉	C₂×C₄	1	2	yes	—
17	C₂₃₉⋊C₈	—	2	2	no	—
18	C₂×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
21	C₈×C₂₃₉	—	2	1	no	—
22	C₈×C₂₃₉	Q₈	1	2	yes	complemented
23	C₂²×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₂₃₉	—	2	1	no	—
25	C₄×C₂₃₉	M₁₆	1	2	yes	—
26	C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
27	C₄×C₂₃₉	#41	1	1	yes	characteristic
28	Dic₂₃₉	—	2	2	no	—
29	Dic₂₃₉	—	2	2	no	—
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	—	239	1	no	—
36	C₄⋊C₈	—	239	2	no	—
37	C₄⋊C₈	—	239	1	no	—
38	#97	—	239	2	no	—
39	C₄×C₈	D₂₃₉	1	1	yes	characteristic
40	C₂×Q₈	—	239	1	no	—
41	C₂×C₈	—	239	2	no	—
42	C₂×C₈	C₂×D₂₃₉	1	1	yes	characteristic
43	C₂×C₈	C₂×D₂₃₉	1	1	yes	characteristic
44	C₄⋊C₄	—	239	2	no	—
45	C₄⋊C₄	—	239	1	no	—
46	C₄²	—	239	2	no	—
47	C₄²	C₂×D₂₃₉	1	1	yes	characteristic
48	Q₈	—	478	2	no	—
49	C₂×C₄	—	239	2	no	—
50	C₂×C₄	—	239	2	no	—
51	C₂×C₄	C₄×D₂₃₉	1	1	yes	characteristic
52	C₂×C₄	C₄×D₂₃₉	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
54	C₈	—	2	1	no	—
55	C₈	—	478	2	no	—
56	C₈	Dic₄₇₈	1	2	yes	complemented
57	C₂²	C₂×C₄×D₂₃₉	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	478	2	no	—
60	C₄	—	478	2	no	—
61	C₄	#16838531	1	2	yes	—
62	C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
63	C₄	#16838548	1	1	yes	characteristic
64	C₂	C₄×Dic₄₇₈	1	1	yes	characteristic
65	C₂	C₂×#16838531	1	1	yes	characteristic
66	C₂	#21439966	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃₉
commutator, socle, prime radical	C₂²×C₂₃₉	C₂²×C₄
Fitting	C₄×C₈×C₂₃₉	C₂
cosocle	C₂×C₄×C₂₃₉	C₂³
hypercenter	C₄×C₈	D₂₃₉
hypocenter	C₂₃₉	#446
LCS	G ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
22. C₈×C₂₃₉	Q₈	semidirect
33. C₂₃₉	#446	semidirect
56. C₈	Dic₄₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₄₇₈	2
C₄⋊(C₂₃₉⋊C₈)	2
Dic₂₃₉⋊C₈	2
#21439870	2
C₄×C₈×C₂₃₉	2
#446	239