Subgroups — #29504504

Order 15296.

70 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	C₄×C₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₇₈	C₄	1	1	yes	characteristic
5	C₂×C₂₃₉⋊C₈	—	2	1	no	—
6	C₄⋊Dic₂₃₉	C₄	1	1	yes	characteristic
7	Dic₂₃₉⋊C₄	—	2	1	no	—
8	C₄×Dic₂₃₉	—	2	1	no	—
9	C₂×C₈×C₂₃₉	—	2	1	no	—
10	C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
11	Dic₄₇₈	—	2	1	no	—
12	Dic₄₇₈	C₈	1	2	yes	complemented
13	C₂×Dic₂₃₉	—	2	1	no	—
14	C₂×Dic₂₃₉	—	2	1	no	—
15	C₂₃₉⋊C₈	—	4	1	no	—
16	C₂×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₃₉	D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₃₉	D₄	1	1	yes	characteristic
19	C₈×C₂₃₉	—	2	2	no	—
20	C₂²×C₂₃₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₂₃₉	—	2	1	no	—
22	C₄×C₂₃₉	C₂×C₈	1	1	yes	characteristic
23	C₄×C₂₃₉	M₁₆	1	1	yes	characteristic
24	C₄×C₂₃₉	SD₁₆	1	1	yes	characteristic
25	C₄×C₂₃₉	Q₁₆	1	1	yes	characteristic
26	Dic₂₃₉	—	2	2	no	—
27	Dic₂₃₉	—	4	1	no	—
28	C₂×C₂₃₉	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×C₂₃₉	#103	1	1	yes	characteristic
30	C₂×C₂₃₉	#104	1	1	yes	characteristic
31	C₂₃₉	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
32	Q₈⋊C₈	—	239	1	no	maximal, Sylow
33	C₄×Q₈	—	239	1	no	—
34	C₄⋊C₈	—	239	1	no	—
35	C₄×C₈	D₂₃₉	1	1	yes	characteristic
36	C₂×Q₈	—	239	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	478	1	no	—
39	C₄⋊C₄	—	478	1	no	—
40	C₄⋊C₄	—	239	1	no	—
41	C₄²	—	478	1	no	—
42	C₄²	C₂×D₂₃₉	1	1	yes	characteristic
43	Q₈	—	478	1	no	—
44	Q₈	—	239	2	no	—
45	C₂×C₄	—	478	1	no	—
46	C₂×C₄	—	478	1	no	—
47	C₂×C₄	C₄×D₂₃₉	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₂₃₉	1	1	yes	characteristic
49	C₂×C₄	D₉₅₆	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₈	—	956	1	no	—
52	C₂²	(C₂×D₂₃₉)⋊C₄	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	956	1	no	—
55	C₄	—	478	2	no	—
56	C₄	C₈×D₂₃₉	1	1	yes	characteristic
57	C₄	#16838531	1	1	yes	characteristic
58	C₄	#16838537	1	1	yes	characteristic
59	C₄	Dic₉₅₆	1	1	yes	characteristic
60	C₂	(C₂×D₂₃₉)⋊C₈	1	1	yes	characteristic
61	C₂	#21439885	1	1	yes	characteristic
62	C₂	#21439888	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₉₅₆
commutator	C₄×C₂₃₉	C₂×C₈
Frattini	C₄²	C₂×D₂₃₉
Fitting	C₄×C₈×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	C₂²⋊C₄
cosocle	C₄²×C₂₃₉	C₂²
hypercenter	C₄×C₈	D₂₃₉
hypocenter	C₂₃₉	Q₈⋊C₈
LCS	G ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
12. Dic₄₇₈	C₈	semidirect
31. C₂₃₉	Q₈⋊C₈	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
C₄×C₈×C₂₃₉	2
Q₈⋊C₈	239