Subgroups — #29504696

Order 15296.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₃₉⋊C₄	C₂	1	3	yes	maximal, complemented
2	C₂×(C₂×D₂₃₉)⋊C₄	C₂	1	3	yes	maximal
3	#95×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₂₃₉	C₂²	1	1	yes	characteristic
5	C₂²×Dic₂₃₉	C₂²	1	3	yes	—
6	Dic₂₃₉⋊C₄	—	2	6	no	—
7	(C₂×D₂₃₉)⋊C₄	—	2	6	no	—
8	C₂²×C₄×C₂₃₉	C₂²	1	3	yes	—
9	C₂²×D₂₃₉	—	2	3	no	—
10	C₂²×D₂₃₉	—	2	3	no	—
11	C₂²×D₂₃₉	Q₈	1	2	yes	—
12	C₂×Dic₂₃₉	—	2	3	no	—
13	C₂×Dic₂₃₉	—	2	3	no	—
14	C₂×Dic₂₃₉	D₄	1	6	yes	—
15	C₂³×C₂₃₉	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₂₃₉	—	2	3	no	—
17	C₂×C₄×C₂₃₉	—	2	3	no	—
18	C₂×C₄×C₂₃₉	—	2	3	no	—
19	C₂²×C₂₃₉	C₂×D₄	1	3	yes	—
20	C₂²×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
21	C₂²×C₂₃₉	#41	1	3	yes	—
22	C₂×D₂₃₉	—	2	6	no	—
23	C₂×D₂₃₉	—	4	3	no	—
24	C₂×D₂₃₉	—	4	1	no	—
25	C₄×C₂₃₉	—	4	3	no	—
26	Dic₂₃₉	—	2	6	no	—
27	C₂×C₂₃₉	#121	1	3	yes	—
28	C₂×C₂₃₉	C₂²⋊Q₈	1	3	yes	—
29	C₂×C₂₃₉	#126	1	1	yes	characteristic
30	D₂₃₉	—	4	2	no	—
31	C₂₃₉	#396	1	1	yes	characteristic, complemented, Sylow
32	#396	—	239	1	no	maximal, Sylow
33	C₂×C₄⋊C₄	—	239	3	no	—
34	C₂×C₂²⋊C₄	—	239	3	no	—
35	#95	D₂₃₉	1	1	yes	characteristic, complemented
36	C₂⁴	—	239	1	no	—
37	C₂²×C₄	—	239	3	no	—
38	C₂²×C₄	C₂×D₂₃₉	1	3	yes	—
39	C₄⋊C₄	—	478	6	no	—
40	C₂²⋊C₄	—	478	6	no	—
41	C₂³	—	478	3	no	—
42	C₂³	—	478	3	no	—
43	C₂³	—	239	2	no	—
44	C₂³	C₂²×D₂₃₉	1	1	yes	characteristic
45	C₂×C₄	—	2	3	no	—
46	C₂×C₄	—	2	3	no	—
47	C₂×C₄	—	2	3	no	—
48	C₂×C₄	—	478	3	no	—
49	C₂×C₄	—	478	3	no	—
50	C₂×C₄	—	239	6	no	—
51	C₂²	—	956	3	no	—
52	C₂²	—	956	1	no	—
53	C₂²	—	478	6	no	—
54	C₂²	D₄×D₂₃₉	1	3	yes	—
55	C₂²	Q₈×D₂₃₉	1	1	yes	characteristic
56	C₂²	#16838548	1	3	yes	—
57	C₄	—	4	3	no	—
58	C₄	—	478	6	no	—
59	C₂	—	956	2	no	—
60	C₂	#21439933	1	3	yes	—
61	C₂	#21439938	1	3	yes	—
62	C₂	#21439954	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₃₉
commutator, socle, cosocle	C₂³×C₂₃₉	C₂³
Fitting	#95×C₂₃₉	C₂
prime radical	C₂³×D₂₃₉	C₂²
hypercenter	#95	D₂₃₉
hypocenter	C₂₃₉	#396
LCS	G ▷ C₂³×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₂₃₉⋊C₄	C₂	semidirect
3. #95×C₂₃₉	C₂	semidirect
31. C₂₃₉	#396	semidirect
35. #95	D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₃₉⋊C₄	2
C₂×(C₂×D₂₃₉)⋊C₄	2
#95×C₂₃₉	2
#396	239