Subgroups — #11928135

Order 1248.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#10597662	C₂	1	1	yes	maximal, characteristic, complemented
4	#88063	—	3	1	no	maximal, Hall
5	C₂²×D₃₉	C₄	1	2	yes	—
6	C₂×C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
7	D₁₅₆	—	2	1	no	—
8	D₁₅₆	—	2	1	no	—
9	C₃×C₈×C₁₃	—	2	1	no	—
10	C₃⋊#1055	—	2	1	no	—
11	C₂×D₅₂	—	3	1	no	—
12	C₁₃×M₁₆	D₃	1	1	yes	characteristic, complemented
13	#6099	—	3	1	no	—
14	C₂²×C₃×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂×D₃₉	—	2	2	no	—
16	C₂×D₃₉	—	4	2	no	—
17	C₃×C₄×C₁₃	D₄	1	1	yes	characteristic
18	C₃×C₄×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×D₁₃	—	3	2	no	—
20	C₂×C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
21	D₅₂	—	6	1	no	—
22	D₅₂	—	6	1	no	—
23	C₈×C₁₃	—	2	1	no	—
24	#1055	—	6	1	no	—
25	#823	—	13	1	no	maximal, Hall
26	C₂×C₃×C₁₃	—	2	1	no	—
27	C₂×C₃×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
28	D₃₉	—	4	2	no	—
29	C₂²×C₁₃	C₄×D₃	1	1	yes	characteristic
30	C₂×D₁₃	—	6	2	no	—
31	C₂×D₁₃	—	12	2	no	—
32	C₄×C₁₃	D₁₂	1	1	yes	characteristic
33	C₄×C₁₃	#67	1	1	yes	characteristic
34	C₂×D₁₂	—	13	1	no	—
35	C₃×M₁₆	D₁₃	1	1	yes	characteristic, complemented
36	#208	—	13	1	no	—
37	C₃×C₁₃	#100	1	1	yes	characteristic, complemented, Hall
38	#100	—	39	1	no	Sylow
39	C₂×C₁₃	—	2	1	no	—
40	C₂×C₁₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
41	D₁₃	—	12	2	no	—
42	C₂²×D₃	—	13	2	no	—
43	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
44	D₁₂	—	26	1	no	—
45	D₁₂	—	26	1	no	—
46	C₃×C₈	—	2	1	no	—
47	C₃⋊C₈	—	26	1	no	—
48	C₂×D₄	—	39	1	no	—
49	M₁₆	—	39	1	no	—
50	M₁₆	D₃₉	1	1	yes	characteristic, complemented
51	C₁₃	#823	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₃	C₄×D₁₃	1	1	yes	characteristic
53	C₂×D₃	—	26	2	no	—
54	C₂×D₃	—	52	2	no	—
55	C₃×C₄	D₅₂	1	1	yes	characteristic
56	C₃×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
57	C₂³	—	39	2	no	—
58	D₄	—	78	1	no	—
59	D₄	—	78	1	no	—
60	C₂×C₄	C₂×D₃₉	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	78	1	no	—
63	C₂×C₃	—	2	1	no	—
64	C₂×C₃	#6103	1	1	yes	characteristic
65	D₃	—	52	2	no	—
66	C₂²	—	78	2	no	—
67	C₂²	—	156	2	no	—
68	C₂²	C₄×D₃₉	1	1	yes	characteristic
69	C₄	D₁₅₆	1	1	yes	characteristic
70	C₄	#64625	1	1	yes	characteristic
71	C₃	#88063	1	1	yes	characteristic, complemented, Sylow
72	C₂	—	2	1	no	—
73	C₂	—	156	2	no	—
74	C₂	#10597666	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10597666
commutator	C₂²×C₃×C₁₃	C₂×C₄
Frattini	C₂×C₄	C₂×D₃₉
Fitting	C₃×C₁₃×M₁₆	C₂
socle	C₂×C₃×C₁₃	C₂²⋊C₄
prime radical	C₂×D₁₅₆	C₂
cosocle	C₂×C₃×C₄×C₁₃	C₂²
hypercenter	M₁₆	D₃₉
hypocenter	C₃×C₁₃	#100
LCS	G ▷ C₂²×C₃×C₁₃ ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₁₃×M₁₆	C₂	semidirect
3. #10597662	C₂	semidirect
12. C₁₃×M₁₆	D₃	semidirect
35. C₃×M₁₆	D₁₃	semidirect
37. C₃×C₁₃	#100	semidirect
50. M₁₆	D₃₉	semidirect
51. C₁₃	#823	semidirect
71. C₃	#88063	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₅₆	2
C₃×C₁₃×M₁₆	2
#10597662	2
#88063	3
#823	13