Subgroups — #26103943

Order 12032.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₄₇⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
2	C₁₆×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄₇×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇⋊C₃₂	—	2	1	no	—
5	C₄×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
6	C₈×C₄₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₄₇	—	2	1	no	—
8	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×C₄₇⋊C₁₆	—	2	1	no	—
10	C₄×C₄₇⋊C₈	C₈	1	1	yes	characteristic
11	C₄×C₄₇⋊C₈	C₈	1	1	yes	characteristic
12	C₄₇⋊C₃₂	—	4	1	no	—
13	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₄₇	Q₈	1	1	yes	characteristic
15	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₄₇	—	4	1	no	—
17	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
19	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
20	C₂×C₄₇⋊C₈	—	2	1	no	—
21	C₂×C₄₇⋊C₈	C₁₆	1	1	yes	characteristic
22	C₂×C₄₇⋊C₈	C₁₆	1	1	yes	characteristic
23	C₁₆×C₄₇	—	2	1	no	—
24	C₁₆×C₄₇	SD₁₆	1	1	yes	characteristic
25	C₁₆×C₄₇	SD₁₆	1	1	yes	characteristic
26	C₄₇⋊C₁₆	—	2	2	no	—
27	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
28	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
29	C₂×C₄×C₄₇	M₃₂	1	1	yes	characteristic
30	C₈×C₄₇	—	2	1	no	—
31	C₈×C₄₇	#106	1	1	yes	characteristic
32	C₈×C₄₇	#108	1	1	yes	characteristic
33	C₄₇⋊C₈	—	2	2	no	—
34	C₄₇⋊C₈	—	2	1	no	—
35	C₄₇⋊C₈	C₃₂	1	2	yes	complemented
36	#7455	—	47	1	no	maximal, Sylow
37	C₂²×C₄₇	C₄⋊C₁₆	1	1	yes	characteristic
38	C₄×C₄₇	—	2	1	no	—
39	C₄×C₄₇	#334	1	1	yes	characteristic
40	C₄×C₄₇	#364	1	1	yes	characteristic
41	C₄×C₄₇	C₂×C₃₂	1	1	yes	characteristic
42	C₄×C₄₇	M₆₄	1	1	yes	characteristic
43	C₄⋊C₃₂	—	47	1	no	—
44	C₄⋊C₃₂	D₄₇	1	1	yes	characteristic
45	C₈×C₁₆	—	47	1	no	—
46	C₂×C₄₇	#1367	1	1	yes	characteristic
47	C₂×C₄₇	C₄⋊C₃₂	1	1	yes	characteristic
48	C₂×C₄₇	#1422	1	1	yes	characteristic
49	C₂×C₃₂	—	2	1	no	—
50	C₂×C₃₂	—	94	1	no	—
51	C₄×C₁₆	—	47	1	no	—
52	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
53	C₈²	—	47	1	no	—
54	C₄₇	#7455	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₁₆	—	94	1	no	—
56	C₂×C₁₆	Dic₉₄	1	1	yes	characteristic
57	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
58	C₄×C₈	—	47	1	no	—
59	C₄×C₈	—	47	1	no	—
60	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
61	C₃₂	—	4	1	no	—
62	C₃₂	—	188	1	no	—
63	C₂×C₈	—	47	1	no	—
64	C₂×C₈	—	47	1	no	—
65	C₂×C₈	—	94	1	no	—
66	C₂×C₈	#10625324	1	1	yes	characteristic
67	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
68	C₄²	C₈×D₄₇	1	1	yes	characteristic
69	C₁₆	—	2	1	no	—
70	C₁₆	—	94	2	no	—
71	C₁₆	#10625335	1	1	yes	characteristic
72	C₁₆	#10625336	1	1	yes	characteristic
73	C₂×C₄	Dic₄₇⋊C₈	1	1	yes	characteristic
74	C₂×C₄	C₁₆×D₄₇	1	1	yes	characteristic
75	C₂×C₄	#13093963	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	47	2	no	—
78	C₈	—	94	2	no	—
79	C₈	—	94	1	no	—
80	C₈	#13093955	1	1	yes	characteristic
81	C₈	#13093959	1	1	yes	characteristic
82	C₂²	Dic₄₇⋊C₁₆	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	#15688873	1	1	yes	characteristic
85	C₄	#15688943	1	1	yes	characteristic
86	C₄	C₃₂×D₄₇	1	1	yes	characteristic
87	C₄	#15688956	1	1	yes	characteristic
88	C₂	#20241451	1	1	yes	characteristic
89	C₂	Dic₄₇⋊C₃₂	1	1	yes	characteristic
90	C₂	#20241585	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂×C₃₂
Frattini	C₄×C₁₆	C₂×D₄₇
Fitting	C₄₇×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₄₇	C₄⋊C₁₆
cosocle	C₄×C₁₆×C₄₇	C₂²
hypercenter	C₄⋊C₃₂	D₄₇
hypocenter	C₄₇	#7455
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

Complements

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

N	H ≅ G/N	type
35. C₄₇⋊C₈	C₃₂	semidirect
54. C₄₇	#7455	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₄₇⋊C₃₂)	2
C₁₆×C₄₇⋊C₈	2
C₄₇×C₄⋊C₃₂	2
#7455	47