Subgroups — #11928963

Order 1248.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10597790	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#10597662	C₂	1	1	yes	maximal, characteristic, complemented
4	#10597657	C₂	1	2	yes	maximal, complemented
5	C₈×D₃₉	C₂	1	2	yes	maximal, complemented
6	#88155	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
8	#64625	C₄	1	2	yes	—
9	D₁₅₆	C₄	1	1	yes	characteristic
10	C₄×D₃₉	C₂²	1	2	yes	—
11	Dic₇₈	C₄	1	1	yes	characteristic
12	C₃×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₃×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₃⋊#1055	C₂²	1	2	yes	—
15	#6127	—	3	1	no	—
16	C₂×C₈×C₁₃	D₃	1	1	yes	characteristic, complemented
17	#6099	—	3	1	no	—
18	#6094	—	3	2	no	—
19	C₈×D₁₃	—	3	2	no	—
20	C₂²×C₃×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×D₃₉	C₂×C₄	1	2	yes	—
22	C₃×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₃×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	Dic₃₉	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
26	C₂²⋊D₁₃	—	3	2	no	—
27	D₅₂	—	3	1	no	—
28	C₄×D₁₃	—	3	2	no	—
29	Dic₂₆	—	3	1	no	—
30	C₈×C₁₃	C₂×D₃	1	1	yes	characteristic
31	C₈×C₁₃	C₂×D₃	1	1	yes	characteristic
32	#1055	—	3	2	no	—
33	#901	—	13	1	no	maximal, Hall
34	C₂×C₃×C₁₃	—	2	1	no	—
35	C₂×C₃×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	D₃₉	—	2	2	no	—
37	C₂²×C₁₃	C₄×D₃	1	1	yes	characteristic
38	C₂×D₁₃	—	3	2	no	—
39	C₄×C₁₃	C₄×D₃	1	1	yes	characteristic
40	C₄×C₁₃	C₂²×D₃	1	1	yes	characteristic
41	Dic₁₃	—	3	2	no	—
42	#235	—	13	1	no	—
43	C₂×C₃×C₈	D₁₃	1	1	yes	characteristic, complemented
44	#208	—	13	1	no	—
45	ΓL(1,𝔽₂₅)	—	13	2	no	—
46	D₃×C₈	—	13	2	no	—
47	C₃×C₁₃	#131	1	1	yes	characteristic, complemented, Hall
48	#131	—	39	1	no	Sylow
49	C₂×C₁₃	—	2	1	no	—
50	C₂×C₁₃	C₂×C₄×D₃	1	1	yes	characteristic
51	D₁₃	—	6	2	no	—
52	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
53	#67	—	13	2	no	—
54	D₁₂	—	13	1	no	—
55	C₄×D₃	—	13	2	no	—
56	Dic₆	—	13	1	no	—
57	C₃×C₈	C₂×D₁₃	1	1	yes	characteristic
58	C₃×C₈	C₂×D₁₃	1	1	yes	characteristic
59	C₃⋊C₈	—	13	2	no	—
60	#41	—	39	1	no	—
61	M₁₆	—	39	1	no	—
62	M₁₆	—	39	2	no	—
63	C₂×C₈	—	39	2	no	—
64	C₂×C₈	D₃₉	1	1	yes	characteristic, complemented
65	C₁₃	#901	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₃	C₄×D₁₃	1	1	yes	characteristic
67	C₂×D₃	—	13	2	no	—
68	C₃×C₄	C₄×D₁₃	1	1	yes	characteristic
69	C₃×C₄	C₂²×D₁₃	1	1	yes	characteristic
70	Dic₃	—	13	2	no	—
71	Q₈	—	39	1	no	—
72	D₄	—	39	2	no	—
73	D₄	—	39	1	no	—
74	C₂×C₄	—	39	2	no	—
75	C₂×C₄	C₂×D₃₉	1	1	yes	characteristic
76	C₈	—	39	2	no	—
77	C₈	C₂×D₃₉	1	1	yes	characteristic
78	C₈	C₂×D₃₉	1	1	yes	characteristic
79	C₂×C₃	—	2	1	no	—
80	C₂×C₃	C₂×C₄×D₁₃	1	1	yes	characteristic
81	D₃	—	26	2	no	—
82	C₂²	—	39	2	no	—
83	C₂²	C₄×D₃₉	1	1	yes	characteristic
84	C₄	—	39	2	no	—
85	C₄	C₄×D₃₉	1	1	yes	characteristic
86	C₄	C₂²×D₃₉	1	1	yes	characteristic
87	C₃	#88155	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	2	1	no	—
89	C₂	—	78	2	no	—
90	C₂	C₂×C₄×D₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₃₉
commutator, socle	C₂×C₃×C₁₃	C₂²×C₄
Frattini	C₄	C₂²×D₃₉
Fitting	C₂×C₃×C₈×C₁₃	C₂
prime radical	#10597790	C₂
cosocle	C₃×C₄×C₁₃	C₂³
hypercenter	C₂×C₈	D₃₉
hypocenter	C₃×C₁₃	#131
LCS	G ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₈×C₁₃	C₂	semidirect
3. #10597662	C₂	semidirect
4. #10597657	C₂	semidirect
5. C₈×D₃₉	C₂	semidirect
16. C₂×C₈×C₁₃	D₃	semidirect
43. C₂×C₃×C₈	D₁₃	semidirect
47. C₃×C₁₃	#131	semidirect
64. C₂×C₈	D₃₉	semidirect
65. C₁₃	#901	semidirect
87. C₃	#88155	semidirect

Maximal subgroups

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

subgroup	index
#10597790	2
C₂×C₃×C₈×C₁₃	2
#10597662	2
#10597657	2
C₈×D₃₉	2
#88155	3
#901	13