Subgroups — #26130985

Order 12032.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245615	C₂	1	4	yes	maximal
2	#20242998	C₂	1	1	yes	maximal, characteristic
3	#20241305	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1713	C₂	1	1	yes	maximal, characteristic
5	#15689528	C₂²	1	2	yes	—
6	#15689474	—	2	2	no	—
7	#15688927	—	2	8	no	—
8	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	C₄₇×#500	—	2	2	no	—
10	C₄₇×#494	C₂²	1	2	yes	—
11	C₈²×C₄₇	C₂²	1	1	yes	characteristic
12	#13094024	—	2	2	no	—
13	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
14	#13093958	—	2	4	no	—
15	#13093949	—	4	4	no	—
16	C₂×Q₁₆×C₄₇	—	2	4	no	—
17	C₄⋊Q₈×C₄₇	—	2	2	no	—
18	#107×C₄₇	—	4	4	no	—
19	C₄×C₈×C₄₇	D₄	1	2	yes	—
20	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₄₇	—	4	2	no	—
22	C₂×Dic₉₄	—	4	2	no	—
23	Q₁₆×C₄₇	—	4	8	no	—
24	C₂×C₈×C₄₇	D₈	1	4	yes	—
25	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
26	C₄⋊C₄×C₄₇	—	4	4	no	—
27	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
28	C₄⋊Dic₄₇	—	4	2	no	—
29	C₄⋊Dic₄₇	—	4	2	no	—
30	C₄₇⋊C₁₆	—	2	4	no	—
31	Q₈×C₄₇	—	8	4	no	—
32	C₂×C₄×C₄₇	—	8	2	no	—
33	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
34	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
35	C₂×Dic₄₇	—	8	2	no	—
36	Dic₉₄	—	8	4	no	—
37	C₈×C₄₇	—	2	4	no	—
38	C₈×C₄₇	#135	1	4	yes	—
39	#12514	—	47	1	no	maximal, Sylow
40	C₂²×C₄₇	#493	1	1	yes	characteristic
41	C₄×C₄₇	—	16	2	no	—
42	C₄×C₄₇	#495	1	2	yes	—
43	C₄×C₄₇	#510	1	4	yes	—
44	Dic₄₇	—	16	2	no	—
45	#2241	—	47	4	no	—
46	#1713	D₄₇	1	1	yes	characteristic
47	#1710	—	47	1	no	—
48	#1311	—	47	1	no	—
49	C₂×C₄₇	#1709	1	1	yes	characteristic
50	C₂×C₄₇	#2251	1	2	yes	—
51	#500	—	2	2	no	—
52	#500	—	47	2	no	—
53	#494	C₂×D₄₇	1	2	yes	—
54	#487	—	94	2	no	—
55	#358	—	94	8	no	—
56	C₄×C₁₆	—	47	2	no	—
57	C₈²	C₂×D₄₇	1	1	yes	characteristic
58	C₄₇	#12514	1	1	yes	characteristic, complemented, Sylow
59	C₂×Q₁₆	—	2	4	no	—
60	C₄⋊Q₈	—	2	2	no	—
61	C₄⋊Q₈	—	94	2	no	—
62	C₂×C₁₆	—	47	4	no	—
63	#107	—	4	4	no	—
64	#107	—	94	4	no	—
65	#103	—	188	4	no	—
66	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
67	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
68	C₂×Q₈	—	4	2	no	—
69	C₂×Q₈	—	188	2	no	—
70	Q₁₆	—	4	8	no	—
71	C₂×C₈	#10625334	1	4	yes	—
72	C₂×C₈	D₄×D₄₇	1	2	yes	—
73	C₄⋊C₄	—	4	4	no	—
74	C₄⋊C₄	—	188	2	no	—
75	C₄⋊C₄	—	188	2	no	—
76	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
77	C₁₆	—	94	4	no	—
78	Q₈	—	8	4	no	—
79	Q₈	—	376	4	no	—
80	C₂×C₄	—	8	2	no	—
81	C₂×C₄	—	376	2	no	—
82	C₂×C₄	#13094020	1	1	yes	characteristic
83	C₂×C₄	C₂×#10625334	1	2	yes	—
84	C₈	—	2	4	no	—
85	C₈	#13094043	1	4	yes	—
86	C₂²	#15689499	1	1	yes	characteristic
87	C₄	—	16	2	no	—
88	C₄	—	752	2	no	—
89	C₄	#15689506	1	2	yes	—
90	C₄	#15689564	1	4	yes	—
91	C₂	#20242993	1	1	yes	characteristic
92	C₂	#20245657	1	2	yes	—

	subgroup	G/H
center	C₂²	#15689499
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1713	C₂
socle, prime radical	C₂²×C₄₇	#493
hypercenter	#1713	D₄₇
hypocenter	C₄₇	#12514
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

Complements

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

N	H ≅ G/N	type
58. C₄₇	#12514	semidirect

Maximal subgroups

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

subgroup	index
#20245615	2
#20242998	2
#20241305	2
C₄₇×#1713	2
#12514	47