Subgroups — #29504957

Order 15296.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21440008	C₂	1	1	yes	maximal, characteristic
2	#21439993	C₂	1	2	yes	maximal, complemented
3	#21439958	C₂	1	1	yes	maximal, characteristic, complemented
4	#21439886	C₂	1	2	yes	maximal, complemented
5	#101×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16838549	—	2	2	no	—
7	#16838548	—	2	2	no	—
8	#16838548	C₂²	1	2	yes	—
9	C₂×Dic₄₇₈	C₂²	1	1	yes	characteristic
10	Q₈×D₂₃₉	—	2	2	no	—
11	Dic₉₅₆	—	2	2	no	—
12	#16838537	—	2	2	no	—
13	Dic₂₃₉⋊C₄	—	2	2	no	—
14	C₄×Dic₂₃₉	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
16	M₁₆×C₂₃₉	C₂²	1	2	yes	—
17	Dic₄₇₈	—	2	2	no	—
18	Dic₄₇₈	—	2	2	no	—
19	Dic₄₇₈	D₄	1	2	yes	—
20	D₉₅₆	—	2	2	no	—
21	D₉₅₆	D₄	1	2	yes	—
22	C₂²⋊D₂₃₉	—	2	2	no	—
23	C₂×Dic₂₃₉	—	2	1	no	—
24	C₂×Dic₂₃₉	—	2	1	no	—
25	C₄×D₂₃₉	—	2	4	no	—
26	C₄×D₂₃₉	—	2	2	no	—
27	Q₈×C₂₃₉	—	2	2	no	—
28	C₂×C₄×C₂₃₉	D₄	1	2	yes	—
29	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
30	C₈×C₂₃₉	—	2	2	no	—
31	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
32	C₂×D₂₃₉	—	2	2	no	—
33	C₄×C₂₃₉	—	2	2	no	—
34	C₄×C₂₃₉	C₂×D₄	1	2	yes	—
35	Dic₂₃₉	—	2	2	no	—
36	Dic₂₃₉	—	2	2	no	—
37	Dic₂₃₉	—	4	1	no	—
38	C₂×C₂₃₉	—	2	1	no	—
39	C₂×C₂₃₉	#120	1	1	yes	characteristic
40	D₂₃₉	—	4	2	no	—
41	C₂₃₉	#456	1	1	yes	characteristic, complemented, Sylow
42	#456	—	239	1	no	maximal, Sylow
43	2₋¹⁺⁴	—	239	1	no	—
44	#137	—	239	2	no	—
45	C₄⋊Q₈	—	239	1	no	—
46	#104	—	239	2	no	—
47	#101	D₂₃₉	1	1	yes	characteristic, complemented
48	#41	—	478	2	no	—
49	#41	—	478	2	no	—
50	#41	—	239	2	no	—
51	C₂×Q₈	—	478	2	no	—
52	C₂×Q₈	—	239	1	no	—
53	C₂×Q₈	C₂×D₂₃₉	1	1	yes	characteristic
54	Q₁₆	—	478	2	no	—
55	SD₁₆	—	478	2	no	—
56	M₁₆	C₂×D₂₃₉	1	2	yes	—
57	C₄⋊C₄	—	478	2	no	—
58	C₄²	—	239	1	no	—
59	Q₈	—	2	2	no	—
60	Q₈	—	478	2	no	—
61	Q₈	—	478	2	no	—
62	Q₈	—	239	2	no	—
63	D₄	—	478	2	no	—
64	D₄	—	478	2	no	—
65	D₄	—	239	2	no	—
66	C₂×C₄	—	478	4	no	—
67	C₂×C₄	—	478	1	no	—
68	C₂×C₄	—	478	1	no	—
69	C₂×C₄	—	478	2	no	—
70	C₂×C₄	D₉₅₆	1	2	yes	—
71	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₂²	—	478	2	no	—
74	C₂²	C₂×D₉₅₆	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	956	1	no	—
77	C₄	—	478	2	no	—
78	C₄	—	478	2	no	—
79	C₄	D₄×D₂₃₉	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	956	2	no	—
82	C₂	#21439930	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21439930
commutator, cosocle	C₂×C₄×C₂₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₃₉
Fitting	#101×C₂₃₉	C₂
socle	C₂×C₂₃₉	#120
prime radical	#21440008	C₂
hypercenter	#101	D₂₃₉
hypocenter	C₂₃₉	#456
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

Complements

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

N	H ≅ G/N	type
2. #21439993	C₂	semidirect
3. #21439958	C₂	semidirect
4. #21439886	C₂	semidirect
5. #101×C₂₃₉	C₂	semidirect
41. C₂₃₉	#456	semidirect
47. #101	D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#21440008	2
#21439993	2
#21439958	2
#21439886	2
#101×C₂₃₉	2
#456	239