Subgroups — #29505111

Order 15296.

96 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	#21439918	C₂	1	1	yes	maximal, characteristic
3	#21439893	C₂	1	1	yes	maximal, characteristic, complemented
4	#21439892	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	C₂²	1	1	yes	characteristic
9	C₂×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
12	Dic₂₃₉⋊C₄	—	2	1	no	—
13	(C₂×D₂₃₉)⋊C₄	—	2	1	no	—
14	(C₂×D₂₃₉)⋊C₄	—	2	1	no	—
15	C₄×Dic₂₃₉	—	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	C₂₃₉⋊C₈	—	2	1	no	—
26	Q₈×C₂₃₉	—	2	1	no	—
27	Q₈×C₂₃₉	—	2	1	no	—
28	C₂×C₄×C₂₃₉	—	2	1	no	—
29	C₂×C₄×C₂₃₉	—	2	1	no	—
30	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
31	C₈×C₂₃₉	—	2	1	no	—
32	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
33	C₂×D₂₃₉	—	2	1	no	—
34	C₂×D₂₃₉	—	2	1	no	—
35	C₂×D₂₃₉	—	2	1	no	—
36	C₄×C₂₃₉	—	4	1	no	—
37	C₄×C₂₃₉	—	4	1	no	—
38	C₄×C₂₃₉	#41	1	1	yes	characteristic
39	C₄×C₂₃₉	#41	1	1	yes	characteristic
40	Dic₂₃₉	—	2	1	no	—
41	Dic₂₃₉	—	4	1	no	—
42	Dic₂₃₉	—	4	1	no	—
43	C₂×C₂₃₉	#123	1	1	yes	characteristic
44	C₂×C₂₃₉	#135	1	1	yes	characteristic
45	C₂×C₂₃₉	#137	1	1	yes	characteristic
46	D₂₃₉	—	4	1	no	—
47	C₂₃₉	#485	1	1	yes	characteristic, complemented, Sylow
48	#485	—	239	1	no	maximal, Sylow
49	C₂²⋊Q₈	—	239	1	no	—
50	#117	—	239	1	no	—
51	#107	—	239	1	no	—
52	#106	—	239	1	no	—
53	#103	—	239	1	no	—
54	#103	D₂₃₉	1	1	yes	characteristic, complemented
55	C₂²⋊C₈	—	239	1	no	—
56	C₂×Q₈	C₂×D₂₃₉	1	1	yes	characteristic
57	C₂²×C₄	—	239	1	no	—
58	C₂×C₈	—	239	1	no	—
59	C₂×C₈	C₂×D₂₃₉	1	1	yes	characteristic
60	C₄⋊C₄	—	478	1	no	—
61	C₄⋊C₄	—	239	1	no	—
62	C₄⋊C₄	—	239	1	no	—
63	C₄⋊C₄	C₂×D₂₃₉	1	1	yes	characteristic
64	C₂²⋊C₄	—	478	1	no	—
65	C₂²⋊C₄	—	478	1	no	—
66	C₄²	—	478	1	no	—
67	C₂³	—	239	1	no	—
68	Q₈	—	2	1	no	—
69	Q₈	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	478	1	no	—
73	C₂×C₄	—	478	1	no	—
74	C₂×C₄	—	478	1	no	—
75	C₂×C₄	—	478	1	no	—
76	C₂×C₄	—	239	1	no	—
77	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	478	1	no	—
80	C₂²	—	478	1	no	—
81	C₂²	—	478	1	no	—
82	C₂²	—	478	1	no	—
83	C₂²	D₄×D₂₃₉	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	956	1	no	—
87	C₄	—	956	1	no	—
88	C₄	—	478	1	no	—
89	C₄	#16838547	1	1	yes	characteristic
90	C₄	#16838548	1	1	yes	characteristic
91	C₂	—	956	1	no	—
92	C₂	#21439942	1	1	yes	characteristic
93	C₂	#21439981	1	1	yes	characteristic
94	C₂	#21439992	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	#485
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. #21439893	C₂	semidirect
4. #21439892	C₂	semidirect
5. #21439884	C₂	semidirect
7. #103×C₂₃₉	C₂	semidirect
47. C₂₃₉	#485	semidirect
54. #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
#21439918	2
#21439893	2
#21439892	2
#21439884	2
(C₂×D₂₃₉)⋊C₈	2
#103×C₂₃₉	2
#485	239