Subgroups — #13094017

Order 1504.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₄₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₄₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇	—	2	1	no	—
16	C₂²×C₄₇	—	2	1	no	—
17	C₂²×C₄₇	—	2	1	no	—
18	C₂²×C₄₇	C₂³	1	1	yes	characteristic
19	C₄×C₄₇	—	2	1	no	—
20	C₄×C₄₇	—	2	1	no	—
21	Dic₄₇	—	2	1	no	—
22	Dic₄₇	—	2	1	no	—
23	Dic₄₇	—	2	1	no	—
24	Dic₄₇	—	2	1	no	—
25	C₂×C₄₇	—	4	1	no	—
26	C₂×C₄₇	#41	1	1	yes	characteristic
27	C₂×C₄₇	#41	1	1	yes	characteristic
28	C₂×C₄₇	#41	1	1	yes	characteristic
29	C₄₇	#126	1	1	yes	characteristic, complemented, Sylow
30	#126	—	47	1	no	maximal, Sylow
31	C₄⋊C₄	—	47	1	no	—
32	C₄⋊C₄	—	47	1	no	—
33	C₄⋊C₄	—	47	1	no	—
34	C₂²⋊C₄	—	47	1	no	—
35	C₂²⋊C₄	—	47	1	no	—
36	C₂²⋊C₄	D₄₇	1	1	yes	characteristic
37	C₄²	—	47	1	no	—
38	C₂³	C₂×D₄₇	1	1	yes	characteristic
39	C₂×C₄	—	47	1	no	—
40	C₂×C₄	—	47	1	no	—
41	C₂×C₄	—	47	1	no	—
42	C₂×C₄	—	47	1	no	—
43	C₂×C₄	C₂×D₄₇	1	1	yes	characteristic
44	C₂×C₄	C₂×D₄₇	1	1	yes	characteristic
45	C₂²	—	2	1	no	—
46	C₂²	—	2	1	no	—
47	C₂²	—	2	1	no	—
48	C₂²	C₂²×D₄₇	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	—	2	1	no	—
51	C₄	—	94	1	no	—
52	C₄	—	94	1	no	—
53	C₄	—	94	1	no	—
54	C₄	—	94	1	no	—
55	C₂	—	4	1	no	—
56	C₂	#10625350	1	1	yes	characteristic
57	C₂	#10625352	1	1	yes	characteristic
58	C₂	#10625352	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂³
Fitting	C₂²⋊C₄×C₄₇	C₂
prime radical	C₂³×C₄₇	C₂²
hypercenter	C₂²⋊C₄	D₄₇
hypocenter	C₄₇	#126
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₄⋊Dic₄₇	C₂	semidirect
5. Dic₄₇⋊C₄	C₂	semidirect
6. Dic₄₇⋊C₄	C₂	semidirect
7. C₄×Dic₄₇	C₂	semidirect
29. C₄₇	#126	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊C₄×C₄₇	2
C₂²⋊Dic₄₇	2
C₄⋊Dic₄₇	2
Dic₄₇⋊C₄	2
C₄×Dic₄₇	2
#126	47