Subgroups — #29505589

Order 15296.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21440007	C₂	1	2	yes	maximal, complemented
2	#21439993	C₂	1	6	yes	maximal, complemented
3	#21439982	C₂	1	3	yes	maximal, complemented
4	C₂×Dic₉₅₆	C₂	1	3	yes	maximal, complemented
5	#131×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16838548	C₂²	1	6	yes	—
7	#16838547	—	2	6	no	—
8	C₂×Dic₄₇₈	C₂²	1	6	yes	—
9	Q₈×D₂₃₉	—	2	2	no	—
10	Dic₉₅₆	C₂²	1	3	yes	—
11	Dic₉₅₆	C₂²	1	6	yes	—
12	#16838537	C₂²	1	6	yes	—
13	D₁₉₁₂	C₂²	1	1	yes	characteristic
14	#41×C₂₃₉	C₂²	1	1	yes	characteristic
15	M₁₆×C₂₃₉	C₂²	1	3	yes	—
16	C₂×C₈×C₂₃₉	C₂²	1	3	yes	—
17	Dic₄₇₈	—	2	6	no	—
18	Dic₄₇₈	C₂³	1	6	yes	—
19	D₉₅₆	C₂³	1	2	yes	—
20	C₂²⋊D₂₃₉	—	2	6	no	—
21	C₂×Dic₂₃₉	—	2	6	no	—
22	C₄×D₂₃₉	—	2	6	no	—
23	Q₈×C₂₃₉	D₄	1	1	yes	characteristic
24	D₄×C₂₃₉	D₄	1	3	yes	—
25	C₂×C₄×C₂₃₉	C₂³	1	3	yes	—
26	C₈×C₂₃₉	C₂³	1	1	yes	characteristic
27	C₈×C₂₃₉	C₂³	1	3	yes	—
28	C₂²×C₂₃₉	C₂×D₄	1	3	yes	—
29	C₂×D₂₃₉	—	2	2	no	—
30	C₄×C₂₃₉	C₂×D₄	1	3	yes	—
31	C₄×C₂₃₉	C₂⁴	1	1	yes	characteristic
32	Dic₂₃₉	—	2	6	no	—
33	C₂×C₂₃₉	—	2	3	no	—
34	C₂×C₂₃₉	C₂²×D₄	1	1	yes	characteristic
35	D₂₃₉	—	4	2	no	—
36	C₂₃₉	#578	1	1	yes	characteristic, complemented, Sylow
37	#578	—	239	1	no	maximal, Sylow
38	2₋¹⁺⁴	—	239	2	no	—
39	#137	—	239	6	no	—
40	#135	—	239	3	no	—
41	C₂×Q₁₆	—	239	3	no	—
42	#131	D₂₃₉	1	1	yes	characteristic, complemented
43	#41	—	478	6	no	—
44	#41	—	239	6	no	—
45	#41	C₂×D₂₃₉	1	1	yes	characteristic
46	C₂×Q₈	—	478	2	no	—
47	C₂×Q₈	—	239	6	no	—
48	Q₁₆	—	239	3	no	—
49	Q₁₆	—	239	6	no	—
50	SD₁₆	—	239	6	no	—
51	D₈	—	239	1	no	—
52	M₁₆	C₂×D₂₃₉	1	3	yes	—
53	C₂×C₈	C₂×D₂₃₉	1	3	yes	—
54	Q₈	—	478	6	no	—
55	Q₈	—	239	6	no	—
56	Q₈	D₉₅₆	1	1	yes	characteristic
57	D₄	—	478	6	no	—
58	D₄	—	239	2	no	—
59	D₄	D₉₅₆	1	3	yes	—
60	C₂×C₄	—	478	6	no	—
61	C₂×C₄	—	478	6	no	—
62	C₂×C₄	C₂²×D₂₃₉	1	3	yes	—
63	C₈	C₂²×D₂₃₉	1	1	yes	characteristic
64	C₈	C₂²×D₂₃₉	1	3	yes	—
65	C₂²	—	478	2	no	—
66	C₂²	C₂×D₉₅₆	1	3	yes	—
67	C₄	—	478	6	no	—
68	C₄	C₂×D₉₅₆	1	3	yes	—
69	C₄	C₂³×D₂₃₉	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	956	2	no	—
72	C₂	C₂²×D₉₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₉₅₆
commutator, cosocle	C₄×C₂₃₉	C₂⁴
Frattini	C₄	C₂³×D₂₃₉
Fitting	#131×C₂₃₉	C₂
socle	C₂×C₂₃₉	C₂²×D₄
hypercenter	#131	D₂₃₉
hypocenter	C₂₃₉	#578
LCS	G ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #21440007	C₂	semidirect
2. #21439993	C₂	semidirect
3. #21439982	C₂	semidirect
4. C₂×Dic₉₅₆	C₂	semidirect
5. #131×C₂₃₉	C₂	semidirect
36. C₂₃₉	#578	semidirect
42. #131	D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#21440007	2
#21439993	2
#21439982	2
C₂×Dic₉₅₆	2
#131×C₂₃₉	2
#578	239