Subgroups — #29505157

Order 15296.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16838535	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₉₁₂	C₂	1	1	yes	maximal, characteristic
3	#21439949	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₁₆×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₉₅₆	C₂²	1	2	yes	—
7	#16838535	—	2	4	no	—
8	D₁₉₁₂	—	2	2	no	—
9	C₂×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
10	(C₂×D₂₃₉)⋊C₄	—	2	4	no	—
11	C₄×Dic₂₃₉	C₂²	1	1	yes	characteristic
12	C₂×Q₈×C₂₃₉	C₂²	1	2	yes	—
13	Q₁₆×C₂₃₉	—	2	2	no	—
14	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₂²×D₂₃₉	—	2	2	no	—
16	D₉₅₆	—	2	2	no	—
17	D₉₅₆	—	2	2	no	—
18	C₂×Dic₂₃₉	D₄	1	2	yes	—
19	C₂₃₉⋊C₈	D₄	1	2	yes	—
20	Q₈×C₂₃₉	—	2	2	no	—
21	Q₈×C₂₃₉	—	2	2	no	—
22	C₂×C₄×C₂₃₉	—	2	2	no	—
23	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
24	C₈×C₂₃₉	D₄	1	2	yes	—
25	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
26	C₂×D₂₃₉	—	4	2	no	—
27	C₂×D₂₃₉	—	4	4	no	—
28	C₄×C₂₃₉	—	4	2	no	—
29	C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
30	C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
31	Dic₂₃₉	—	2	2	no	—
32	C₂×C₂₃₉	#127	1	1	yes	characteristic
33	C₂×C₂₃₉	#135	1	2	yes	—
34	D₂₃₉	—	8	2	no	—
35	C₂₃₉	#495	1	1	yes	characteristic, complemented, Sylow
36	#495	—	239	1	no	maximal, Sylow
37	C₂×Q₁₆	D₂₃₉	1	1	yes	characteristic, complemented
38	C₂×SD₁₆	—	239	2	no	—
39	C₂×D₈	—	239	1	no	—
40	#124	—	239	2	no	—
41	C₄×C₈	—	239	1	no	—
42	C₂×Q₈	C₂×D₂₃₉	1	2	yes	—
43	C₂×D₄	—	239	2	no	—
44	Q₁₆	—	2	2	no	—
45	SD₁₆	—	478	4	no	—
46	D₈	—	478	2	no	—
47	C₂×C₈	—	239	1	no	—
48	C₂×C₈	C₂×D₂₃₉	1	1	yes	characteristic
49	C₂²⋊C₄	—	478	4	no	—
50	C₄²	—	239	1	no	—
51	C₂³	—	478	2	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	2	2	no	—
54	D₄	—	478	2	no	—
55	D₄	—	478	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	239	2	no	—
58	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
59	C₈	—	239	2	no	—
60	C₈	C₂²⋊D₂₃₉	1	2	yes	—
61	C₂²	—	956	2	no	—
62	C₂²	—	956	4	no	—
63	C₂²	D₄×D₂₃₉	1	1	yes	characteristic
64	C₄	—	4	2	no	—
65	C₄	—	478	2	no	—
66	C₄	D₄×D₂₃₉	1	1	yes	characteristic
67	C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
68	C₂	—	1912	2	no	—
69	C₂	#21439956	1	1	yes	characteristic
70	C₂	#21439983	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₃₉
commutator, cosocle	C₂×C₄×C₂₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₃₉
Fitting	C₂×Q₁₆×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₂×D₄
prime radical	C₂×D₁₉₁₂	C₂
hypercenter	C₂×Q₁₆	D₂₃₉
hypocenter	C₂₃₉	#495
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16838535	C₂	semidirect
3. #21439949	C₂	semidirect
4. C₈×Dic₂₃₉	C₂	semidirect
5. C₂×Q₁₆×C₂₃₉	C₂	semidirect
35. C₂₃₉	#495	semidirect
37. C₂×Q₁₆	D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16838535	2
C₂×D₁₉₁₂	2
#21439949	2
C₈×Dic₂₃₉	2
C₂×Q₁₆×C₂₃₉	2
#495	239