Subgroups — #29505022

Order 15296.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16838531	C₂	1	1	yes	maximal, characteristic, complemented
2	#21439931	C₂	1	2	yes	maximal, complemented
3	#21439892	C₂	1	1	yes	maximal, characteristic, complemented
4	#21439881	C₂	1	2	yes	maximal, complemented
5	C₂×D₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊D₂₃₉	—	2	2	no	—
7	C₂×C₄×D₂₃₉	C₂²	1	1	yes	characteristic
8	#16838531	—	2	2	no	—
9	C₂×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
11	C₂²⋊Dic₂₃₉	—	2	2	no	—
12	C₂×D₄×C₂₃₉	C₂²	1	2	yes	—
13	D₈×C₂₃₉	—	2	2	no	—
14	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₂²×D₂₃₉	D₄	1	1	yes	characteristic
16	C₂²⋊D₂₃₉	—	4	4	no	—
17	C₂×Dic₂₃₉	—	2	2	no	—
18	C₂×Dic₂₃₉	D₄	1	1	yes	characteristic
19	C₄×D₂₃₉	—	2	1	no	—
20	C₄×D₂₃₉	—	2	1	no	—
21	C₂₃₉⋊C₈	—	2	1	no	—
22	C₂³×C₂₃₉	—	2	2	no	—
23	D₄×C₂₃₉	—	2	2	no	—
24	D₄×C₂₃₉	—	2	2	no	—
25	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
26	C₈×C₂₃₉	D₄	1	2	yes	complemented
27	C₂²×C₂₃₉	—	4	2	no	—
28	C₂²×C₂₃₉	—	4	2	no	—
29	C₂²×C₂₃₉	—	4	2	no	—
30	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
31	C₂×D₂₃₉	—	2	1	no	—
32	C₂×D₂₃₉	—	2	2	no	—
33	C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
34	C₄×C₂₃₉	#41	1	1	yes	characteristic
35	Dic₂₃₉	—	2	1	no	—
36	Dic₂₃₉	—	4	2	no	—
37	C₂×C₂₃₉	—	8	2	no	—
38	C₂×C₂₃₉	#121	1	1	yes	characteristic
39	C₂×C₂₃₉	AGL(1,ℤ/8)	1	2	yes	—
40	D₂₃₉	—	4	1	no	—
41	C₂₃₉	#469	1	1	yes	characteristic, complemented, Sylow
42	#469	—	239	1	no	maximal, Sylow
43	C₂×D₈	D₂₃₉	1	1	yes	characteristic, complemented
44	C₂×M₁₆	—	239	1	no	—
45	#121	—	239	2	no	—
46	#106	—	239	1	no	—
47	#102	—	239	2	no	—
48	C₂×D₄	—	478	2	no	—
49	C₂×D₄	C₂×D₂₃₉	1	2	yes	—
50	C₂²×C₄	—	239	1	no	—
51	D₈	—	2	2	no	—
52	M₁₆	—	478	2	no	—
53	C₂×C₈	—	239	1	no	—
54	C₂×C₈	C₂×D₂₃₉	1	1	yes	characteristic
55	C₄⋊C₄	—	239	2	no	—
56	C₂²⋊C₄	—	478	2	no	—
57	C₂³	—	2	2	no	—
58	C₂³	—	239	1	no	—
59	D₄	—	2	2	no	—
60	D₄	—	2	2	no	—
61	D₄	—	956	4	no	—
62	C₂×C₄	—	478	2	no	—
63	C₂×C₄	—	478	1	no	—
64	C₂×C₄	—	478	1	no	—
65	C₂×C₄	—	239	1	no	—
66	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
67	C₈	—	478	1	no	—
68	C₈	C₂²⋊D₂₃₉	1	2	yes	complemented
69	C₂²	—	4	2	no	—
70	C₂²	—	4	2	no	—
71	C₂²	—	4	2	no	—
72	C₂²	—	478	1	no	—
73	C₂²	—	478	2	no	—
74	C₂²	D₄×D₂₃₉	1	1	yes	characteristic
75	C₄	—	956	2	no	—
76	C₄	—	478	1	no	—
77	C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
78	C₄	#16838547	1	1	yes	characteristic
79	C₂	—	8	2	no	—
80	C₂	—	956	1	no	—
81	C₂	#21439931	1	1	yes	characteristic
82	C₂	#21439984	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₃₉
commutator, cosocle	C₂×C₄×C₂₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₃₉
Fitting	C₂×D₈×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₂×D₄
hypercenter	C₂×D₈	D₂₃₉
hypocenter	C₂₃₉	#469
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16838531	C₂	semidirect
2. #21439931	C₂	semidirect
3. #21439892	C₂	semidirect
4. #21439881	C₂	semidirect
5. C₂×D₈×C₂₃₉	C₂	semidirect
26. C₈×C₂₃₉	D₄	semidirect
41. C₂₃₉	#469	semidirect
43. C₂×D₈	D₂₃₉	semidirect
68. C₈	C₂²⋊D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16838531	2
#21439931	2
#21439892	2
#21439881	2
C₂×D₈×C₂₃₉	2
#469	239