Subgroups — #26131019

Order 12032.

238 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245670	C₂	1	2	yes	maximal
2	#20245663	C₂	1	2	yes	maximal
3	#20242998	C₂	1	2	yes	maximal
4	C₈×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#15689528	C₂²	1	4	yes	—
6	#15689474	—	2	4	no	—
7	#15688951	—	2	4	no	—
8	#15688949	—	2	4	no	—
9	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
11	C₈²×C₄₇	C₂²	1	1	yes	characteristic
12	#13094024	—	2	4	no	—
13	#13093958	—	2	4	no	—
14	#13093958	—	2	4	no	—
15	#13093949	—	4	8	no	—
16	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
17	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
18	C₄×C₈×C₄₇	D₄	1	2	yes	—
19	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
20	C₂×Dic₉₄	—	4	4	no	—
21	C₂×C₈×C₄₇	D₈	1	4	yes	—
22	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
23	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
24	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
25	C₄⋊Dic₄₇	—	4	4	no	—
26	C₄⋊Dic₄₇	—	4	4	no	—
27	C₁₆×C₄₇	Q₁₆	1	4	yes	—
28	C₁₆×C₄₇	Q₁₆	1	4	yes	—
29	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
30	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
31	C₂×Dic₄₇	—	8	4	no	—
32	Dic₉₄	—	8	8	no	—
33	C₈×C₄₇	—	2	4	no	—
34	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
35	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
36	#12522	—	47	1	no	maximal, Sylow
37	C₂²×C₄₇	#493	1	1	yes	characteristic
38	C₄×C₄₇	#494	1	1	yes	characteristic
39	C₄×C₄₇	#494	1	1	yes	characteristic
40	C₄×C₄₇	#508	1	4	yes	—
41	Dic₄₇	—	16	4	no	—
42	#2254	—	47	2	no	—
43	#2252	—	47	2	no	—
44	#1710	—	47	2	no	—
45	C₈×C₁₆	D₄₇	1	1	yes	characteristic
46	C₂×C₄₇	#1713	1	1	yes	characteristic
47	C₂×C₄₇	#2249	1	2	yes	—
48	#500	—	47	4	no	—
49	#487	—	94	4	no	—
50	#367	—	94	4	no	—
51	#366	—	94	4	no	—
52	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
53	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
54	C₈²	C₂×D₄₇	1	1	yes	characteristic
55	C₄₇	#12522	1	1	yes	characteristic, complemented, Sylow
56	C₄⋊Q₈	—	94	4	no	—
57	C₂×C₁₆	D₁₈₈	1	2	yes	—
58	C₂×C₁₆	D₁₈₈	1	2	yes	—
59	#107	—	94	4	no	—
60	#107	—	94	4	no	—
61	#103	—	188	8	no	—
62	C₄×C₈	D₁₈₈	1	2	yes	—
63	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
64	C₂×Q₈	—	188	4	no	—
65	C₂×C₈	D₃₇₆	1	4	yes	—
66	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
67	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
68	C₄⋊C₄	—	188	4	no	—
69	C₄⋊C₄	—	188	4	no	—
70	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
71	C₁₆	Dic₁₈₈	1	4	yes	—
72	C₁₆	Dic₁₈₈	1	4	yes	—
73	Q₈	—	376	8	no	—
74	C₂×C₄	—	376	4	no	—
75	C₂×C₄	#13094021	1	1	yes	characteristic
76	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
77	C₈	—	2	4	no	—
78	C₈	C₂×Dic₁₈₈	1	2	yes	—
79	C₈	C₂×Dic₁₈₈	1	2	yes	—
80	C₂²	#15689500	1	1	yes	characteristic
81	C₄	—	752	4	no	—
82	C₄	#15689503	1	1	yes	characteristic
83	C₄	#15689503	1	1	yes	characteristic
84	C₄	#15689555	1	4	yes	—
85	C₂	#20243006	1	1	yes	characteristic
86	C₂	#20245649	1	2	yes	—

	subgroup	G/H
center	C₂²	#15689500
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₈×C₁₆×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	#493
hypercenter	C₈×C₁₆	D₄₇
hypocenter	C₄₇	#12522
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
55. C₄₇	#12522	semidirect

Maximal subgroups

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

subgroup	index
#20245670	2
#20245663	2
#20242998	2
C₈×C₁₆×C₄₇	2
#12522	47