Subgroups — #29505151

Order 15296.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16838538	C₂	1	4	yes	maximal
2	#21439960	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₇₈	C₂²	1	2	yes	—
6	#16838538	—	2	8	no	—
7	C₂×C₂₃₉⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₂₃₉	—	2	2	no	—
9	C₂×Q₈×C₂₃₉	C₂²	1	2	yes	—
10	C₄⋊C₄×C₂₃₉	—	2	2	no	—
11	C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
12	Dic₄₇₈	—	2	4	no	—
13	C₂×Dic₂₃₉	—	2	2	no	—
14	C₂₃₉⋊C₈	D₄	1	4	yes	—
15	Q₈×C₂₃₉	—	2	4	no	—
16	C₂×C₄×C₂₃₉	—	2	2	no	—
17	C₂×C₄×C₂₃₉	D₄	1	2	yes	—
18	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
19	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
20	C₄×C₂₃₉	—	4	2	no	—
21	C₄×C₂₃₉	Q₁₆	1	4	yes	complemented
22	C₄×C₂₃₉	C₂×D₄	1	2	yes	—
23	Dic₂₃₉	—	4	2	no	—
24	C₂×C₂₃₉	#127	1	1	yes	characteristic
25	C₂×C₂₃₉	C₂×Q₁₆	1	2	yes	—
26	C₂₃₉	#494	1	1	yes	characteristic, complemented, Sylow
27	#494	—	239	1	no	maximal, Sylow
28	C₂×Q₁₆	—	239	4	no	—
29	C₄⋊Q₈	—	239	1	no	—
30	C₄⋊Q₈	D₂₃₉	1	1	yes	characteristic
31	C₄×C₈	—	239	1	no	—
32	C₂×Q₈	—	239	2	no	—
33	C₂×Q₈	C₂×D₂₃₉	1	2	yes	—
34	Q₁₆	—	478	8	no	—
35	C₂×C₈	—	239	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	478	2	no	—
38	C₄²	C₂×D₂₃₉	1	1	yes	characteristic
39	Q₈	—	2	4	no	—
40	Q₈	—	478	4	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	478	2	no	—
43	C₂×C₄	C₂²⋊D₂₃₉	1	2	yes	—
44	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
45	C₈	—	239	4	no	—
46	C₂²	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
47	C₄	—	4	2	no	—
48	C₄	—	956	2	no	—
49	C₄	#16838538	1	4	yes	complemented
50	C₄	D₄×D₂₃₉	1	2	yes	—
51	C₂	#21439956	1	1	yes	characteristic
52	C₂	C₂×#16838538	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃₉
commutator, cosocle	C₂×C₄×C₂₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₃₉
Fitting	C₄⋊Q₈×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	C₂×D₄
hypercenter	C₄⋊Q₈	D₂₃₉
hypocenter	C₂₃₉	#494
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
21. C₄×C₂₃₉	Q₁₆	semidirect
26. C₂₃₉	#494	semidirect
49. C₄	#16838538	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16838538	2
#21439960	2
C₄×C₂₃₉⋊C₈	2
C₄⋊Q₈×C₂₃₉	2
#494	239