Subgroups — #26103929

Order 12032.

108 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	#20241460	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#2252	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇⋊C₃₂	—	2	1	no	—
5	#15688876	—	2	1	no	—
6	C₄₇×#500	—	2	1	no	—
7	C₄₇×#366	—	2	1	no	—
8	C₄₇×#366	C₄	1	2	yes	—
9	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
10	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
11	C₄₇⋊C₃₂	—	4	1	no	—
12	C₄⋊Q₈×C₄₇	—	4	1	no	—
13	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
15	#107×C₄₇	—	2	2	no	—
16	#107×C₄₇	—	4	1	no	—
17	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
18	C₂×Q₈×C₄₇	—	8	1	no	—
19	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
21	C₄⋊C₄×C₄₇	—	4	2	no	—
22	C₄⋊C₄×C₄₇	—	8	1	no	—
23	C₄²×C₄₇	D₈	1	1	yes	characteristic
24	C₂×C₄₇⋊C₈	—	8	1	no	—
25	C₁₆×C₄₇	—	2	1	no	—
26	C₁₆×C₄₇	SD₁₆	1	1	yes	characteristic
27	C₁₆×C₄₇	Q₁₆	1	1	yes	characteristic
28	Q₈×C₄₇	—	16	1	no	—
29	Q₈×C₄₇	—	16	1	no	—
30	C₂×C₄×C₄₇	—	8	2	no	—
31	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
32	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
33	C₂×C₄×C₄₇	SD₃₂	1	1	yes	characteristic
34	C₈×C₄₇	—	2	1	no	—
35	C₈×C₄₇	#101	1	1	yes	characteristic
36	C₈×C₄₇	#103	1	1	yes	characteristic
37	C₄₇⋊C₈	—	16	1	no	—
38	#7450	—	47	1	no	maximal, Sylow
39	C₂²×C₄₇	#357	1	1	yes	characteristic
40	C₄×C₄₇	—	2	1	no	—
41	C₄×C₄₇	—	16	2	no	—
42	C₄×C₄₇	#332	1	1	yes	characteristic
43	C₄×C₄₇	#362	1	1	yes	characteristic
44	C₄×C₄₇	D₃₂	1	1	yes	characteristic
45	C₄×C₄₇	SD₆₄	1	1	yes	characteristic
46	#2252	D₄₇	1	1	yes	characteristic
47	C₄⋊C₃₂	—	47	1	no	—
48	#1371	—	47	1	no	—
49	C₂×C₄₇	#1364	1	1	yes	characteristic
50	C₂×C₄₇	#1415	1	1	yes	characteristic
51	C₂×C₄₇	#1420	1	1	yes	characteristic
52	#500	—	2	1	no	—
53	C₂×C₃₂	—	94	1	no	—
54	#366	—	2	1	no	—
55	#366	Dic₄₇	1	2	yes	—
56	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
57	#335	—	94	1	no	—
58	C₄₇	#7450	1	1	yes	characteristic, complemented, Sylow
59	C₄⋊Q₈	—	4	1	no	—
60	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
61	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
62	#107	—	2	2	no	—
63	#107	—	4	1	no	—
64	C₄⋊C₈	—	188	1	no	—
65	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
66	C₃₂	—	188	1	no	—
67	C₂×Q₈	—	8	1	no	—
68	C₂×C₈	—	376	1	no	—
69	C₂×C₈	#10625335	1	1	yes	characteristic
70	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
71	C₄⋊C₄	—	4	2	no	—
72	C₄⋊C₄	—	8	1	no	—
73	C₄²	#10625334	1	1	yes	characteristic
74	C₁₆	—	2	1	no	—
75	C₁₆	#10625336	1	1	yes	characteristic
76	C₁₆	#10625337	1	1	yes	characteristic
77	Q₈	—	16	1	no	—
78	Q₈	—	16	1	no	—
79	C₂×C₄	—	8	2	no	—
80	C₂×C₄	#13093945	1	1	yes	characteristic
81	C₂×C₄	#13093965	1	1	yes	characteristic
82	C₂×C₄	#13093968	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	752	1	no	—
85	C₈	#13093943	1	1	yes	characteristic
86	C₈	#13093948	1	1	yes	characteristic
87	C₂²	#15688924	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	16	2	no	—
90	C₄	#15688869	1	1	yes	characteristic
91	C₄	#15688939	1	1	yes	characteristic
92	C₄	#15688958	1	1	yes	characteristic
93	C₄	#15688961	1	1	yes	characteristic
94	C₂	#20241443	1	1	yes	characteristic
95	C₂	#20241566	1	1	yes	characteristic
96	C₂	#20241581	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15688924
commutator	C₂×C₁₆×C₄₇	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₄₇
Fitting	C₄₇×#2252	C₂
socle, prime radical	C₂²×C₄₇	#357
cosocle	C₄×C₁₆×C₄₇	C₂²
hypercenter	#2252	D₄₇
hypocenter	C₄₇	#7450
LCS	G ▷ C₂×C₁₆×C₄₇ ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

Complements

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

N	H ≅ G/N	type
58. C₄₇	#7450	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
#20241460	2
C₄₇×#2252	2
#7450	47