Subgroups — #29505106

Order 15296.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21439936	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×D₂₃₉	C₂	1	1	yes	maximal, characteristic
3	#21439894	C₂	1	1	yes	maximal, characteristic, complemented
4	#21439893	C₂	1	1	yes	maximal, characteristic, complemented
5	#21439884	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₂₃₉)⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#103×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₂₃₉	—	2	1	no	—
9	C₂×C₄×D₂₃₉	C₂²	1	1	yes	characteristic
10	C₂×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
13	Dic₂₃₉⋊C₄	—	2	1	no	—
14	Dic₂₃₉⋊C₄	—	2	1	no	—
15	(C₂×D₂₃₉)⋊C₄	—	2	1	no	—
16	C₂×Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
19	C₂²×D₂₃₉	D₄	1	1	yes	characteristic
20	C₂×Dic₂₃₉	—	2	1	no	—
21	C₂×Dic₂₃₉	—	2	1	no	—
22	C₂×Dic₂₃₉	D₄	1	1	yes	characteristic
23	C₄×D₂₃₉	—	2	1	no	—
24	C₄×D₂₃₉	—	2	1	no	—
25	C₄×D₂₃₉	—	4	2	no	—
26	C₂₃₉⋊C₈	—	2	1	no	—
27	Q₈×C₂₃₉	—	2	1	no	—
28	Q₈×C₂₃₉	—	2	1	no	—
29	C₂×C₄×C₂₃₉	—	2	1	no	—
30	C₂×C₄×C₂₃₉	—	2	1	no	—
31	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
32	C₈×C₂₃₉	—	2	1	no	—
33	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
34	C₂×D₂₃₉	—	2	1	no	—
35	C₂×D₂₃₉	—	2	1	no	—
36	C₂×D₂₃₉	Q₁₆	1	2	yes	—
37	C₄×C₂₃₉	—	4	1	no	—
38	C₄×C₂₃₉	—	4	1	no	—
39	C₄×C₂₃₉	#41	1	1	yes	characteristic
40	C₄×C₂₃₉	#41	1	1	yes	characteristic
41	Dic₂₃₉	—	2	1	no	—
42	Dic₂₃₉	—	4	1	no	—
43	Dic₂₃₉	—	4	1	no	—
44	C₂×C₂₃₉	#123	1	1	yes	characteristic
45	C₂×C₂₃₉	C₂×Q₁₆	1	1	yes	characteristic
46	C₂×C₂₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
47	D₂₃₉	—	2	2	no	—
48	C₂₃₉	#484	1	1	yes	characteristic, complemented, Sylow
49	#484	—	239	1	no	maximal, Sylow
50	C₂²⋊Q₈	—	239	1	no	—
51	C₂×C₄⋊C₄	—	239	1	no	—
52	#107	—	239	1	no	—
53	#107	—	239	1	no	—
54	#103	—	239	1	no	—
55	#103	D₂₃₉	1	1	yes	characteristic, complemented
56	C₂²⋊C₈	—	239	1	no	—
57	C₂×Q₈	C₂×D₂₃₉	1	1	yes	characteristic
58	C₂²×C₄	—	478	1	no	—
59	C₂²×C₄	—	239	1	no	—
60	C₂×C₈	—	239	1	no	—
61	C₂×C₈	C₂×D₂₃₉	1	1	yes	characteristic
62	C₄⋊C₄	—	478	1	no	—
63	C₄⋊C₄	—	478	1	no	—
64	C₄⋊C₄	—	239	1	no	—
65	C₄⋊C₄	—	239	1	no	—
66	C₄⋊C₄	C₂×D₂₃₉	1	1	yes	characteristic
67	C₂²⋊C₄	—	478	1	no	—
68	C₂³	—	239	1	no	—
69	Q₈	—	2	1	no	—
70	Q₈	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	956	2	no	—
74	C₂×C₄	—	478	1	no	—
75	C₂×C₄	—	478	1	no	—
76	C₂×C₄	—	478	1	no	—
77	C₂×C₄	—	478	1	no	—
78	C₂×C₄	—	239	1	no	—
79	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	478	1	no	—
82	C₂²	—	478	1	no	—
83	C₂²	—	478	1	no	—
84	C₂²	—	239	2	no	—
85	C₂²	D₄×D₂₃₉	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	956	1	no	—
89	C₄	—	956	1	no	—
90	C₄	—	478	1	no	—
91	C₄	#16838547	1	1	yes	characteristic
92	C₄	#16838548	1	1	yes	characteristic
93	C₂	—	478	2	no	—
94	C₂	#21439942	1	1	yes	characteristic
95	C₂	Q₁₆×D₂₃₉	1	1	yes	characteristic
96	C₂	#21439987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₃₉
commutator, cosocle	C₂×C₄×C₂₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₃₉
Fitting	#103×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₂×D₄
prime radical	C₂²×D₂₃₉	D₄
hypercenter	#103	D₂₃₉
hypocenter	C₂₃₉	#484
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
3. #21439894	C₂	semidirect
4. #21439893	C₂	semidirect
5. #21439884	C₂	semidirect
7. #103×C₂₃₉	C₂	semidirect
48. C₂₃₉	#484	semidirect
55. #103	D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#21439936	2
C₄⋊C₄×D₂₃₉	2
#21439894	2
#21439893	2
#21439884	2
(C₂×D₂₃₉)⋊C₈	2
#103×C₂₃₉	2
#484	239