Subgroups — #26110012

Order 12032.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241260	C₂	1	3	yes	maximal
2	#20241253	C₂	1	3	yes	maximal
3	C₄₇×#1455	C₂	1	1	yes	maximal, characteristic
4	#15689180	C₂²	1	3	yes	—
5	#15688991	C₂²	1	1	yes	characteristic
6	C₄×M₁₆×C₄₇	C₂²	1	3	yes	—
7	C₄₇×#322	C₄	1	3	yes	complemented
8	C₈²×C₄₇	C₄	1	1	yes	characteristic, complemented
9	C₄⋊(C₄₇⋊C₈)	—	2	3	no	—
10	C₂²⋊(C₄₇⋊C₈)	—	2	3	no	—
11	#13093935	—	2	3	no	—
12	C₄₇⋊#95	—	2	3	no	—
13	C₂×M₁₆×C₄₇	—	2	3	no	—
14	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
15	#97×C₄₇	—	2	3	no	—
16	C₄×C₈×C₄₇	C₂×C₄	1	3	yes	—
17	C₄×C₈×C₄₇	C₂×C₄	1	3	yes	—
18	C₂²×C₄×C₄₇	#41	1	3	yes	—
19	C₂²×Dic₄₇	—	4	1	no	—
20	M₁₆×C₄₇	—	4	6	no	—
21	C₂×C₈×C₄₇	—	2	3	no	—
22	C₂×C₈×C₄₇	—	2	3	no	—
23	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
24	C₄²×C₄₇	#41	1	3	yes	—
25	C₂×C₄₇⋊C₈	—	4	3	no	—
26	C₂³×C₄₇	#126	1	1	yes	characteristic
27	C₂×C₄×C₄₇	—	4	3	no	—
28	C₂×C₄×C₄₇	#117	1	3	yes	—
29	C₂×C₄×C₄₇	#126	1	3	yes	—
30	C₂×Dic₄₇	—	8	3	no	—
31	C₈×C₄₇	—	2	6	no	—
32	C₈×C₄₇	—	4	3	no	—
33	C₄₇⋊C₈	—	8	3	no	—
34	#8804	—	47	1	no	maximal, Sylow
35	C₂²×C₄₇	—	2	3	no	—
36	C₂²×C₄₇	#383	1	1	yes	characteristic
37	C₄×C₄₇	—	2	3	no	—
38	C₄×C₄₇	—	2	3	no	—
39	Dic₄₇	—	16	1	no	—
40	#1455	D₄₇	1	1	yes	characteristic
41	#1291	—	47	3	no	—
42	#1288	—	47	3	no	—
43	C₂×C₄₇	—	4	1	no	—
44	C₂×C₄₇	#1841	1	3	yes	—
45	#432	—	47	3	no	—
46	C₄×M₁₆	C₂×D₄₇	1	3	yes	—
47	#383	—	47	1	no	—
48	#322	Dic₄₇	1	3	yes	complemented
49	C₈²	Dic₄₇	1	1	yes	characteristic, complemented
50	C₄₇	#8804	1	1	yes	characteristic, complemented, Sylow
51	C₂×M₁₆	—	2	3	no	—
52	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
53	C₄⋊C₈	—	94	3	no	—
54	C₂²⋊C₈	—	94	3	no	—
55	#97	—	2	3	no	—
56	#97	—	94	3	no	—
57	C₄×C₈	C₂×Dic₄₇	1	3	yes	—
58	C₄×C₈	C₂×Dic₄₇	1	3	yes	—
59	#95	—	94	3	no	—
60	C₂²×C₄	—	188	1	no	—
61	C₂²×C₄	#10625350	1	3	yes	—
62	M₁₆	—	4	6	no	—
63	C₂×C₈	—	2	3	no	—
64	C₂×C₈	—	2	3	no	—
65	C₂×C₈	—	188	3	no	—
66	C₄²	#10625350	1	3	yes	—
67	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
68	C₂³	#13094018	1	1	yes	characteristic
69	C₂×C₄	—	4	3	no	—
70	C₂×C₄	—	376	3	no	—
71	C₂×C₄	#13093983	1	3	yes	—
72	C₂×C₄	#13094018	1	3	yes	—
73	C₈	—	2	6	no	—
74	C₈	—	4	3	no	—
75	C₈	—	376	3	no	—
76	C₂²	—	2	3	no	—
77	C₂²	#15688991	1	1	yes	characteristic
78	C₄	—	2	3	no	—
79	C₄	—	2	3	no	—
80	C₄	—	752	1	no	—
81	C₂	—	4	1	no	—
82	C₂	#20243537	1	3	yes	—

	subgroup	G/H
center	C₂²	#15688991
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1455	C₂
socle	C₂²×C₄₇	#383
prime radical	C₂³×C₄₇	#126
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1455	D₄₇
hypocenter	C₄₇	#8804
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
7. C₄₇×#322	C₄	semidirect
8. C₈²×C₄₇	C₄	semidirect
48. #322	Dic₄₇	semidirect
49. C₈²	Dic₄₇	semidirect
50. C₄₇	#8804	semidirect

Maximal subgroups

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

subgroup	index
#20241260	2
#20241253	2
C₄₇×#1455	2
#8804	47