Subgroups — #26131111

Order 12032.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244993	C₂	1	3	yes	maximal
2	#20244985	C₂	1	3	yes	maximal
3	C₄₇×#1747	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093964	C₂²	1	3	yes	—
5	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
6	#15688901	C₄	1	12	yes	complemented
7	#15688899	C₄	1	6	yes	complemented
8	C₄×C₄₇⋊C₁₆	C₄	1	6	yes	complemented
9	C₄₇×#405	C₂²	1	3	yes	—
10	C₄₇×#338	C₄	1	1	yes	characteristic
11	C₄₇×H(3,ℤ/4)	C₄	1	3	yes	—
12	#13093964	C₂×C₄	1	12	yes	—
13	C₂×C₄₇⋊C₁₆	C₂×C₄	1	3	yes	—
14	C₂×C₄₇⋊C₁₆	C₂×C₄	1	3	yes	—
15	C₂×C₄₇⋊C₁₆	C₂×C₄	1	6	yes	—
16	C₂×M₁₆×C₄₇	C₂×C₄	1	3	yes	—
17	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
18	#117×C₄₇	C₂×C₄	1	3	yes	—
19	#97×C₄₇	C₂×C₄	1	6	yes	—
20	C₄×C₈×C₄₇	C₂×C₄	1	6	yes	—
21	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
22	M₁₆×C₄₇	—	2	6	no	—
23	C₂×C₈×C₄₇	C₄²	1	6	yes	—
24	C₂×C₈×C₄₇	C₂²×C₄	1	3	yes	—
25	C₂×C₈×C₄₇	C₂²×C₄	1	3	yes	—
26	C₄⋊C₄×C₄₇	—	2	3	no	—
27	C₂²⋊C₄×C₄₇	—	2	3	no	—
28	C₄²×C₄₇	C₄²	1	6	yes	—
29	C₄₇⋊C₁₆	C₄²	1	1	yes	characteristic
30	C₄₇⋊C₁₆	C₄²	1	3	yes	—
31	C₄₇⋊C₁₆	C₄²	1	12	yes	—
32	C₂³×C₄₇	#131	1	1	yes	characteristic
33	C₂×C₄×C₄₇	—	2	6	no	—
34	C₂×C₄×C₄₇	C₂×C₄²	1	3	yes	—
35	C₂×C₄×C₄₇	#131	1	3	yes	—
36	C₈×C₄₇	—	2	6	no	—
37	C₈×C₄₇	C₂×C₄²	1	3	yes	—
38	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
39	#12542	—	47	1	no	maximal, Sylow
40	C₂²×C₄₇	—	4	1	no	—
41	C₂²×C₄₇	#405	1	3	yes	—
42	C₄×C₄₇	—	4	6	no	—
43	C₄×C₄₇	C₄³	1	1	yes	characteristic
44	C₄×C₄₇	#405	1	3	yes	—
45	#2110	—	47	3	no	—
46	#2108	—	47	3	no	—
47	#1747	D₄₇	1	1	yes	characteristic
48	C₂×C₄₇	—	2	3	no	—
49	C₂×C₄₇	#1726	1	1	yes	characteristic
50	C₂×M₃₂	—	47	3	no	—
51	C₂²×C₁₆	—	47	1	no	—
52	#405	C₂×D₄₇	1	3	yes	—
53	#347	—	47	12	no	—
54	#346	—	47	6	no	—
55	C₄×C₁₆	—	47	6	no	—
56	#338	Dic₄₇	1	1	yes	characteristic
57	H(3,ℤ/4)	Dic₄₇	1	3	yes	—
58	C₄₇	#12542	1	1	yes	characteristic, complemented, Sylow
59	C₂×M₁₆	C₂×Dic₄₇	1	3	yes	—
60	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
61	#117	C₂×Dic₄₇	1	3	yes	—
62	M₃₂	—	47	12	no	—
63	C₂×C₁₆	—	47	3	no	—
64	C₂×C₁₆	—	47	6	no	—
65	C₂×C₁₆	—	47	3	no	—
66	#97	C₄×D₄₇	1	6	yes	—
67	C₄×C₈	C₄×D₄₇	1	6	yes	—
68	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
69	M₁₆	—	2	6	no	—
70	C₂×C₈	C₄×Dic₄₇	1	6	yes	—
71	C₂×C₈	C₂×C₄×D₄₇	1	3	yes	—
72	C₂×C₈	C₂×C₄×D₄₇	1	3	yes	—
73	C₄⋊C₄	—	2	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₄²	C₄×Dic₄₇	1	6	yes	—
76	C₁₆	—	47	12	no	—
77	C₁₆	—	47	1	no	—
78	C₁₆	—	47	3	no	—
79	C₂³	#13094031	1	1	yes	characteristic
80	C₂×C₄	—	2	6	no	—
81	C₂×C₄	C₂×C₄×Dic₄₇	1	3	yes	—
82	C₂×C₄	#13094031	1	3	yes	—
83	C₈	—	2	6	no	—
84	C₈	C₄²×D₄₇	1	3	yes	—
85	C₈	C₄²×D₄₇	1	1	yes	characteristic
86	C₂²	—	4	1	no	—
87	C₂²	#15689079	1	3	yes	—
88	C₄	—	4	6	no	—
89	C₄	C₄²×Dic₄₇	1	1	yes	characteristic
90	C₄	#15689079	1	3	yes	—
91	C₂	—	2	3	no	—
92	C₂	#20243075	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄²×D₄₇
commutator	C₄×C₄₇	C₄³
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1747	C₂
socle	C₂×C₄₇	#1726
prime radical	C₂³×C₄₇	#131
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1747	D₄₇
hypocenter	C₄₇	#12542
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ #1747

Complements

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

N	H ≅ G/N	type
6. #15688901	C₄	semidirect
7. #15688899	C₄	semidirect
8. C₄×C₄₇⋊C₁₆	C₄	semidirect
58. C₄₇	#12542	semidirect

Maximal subgroups

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

subgroup	index
#20244993	2
#20244985	2
C₄₇×#1747	2
#12542	47