Subgroups — #15689145

Order 3008.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625324	C₂	1	2	yes	maximal
2	C₄²×D₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₄₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄₇	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄₇	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₄₇	C₂²	1	2	yes	—
10	C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄×Dic₄₇	C₄	1	1	yes	characteristic
12	C₄×Dic₄₇	C₄	1	1	yes	characteristic
13	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄₇⋊C₈	C₂²	1	2	yes	—
15	#10625324	—	2	4	no	—
16	C₂²×D₄₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
20	C₂×Dic₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₄₇	C₂×C₄	1	1	yes	characteristic
23	C₄×D₄₇	—	2	2	no	—
24	C₄×D₄₇	—	2	2	no	—
25	C₄×D₄₇	D₄	1	2	yes	—
26	C₄×D₄₇	Q₈	1	2	yes	—
27	C₈×C₄₇	—	2	2	no	—
28	C₄₇⋊C₈	—	2	2	no	—
29	C₂²×C₄₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×D₄₇	—	2	1	no	—
31	C₂×D₄₇	—	2	1	no	—
32	C₂×D₄₇	C₄⋊C₄	1	2	yes	—
33	C₄×C₄₇	—	2	1	no	—
34	C₄×C₄₇	M₁₆	1	2	yes	complemented
35	C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
37	Dic₄₇	—	2	1	no	—
38	Dic₄₇	C₄⋊C₄	1	2	yes	—
39	Dic₄₇	M₁₆	1	2	yes	complemented
40	C₂×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄₇	C₂×M₁₆	1	1	yes	characteristic
42	C₂×C₄₇	C₂×M₁₆	1	1	yes	characteristic
43	D₄₇	—	2	2	no	—
44	#423	—	47	1	no	maximal, Sylow
45	C₄₇	#423	1	1	yes	characteristic, complemented, Sylow
46	C₂×M₁₆	—	47	2	no	—
47	C₂×C₄²	—	47	1	no	—
48	C₄⋊C₈	—	47	2	no	—
49	C₄⋊C₈	—	47	1	no	—
50	C₄⋊C₈	D₄₇	1	1	yes	characteristic, complemented
51	C₂²×C₄	—	47	1	no	—
52	C₂²×C₄	—	47	1	no	—
53	C₂²×C₄	—	47	1	no	—
54	M₁₆	—	94	4	no	—
55	C₂×C₈	—	47	2	no	—
56	C₂×C₈	C₂×D₄₇	1	2	yes	—
57	C₄²	—	47	1	no	—
58	C₄²	—	47	1	no	—
59	C₄²	—	47	1	no	—
60	C₄²	C₂×D₄₇	1	1	yes	characteristic
61	C₂³	—	47	1	no	—
62	C₂×C₄	—	47	1	no	—
63	C₂×C₄	—	47	1	no	—
64	C₂×C₄	—	47	2	no	—
65	C₂×C₄	—	47	2	no	—
66	C₂×C₄	—	47	1	no	—
67	C₂×C₄	—	94	2	no	—
68	C₂×C₄	—	94	2	no	—
69	C₂×C₄	C₄×D₄₇	1	1	yes	characteristic
70	C₂×C₄	C₄×D₄₇	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₈	—	94	2	no	—
74	C₂²	—	47	2	no	—
75	C₂²	—	94	1	no	—
76	C₂²	—	94	1	no	—
77	C₂²	C₂×C₄×D₄₇	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	47	2	no	—
80	C₄	—	47	2	no	—
81	C₄	—	94	1	no	—
82	C₄	#10625324	1	2	yes	complemented
83	C₄	D₄×D₄₇	1	1	yes	characteristic
84	C₄	Q₈×D₄₇	1	1	yes	characteristic
85	C₂	—	94	2	no	—
86	C₂	C₄⋊C₄×D₄₇	1	1	yes	characteristic
87	C₂	M₁₆×D₄₇	1	1	yes	characteristic
88	C₂	C₂×#10625324	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₇
commutator, socle	C₂²×C₄₇	C₂²×C₄
Fitting	C₄⋊C₈×C₄₇	C₂
prime radical	C₂²×D₄₇	C₂×C₄
cosocle	C₂×C₄×C₄₇	C₂³
hypercenter	C₄⋊C₈	D₄₇
hypocenter	C₄₇	#423
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₄₇⋊C₈)	C₂	semidirect
4. Dic₄₇⋊C₈	C₂	semidirect
5. C₄⋊C₈×C₄₇	C₂	semidirect
34. C₄×C₄₇	M₁₆	semidirect
39. Dic₄₇	M₁₆	semidirect
45. C₄₇	#423	semidirect
50. C₄⋊C₈	D₄₇	semidirect
82. C₄	#10625324	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625324	2
C₄²×D₄₇	2
C₄⋊(C₄₇⋊C₈)	2
Dic₄₇⋊C₈	2
C₄⋊C₈×C₄₇	2
#423	47