Subgroups — #26104092

Order 12032.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245681	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇⋊C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689539	C₄	1	1	yes	characteristic
5	#15688957	C₄	1	1	yes	characteristic
6	#15688956	—	2	1	no	—
7	C₃₂×D₄₇	—	2	1	no	—
8	C₄₇⋊C₆₄	—	2	1	no	—
9	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic
10	C₄₇×C₆₄	—	2	1	no	—
11	#13094030	C₈	1	1	yes	characteristic
12	#13093964	C₈	1	1	yes	characteristic
13	#13093963	—	2	1	no	—
14	C₁₆×D₄₇	—	2	1	no	—
15	C₄₇⋊C₃₂	—	2	1	no	—
16	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
17	C₃₂×C₄₇	D₄	1	1	yes	characteristic
18	C₃₂×C₄₇	D₄	1	1	yes	characteristic
19	#10625350	C₁₆	1	1	yes	characteristic
20	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
21	#10625329	C₁₆	1	1	yes	characteristic
22	#10625324	—	2	1	no	—
23	C₈×D₄₇	—	2	1	no	—
24	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₁₆×C₄₇	M₁₆	1	1	yes	characteristic
26	C₄₇⋊C₁₆	—	2	1	no	—
27	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
28	C₂²⋊D₄₇	—	2	1	no	—
29	D₁₈₈	C₃₂	1	1	yes	characteristic
30	C₄×D₄₇	—	2	1	no	—
31	Dic₉₄	C₃₂	1	1	yes	characteristic
32	C₈×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₈×C₄₇	M₃₂	1	1	yes	characteristic
34	C₄₇⋊C₈	—	2	1	no	—
35	#7514	—	47	1	no	maximal, Sylow
36	C₂²×C₄₇	M₆₄	1	1	yes	characteristic
37	C₂×D₄₇	—	2	1	no	—
38	C₄×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
39	C₄×C₄₇	C₂×C₃₂	1	1	yes	characteristic
40	Dic₄₇	—	2	1	no	—
41	#2258	—	47	1	no	—
42	M₁₂₈	D₄₇	1	1	yes	characteristic, complemented
43	C₂×C₆₄	—	47	1	no	—
44	C₂×C₄₇	—	2	1	no	—
45	C₂×C₄₇	C₂²⋊C₃₂	1	1	yes	characteristic
46	D₄₇	—	4	1	no	—
47	#504	—	47	1	no	—
48	M₆₄	—	47	1	no	—
49	M₆₄	—	94	1	no	—
50	C₂×C₃₂	—	94	1	no	—
51	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
52	C₆₄	—	2	1	no	—
53	C₆₄	—	94	1	no	—
54	C₄₇	#7514	1	1	yes	characteristic, complemented, Sylow
55	#131	—	47	1	no	—
56	M₃₂	—	47	1	no	—
57	M₃₂	—	94	1	no	—
58	C₂×C₁₆	—	94	1	no	—
59	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
60	C₃₂	—	94	1	no	—
61	C₃₂	D₁₈₈	1	1	yes	characteristic
62	C₃₂	C₂²⋊D₄₇	1	1	yes	characteristic
63	#41	—	47	1	no	—
64	M₁₆	—	47	1	no	—
65	M₁₆	—	94	1	no	—
66	C₂×C₈	—	94	1	no	—
67	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
68	C₁₆	—	94	1	no	—
69	C₁₆	#10625324	1	1	yes	characteristic
70	C₁₆	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
71	Q₈	—	47	1	no	—
72	D₄	—	47	1	no	—
73	D₄	—	94	1	no	—
74	C₂×C₄	—	94	1	no	—
75	C₂×C₄	C₁₆×D₄₇	1	1	yes	characteristic
76	C₈	—	94	1	no	—
77	C₈	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
78	C₈	#13093963	1	1	yes	characteristic
79	C₂²	—	94	1	no	—
80	C₂²	#15688956	1	1	yes	characteristic
81	C₄	—	94	1	no	—
82	C₄	(C₂×D₄₇)⋊C₁₆	1	1	yes	characteristic
83	C₄	C₃₂×D₄₇	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	188	1	no	—
86	C₂	(C₂×D₄₇)⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₄₇
Fitting	C₄₇×M₁₂₈	C₂
socle	C₂×C₄₇	C₂²⋊C₃₂
prime radical	#10625350	C₁₆
cosocle	C₂×C₃₂×C₄₇	C₂²
hypercenter	M₁₂₈	D₄₇
hypocenter	C₄₇	#7514
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₇⋊C₆₄	C₂	semidirect
3. C₄₇×M₁₂₈	C₂	semidirect
42. M₁₂₈	D₄₇	semidirect
54. C₄₇	#7514	semidirect

Maximal subgroups

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

subgroup	index
#20245681	2
C₂×C₄₇⋊C₆₄	2
C₄₇×M₁₂₈	2
#7514	47