Subgroups — #29505061

Order 15296.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21439959	C₂	1	1	yes	maximal, characteristic, complemented
2	#21439922	C₂	1	1	yes	maximal, characteristic
3	#21439891	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₂₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#21439882	C₂	1	1	yes	maximal, characteristic
6	#21439880	C₂	1	1	yes	maximal, characteristic
7	#106×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₄₇₈	—	2	1	no	—
9	C₂×D₉₅₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₃₉	—	2	1	no	—
11	C₂×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
13	Dic₂₃₉⋊C₄	—	2	1	no	—
14	(C₂×D₂₃₉)⋊C₄	—	2	1	no	—
15	C₄×Dic₂₃₉	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
19	C₂²×D₂₃₉	—	2	1	no	—
20	Dic₄₇₈	—	4	2	no	—
21	D₉₅₆	—	2	1	no	—
22	D₉₅₆	Q₈	1	2	yes	complemented
23	C₂×Dic₂₃₉	—	2	1	no	—
24	C₂×Dic₂₃₉	D₄	1	1	yes	characteristic
25	C₂×Dic₂₃₉	D₄	1	1	yes	characteristic
26	C₄×D₂₃₉	—	4	2	no	—
27	C₂₃₉⋊C₈	—	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	2	no	—
34	C₂×D₂₃₉	—	4	1	no	—
35	C₂×D₂₃₉	—	4	1	no	—
36	C₄×C₂₃₉	—	4	1	no	—
37	C₄×C₂₃₉	—	4	1	no	—
38	C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
39	C₄×C₂₃₉	#41	1	1	yes	characteristic
40	Dic₂₃₉	—	2	1	no	—
41	Dic₂₃₉	—	4	1	no	—
42	Dic₂₃₉	SD₁₆	1	2	yes	—
43	C₂×C₂₃₉	C₂²⋊Q₈	1	1	yes	characteristic
44	C₂×C₂₃₉	C₂×SD₁₆	1	1	yes	characteristic
45	C₂×C₂₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
46	D₂₃₉	—	4	2	no	—
47	C₂₃₉	#476	1	1	yes	characteristic, complemented, Sylow
48	#476	—	239	1	no	maximal, Sylow
49	C₄⋊Q₈	—	239	1	no	—
50	C₄×D₄	—	239	1	no	—
51	#106	—	239	1	no	—
52	#106	D₂₃₉	1	1	yes	characteristic, complemented
53	C₄⋊C₈	—	239	1	no	—
54	#102	—	239	1	no	—
55	#102	—	239	1	no	—
56	C₂×Q₈	—	478	1	no	—
57	C₂×D₄	—	239	1	no	—
58	C₂²×C₄	—	478	1	no	—
59	C₂×C₈	—	239	1	no	—
60	C₂×C₈	C₂×D₂₃₉	1	1	yes	characteristic
61	C₄⋊C₄	—	478	1	no	—
62	C₄⋊C₄	—	239	1	no	—
63	C₄⋊C₄	C₂×D₂₃₉	1	1	yes	characteristic
64	C₄⋊C₄	C₂×D₂₃₉	1	1	yes	characteristic
65	C₂²⋊C₄	—	478	1	no	—
66	C₄²	—	239	1	no	—
67	C₂³	—	478	1	no	—
68	Q₈	—	956	2	no	—
69	D₄	—	478	1	no	—
70	D₄	—	239	2	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	—	239	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	—	956	1	no	—
81	C₂²	—	956	1	no	—
82	C₂²	—	478	2	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	—	478	1	no	—
88	C₄	—	239	2	no	—
89	C₄	Q₈×D₂₃₉	1	1	yes	characteristic
90	C₄	#16838548	1	1	yes	characteristic
91	C₂	—	956	2	no	—
92	C₂	#21439938	1	1	yes	characteristic
93	C₂	SD₁₆×D₂₃₉	1	1	yes	characteristic
94	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	#106×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₂×D₄
prime radical	C₂×D₉₅₆	C₂²
hypercenter	#106	D₂₃₉
hypocenter	C₂₃₉	#476
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
1. #21439959	C₂	semidirect
3. #21439891	C₂	semidirect
4. Dic₂₃₉⋊C₈	C₂	semidirect
7. #106×C₂₃₉	C₂	semidirect
22. D₉₅₆	Q₈	semidirect
47. C₂₃₉	#476	semidirect
52. #106	D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#21439959	2
#21439922	2
#21439891	2
Dic₂₃₉⋊C₈	2
#21439882	2
#21439880	2
#106×C₂₃₉	2
#476	239