Subgroups — #15688885

Order 3008.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×C₄₇	—	2	1	no	—
5	C₂²×C₄×C₄₇	C₂²	1	1	yes	characteristic
6	C₂²×Dic₄₇	—	2	1	no	—
7	C₄⋊C₄×C₄₇	—	2	1	no	—
8	C₄⋊C₄×C₄₇	C₄	1	2	yes	complemented
9	C₄⋊Dic₄₇	—	2	1	no	—
10	C₄⋊Dic₄₇	C₄	1	2	yes	complemented
11	C₂×C₄₇⋊C₈	—	2	1	no	—
12	C₂×C₄₇⋊C₈	—	2	1	no	—
13	C₂×C₄₇⋊C₈	C₄	1	2	yes	complemented
14	C₂³×C₄₇	D₄	1	1	yes	characteristic
15	C₂×C₄×C₄₇	—	2	2	no	—
16	C₂×C₄×C₄₇	—	4	1	no	—
17	C₂×C₄×C₄₇	—	4	1	no	—
18	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₇	Q₈	1	1	yes	characteristic
24	C₂×Dic₄₇	—	2	2	no	—
25	C₂×Dic₄₇	—	4	1	no	—
26	C₂×Dic₄₇	—	4	1	no	—
27	C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄₇	D₈	1	1	yes	characteristic
32	C₂²×C₄₇	SD₁₆	1	1	yes	characteristic
33	C₂²×C₄₇	SD₁₆	1	1	yes	characteristic
34	C₂²×C₄₇	Q₁₆	1	1	yes	characteristic
35	C₄×C₄₇	—	4	2	no	—
36	C₄×C₄₇	C₄²	1	1	yes	characteristic
37	C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
39	C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
40	Dic₄₇	—	4	2	no	—
41	C₂×C₄₇	#95	1	1	yes	characteristic
42	C₂×C₄₇	#102	1	1	yes	characteristic
43	C₂×C₄₇	#102	1	1	yes	characteristic
44	C₂×C₄₇	#103	1	1	yes	characteristic
45	C₂×C₄₇	#103	1	1	yes	characteristic
46	C₂×C₄₇	#106	1	1	yes	characteristic
47	C₂×C₄₇	#107	1	1	yes	characteristic
48	#340	—	47	1	no	maximal, Sylow
49	C₄₇	#340	1	1	yes	characteristic, complemented, Sylow
50	C₂²×C₈	—	47	1	no	—
51	C₂×C₄⋊C₄	—	47	1	no	—
52	C₂×C₄⋊C₄	D₄₇	1	1	yes	characteristic
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	—	94	1	no	—
55	C₂²×C₄	C₂×D₄₇	1	1	yes	characteristic
56	C₂×C₈	—	47	2	no	—
57	C₂×C₈	—	94	1	no	—
58	C₂×C₈	—	94	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	47	2	no	—
61	C₄⋊C₄	—	94	1	no	—
62	C₄⋊C₄	Dic₄₇	1	2	yes	complemented
63	C₂³	C₂²⋊D₄₇	1	1	yes	characteristic
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	4	1	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	—	94	2	no	—
68	C₂×C₄	—	188	1	no	—
69	C₂×C₄	—	188	1	no	—
70	C₂×C₄	Dic₉₄	1	1	yes	characteristic
71	C₂×C₄	C₄×D₄₇	1	1	yes	characteristic
72	C₂×C₄	C₄×D₄₇	1	1	yes	characteristic
73	C₂×C₄	D₁₈₈	1	1	yes	characteristic
74	C₂×C₄	C₂×Dic₄₇	1	1	yes	characteristic
75	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
76	C₈	—	94	2	no	—
77	C₂²	Dic₄₇⋊C₄	1	1	yes	characteristic
78	C₂²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
79	C₂²	#10625334	1	1	yes	characteristic
80	C₂²	#10625335	1	1	yes	characteristic
81	C₂²	#10625336	1	1	yes	characteristic
82	C₂²	#10625337	1	1	yes	characteristic
83	C₂²	C₂²⋊Dic₄₇	1	1	yes	characteristic
84	C₄	—	4	2	no	—
85	C₄	—	188	2	no	—
86	C₄	C₄×Dic₄₇	1	1	yes	characteristic
87	C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
88	C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
89	C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
90	C₂	C₄₇⋊#95	1	1	yes	characteristic
91	C₂	#13093944	1	1	yes	characteristic
92	C₂	#13093945	1	1	yes	characteristic
93	C₂	#13093947	1	1	yes	characteristic
94	C₂	#13093948	1	1	yes	characteristic
95	C₂	#13093955	1	1	yes	characteristic
96	C₂	#13093957	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₄²
Frattini	C₂²×C₄	C₂×D₄₇
Fitting	C₂×C₄⋊C₄×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	D₄
cosocle	C₂²×C₄×C₄₇	C₂²
hypercenter	C₂×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#340
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
8. C₄⋊C₄×C₄₇	C₄	semidirect
10. C₄⋊Dic₄₇	C₄	semidirect
13. C₂×C₄₇⋊C₈	C₄	semidirect
49. C₄₇	#340	semidirect
62. C₄⋊C₄	Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄₇⋊C₈	2
C₂×C₄⋊Dic₄₇	2
C₂×C₄⋊C₄×C₄₇	2
#340	47