Subgroups — #26132680

Order 12032.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241466	C₂	1	6	yes	maximal
2	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄₇⋊C₁₆	C₂²	1	4	yes	—
4	C₂×#97×C₄₇	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
6	C₄₇×#376	C₄	1	1	yes	characteristic
7	C₂×C₄₇⋊C₁₆	—	2	12	no	—
8	C₂²×C₈×C₄₇	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₄₇	C₂×C₄	1	3	yes	—
11	#97×C₄₇	—	2	6	no	—
12	C₄×C₈×C₄₇	—	2	6	no	—
13	#95×C₄₇	C₈	1	4	yes	—
14	C₂²×C₄×C₄₇	C₂×C₈	1	6	yes	—
15	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
16	C₂×C₈×C₄₇	—	2	3	no	—
17	C₂×C₈×C₄₇	—	2	3	no	—
18	C₂×C₈×C₄₇	M₁₆	1	6	yes	—
19	C₂×C₈×C₄₇	#41	1	6	yes	—
20	C₄²×C₄₇	—	2	3	no	—
21	C₄²×C₄₇	—	2	3	no	—
22	C₄₇⋊C₁₆	—	4	4	no	—
23	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
24	C₂×C₄×C₄₇	—	2	6	no	—
25	C₂×C₄×C₄₇	—	2	12	no	—
26	C₂×C₄×C₄₇	#117	1	3	yes	—
27	C₂×C₄×C₄₇	C₂×M₁₆	1	3	yes	—
28	C₈×C₄₇	—	2	6	no	—
29	C₈×C₄₇	#126	1	4	yes	—
30	#12895	—	47	1	no	maximal, Sylow
31	C₂²×C₄₇	#431	1	3	yes	—
32	C₂²×C₄₇	#504	1	4	yes	—
33	C₄×C₄₇	—	4	6	no	—
34	C₄×C₄₇	#383	1	1	yes	characteristic
35	C₄×C₄₇	#432	1	3	yes	—
36	#1726	D₄₇	1	1	yes	characteristic
37	#1374	—	47	6	no	—
38	C₂×C₄₇	#1839	1	1	yes	characteristic
39	C₂×C₄₇	#2164	1	6	yes	—
40	C₂²×C₁₆	—	47	4	no	—
41	C₂×#97	Dic₄₇	1	3	yes	—
42	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
43	#376	Dic₄₇	1	1	yes	characteristic
44	C₄₇	#12895	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₈	C₂×Dic₄₇	1	3	yes	—
46	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
47	C₂×C₄²	C₂×Dic₄₇	1	3	yes	—
48	C₂×C₁₆	—	94	12	no	—
49	#97	—	2	6	no	—
50	C₄×C₈	—	2	6	no	—
51	#95	C₄₇⋊C₈	1	4	yes	—
52	C₂²×C₄	C₂×C₄₇⋊C₈	1	6	yes	—
53	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
54	C₂×C₈	—	2	3	no	—
55	C₂×C₈	—	2	3	no	—
56	C₂×C₈	#10625329	1	6	yes	—
57	C₂×C₈	#10625350	1	6	yes	—
58	C₄²	—	2	3	no	—
59	C₄²	—	2	3	no	—
60	C₁₆	—	188	4	no	—
61	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
62	C₂×C₄	—	2	12	no	—
63	C₂×C₄	—	2	6	no	—
64	C₂×C₄	#13093983	1	3	yes	—
65	C₂×C₄	C₂×#10625329	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	#13094018	1	4	yes	—
68	C₂²	#15689176	1	3	yes	—
69	C₂²	#15689540	1	4	yes	—
70	C₄	—	4	6	no	—
71	C₄	#15688991	1	1	yes	characteristic
72	C₄	#15689180	1	3	yes	—
73	C₂	#20243531	1	1	yes	characteristic
74	C₂	#20245195	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂²×C₈
Fitting	C₄₇×#1726	C₂
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#12895
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
44. C₄₇	#12895	semidirect

Maximal subgroups

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

subgroup	index
#20241466	2
C₄₇×#1726	2
#12895	47